Development of Roundabout Crash Prediction Models and Methods (2019)

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53 Crash Prediction Model Development Approach This chapter describes the approach taken, including specific considerations, in developing the crash prediction models. The following sections present the specifics of how the planning-level, intersection-level, and leg-level crash prediction models were developed, including the modeling framework, data used, and specific statistical approach. It is organized into the following main sections: • 5.1 Planning-Level Crash Prediction Models, • 5.2 Intersection-Level Crash Prediction Models for Design, • 5.3 Leg-Level Crash Prediction Models for Design, • 5.4 Practitioner Validation, • 5.5 Effect of Driver Learning Curve on Roundabout Safety Performance, and • 5.6 References. Chapter 5 is written for individuals who want to under- stand how the models were developed and how driver learn- ing curve was evaluated. Chapter 6 is a shorter presentation that discusses the final models themselves and the findings from the effect of driver learning-curve evaluation. 5.1 Planning-Level Crash Prediction Models 5.1.1 Introduction This section describes the safety prediction models for roundabouts developed for planning or network screening applications. Models were developed with only annual average daily traffic (AADT) predictor variables, and in some cases, additional models were developed including variables from a select list of additional variables that may be known at the planning stage of roundabout construction. Each model pre- dicts the average crash frequency of one roundabout, inclusive of crashes within the circulating roadway and those crashes on the roundabout legs that are considered related to the roundabout (i.e., the leg geometry or operation was a likely contributing factor in the crash). The model development builds on the effort for the mod- els estimated for NCHRP Report 572 (Rodegerdts et al., 2007) and NCHRP Report 672 (Rodegerdts et al., 2010), which were based on a limited sample of 90 roundabouts. The majority of these were single-lane roundabout sites in urban or sub- urban environments, as shown in Figure 5-1. The NCHRP Report 572 models, which were for rural and urban environments combined, had total entering AADT as the only variable and had a constant AADT parameter (expo- nent) and separate multipliers for roundabouts with various combinations of numbers of legs and circulating lanes. The expectation was that the earlier models could be improved substantially by potentially • Estimating separate models for urban and rural round- abouts, • Estimating separate models (with separate AADT param- eters) for single and multilane roundabouts, • Including more variables, and • Including major and minor road AADT as separate terms. This section is organized into the following subsections: • Predictive Model Framework; • Database Summary; and • Model Development. 5.1.2 Predictive Model Framework The intersection-level models for planning or network screening applications include two categories for area type, two categories for number of circulating lanes, and three severity categories. In combination, there were 12 (= 2 × 2 × 3) potential intersection-level models. These models are identified in the following list: • Rural single lane, total; • Rural single lane, KABC; C H A P T E R 5

54 characteristics under consideration for the planning and net- work screening level models as well as the crash characteris- tics in terms of crashes per million entering vehicles (MEV). It is apparent that urban/rural, number of legs, and number of circulating lanes impact the crash rate, with higher crash rates in rural areas, at four-leg roundabouts, and at multilane roundabouts. Caution should be taken, however, in inter- preting the crash rates. Crash frequency may not have a linear relationship with traffic volumes. 5.1.4 Model Development The planning-level models were developed in the follow- ing six steps: 1. Investigate if separate models are possible for all four site types defined by area type and the number of circulating lanes while withholding a validation sample, 2. Develop preliminary models using only AADT and site- type variables, 3. Develop preliminary models with additional predictor variables, 4. Validate the preliminary models developed in Steps 2 and 3, 5. Investigate alternate model forms for the models and over- dispersion parameters, and 6. Develop the final models using all sites. In Step 1, to test whether it would be possible to estimate models for all site types while withholding a validation sam- ple, a random sample of 20 sites was taken out of the data, and simple models using only major and minor road AADTs and the number of legs were estimated. This was done repeat- edly for each site type, and the randomization of the process ensured a different group of sites for each model estimated. This served to provide some assurance that the success of model estimation was not dependent on the sites used, and the models were not highly dependent on which 20 sites were withheld for validation. The results of this process were reassuring in that models were successfully estimated, and the parameter estimates remained consistent. It was also concluded that the number of rural multilane sites was small, resulting in larger standard errors for model coefficients than desired, and thus these sites would be combined with the rural single-lane sites for further model development. In addition to traffic volumes, possible predictor variables considered the following: • Area type (urban versus rural), • Ramp terminal versus nonramp terminal, • Rural single lane, property damage–only (PDO); • Rural multilane, total; • Rural multilane, KABC; • Rural multilane, PDO; • Urban single lane, total; • Urban single lane, KABC; • Urban single lane, PDO; • Urban multilane, total; • Urban multilane, KABC; and • Urban multilane, PDO. Bicycle and pedestrian–involved crashes are not included in the crash types modeled because of the paucity of these crashes. The predicted average crash frequency for each site is computed using a prediction model. The general form of the generalized linear safety predictive model is shown below. Equation 5-1 1 1 . . .N exp MAJAADT MINAADT expa b c x nxn= ( )β + β where N = predicted average crash frequency, crashes/yr; MAJAADT = AADT on the major road; MINAADT = AADT on the minor road; X1 . . . Xn = a series of predictor variables; a, b, c, β1 . . . βn = are estimated parameters; and k = the estimated overdispersion parameter. Alternative model forms were considered but not adopted. 5.1.3 Database Summary The database used for this modeling effort included 355 sites. Tables 5-1, 5-2, and 5-3 summarize the roundabout Figure 5-1. Summary characteristics of roundabouts used for models for NCHRP Report 572.

Area Type Circulating Lanes Number of Legs Number of Sites Inscribed Circle Diameter, ft. Sites at a Ramp Terminal Major Road Speed Limit Minor Road Speed Limit Major Road AADT Minor Road AADT Average Stdev Average Stdev Average Stdev Average Stdev Average Stdev Rural 1 3 19 120.0 21.1 0 36.3 9.7 32.9 10.8 8,128 4,033 4,017 2,658 4 58 134.1 27.1 5 44.4 8.7 39.8 10.3 6,988 3,414 3,101 2,027 2 3 7 173.3 39.8 0 47.5 4.2 37.9 9.8 13,110 7,196 5,391 5,198 4 21 172.7 30.7 3 46.5 7.3 39.9 6.5 13,123 6,358 6,298 3,808 Urban 1 3 50 118.5 20.7 1 33.9 8.2 31.4 8.0 7,267 4,100 4,348 2,917 4 108 118.7 25.5 6 34.9 31.7 31.7 8.1 7,882 3,892 3,890 2,455 2 3 28 178.2 54.1 1 37.8 9.0 34.8 8.2 10,531 5,672 5,228 3,914 4 64 184.4 58.7 14 36.6 7.7 31.7 6.6 1,2229 5,835 5,992 4,417 Table 5-1. Site characteristics of planning-level roundabout data. Area Type Circulating Lanes Number of Legs Number of Sites Inscribed Circle Diameter, ft. Major Road Speed Limit Minor Road Speed Limit Major Road AADT Minor Road AADT min max min max min max min max min max Rural 1 3 19 65 152 25 55 10 55 1,000 15,400 409 8,767 4 58 58 236 25 55 15 55 1,000 17,560 280 9,198 2 3 7 97 206 40 50 25 50 1,000 22,050 476 14,095 4 21 110 268 30 55 25 50 5,010 26,366 390 13,750 Urban 1 3 50 64 156 10 55 10 55 675 17,369 416 14,000 4 108 59 259 20 55 15 55 1,000 19,733 340 11,239 2 3 28 106 358 20 50 20 50 1,000 21,768 500 15,108 4 64 107 426 20 55 20 55 1,000 28,333 500 19,371 Table 5-2. Validity ranges of planning-level roundabout data.

56 • Number of circulating lanes, • Number of legs, • Number of entering lanes per leg, • Number of exiting lanes per leg, • Posted speed limit, and • Inscribed circle diameter. The development of the models is discussed in detail in the following subsections. Each subsection describes the initial model developed, the validation of the model, and the final model developed using the initial estimation and the valida- tion data. 5.1.4.1 Models for Rural Roundabouts Validation Models for Rural Roundabouts. The models developed for rural roundabouts for validation took the model form shown below. Equation 5-2 N exp MAJAADT MINAADT exp a STATE b c d NUMBERLEGS e CIRCLANES f MAJSPD = ( ) + × + × + × where N = predicted average crash frequency, crashes/yr; STATE = an additive intercept term dependent on the geographic state a roundabout resides in; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4-leg; CIRCLANES = 1 if a single-lane roundabout; 0 if more than 1 circulating lane; and MAJSPD = posted speed on the major road (mph). The models with only traffic volume and site-type vari- ables did not include the variable MAJSPD. Although it is perhaps more intuitive that the posted speed variable should be treated as categorical, it was found in the modeling that the results were more consistent when this is treated as a continu- ous variable. Tables 5-4 to 5-6 present the parameter estimates and stan- dard errors (in brackets) for the models developed for each crash type, as well as the negative binomial overdispersion parameters, k. Models were estimated with and without the Area Type Circulating Lanes Number of Legs Number of Sites Total Crashes Per MEV KABC Crashes Per MEV PDO Crashes Per MEV Average Stdev Average Stdev Average Stdev Rural 1 3 19 0.47 0.84 0.08 0.13 0.38 0.74 4 58 0.49 0.47 0.10 0.11 0.39 0.43 2 3 7 0.95 0.87 0.09 0.11 0.85 0.86 4 21 1.20 1.12 0.18 0.19 1.02 0.96 Urban 1 3 50 0.41 0.73 0.12 0.31 0.29 0.46 Table 5-3. Crash characteristics of planning-level roundabout data. Model ID a STATE b c d e f k TOT1 -6.3159 (1.5168) no 0.4502 (0.1669) 0.4907 (0.1200) -0.8021 (0.2283) -1.0728 (0.2198) n/a 0.6202 (0.1101) TOT2 -6.7810 (1.6080) yes 0.5336 (0.1791) 0.4714 (0.1181) -0.7987 (0.2340) -0.8977 (0.2336) n/a 0.5353 (0.0969) TOT3 -10.3730 (2.0933) no 0.8145 (0.2067) 0.4529 (0.1220) -0.8047 (0.2406) -0.6840 (0.2528) 0.0237 (0.0117) 0.5622 (0.1028) TOT4 -10.6281 (2.1861) yes 0.8650 (0.2178) 0.4381 (0.1218) -0.7875 (0.2437) -0.5532 (0.2656) 0.0199 (0.0127) 0.4946 (0.0925) Table 5-4. Planning-level validation models: total crash models for rural roundabouts.

57 STATE term to assess the impact on model accuracy. In the tables, the column, STATE, indicates whether or not the vari- able was included in the model. The parameter estimates for STATE are not included in the tables because the purpose of the validation models is to assess how well the model can be estimated to the validation dataset so the intercept terms for each state in the data being modeled are immaterial. It was observed, in any case, that the intercept terms by state are often not statistically different from each other. The next step was to validate the models using the sites kept for validation. Each model was calibrated using the Highway Safety Manual (HSM) calibration procedure prior to calculat- ing goodness-of-fit statistics. The models including MAJSPD were not tested; their improvement over the models without MAJSPD were relatively small, as evidenced by the modest reduction in the estimated overdispersion parameters, and it was desired to keep all models as similar as possible for com- parison purposes. Table 5-7 provides the overall measures calculated. For each model, the total number of observed and predicted crashes prior to calibration is provided along with the calibration fac- tor, its variance V(C), and its coefficient of variation CV(C). Also provided are the goodness-of-fit statistics calculated after applying the calibration factor, including the modified R2, the mean absolute deviation, and the calibrated over dispersion parameter. Model ID a STATE b c d e f k FI1 -11.5905 (2.3712) no 0.9696 (0.2491) 0.3140 (0.1434) -1.0336 (0.2998) -0.4564 (0.2847) n/a 0.5735 (0.1638) FI2 -11.1043 (2.5545) yes 0.8737 (0.2634) 0.3838 (0.1407) -1.1716 (0.3112) -0.5540 (0.2952) n/a 0.4301 (0.1401) Table 5-5. Planning-level validation models: fatal-and-injury crash models for rural roundabouts. Model ID a STATE b c d e f k PDO1 -5.9415 (1.6969) no 0.3494 (0.1882) 0.5386 (0.1367) -0.7886 (0.2592) -1.1965 (0.2496) n/a 0.8045 (0.1474) PDO2 -6.9746 (1.7487) yes 0.5156 (0.1984) 0.4975 (0.1328) -0.7605 (0.2566) -0.9835 (0.2539) n/a 0.6373 (0.1217) PDO3 -10.2883 (2.3891) no 0.7401 (0.2386) 0.4718 (0.1410) -0.7843 (0.2741) -0.7891 (0.2908) 0.0247 (0.0132) 0.7487 (0.1422) PDO4 -11.3352 (2.4390) yes 0.8998 (0.2488) 0.4672 (0.1389) -0.7445 (0.2681) -0.6084 (0.2926) 0.0202 (0.0141) 0.5969 (0.1186) Table 5-6. Planning-level validation models: PDO crash models for rural roundabouts. Model Name Total Observed Crashes Total Predicted Crashes Calibration Factor Modiied R2 Mean Absolute Deviation Overdispersion Value V(C) CV(C) TOT1 206 180.66 1.14 0.34 8.54 1.16 0.23 0.42 TOT2 206 244.94 0.84 0.37 8.44 1.20 0.13 0.43 FI1 38 45.20 0.84 0.67 1.38 0.38 0.05 0.26 FI2 38 48.26 0.79 0.64 1.38 0.36 0.04 0.26 PDO1 168 137.93 1.22 0.29 7.60 1.27 0.31 0.46 PDO2 168 196.01 0.86 0.34 7.46 1.33 0.16 0.47 Table 5-7. Validation results for planning-level rural roundabout models: overall measures.

58 The results in Table 5-7 show little difference between the models estimated with or without the STATE variable. The cali- brated overdispersion parameters are larger than for the origi- nal models for total and PDO crashes but are a reasonable size. For fatal-and-injury (FI) crashes, they are actually smaller. Table 5-8 provides two measures taken from the CURE plots developed for each model. A CURE plot is a graph of the cumu- lative residuals (observed minus predicted crashes) against a variable of interest sorted in ascending order (e.g., major road traffic volume). CURE plots provide a visual representation of GOF over the range of a given variable, and help identify potential concerns such as the following: • Long trends: Long trends in the CURE plot (increasing or decreasing) indicate regions of bias that should be recti- fied through improvement to the safety performance factor (SPF) either by the addition of new variables or by a change of functional form. • Percent exceeding the confidence limits: Cumulative resid- uals outside the confidence limits indicate a poor fit over that range in the variable of interest. Cumulative residuals frequently outside the confidence limits indicate notable bias in the SPF. • Vertical changes: Large vertical changes in the CURE plot are potential indicators of outliers, which require further examination. CURE plots were constructed for MAJAADT, MINAADT, MAJSPD, MINSPD (posted speed on minor road), and the predicted number of crashes at each site (FITTED VALUE). These measures include the maximum absolute value of the CURE plots deviation from 0 and the percentage of observa- tions that are outside the two standard deviation confidence boundaries on the CURE plot. The results in Table 5-8 indicate that for total crashes the results are close between the models with and without the STATE variable, with some preference for the model with- out STATE. This is reversed for PDO crashes. Overall, the results are reasonable with 5% or fewer of the observations outside the confidence boundaries, which would be 1 out of the 20 observations. For FI crashes, the two models perform similarly, but the overall fit is worse than for the other crash types, with up to 60% of sites (12 of 20 observations) outside the confidence boundaries for MAJAADT. The overall assessment of the validation of the rural roundabout models is that the model form selected for the validation models performs reasonably well and could be retained for developing the final models. The models includ- ing the STATE term perform similarly to those without. Considering that the final models will need to be calibrated to any jurisdiction prior to application, and the relative low frequency of sites per jurisdiction it seemed reasonable to not include STATE in the final model development. Investigation of Nonlinear Models and Nonconstant Over dispersion Parameters for Rural Roundabouts. In this step, it was investigated whether improvements could be made to the model form to improve the model fit by consider- ing nonlinear models. The CURE plots developed for the vali- dation models were used to assess ranges of the independent variables where the model is either over- or underpredicting crashes and then to discern a possible alternate model form. This investigation did not reveal any preferable model forms. It was also investigated whether the overdispersion param- eter should be modeled as a function of site characteristics or a constant value. Several models were considered and attempted before ultimately selecting a function that is dependent on the total entering AADT. To assess any improvement in fit, the Akaike information criterion (AIC) values of the validation models were compared to the same model calibrated using an overdispersion parameter that is expressed as the following function: Equation 5-3overdispersion expa b ln AADT= ( )+ × Model Name Variable Name Max Absolute CURE Deviation % CURE Deviation TOT1 [MAJSPD] 23.89 5% TOT1 [MINSPD] 29.40 5% TOT1 [MAJAADT] 35.88 0% TOT1 [MINAADT] 46.09 0% TOT1 FITTED VALUE 34.75 0% TOT2 [MAJSPD] 33.32 5% TOT2 [MINSPD] 27.42 5% TOT2 [MAJAADT] 37.65 5% TOT2 [MINAADT] 44.79 5% TOT2 FITTED VALUE 35.99 0% FI1 [MAJSPD] 5.55 5% FI1 [MINSPD] 5.78 5% FI1 [MAJAADT] 11.45 60% FI1 [MINAADT] 6.97 5% FI1 FITTED VALUE 8.26 20% FI2 [MAJSPD] 5.45 5% FI2 [MINSPD] 5.54 5% FI2 [MAJAADT] 11.19 50% FI2 [MINAADT] 7.45 10% FI2 FITTED VALUE 8.21 20% PDO1 [MAJSPD] 34.95 5% PDO1 [MINSPD] 29.29 5% PDO1 [MAJAADT] 25.00 0% PDO1 [MINAADT] 39.04 5% PDO1 FITTED VALUE 30.75 5% PDO2 [MAJSPD] 31.77 5% PDO2 [MINSPD] 26.36 0% PDO2 [MAJAADT] 28.31 5% PDO2 [MINAADT] 37.87 5% PDO2 FITTED VALUE 29.25 0% Table 5-8. Validation results for planning-level rural roundabout models: CURE plot results.

59 The AIC measure reflects the overall fit of the model and penalizes for the addition of parameters, and thus selects a model that fits well but has a minimum number of param- eters. AIC is not typically used as a goodness-of-fit measure but can be used to compare the relative fit of alternate models. The lower value of AIC is preferred. Table 5-9 shows the results for total and FI crash models. The results from models for PDO crashes were not estimated because of the expectation that they would closely follow the total crash results. For both models, the results indicate that the overdispersion parameter increases as entering AADT increases; however, the parameter estimates for total crashes are not statistically significant, while the AADT parameter is significant at the 90th percent confidence limit for FI crashes. When comparing two models to assess whether the addi- tion of a parameter significantly increases the model fit, the AIC value can be used. A minimum reduction of 10 is often considered necessary to conclude that the additional param- eter should be kept. Otherwise the model may be overfit to the data. The results show that for total crashes the over- dispersion as a function increases the AIC value by 1.9 and for FI crashes reduces the AIC value of 1.9. Therefore, it was concluded that the overdispersion parameter should be kept constant for rural roundabouts. Final Models for Rural Roundabouts. The final models developed for rural roundabouts, which were estimated with the combined estimation and validation datasets, took the model from shown below. For the final models, the vari- able for major road speed limit was not included due to its low statistical significance. Although the validation sample indicated that including the STATE variable did not signifi- cantly improve model performance it was found that when estimating the models with the combined dataset (initial esti- mation plus validation data) the exclusion of STATE resulted in a large standard error for the major road AADT variable. For this reason, STATE was included in the model. Equation 5-4 N exp MAJAADT MINAADT exp a STATE b c d NUMBERLEGS e CIRCLANES = ( ) + × + × where N = predicted average crash frequency, crashes/yr; STATE = an additive intercept term dependent on the geographic state a roundabout resides in; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4-legs; and CIRCLANES = 1 if a single-lane roundabout; 0 if more than 1 circulating lane. Table 5-10 presents the parameter estimates and stan- dard errors (in brackets) for the models developed for each crash type, as well as the negative binomial overdispersion parameter, k. 5.1.4.2 Models for Urban Single-Lane Roundabouts Validation Models for Urban Single-Lane Roundabouts. The models developed for urban single-lane roundabouts for Crash Type AIC Constant AIC Function a b Total 721.3 723.2 -1.3385 (2.8289) 0.1057 (0.2995) FI 411.9 410.0 -11.9886 (6.3555) 1.1806 (0.6530) Table 5-9. Planning-level rural roundabout models: comparison of alternate overdispersion specifications. Model ID a b c d e k TOTAL -5.3299 (1.5425) 0.3356 (0.1767) 0.5142 (0.1165) -0.6854 (0.2244) -0.9375 (0.2309) 0.6292 (0.1005) FI -10.4848 (2.2682) 0.7756 (0.2356) 0.4239 (0.1312) -1.0080 (0.2827) -0.5506 (0.2658) 0.4424 (0.1288) PDO -5.4115 (1.6602) 0.2980 (0.1939) 0.5463 (0.1280) -0.7104 (0.2430) -1.0192 (0.2488) 0.7284 (0.1192) Table 5-10. Final planning-level models for rural roundabouts.

60 validation took the model form below. No other indepen- dent variables were successfully included in any of the models attempted. Equation 5-5 N exp MAJAADT MINAADT expa STATE b c d NUMBERLEGS= ( )+ × where N = predicted average crash frequency, crashes/yr; STATE = an additive intercept term dependent on the geographic state a roundabout resides in; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; and NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4-legs. Tables 5-11 through 5-13 present the parameter estimates and standard errors (in brackets) for the models developed for each crash type. Models were estimated with and without the STATE term to assess the impact on model accuracy. In the tables, the STATE column indicates whether or not the variable was included in the model. The parameter estimates for STATE are not included in the table because the purpose of the validation models is to assess how well the model is esti- mated to the validation dataset, making the intercept terms for each state in the data being modeled immaterial. It was observed that the intercept terms by state are often not statis- tically different from each other. Standard error estimates for MAJAADT are higher than desired, but the magnitude and direction of effect seem reasonable. The next step was to validate the models using the sites kept for validation. Each model was calibrated using the HSM calibration procedure prior to calculating goodness-of-fit statistics. Table 5-14 provides the overall measures calculated. For each model, the total number of observed and predicted crashes prior to calibration is provided along with the cali- bration factor, its variance V(C), and its coefficient of varia- tion CV(C). Also provided are the goodness-of-fit statistics calculated after applying the calibration factor, including the modified R2, mean absolute deviation, and the calibrated overdispersion parameter. The results in Table 5-14 show little difference between the models estimated with or without the STATE variable. The calibrated overdispersion parameters are actually smaller than for the original models. Table 5-15 provides two measures taken from the CURE plots developed for each model. CURE plots were con- structed for MAJAADT, MINAADT, MAJSPD, MINSPD (posted speed on minor road) and the predicted number of crashes at each site (FITTED VALUE). Although posted speed limit was not included in the models for urban single-lane roundabouts, it was of interest to see if there was any bias in the predictions across those variables. These measures include the maximum absolute value of the CURE plots deviation from 0 and the percentage of observations that are outside the two standard deviation confidence boundaries on the CURE plot. The results in Table 5-15 indicate that for total and PDO crashes the model with STATE performs slightly better, while for FI crashes the model without STATE performs slightly better. There is some bias evident in the models, Model ID a STATE b c d k TOT1 -4.3779 (1.2131) no 0.2631 (0.1760) 0.3333 (0.1345) -0.7149 (0.1745) 0.6402 (0.0968) TOT2 -4.2115 (1.1713) yes 0.2568 (0.1634) 0.3553 (0.1310) -0.5799 (0.1657) 0.5027 (0.0808) Table 5-11. Planning-Level validation models: total crash models for urban single-lane roundabouts. Model ID a STATE b c d k FI1 -7.0464 (1.4146) no 0.3901 (0.2030) 0.2939 (0.1503) -0.6039 (0.2001) 0.3636 (0.1097) FI2 -6.1984 (1.4304) yes 0.2904 (0.2021) 0.3330 (0.1529) -0.5068 (0.2014) 0.3116 (0.1006) Table 5-12. Planning-level validation models: FI crash models for urban single-lane roundabouts.

61 with a maximum cure deviation of 25%, which would be 5 of the 20 observations. The overall assessment of the validation of the urban single- lane roundabout models is that the model form selected for the validation models is performing reasonably well, although some bias is evidenced across the AADT and posted speed variables. The models including the STATE term perform similarly to those without. Considering that the final models will need to be calibrated to any jurisdiction prior to applica- tion and the relative low frequency of sites per jurisdiction, it seemed reasonable to not include STATE in the final model development. Investigation of Nonlinear Models and Nonconstant Over dispersion Parameters for Urban Single-Lane Round- abouts. This step investigated whether improvements could be made to the model form to improve the model fit by con- sidering nonlinear models. The CURE plots developed for the validation models were used to assess ranges of the inde- pendent variables where the model is either over- or under- predicting crashes and then to discern a possible alternate model form. This investigation did not reveal any preferable model forms. It was also investigated whether the overdispersion param- eter should be modeled as a function of site characteristics as opposed to a constant value. The model in Equation 5-3 was again selected. Table 5-16 shows the results for total and FI crash models. For both models the results indicate that the overdispersion parameter decreases as entering AADT increases, which is opposite to the indication for rural roundabouts. Consider- ing the AIC values, the results show that for total crashes the overdispersion as a function decreases the AIC value by 8.3 and for FI crashes decreases the AIC value by 0.3. Therefore, it is concluded that the overdispersion parameter should be kept constant for urban single-lane roundabouts. Final Models for Urban Single-Lane Roundabouts. The final models developed for urban single-lane roundabouts, which were estimated with the combined estimation and validation datasets, are of the form show below. Equation 5-6 N exp MAJAADT MINAADT expa b c d NUMBERLEGS= ( )× where N = predicted average crash frequency, crashes/yr; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; and NUMBERLEGS = 1 if a 3-leg roundabout; 0 if a 4-leg roundabout. Model ID a STATE b c d k PDO1 -4.9006 (1.3509) no 0.2452 (0.1973) 0.3613 (0.1512) -0.7993 (0.1925) 0.7696 (0.1167) PDO2 -4.3879 (1.2836) yes 0.2469 (0.1800) 0.3656 (0.1453) -0.6467 (0.1814) 0.5699 (0.0948) Table 5-13. Planning-level validation models: PDO crash models for urban single-lane roundabouts. Model Name Total Observed Crashes Total Predicted Crashes Calibration Factor Modiied R2 Mean Absolute Deviation Overdispersion Value V(C) CV(C) TOT1 203 202.23 1.00 0.40 5.69 0.35 0.05 0.22 TOT2 203 356.95 0.57 0.39 5.77 0.36 0.02 0.22 FI1 46 42.19 1.09 0.82 1.39 0.16 0.05 0.20 FI2 46 56.04 0.82 0.80 1.40 0.18 0.03 0.21 PDO1 157 162.59 0.97 0.26 5.48 0.45 0.07 0.26 PDO2 157 294.11 0.53 0.26 5.51 0.47 0.02 0.27 Table 5-14. Validation results for planning-level urban single-lane roundabout models: CURE plot results.

62 Model Name Variable Name Max Absolute CURE Deviation % CURE Deviation TOT1 [MAJSPD] 37.65 20% TOT1 [MINSPD] 37.65 15% TOT1 [MAJAADT] 20.32 5% TOT1 [MINAADT] 45.81 25% TOT1 FITTED VALUE 20.23 5% TOT2 [MAJSPD] 35.09 10% TOT2 [MINSPD] 35.09 10% TOT2 [MAJAADT] 20.96 0% TOT2 [MINAADT] 47.37 20% TOT2 FITTED VALUE 20.09 0% FI1 [MAJSPD] 6.16 5% FI1 [MINSPD] 6.44 20% FI1 [MAJAADT] 6.58 15% FI1 [MINAADT] 7.05 10% FI1 FITTED VALUE 5.69 20% FI2 [MAJSPD] 6.60 10% FI2 [MINSPD] 6.94 20% FI2 [MAJAADT] 7.25 15% FI2 [MINAADT] 7.13 10% FI2 FITTED VALUE 5.95 20% PDO1 [MAJSPD] 32.53 5% PDO1 [MINSPD] 32.53 5% PDO1 [MAJAADT] 22.88 5% PDO1 [MINAADT] 46.95 20% PDO1 FITTED VALUE 24.32 5% PDO2 [MAJSPD] 33.09 5% PDO2 [MINSPD] 33.09 5% PDO2 [MAJAADT] 22.42 5% PDO2 [MINAADT] 47.10 25% PDO2 FITTED VALUE 23.44 5% Table 5-15. Validation results for urban single-lane roundabouts: CURE plot results. Crash Type AIC Constant AIC Function a b Total 1008.3 1000.0 -6.1839 (2.0364) -0.7254 (0.2203) FI 569.5 569.2 8.1710 (5.0191) -0.9794 (0.5405) Table 5-16. Planning-level urban single-lane roundabout models: comparison of alternate overdispersion specifications.

63 Table 5-17 presents the parameter estimates and standard errors (in brackets) for the models developed for each crash type, as well as the negative binomial overdispersion param- eters, k. 5.1.4.3 Models for Urban Multilane Roundabouts Validation Models for Urban Multilane Roundabouts. The models developed for urban multilane roundabouts for validation took the model form in Equation 5-5 for total and PDO crashes and Equation 5-7 for FI crashes. No other inde- pendent variables were successfully included in any of the models attempted. Equation 5-7 N exp MAJAADT MINAADT expa STATE b d NUMBERLEGS( )= + ( )+ × where N = predicted average crash frequency, crashes/yr; STATE = an additive intercept term dependent on the geographic state a roundabout resides in; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; and NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4 legs. Tables 5-18 to 5-20 present the parameter estimates and standard errors (in brackets) for the models developed for each crash type. Models were estimated with and without the STATE term to assess the impact on model accuracy. In the tables, the STATE column indicates whether or not the variable was included in the model. The parameter estimates for STATE are not included in the table because the purpose of the validation models is to assess how well the model is estimated to the validation dataset, so the intercept terms for each state in the data being modeled are immaterial. It was observed that the intercept terms by state are often not sta- tistically different from each other. The parameter estimates for the AADT variables change noticeably depending on whether STATE is included in the models, and the over- dispersion parameter is significantly smaller when STATE is included. The next step was to validate the models using the sites kept for validation. Each model was calibrated using the HSM calibration procedure prior to calculating goodness-of-fit statistics. Table 5-21 provides the overall measures calculated. For each model, the total number of observed and predicted crashes prior to calibration is provided along with the cali- bration factor, its variance V(C), and its coefficient of varia- tion CV(C). Also provided are the goodness-of-fit statistics calculated after applying the calibration factor, including the modified R2, mean absolute deviation, and the calibrated overdispersion parameter. The results in Table 5-21 show little difference between the models estimated with or without the STATE variable. There may be some preference for the without STATE models, which have a lower calibrated overdispersion parameter for total and PDO crashes. Table 5-22 provides two measures taken from the CURE plots developed for each model. CURE plots were constructed for MAJAADT, MINAADT, MAJSPD, MINSPD (posted speed on minor road), and the predicted number of crashes at each site (FITTED VALUE). Although posted speed limit was not included in the models for urban multilane roundabouts, it was of interest to see if there was any bias in the predictions across those variables. These measures include the maximum absolute value of the CURE plots deviation from 0 and the percentage of observations that are outside the two standard deviation confidence boundaries on the CURE plot. Model ID a b c d k TOT -5.6049 (1.0533) 0.3274 (0.1479) 0.3960 (0.1157) -0.8681 (0.1489) 0.5030 (0.0727) FI -8.6597 (1.3337) 0.5271 (0.1886) 0.3505 (0.1374) -0.7317 (0.1822) 0.3290 (0.0908) PDO -5.5319 (1.1659) 0.2653 (0.1650) 0.4294 (0.1304) -0.9260 (0.1656) 0.6064 (0.0878) Table 5-17. Final planning-level models for urban single-lane roundabouts. Model ID a STATE b c d k TOT1 -8.9177 (1.5905) no 0.8297 (0.2082) 0.3348 (0.1441) -0.9003 (0.2590) 0.8793 (0.1552) TOT2 -9.8548 (1.4215) yes 1.0873 (0.1623) 0.1743 (0.1147) -0.9154 (0.1947) 0.3729 (0.0804) Table 5-18. Planning-level validation models: total crash models for urban multilane roundabouts.

64 The results in Table 5-22 indicate that for PDO crashes the model with STATE performs slightly better, while for total and FI crashes the models, with and without STATE per- form similarly with no consistency in which performs bet- ter across the different variables tested. There is some bias evident in the models with a maximum cure deviation of 35% for MAJAADT with the FI models, which would be 7 of the 20 observations. Potential changes to the model form were considered to remove the bias as discussed in the next subsection. The overall assessment of the validation of the urban multilane roundabout models is that the model form selected for the validation models is performing well, although some bias is evidenced across the AADT and posted speed variables. The models including the STATE term perform similarly to those without. Considering that the final models will need to be calibrated to any jurisdiction prior to application, and the relative low frequency of sites per jurisdiction, it seemed rea- sonable to not include STATE in the final model development. Investigation of Nonlinear Models and Nonconstant Overdispersion Parameters for Urban Multilane Round- abouts. This step investigated whether improvements could be made to the model form to improve the model fit by considering nonlinear models. The CURE plots devel- oped for the validation models were used to assess ranges of the independent variables where the model is either over- or underpredicting crashes and to discern a possible alternate model form. This investigation did not reveal any preferable model forms. It was also investigated whether the overdispersion param- eter should be modeled as a function of site characteristics as Model ID a STATE b c d k FI1 -11.3619 (1.9513) yes 1.1409 (0.1987) n/a -0.6114 (0.2543) 0.5081 (0.1491) FI2 -10.4258 (2.0338) no 1.0668 (0.2077) n/a -0.6040 (0.2387) 0.3294 (0.1134) Table 5-19. Planning-level validation models: FI crash models for urban multilane roundabouts. Model ID a STATE b c d k PDO1 -9.3265 (1.6915) no 0.8168 (0.2175) 0.3778 (0.1511) -0.9554 (0.2767) 0.9862 (0.1744) PDO2 -10.4621 (1.4066) yes 1.0886 (0.1587) 0.2263 (0.1133) -1.0064 (0.1912) 0.3218 (0.0731) Table 5-20. Planning-level validation models: PDO crash models for urban multilane roundabouts. Model Name Total Observed Crashes Total Predicted Crashes Calibration Factor Modiied R2 Mean Absolute Deviation Overdispersion Value V(C) CV(C) TOT1 804 647.46 1.24 0.55 28.40 2.07 0.62 0.63 TOT2 804 732.99 1.10 0.56 28.66 2.47 0.58 0.69 FI1 138 98.64 1.40 0.68 4.27 0.86 0.27 0.37 FI2 138 120.13 1.15 0.66 4.35 0.86 0.19 0.37 PDO1 666 648.79 1.03 0.47 25.75 1.59 0.35 0.57 PDO2 666 630.57 1.06 0.54 24.90 2.86 0.66 0.77 Table 5-21. Validation results for planning-level urban multilane roundabout models: CURE plot results.

65 Model Name Variable Name Max Absolute CURE Deviation % CURE Deviation TOT1 [MAJSPD] 107.59 0% TOT1 [MINSPD] 137.84 5% TOT1 [MAJAADT] 169.32 5% TOT1 [MINAADT] 85.21 0% TOT1 FITTED VALUE 196.25 5% TOT2 [MAJSPD] 95.08 5% TOT2 [MINSPD] 140.25 5% TOT2 [MAJAADT] 158.93 0% TOT2 [MINAADT] 107.93 5% TOT2 FITTED VALUE 202.08 5% FI1 [MAJSPD] 15.08 0% FI1 [MINSPD] 15.08 0% FI1 [MAJAADT] 26.98 35% FI1 [MINAADT] 20.62 5% FI1 FITTED VALUE 30.95 10% FI2 [MAJSPD] 15.82 5% FI2 [MINSPD] 15.82 5% FI2 [MAJAADT] 28.13 35% FI2 [MINAADT] 21.88 0% FI2 FITTED VALUE 31.91 10% PDO1 [MAJSPD] 101.06 5% PDO1 [MINSPD] 118.75 5% PDO1 [MAJAADT] 141.67 5% PDO1 [MINAADT] 83.11 5% PDO1 FITTED VALUE 194.49 10% PDO2 [MAJSPD] 89.59 5% PDO2 [MINSPD] 123.85 0% PDO2 [MAJAADT] 131.73 0% PDO2 [MINAADT] 78.41 0% PDO2 FITTED VALUE 165.06 0% Table 5-22. Validation results for urban multilane roundabouts: CURE plot results. opposed to a constant value. The model in Equation 5-3 was again selected. Table 5-23 shows the results for total and FI crash models. For both models, the results indicate that the overdispersion parameter decreases as entering AADT increases, which is opposite to rural roundabouts but consistent with urban single-lane roundabouts. Considering the AIC values, the results show that for total crashes the overdispersion as a function model decreases the AIC value by 2.1 and for FI crashes increases the AIC value of 0.8. Therefore, it is concluded that the overdispersion parameter should be kept constant for urban multilane roundabouts. Final Models for Urban Multilane Roundabouts. The final models developed for urban multilane roundabouts, which were estimated with the combined estimation and validation datasets, are of the form shown in Equation 5-6. Equation 5-8 N exp MAJAADT MINAADT expa b c d NUMBERLEGS= ( )×

66 where N = predicted average crash frequency, crashes/yr; MAJAADT = total entering AADT on major road; MINAADT = total entering AADT on minor road; and NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4 legs. Table 5-24 presents the parameter estimates and stan- dard errors (in brackets) for the models developed for each crash type, as well as the negative binomial overdispersion parameters, k. 5.2 Intersection-Level Crash Prediction Models for Design 5.2.1 Introduction This section describes the development of a set of models for predicting a roundabout’s average crash frequency in total and by crash type or severity. The models are intended to be used to evaluate alternative roundabout features or design elements, as may be considered during the preliminary design or final design stages of the project development process. The models are calibrated to predict crashes of all types except for vehicle–pedestrian and vehicle–bicycle crashes. There were insufficient vehicle–pedestrian crashes represented in the assembled database to support the development of a model for predicting the frequency of these crashes. Similarly, there were insufficient vehicle–bicycle crashes in the database to support the development of a model for predicting vehicle– bicycle crash frequency. Seven subsections follow this introductory section. The titles of these subsections are provided in the following list. • 5.2.2 Predictive Model Framework, • 5.2.3 Crash Frequency Prediction Database Summary, • 5.2.4 Crash Frequency Prediction Model Development, • 5.2.5 Crash Severity Prediction Database Summary, • 5.2.6 Crash Severity Prediction Model Development, • 5.2.7 Crash-Type Prediction Database Summary, and • 5.2.8 Crash-Type Distribution Table Development. The first subsection provides some background informa- tion related to the development of roundabout safety predic- tion models. The second and third subsections describe the data and the process, respectively, used to calibrate the crash frequency prediction models. The fourth and fifth subsections describe the data and the process, respectively, used to develop the severity distribution models. The sixth and seventh sub- sections describe the data and the process, respectively, used to develop the crash-type prediction table. 5.2.2 Predictive Model Framework Crash severity is described in Section 5.2 using the follow- ing five crash severity categories: fatal (K), incapacitating- injury (A), nonincapacitating-injury (B), possible-injury (C), and PDO (O) crashes. This severity description is often referred to as the “KABCO” scale. The FI crash severity com- bination discussed in Section 5.2 includes the categories: K, A, B, and C. As described in Section 3.1.2, the objective of the model development activities was to develop a set of models for predicting the average crash frequency (by severity category) associated with a specific roundabout. One subset of models is used to predict the average crash frequency. A second sub- set of models is used to predict the crash severity distribution. Within the aforementioned first subset of models (those predicting average crash frequency), one group of models was calibrated to predict FI crash frequency. A second group Crash Type AIC (Constant) AIC (Function) a b Total 781.4 779.3 4.6696 (2.2967) -0.4835 (0.2391) FI 459.2 460.0 4.5357 (4.1700) -0.5154 (0.4269) Table 5-23. Planning-level urban multilane roundabout models: comparison of alternate overdispersion specifications. Model ID a b c d k TOT -5.6642 (1.2790) 0.5210 (0.1545) 0.2905 (0.1198) -0.4610 (0.2357) 0.9263 (0.1371) FI -10.3369 (1.7505) 0.9134 (0.2129) 0.1937 (0.1248) -0.5131 (0.2261) 0.5611 (0.1398) PDO -5.7669 (1.3664) 0.4954 (0.1640) 0.3098 (0.1274) -0.4618 (0.2527) 1.0642 (0.1583) Table 5-24. Final planning-level models for urban multilane roundabouts.

67 of models was calibrated to predict PDO crash frequency. This approach for modeling FI crash frequency is in contrast to that used by some researchers who developed models for other severity combinations, such as the models developed by Rodegerdts et al. (2007) that predict the frequency of the K, A, and B categories combined. The models in the second subset (those predicting crash severity distribution) were calibrated to predict the dis- tribution of K, A, B, and C crashes (excluding O crashes). The two subsets of models can be used together to predict the frequency of each of the five crash severity levels in the KABCO scale. 5.2.2.1 Crash Frequency Prediction Models The crash frequency prediction models developed for this research project follow the structure used in the Part C chap- ters of the HSM (AASHTO, 2010). In this regard, they consist of an SPF and one or more crash modification factors (CMFs). In theory, the SPF describes the exposure to crash events by relating traffic volume to crash frequency. The CMFs indi- vidually account for the influence of a traffic characteristic, geometric element, or traffic control feature on crash risk. The SPF is multiplied by the CMFs to predict annual average crash frequency for a specific site. The SPFs were developed using a regression model with covariates that relate crash frequency with various site charac- teristics. The covariates were used to establish representative base conditions for the SPF (following the guidance provided in Appendix A of HSM Part C). The covariates were also used to infer CMFs for those specific site characteristics that were found to have a plausible and statistically valid association with crash frequency. Each of the crash frequency prediction models is devel- oped to predict the average crash frequency of one round- about. The predicted frequency includes crashes whose location is described as being within 250 ft of the round- about (measured back from the yield line) and either iden- tified as “at-intersection” or “intersection-related” or not identified as any of the following: “at driveway,” “signal control,” or “stop-sign control.” Thus, the predictive models are called intersection-level models because they predict the frequency of crashes at or related to the roundabout. Most roundabouts can be characterized as having either three or four legs. Most can also be characterized as having one or two circulating lanes. Each combination of lanes and legs has different geometric characteristics (e.g., number of conflict points between entering and circulating vehicles) that are likely to influence safety. Ideally one model would be developed to separately address each combination of legs and lanes. This approach would likely provide very reliable model predictions. Recent research indicates that the underlying causal mech- anisms for FI crashes are sometimes different from those for PDO crashes. Some roadway geometric elements can influ- ence FI crash occurrence more than they influence PDO crash occurrence (and vice versa). The functional relation- ship between a geometric element’s dimension and crash fre- quency may have a different shape for FI crashes than it does for PDO crashes. Plotted relationships showing FI and PDO crash frequency as a function of variables (e.g., volume) indi- cate differences in slope and curvature among the two sever- ity categories. For these reasons, many of the more recently developed models for Part C of the HSM have included sepa- rate models for FI and PDO crashes. There are two options for obtaining an estimate of total crash frequency. One option is to add the predicted FI and PDO crash frequencies. This option is available when both FI and PDO models have been developed. The second option is to separately develop a model for predicting total (= FI + PDO) crash frequency. However, this option may produce SPFs and CMFs whose individual pre- dictions do not compare well to those obtained from their counterparts in FI and PDO models. This potential limita- tion of the second option occurs when a suboptimal func- tional form is used for the total crash model. Elvik (2011) discusses the potential for biased estimates and misleading conclusions when a suboptimal model form is calibrated. After considering the issues described in the previous para- graphs, it was determined that the best approach for devel- oping the roundabout models was to calibrate a predictive model for each of two severity categories (FI and PDO). The estimates from these two models would be added to obtain an estimate of total crash frequency. In summary, eight inter- section-level models were developed (= 2 leg combinations × 2 circulating lane combinations × 2 severity categories). These models are identified in the following list: • One circulating lane, three legs, KABC; • One circulating lane, three legs, PDO; • One circulating lane, four legs, KABC; • One circulating lane, four legs, PDO; • Two circulating lanes, three legs, KABC; • Two circulating lanes, three legs, PDO; • Two circulating lanes, four legs, KABC; and • Two circulating lanes, four legs, PDO. The development of these crash frequency prediction models is described in Section 5.2.4. Ideally, one set of eight models would be developed to predict roundabout crash frequency (excluding vehicle– pedestrian and vehicle–bicycle crashes). A second set of eight models would be developed to predict vehicle–pedestrian crash frequency, and a third set of eight models would be

68 developed to predict vehicle–bicycle crash frequency. How- ever, the frequency of vehicle–pedestrian and vehicle–bicycle crashes at roundabouts is typically very small. In fact, their number was so small in the assembled database that it was not possible to develop models for predicting these crash types. The number of pedestrian- and bicycle-related crashes in the assembled database is described in Section 5.2.3. 5.2.2.2 Crash Severity Distribution Prediction Models Crash severity distribution prediction models were devel- oped to predict the proportion of crashes by severity category. In application, these proportions would be used by the ana- lyst with the crash frequency prediction models (described in the previous subsection) to predict the frequency of each crash severity category. Specifically, the crash severity distri- bution models are used to predict the proportion of K, A, B, and C crashes. Then, these proportions are used with the pre- dicted FI crash frequency to estimate the frequency of K, A, B, and C crashes. The development of the severity distribution models is described in Section 5.2.6. The crash severity distribution prediction model is referred to as a “severity distribution function” (SDF). It is repre- sented as a logit regression model that includes variables for various geometric design elements and traffic control fea- tures that are correlated with the severity of a crash. Models of this type were developed for the freeway safety prediction model in Chapter 19 of the HSM 2014 Supplement (Bonneson et al., 2012). 5.2.3 Crash Frequency Prediction Database Summary This section describes the data used to calibrate the crash frequency prediction models. The first subsection lists the variables in the database. The second subsection summa- rizes the geometric design, traffic, and crash characteristics associated with the sites in the database. The last subsection presents the findings from an exploratory analysis of trends in the data. 5.2.3.1 Database Variables This section summarizes selected variables in the database assembled for the project. These data are listed in Table 5-25. The area type, traffic volume, and crash data were obtained from various public agencies responsible for each round- about study site. All other data elements listed in Table 5-25 were obtained from aerial imagery for the years represented by the crash data. The StartDate and EndDate variables were used to define the evaluation period for each roundabout in the database. The “evaluation period” defines the period during which the roundabout’s crash history was evaluated. This period was defined to include one or more consecutive calendar years. The AADT volume was obtained for each roundabout for the years associated with its evaluation period. This volume was obtained for each leg of the roundabout when avail- able. To be retained in the database, each roundabout was required to have AADT volume for one major road leg and for one minor road leg for one or more years in the evalua- tion period. In those cases where the AADT for a given leg was not available for all years of the evaluation period, it was estimated using the procedure described in Part C of the HSM (AASHTO, 2010) (also the procedure in Step 3 of the predictive method). 5.2.3.2 Database Summary This subsection summarizes the data assembled for the purpose of calibrating the predictive models. Initially, the geometric and traffic characteristics are summarized. Then, the crash data are summarized. The purpose of this summary is to provide information about the range of data included in the database and to pro- vide some insight to guide the development of the predictive model form. The discussion in this subsection is not intended to indicate conclusive results or recommendations. The rec- ommended predictive models (and associated trends) are documented in Section 6.1.2. The database assembled for the project includes 355 round- abouts. However, data for 28 roundabouts were removed from the database due to anomalies in the data. Three round- abouts were removed because findings from a preliminary examination of the data indicated the three roundabouts had an exceptionally high FI crash rate and thus may not be representative of the population of roundabouts. Similarly, one roundabout was removed because it had an usually high FI crash rate and an usually high PDO crash rate. Finally, 24 roundabouts were removed because they had unusually high year-to-year variation in the AADT volume of one or more legs. Twenty-eight roundabouts were removed from the database, leaving 327 roundabouts for model development. Geometric and Traffic Characteristics. The database included data for 327 roundabout study sites. The distribu- tion of these sites by number of circulating lanes, number of legs, and area type is provided in Table 5-26. The first column of Table 5-26 separates the database into roundabouts with one or two circulating lanes. The sites indi- cated to have one circulating lane have one circulating lane conflicting with each leg. In contrast, only 22 of the 115 round- abouts identified as having two circulating lanes actually have two circulating lanes conflicting with each leg. Most of the

69 115 roundabouts identified as two circulating lanes actually have some combination of one and two circulating lanes within the roundabout. The inscribed circle diameter describes the circle that best fits the outside edge of the circulating lanes. This diameter does not vary significantly among rural and urban round- abouts. It averages 124 ft for roundabouts with one circu- lating lane and 177 ft for roundabouts with two circulating lanes. This trend confirms that inscribed circle diameter is correlated with the number of circulating lanes. There are 25 roundabouts in the database that have a drive- way providing direct access to the circulating roadway. The driveway is not considered to be a leg because it does not have a splitter island. For this reason, the driveway was not counted toward the total number of legs cited in column 2 of Table 5-26 (i.e., a roundabout with three public street legs and a driveway was recorded as a three-leg roundabout). Nineteen roundabouts have one leg that serves only out- bound traffic. These roundabouts were often (but not always) operating as a crossroad-ramp terminal at an interchange. In N_bk_i Count of vehicle–bicycle crashes that are fatal or injury N_bk_pdo Count of vehicle–bicycle crashes that are PDO Category Variable Description Descriptive Name_Leg Route number or street name (by leg) State State in which roundabout is located (postal code) Area_Type Urban/suburban or rural RampTerm One or more roundabout legs is a ramp to or from a freeway StartDate Starting date of the evaluation period EndDate Ending date of the evaluation period Roadway NumberLegs Number of legs at roundabout (3 or 4) CirculatingLanes Number of circulating lanes in the roundabout (1 or 2) (by leg) EnteringLanes Number of entering lanes (by leg) ExitingLanes Number of exiting lanes (by leg) Bypass Presence of a right-turn bypass lane (yes/no) (by leg) EntryWidth Width of the approaching lanes (plus shoulders, if provided) (by leg) Angle Angle to next leg going in counterclockwise direction (by leg) ICD Inscribed circle diameter CirculatingWidth Width of circulating lanes (plus shoulders, if provided) (by leg) Trafˆic volume AADT_Year (1...n) Average daily traffic for years 1 to n of evaluation period (by leg) Other Luminaires Number of luminaires within 250 ft of roundabout (by leg) NumberAccess Number of access points within 250 ft of roundabout (by leg) Crash N_k Count of fatal crashes during evaluation period N_a Count of incapacitating-injury crashes during evaluation period N_b Count of nonincapacitating-injury crashes during evaluation period N_c Count of possible-injury crashes during evaluation period N_abc Count of injury crashes (some agencies do not break out by A, B, C) N_pdo Count of PDO crashes during evaluation period N_pd_ˆi Count of vehicle–pedestrian crashes that are fatal or injury N_pd_pdo Count of vehicle–pedestrian crashes that are PDO Table 5-25. Summary of data attributes and sources.

70 this case, the one-way outbound lane corresponded to an entrance ramp for the freeway. A total of 44 roundabouts have a right-turn bypass lane on one or more legs. About 75 percent of these roundabouts are located in urban or suburban areas. These bypass lanes represent a mixture of control types (e.g., add-lane, merge, and yield). A total of 307 roundabouts have lighting on one or more legs. The 20 roundabouts without lighting are nearly evenly split between urban and rural areas. Table 5-27 provides summary statistics for entry width in columns 4 and 5. Entry width represents the width of the traffic lane (or lanes) entering the roundabout. It was mea- sured from the splitter island curb to either the curb face at the outside edge of the travel way or, if there is no curb at the outside edge, to the paved edge of traveled way. As such, this measurement includes the width of all entering lanes and the shoulder (if present). At a few roundabouts, there is a leg with one entering lane, a right-turn bypass lane, and measurement method “b” was applied (i.e., the bypass lane did not have a raised-curb island separating the entering lane and the bypass lane). In these instances, the measured entry width included the width of the bypass lane. To extract the bypass lane width, the recorded entry width was divided by 2 Circu- lating Lanes Number of Legs Area Type Number of Round- abouts Inscribed Circle Diameter, ft Number of Roundabouts with... Driveway on Cir. Roadway 1-Way Outbound Legs Right-Turn Bypass Lane Lighting on 1 or More Legs Average Std. Dev. 1 3 R 19 120 21 4 0 0 17 U 42 119 20 9 1 3 39 4 R 57 135 25 2 3 5 51 U 94 120 26 3 5 15 88 2 3 R 7 173 40 1 0 4 7 U 27 178 55 6 0 5 26 4 R 20 172 31 0 1 2 20 U 61 184 60 0 9 10 59 Grand Total 327 144 46 25 19 44 307 NOTE: Area type: R = rural; U = urban or suburban. Table 5-26. Database sample size and summary of roundabout characteristics. Circu- lating Lanes Number of Legs Area Type Entry Width, ft Number of Access Points on a Leg Circulating Lane Width, ft/lane Average Std. Dev. Average Std. Dev. Average Std. Dev. 1 3 R 20.6 3.0 1.1 1.3 21.7 4.0 U 18.9 4.1 0.9 1.4 18.5 2.8 4 R 20.2 3.7 0.8 1.1 21.6 3.5 U 18.4 3.4 1.6 1.7 20.5 5.3 2 3 R 27.0 5.7 0.1 0.3 24.6 7.1 U 27.0 5.0 0.3 0.9 21.1 6.7 4 R 28.0 4.8 0.8 1.1 19.7 6.4 U 26.0 4.8 1.2 1.7 18.9 5.4 Grand Total 21.8 5.5 1.1 1.5 20.3 5.2 NOTE: Area type: R = rural; U = urban or suburban. Table 5-27. Database summary of roundabout characteristics.

71 (= 1 entering lane + 1 bypass lane) to obtain a more accurate estimate of the actual entry width. Entry width for roundabouts with one circulating lane averaged 19.6 ft. In contrast, the average entry width for roundabouts with two circulating lanes is 27.0 ft. The entry width for roundabouts in urban or suburban areas is one to 2 feet smaller than that for rural areas. There are 256 roundabouts with a driveway or unsignalized access point on one or more legs. The typical roundabout has about one access point on each leg. The average circulating lane width represents the width of the circulating lanes conflicting with the corresponding leg, divided by the number of circulating lanes conflicting with the same leg. The average circulating lane width is 20.3 ft/lane. There is a tendency for roundabouts in urban or suburban areas to have a circulating lane width that is 2 to 3 feet (per lane) narrower than for rural areas. Table 5-28 provides a summary of the AADT volume esti- mates for the major and minor road legs. These AADTs were determined using the average AADT for each roundabout leg (i.e., an average AADT for the evaluation period associated with the site). The trends in Table 5-28 indicate that roundabouts with two circulating lanes have 50 to 60% more traffic than those with one circulating lane. For roundabouts with two circulat- ing lanes, those in rural areas tend to have about 10% more volume than those in urban or suburban areas. Crash Characteristics. The distribution of reported crashes is shown in Table 5-29. Vehicle–pedestrian and vehicle–bicycle crashes are not included. The crashes are categorized by state, number of circulating lanes, and number of legs. There are 327 roundabout study sites col- lectively representing 10 states and Ontario, Canada. These roundabouts experienced 1,111 FI crashes and 5,124 PDO crashes. The number of consecutive years for which crash data were obtained is referred to as the “evaluation period.” Collectively, the evaluation periods at the roundabouts in the database varied from 1 to 15 years, with a median duration of 7 years. The AADT entering each roundabout during the evalua- tion period was used to compute the roundabout’s exposure in MEV. The total entering volume for a roundabout was computed by summing the entering volume for each leg. Entering volume is correlated with the number of legs at the roundabout. The volume increases with an increase in the number of legs that support entry movements. For this reason, the use of entering volume to compute crash rate is intended to minimize differences in crash rate between three- and four-leg roundabouts. The crash rate shown in Table 5-29 was computed as the sum of the crashes divided by the sum of the MEV for all roundabouts represented in the row. The total crash rate of 0.58 crashes per MEV is shown in the last row of Table 5-29. It can be compared to the value of 0.75 crashes per MEV reported by Rodegerdts et al. (2007, Table 69). The lower crash rate in Table 5-29 may reflect the improvements in roundabout design practice that have occurred since 2007. The variability in the PDO crash rates listed in Table 5-29 is larger than the variability in the FI rates (the coefficient of variation for PDO is 0.87, the coefficient of variation for FI is 0.63). This variation is likely to be partly a result of differ- ences in reporting threshold between jurisdictions. Further examination of this data indicated that the variation of PDO Circulating Lanes Number of Legs Area Type Average Daily Trafic, veh/d Major Road Minor Road Average Std. Deviation Average Std. Deviation 1 3 R 8,128 4,033 4,017 2,658 U 8,056 3,941 4,775 2,909 4 R 7,025 3,433 3,143 2,019 U R U 8,155 3,902 4,068 2,369 2 3 13,110 7,196 5,391 5,198 10,884 5,458 5,384 3,899 4 R 12,461 5,732 6,107 3,802 U 12,328 5,880 5,946 4,479 Grand Total 9,317 5,024 4,607 3,320 NOTE: Area type: R = rural; U = urban or suburban. Table 5-28. Database summary traffic volume.

72 State Number of Legs Number of Sites Total Years FI Crashes PDO Crashes Total MEV Crash Rate, cr/MEV FI PDO Total CA 1 3 1 7.0 2 2 17 0.12 0.12 0.24 2 4 3 21.0 49 163 161 0.30 1.01 1.32 FL 1 3 15 120.0 38 64 494 0.08 0.13 0.21 4 30 259.1 115 249 1,318 0.09 0.19 0.28 2 3 5 43.0 17 20 211 0.08 0.09 0.18 4 13 86.0 79 157 541 0.15 0.29 0.44 KS 2 3 1 7.0 0 5 24 0.00 0.21 0.21 4 5 33.0 9 129 144 0.06 0.90 0.96 MI 1 3 5 13.0 1 16 23 0.04 0.68 0.73 4 20 87.0 22 205 358 0.06 0.57 0.63 2 3 7 26.0 10 92 114 0.09 0.81 0.90 4 10 37.0 19 161 191 0.10 0.84 0.94 MN 1 3 3 21.0 1 5 33 0.03 0.15 0.18 4 13 85.0 23 80 276 0.08 0.29 0.37 2 3 5 28.0 5 17 85 0.06 0.20 0.26 4 6 39.0 21 126 280 0.08 0.45 0.53 NC 1 3 10 70.0 5 20 226 0.02 0.09 0.11 4 16 141.0 29 121 475 0.06 0.25 0.32 2 4 1 10.0 2 21 55 0.04 0.38 0.42 NY 1 3 5 37.0 7 27 202 0.03 0.13 0.17 4 8 62.0 17 85 289 0.06 0.29 0.35 2 3 1 7.0 6 46 58 0.10 0.79 0.90 4 5 41.0 97 439 453 0.21 0.97 1.18 ON 2 3 4 20.0 27 147 177 0.15 0.83 0.99 4 3 15.0 25 283 142 0.18 2.00 2.17 PA 1 3 3 21.0 0 5 54 0.00 0.09 0.09 4 6 38.0 11 31 191 0.06 0.16 0.22 WA 1 3 16 145.1 25 115 588 0.04 0.20 0.24 4 24 232.1 71 315 1,000 0.07 0.31 0.39 2 3 5 41.0 16 60 173 0.09 0.35 0.44 4 12 96.0 168 961 699 0.24 1.37 1.61 WI 1 3 3 18.0 2 13 35 0.06 0.37 0.42 4 34 201.1 90 412 813 0.11 0.51 0.62 2 3 6 34.0 10 63 139 0.07 0.45 0.53 4 23 109.0 92 469 644 0.14 0.73 0.87 Grand Total 327 2,250.6 1,111 5,124 10,681 0.10 0.48 0.58 Circulating Lanes Table 5-29. Crash data summary by state.

73 crash rates also varies within each jurisdiction. This variation may be the result of differences among enforcement agencies within the jurisdiction with regard to the degree to which they adhere to their jurisdiction’s legal reporting threshold. A criterion was established that access point–related crashes occurring within 250 ft of the roundabout yield line would be excluded from the database. However, there were 291 roundabouts where access point–related crashes could not be identified in the associated crash data. Thus, the crash data for these roundabouts included an unknown number of access point–related crashes. A condensed summary of the reported crashes in the database is shown in Table 5-30. The crashes are catego- rized by number of circulating lanes and number of legs. Separate summaries are provided for vehicle-only, vehicle– pedestrian, and vehicle–bicycle crashes. The number of vehicle–pedestrian and vehicle–bicycle crashes was deter- mined to be insufficient for the purpose of developing pre- dictive models for these crash types. 5.2.3.3 Exploratory Data Analysis As a precursor to model development, the database was examined graphically to identify the possible association between specific site characteristics and FI crash rate. The insights obtained from this examination were used to deter- mine which characteristics are likely candidates for repre- sentation in the model as a CMF and guide the functional form development for individual CMFs. The discussion in this subsection is not intended to indicate conclusive results or recommendations. The recommended predictive models are documented in Section 6.1.2. The paragraphs to follow describe the findings from the graphical examination of each geometric design element. For each examination, one value of the design element is used to represent one roundabout. For those design elements that describe the geometry of a roundabout leg, the leg AADT is used to compute a volume-weighted average value for the roundabout. For example, a four-leg roundabout has the fol- lowing entry width and AADT for each leg: 20 ft – 1,000 veh/d, 22 ft – 1,000 veh/d, 24 ft – 1,200 veh/d, and 26 ft – 1,200 veh/d. The volume-weighted average entry width is computed as 23 ft = (20 ×1,000 + 22 × 1,000 + 24 × 1,200 + 26 × 1,200)/ (1,000 + 1,000 + 1,200 + 1,200). This approach is intended to recognize that the design elements of the higher-volume legs are more likely to influence roundabout crash frequency. Identifying trends in a figure with 327 data points (one point for each roundabout) was often difficult because of the large number of clustered data points. To improve the exami- nation of trends in the data, the data were sorted by the design element dimension to form groups of roundabouts with sim- ilar dimension value. Then, the total crash frequency and the total MEV were computed for each group. These two values were then used to compute the group crash rate. Each group was sized to have the same total exposure. It was determined that a total exposure of 320 million entering vehicles would produce 34 data points. It was subjectively determined that limiting each figure to 34 data points would minimize data overlap and facilitate an examination of trends in the data. Entry Width. The findings from the examination of entry width are shown in Figure 5-2a. The trend line shown is a line of best fit to the data points based on linear regres- sion. This figure shows the relationship between entry width and FI crash rate for 34 roundabout groups. It indicates that crash rate increases with an increase in entry width. This trend is consistent with the CMF value for entry width shown in Figure 2-3. Figure 5-2b shows the relationship between entry width and crash rate when the data are regrouped according to the number of circulating lanes. Only data for roundabouts in urban and suburban areas are shown. The trends shown in this figure suggest that crash rate increases with an increase in the number of circulating lanes and decreases when entry Circu- lating Lanes Number of Legs Number of Sites Total Years Vehicle-Only Vehicle–Pedestrian Vehicle–Bicycle Total CrashesFI Crashes PDO Crashes FI Crashes PDO Crashes FI Crashes PDO Crashes 1 3 61 452 81 267 1 0 6 0 355 4 151 1105 378 1498 5 0 27 5 1913 2 3 34 206 91 450 0 0 4 1 546 4 81 487 561 2909 12 0 21 1 3504 Grand Total 327 2251 1111 5124 18 0 58 7 6318 Table 5-30. Crash data summary.

74 width is increased. Further examination of the data indicated that there is negligible correlation between entry width and traffic volume. Thus, the trend found in Figure 5-2a is likely a reflection of the correlation between entry width and num- ber of circulating lanes for roundabouts in urban areas. In other words, the number of circulating lanes has a much stronger impact on crash rate than the width of entry lanes. However, once the number of circulating lanes is known, there appears to be a trend that urban roundabout crash rate decreases with an increase in the width of the entry lanes. This trend may be a reflection of the ability for drivers in the wider entries to make slight adjustments in path alignment to avoid an impending crash. Figure 5-2c shows the relationship between entry width and crash rate for rural roundabouts. The trend shown in this figure suggests that crash rate increases with an increase in the number of circulating lanes and with an increase in entry width. The differences in trends between the urban and rural areas may be a reflection of the higher speeds inherent to roundabouts in rural areas (and may not be a true reflection of the effect of entry width). In summary, the trends shown suggest that roundabouts with two circulating lanes have a higher crash rate than those with one circulating lane. For urban roundabouts with a given number of circulating lanes, the trends suggest that crash rate decreases with an increase in entry width. This trend for rural roundabouts is contrary to that for urban roundabouts and likely reflects the effect of other unmeasured factors (e.g., speed) that are correlated with entry width and, if accounted for, could reveal a trend for entry width that is similar to that for urban roundabouts. Angle to Next Leg. The findings from the examination of angle to the next leg are shown in Figure 5-3. The trend line shown is a line of best fit to the data points based on linear regression. This figure shows the relationship between angle and FI crash rate for 34 roundabout groups. The data shown suggest that crash rate decreases with an increase in angle. a. Combined data. b. Urban roundabouts. c. Rural roundabouts. Figure 5-2. Examination of entry width.

75 This trend is consistent with the CMF value for angle to the next leg shown in Figure 2-4. Figure 5-3b shows the relationship between angle to the next leg and crash rate when the data are regrouped accord- ing to the number of legs and number of circulating lanes. Only data for roundabouts in urban and suburban areas are shown (when subdivided by legs and lanes, there was insuf- ficient data for rural roundabouts to discern trends). The trends shown in this figure suggest that crash rate increases with an increase in number of legs and that crash rate is not likely correlated with angle. Thus, the trend found in Fig- ure 5-3a is likely a reflection of the correlation between angle and number of legs for roundabouts in urban areas. Figure 5-3c shows the relationship between angle and crash rate for roundabouts with two circulating lanes. Only data for roundabouts in urban and suburban areas are shown. The trend shown in this figure suggests that crash rate may increase at three-leg roundabouts with an increase in angle. The trend is not clear for four-leg roundabouts. In summary, the trends shown in Figure 5-3 suggest that roundabouts with four legs tend to have a slightly higher crash rate than those with three legs. They also suggest that crash rate at urban roundabouts is not likely influenced by the angle to the next leg. Deviation from 90 Degrees. The angle between legs was evaluated a second time by computing the deviation of the leg from 90 degrees. At three-leg roundabouts, the deviation from 90 and 180 degrees was computed. This examination was undertaken because experience with roundabout opera- tion indicated that crash frequency sometimes was higher when the deviation was about 45 degrees. In contrast, crash frequency was lower when the deviation was about 90 degrees (or 180 degrees at three-leg intersections). Deviation was computed using the following equation. : , 90 , 90 , 90 Equation 5-9 D smaller of mod A mod Ak k k[ ] [ ]( )= − b. Roundabouts with one circulating lane.a. Combined data. c. Roundabouts with two circulating lanes. Figure 5-3. Examination of angle to next leg.

76 where Dk = deviation for leg k, degrees; and Ak = angle to the next leg for leg k, degrees. Representative values for deviation are listed in Table 5-31. The findings from this examination are shown in Fig- ure 5-4. The trend lines shown represent lines of best fit to the data points based on linear regression. This figure shows the relationship between angle and FI crash rate for 23 roundabout groups. The data shown suggest that crash rate decreases slightly with an increase in deviation. How- ever, none of the slopes are significantly different from zero. Thus, it is concluded that the crash rate at the roundabouts represented in the database is not significantly influenced by deviation from 90 degrees. Inscribed Circle Diameter. The findings from the exam- ination of inscribed circle diameter are shown in Figure 5-5. The trend lines shown are lines of best fit to the data points based on linear regression. Figure 5-5a shows the relation- ship between diameter and FI crash rate for 34 roundabout groups. It indicates that crash rate increases with an increase in diameter. This trend is consistent with the CMF value for inscribed circle diameter shown in Figure 2-5. Figure 5-5b shows the relationship between inscribed circle diameter and crash rate when the data are regrouped according to the number of circulating lanes. Only data for roundabouts in urban and suburban areas are shown. The trends shown in this figure suggest that crash rate increases with an increase in the number of circulating lanes and decreases when the diam- eter is increased. Further examination of the data indicated that there is negligible correlation between diameter and traf- fic volume. Thus, the trend found in Figure 5-5a is likely a reflection of the correlation between inscribed circle diameter and number of circulating lanes. Figure 5-5c shows the relationship between inscribed circle diameter and crash rate for roundabouts in rural areas. The trend shown in this figure is consistent with that in Figure 5-5b. It suggests that crash rate decreases with an increase in diameter. In summary, the trends shown suggest that roundabouts with two circulating lanes have a higher crash rate than those with one circulating lane. They also suggest that crash rate decreases as inscribed circle diameter increases. Circulating Width. The findings from the examination of circulating width are shown in Figure 5-6. The trend lines shown are lines of best fit to the data points based on linear regression. Figure 5-6a shows the relationship between width and FI crash rate for 34 roundabout groups. It indicates that crash rate increases with an increase in circulating width. This trend is consistent with the CMF value for circulating width shown in Figure 2-6. Figure 5-6b shows the relationship between circulating lane width and crash rate when the data are regrouped according to the number of circulating lanes. For this figure, the circu- lating width is divided by the number of lanes to facilitate the examination on a lane width basis. Only data for roundabouts Deviation for Selected Angle-to-Next-Leg Values, degrees Angle: 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Deviation: 0 10 20 30 40 40 30 20 10 0 10 20 30 40 40 30 20 10 0 Table 5-31. Deviation for selected angles. a. Roundabouts with one circulating lane. b. Roundabouts with two circulating lanes. Figure 5-4. Examination of deviation from 90 degrees.

77 in urban and suburban areas are shown. The trends in this figure suggest that crash rate increases with an increase in the number of circulating lanes and is not likely correlated with circulating width per lane. Thus, the trend found in Fig- ure 5-6a is likely a reflection of the correlation between circu- lating width and number circulating lanes. Figure 5-6c shows the relationship between circulating lane width and crash rate for roundabouts in rural areas. The trends shown in this figure are consistent with those in Figure 5-6b in that crash rate increases with the number of circulating lanes, but it is not dependent on circulating width per lane. In summary, the trends shown suggest that roundabouts with two circulating lanes have a higher crash rate than those with one circulating lane. There does not appear to be a con- sistent relationship between crash rate and circulating width per lane. Roundabout Lighting. The findings from the examina- tion of roundabout lighting are shown in Figure 5-7. The trend lines shown are lines of best fit to the data points based on linear regression. Figure 5-7a shows the relationship between the number of luminaires present per leg and crash rate when the data are grouped according to the number of circulating lanes. Only data for roundabouts in urban and suburban areas are shown. The trends shown in Figure 5-7a suggests that crash rate is not likely correlated with the number of luminaires present. These trends are not consistent with the CMF value for the effect of lighting presence at traditional intersection configu- rations, as discussed in the previous subsection titled Crash Modification Factors. It is likely that this finding is a reflec- tion of the presence of one or more confounding variables that are correlated with lighting presence (e.g., speed limit) and also influence crash rate. Figure 5-7b shows the relationship between number of luminaires present and crash rate for rural roundabouts. The trends shown in this figure suggest that crash rate decreases as the number of luminaires present increases, however, the rate of decrease may vary with the number of circulating lanes. This trend is consistent with the CMF value for the a. Combined data. b. Urban roundabouts. c. Rural roundabouts. Figure 5-5. Examination of inscribed circle diameter.

78 a. Combined data. b. Urban roundabouts. c. Rural roundabouts. Figure 5-6. Examination of circulating width. a. Urban roundabouts. b. Rural roundabouts. Figure 5-7. Examination of lighting presence.

79 effect of lighting presence at traditional intersection con- figurations, as discussed in Section 2.6.2. As shown in Table 5-26, almost all roundabouts included in the assembled database have some lighting. This trend is believed to be consistent with the collective set of round- abouts in the United States. It likely reflects the recommen- dations in most authoritative roundabout design reference documents. Number of Access Points. The findings from the exami- nation of access point presence are shown in Figure 5-8. The trend lines shown are lines of best fit to the data points based on linear regression. Figure 5-8a shows the relationship between the number of access points per leg and crash rate when the data are grouped according to the number of circu- lating lanes. Only data for roundabouts in urban and subur- ban areas are shown. The trends shown in Figure 5-8a suggest that crash rate increases when the number of access points increases. This trend is consistent with the CMF value for the effect of access points at crossroad-ramp terminals, as shown in Table 2-11. Figure 5-8b shows the relationship between number of access points present and crash rate for rural roundabouts. The trend shown in this figure for roundabouts with two cir- culating lanes is consistent with those in Figure 5-8a. Spe- cifically, it suggests that crash rate increases for roundabouts with two circulating lanes when the number of access points is increased. In contrast, the trend for roundabouts with one circulating lane suggests that crash rate decreases when the number of access points increases. Based on both figures, it appears that an increase in the num- ber of access points is associated with an increase in crash rate. This trend is consistent with the CMF shown in Table 2-11. The trend for rural roundabouts with one circulating lane is counterintuitive and may reflect the presence of one or more confounding variables that are correlated with access point presence and also influence crash rate. It is noted that these trends are based on only the 256 roundabouts for which access point–related crashes were included in the database. Outbound Leg. The findings from the examination of one-way outbound-leg presence are shown in Table 5-32. Legs that serve only outbound traffic are often found at round- abouts operating as a crossroad-ramp terminal at an inter- change. The first column of the table indicates the proportion of total leg AADT that exists on the outbound-only leg. Thus, a roundabout with four legs that each has a leg AADT of 1,000 veh/d, and with one of these legs serving only out- bound traffic, has a proportion of 0.25 (= 1,000/[1,000 + 1,000 + 1,000 + 1,000]). Those roundabouts with an outbound-only leg are repre- sented in the last two rows of Table 5-32. A comparison of the crash rates in the second column indicates that round- abouts with an outbound-only leg have a lower crash rate than those roundabouts that do not have an outbound-only leg. The last column of the table indicates the implied CMF associated with outbound-leg presence. The CMF value in a given row is computed as a ratio of the crash rate in that row to the crash rate in the first row (e.g., 0.62 = 0.070/0.114). Right-Turn Bypass Lane. The findings from the exami- nation of right-turn bypass lane presence are shown in Fig- ure 5-9. The x-axis indicates the proportion of the leg AADTs that exists on the legs with a right-turn bypass lane. Thus, a roundabout with four legs that each has a leg AADT of 1,000 veh/d, with two of these legs having a right-turn bypass lane, has a proportion of 0.50 (= [1,000 + 1,000]/[1,000 + 1,000 + 1,000 + 1,000]). This proportion increases with an increase in the number of legs that have a bypass lane. The trend line in Figure 5-9 indicates that crash rate decreases a. Urban roundabouts. b. Rural roundabouts. Figure 5-8. Examination of number of access points.

80 with an increase in the proportion of total leg AADT associ- ated with legs having a right-turn bypass lane. 5.2.4 Crash Frequency Prediction Model Development This section describes the activities undertaken to develop the models for predicting crash frequency (excluding vehicle– pedestrian and vehicle–bicycle crashes). The subsections to follow provide a description of the basic model form, an overview of the modeling approach, an overview of the sta- tistical analysis methods, and a discussion of the findings from the model calibration and validation activities. 5.2.4.1 Predictive Model Form A predictive model consists of an SPF and several CMFs. The SPF is used to estimate the average crash frequency for a generic site whose attributes are consistent with the SPF’s stated base conditions. The CMFs are used to adjust the SPF estimate when the attributes of the subject site are not consis- tent with the base conditions. The general form of the inter- section safety predictive models used in the Part C chapters of the HSM is shown in the following equation. . . . Equation 5-101N C N CMF CMFSPF n( )= × × × × where N = predicted average crash frequency, crashes/yr; C = local calibration factor; NSPF = predicted average crash frequency for base condi- tions, crashes/yr; CMFi = CMF for traffic characteristic, geometric element, or traffic control feature i (i = 1 to n); and n = total number of CMFs. Equation 5-10 includes a local calibration factor, an SPF, and several CMFs. The SPF is used to estimate the average crash frequency for a site with base conditions as a function of AADT. The CMFs are used to account for conditions at the subject site having one or more conditions that vary from the base conditions. The local calibration factor is used to account for differences between the jurisdiction for which the model was developed and the jurisdiction in which the subject site is located. Each of these equation components are described in more detail in Part C of the HSM. The model structure described by Equation 5-10 does not recognize the somewhat independent nature of each roundabout leg. In this regard, a change can be made to the geometry of one or more legs without directly affecting the crash frequency on a different leg to any significant degree. Similarly, the traffic characteristics of one leg can be differ- ent from that of another leg, which can result in the two legs having a different level of contribution to the overall crash frequency of the roundabout. An alternative model form was formulated for computing the average crash frequency for the overall roundabout (i.e., including all legs and the circulating roadway). The model was derived by spatially disaggregating the intersection into its individual legs (and their associated portion of the cir- culating roadway). With this approach, Equation 5-10 was expanded to the following form to more accurately reflect differences in traffic characteristics, geometric elements, and traffic control features among roundabout legs. . . . . . . . . . . . . Equation 5-11 1 1,1 1, 2 2,1 2, ,1 , N C N p CMF CMF p CMF CMF p CMF CMF p SPF n n m m m n [ ] ( ) ( ) ( ) = × × × × + × × + + × × AADT on Leg with 1-Way Outbound Trafic, as a Proportion of Total Leg AADT FI Crash Rate, cr/MEV Implied CMF 0.000 (no outbound-only leg present) 0.114 1.00 0.113 0.070 0.62 0.301 0.043 0.38 Table 5-32. Examination of outbound-leg presence. Figure 5-9. Examination of presence of a right-turn bypass lane.

81 where Np = predicted average crash frequency, crashes/yr; C = local calibration factor; NSPF = predicted average crash frequency for base condi- tions on all legs, crashes/yr; pj = proportion of crashes associated with round- about leg j (j = 1 to m); CMFj,i = CMF for traffic characteristic, geometric element, or traffic control feature i on leg j (i = 1 to n; j = 1 to m); and m = total number of legs (3 or 4), legs. All other variables are as previously defined. The SPF in this model predicts the crash frequency for base conditions, NSPF. It is a function of the traffic volume on the intersecting roadways. The proportion of crashes on each leg j, pj, must add to 1.0. From a practical perspective, the distribution of crashes among the legs may not always be known (e.g., for a pro- posed intersection). To avoid this issue, it is rationalized that the proportion of intersection crashes associated with one leg can be estimated using the following equation. . . . Equation 5-12 1 2 p AADT AADT AADT AADT j j m = + + + where pj = proportion of total leg traffic volume associated with roundabout leg j (j = 1 to m); and AADTj = AADT volume for roundabout leg j (j = 1 to m). Equation 5-11 (with Equation 5-12) describes the predic- tive model form used to develop the roundabout crash fre- quency prediction models. 5.2.4.2 Modeling Approach This section includes four subsections that collectively describe the researcher’s approach to model development. The first subsection describes the model development pro- cess. The second subsection describes the techniques used to minimize the influence of correlated variables. The third sub section describes the techniques used to develop the desired eight lane-leg-severity combinations (described in Section 5.2.2.1), while recognizing that the number of sites for some combinations is relatively small. The last subsection describes the procedure used to compute an estimate of the inverse dispersion parameter that is not biased by the small number of sites. Model Development Process. The development of the SPFs and CMFs in the predictive model form focused on the use of FI crash data. There are two reasons for this focus. The first reason relates to the variation in crash reporting threshold between and within jurisdictions. An evaluation of PDO crash rates in the database indicated a wide variation in their representation in the crash distribution for each of the study jurisdictions. This variation is likely to be the result of differences in the legal reporting threshold between jurisdic- tions and differences in the level of adherence to this threshold within jurisdictions. This variation in PDO crash represen- tation can cloud the search for association between database variables and crash frequency. In contrast, FI crashes are more consistently reported among jurisdictions and thus provide a more reliable basis for model structure development. A second reason for the focusing model development on FI crashes (as opposed to total crashes) is that developing models using total crashes may increase the potential for creating suboptimal formulations for the SPFs and CMF functions. A suboptimal formulation will bias in the model predictions at some factor levels. Recent research indicates that the underlying causal mecha- nisms for FI crashes are different from those for PDO crashes. These important differences can be overlooked when a total crash model is developed. In contrast, they are more likely to be represented in the SPF and CMF functions when separate models are developed for FI crashes and for PDO crashes. Elvik (2011) discusses the potential for biased estimates and misleading conclusions when the wrong model form is calibrated. For this project, the FI prediction model was developed first. Then, the PDO model was developed. The minimal influence of reporting threshold variation on the FI data ensured that it would provide the most insight into the model development process. Once the FI model was developed, all statistically valid CMFs in the FI model were considered dur- ing the development of the PDO model. The retention of a CMF in the FI and PDO models was based on consider- ation of its regression coefficients and overall model fit (i.e., p-value, direction of effect, practical significance, AIC value). In this manner, the FI model and PDO model tended to use a similar structure and have common CMFs, while allowing the regression coefficients to be individually fit to the respec- tive crash severity category. Correlated Variables. A preliminary analysis of the data- base indicated that some of the site characteristic variables were correlated with roundabout location. For example, roundabouts for some jurisdictions were mostly located in rural areas, while those in other jurisdictions were located in urban areas. As a result of this correlation, the model develop- ment process required two stages. In the first stage, the regres- sion model included only site characteristics variables (it did not include variables specific to the jurisdictions). The regres- sion coefficient for each site characteristic variable was exam- ined for magnitude, direction, and statistical significance.

82 If the magnitude, direction, and significance were acceptable, then the variable was retained in the model. At the conclusion of the first stage, the calibrated regression model included only variables whose coefficients were considered acceptable. During the second stage of the model development pro- cess, the calibrated regression model from the first stage was expanded to include one or more jurisdiction-specific indi- cator variables. The coefficient associated with this variable would serve to adjust the model prediction (similar to a local calibration factor) for those sites in a jurisdiction that had significantly more or less frequent crashes than the other jurisdictions. One jurisdiction-specific indicator variable was added to the model. This process was repeated for all jurisdictions represented in the database. If the coefficient for a given jurisdiction-specific indicator was found to be statisti- cally significant, and if it did not notably alter the magnitude, direction, or significance of any site characteristic variable, then it was retained in the model. Combined Regression Modeling. As described in Sec- tion 5.2.2.1, two sets of four intersection-level models were planned for development. One set would predict FI crash fre- quency and the other set would predict PDO crash frequency. Each set of four models included two combinations of cir- culating lanes (one and two lanes) and two combinations of legs (three and four legs). A preliminary regression analysis of the data indicated that the site sample size for some of the four model combinations was too small to develop a reliable model. To overcome this issue, it was decided that separate models would be independently developed for each of the two combinations of circulating lanes, and that a com- bined modeling approach would be needed to develop semi- independent models for the two combinations of legs. Thus, the model development plan was revised to include develop- ment of the four models identified in the following list: • One circulating lane with SPF for three legs, SPF for four legs, and common CMFs, KABC; • One circulating lane with SPF for three legs, SPF for four legs, and common CMFs, PDO; • Two circulating lanes with SPF for three legs, SPF for four legs, and common CMFs, KABC; and • Two circulating lanes with SPF for three legs, SPF for four legs, and common CMFs, PDO. The general form of each model is shown in the following equations. . . . . . . . . . Equation 5-13 3 ,3 1 1,1 1, 2 2,1 2, 3 3,1 3, N N p CMF CMF p CMF CMF p CMF CMF SPF n n n [ ( ) ( ) ( ) = × × × + × × + × × and . . . . . . . . . . . . Equation 5-14 4 ,4 1 1,1 1, 2 2,1 2, 3 3,1 3, 4 4,1 4, N N p CMF CMF p CMF CMF p CMF CMF p CMF CMF SPF n n n n [ ] ( ) ( ) ( ) ( ) = × × × + × × + × × + × × where Nm = predicted average crash frequency for roundabout with m legs (m = 3, 4), crashes/yr; NSPF,m = predicted average crash frequency for base con- ditions on all legs for roundabout with m legs, crashes/yr; pj = proportion of total leg traffic volume associated with roundabout leg j (j = 1 to m); and CMFj,i = CMF for traffic characteristic, geometric ele- ment, or traffic control feature i on leg j (i = 1 to n; j = 1 to m). The SPFs associated with these models are defined as Equation 5-15,3 0,3 ,3 3 1000N eSPF b bAADT LN EntAADT= ( )+ and Equation 5-16,4 0,4 ,4 4 1000N eSPF b bAADT LN EntAADT= ( )+ with 2 Equation 5-173 1 2 3 EntAADT AADT AADT AADT= + + 2 Equation 5-18 4 1 2 3 4 EntAADT AADT AADT AADT AADT= + + + where EntAADTm = entering AADT for roundabout with m legs (m = 3, 4), veh/d; AADTj = AADT volume for roundabout leg j, veh/d; and bi = calibration coefficient for condition i. All other variables are as previously defined. The CMFs in Equation 5-13 are shown to be the same as the CMFs in Equation 5-14. That is, the CMF for a given traffic characteristic, geometric element, or traffic control feature i is the same in each equation. The calibration coefficient asso- ciated with each CMF is also the same. Therefore, if the CMF for a given characteristic, element, or feature is a function of

83 variables (e.g., entry width) and the variables have the same value at a three-leg roundabout and at a four-leg roundabout, then the CMF value is the same for both roundabouts. This approach recognizes that some characteristics, elements, or features have a similar influence on crash frequency, regard- less of whether the roundabout has three or four legs. Equations 5-13 and 5-14 are shown to use common CMFs. However, before it was determined that a common CMF was appropriate, a preliminary regression analysis was under- taken to determine if the regression coefficient associated with each CMF was significantly different when applied only to three-leg roundabouts, relative to when it was applied only to four-leg roundabouts. If the two coefficients were not sig- nificantly different, then one coefficient was used for both three- and four-leg roundabouts (to create a common CMF). The use of common CMFs has the advantage of maximizing the sample size available to calibrate the CMF. In this man- ner, the data for both three- and four-leg roundabouts are pooled to provide a more efficient estimate of the CMF coef- ficient. The use of common CMFs in two models required the use of a combined regression modeling approach. With this approach, the regression analysis evaluated both models simultaneously and used the total log-likelihood statistic to determine the best-fit calibration coefficients. The regression analysis is described in more detail in subsequent sections. Inverse Dispersion Parameter. It was assumed that roundabout crash frequency is Poisson distributed and the distribution of the mean crash frequency for a group of simi- lar roundabouts is gamma distributed. In this manner, the distribution of crashes for a group of similar roundabouts can be described by the negative binomial distribution. The variance of this distribution is computed using the following equation. Equation 5-19 2 V X yN yN K [ ] ( )= + where V[X] = crash frequency variance for a group of similar locations, crashes2; N = predicted average crash frequency, crashes/yr; X = reported crash count for y years, crashes; y = time interval during which X crashes were reported (i.e., evaluation period), yr; and K = inverse dispersion parameter (= 1/k, where k = overdispersion parameter). Research by Lord (2006) indicates that databases with low sample mean values and small sample size may not exhibit the variability associated with crash data of similar sites, as described by Equation 5-19. Rather, the regression model may overexplain some of the random variability in a small database, or the low sample mean may introduce instability in the model coefficients. Lord (2006) explored the effect of sample size on the vari- ability of crash data through the use of simulation. Specifi- cally, he simulated the crash frequency for three database sizes (50, 100, and 1,000 sites), each with three different aver- age crash frequencies (0.5, 1.0, and 10 crashes per site), and three inverse dispersion parameters (0.5, 1, and 2). One com- plete set of simulations consisted of the nine combinations of average crash frequency and dispersion parameter, each sim- ulated once for each site yielding 10,350 (= 3 × 3 × [50 + 100 + 1,000]) site crash frequency estimates. A total of 30 repli- cations were conducted yielding 310,500 site estimates. A maximum-likelihood technique was used to estimate the dispersion parameter for each of the 27 combinations. This process was repeated 30 times to yield 30 estimates of K for each of the 27 combinations. The average value of K for each combination is shown in Figure 5-10. As shown in Figure 5-10, the estimated inverse dispersion parameter for a given database is nearly equal to the speci- fied (i.e., true) inverse dispersion parameter, provided that the database contains many sites collectively experiencing 1,000 or more crashes. However, the estimated dispersion parameter is larger than the true value when the number of sites and associ- ated crashes is small. The inverse dispersion parameter was computed for each of the predictive models described in a subsequent subsection. Table 5-33 lists the computed values in column 4. A cursory inspection of these parameters suggests that those values for the three-leg FI models (i.e., 5.71 and 4.62) are much larger than the parameter values for the other models. In general, a larger inverse dispersion parameter suggests that the sites are more similar in the set of characteristics that determine their crash frequency. The larger inverse dispersion values in column 4 suggest that the three-leg sites are very similar in terms of FI crash Figure 5-10. Simulation results for the inverse dispersion parameter.

84 history. Yet this high level of similarity is not evident in the original inverse dispersion parameter values for the PDO crash history at the same three-leg sites (2.07, 0.98). It is likely that the sites are not as similar as the FI crash data sug- gest. Rather, the PDO crashes (being more frequent) tell us more about the three-leg site similarity and that the disper- sion parameter based on FI crashes is biased to a large value because of the relatively small number of FI crashes for these sites. If more years of crash data were available for these sites, the dispersion parameters for the FI models would likely converge to a value more similar to that for the PDO models. The pattern described for the three-leg sites is also evi- dent in the table for four-leg sites. The dispersion param- eters for the FI models are larger than those for the PDO models. However, the extent to which the parameters for the FI models exceed those for the PDO models is smaller (than that for the three-leg models) because of a larger number of crashes associated with the four-leg sites (relative to the three-leg sites). Bonneson et al. (2006) used the concepts identified by Lord (2006) to develop an equation for computing an adjusted inverse dispersion parameter. The adjustment was intended to minimize any bias due to small sample size. The following equation was derived to relate the original and true inverse dispersion parameters based on observed trends in the data reported by Lord (2006). 17.2 Equation 5-20 2 K K K n p m r t t ( ) = + − where Kr = original inverse dispersion parameter obtained from database analysis; Kt = true (adjusted) inverse dispersion parameter; n = number of observations (i.e., segments, intersections, or roundabouts in database); p = number of model variables; and m = average number of crashes per observation (= total crashes in database/n). The constant 17.2 in Equation 5-20 represents an empiri- cal adjustment derived through weighted regression analysis (sk = 2.45, p = 0.0001). Equation 5-20 was algebraically manip- ulated to yield the following relationship for estimating the true (adjusted) inverse dispersion parameter, given the other variables as input values. 68.8 34.5 Equation 5-21 2 2 K n p m K n p m n p m t r( ) ( ) ( )= − + − − − Equation 5-21 was used to estimate the true (adjusted) inverse dispersion parameter for each of the models described in Section 5.2.4. All subsequent references to the inverse dis- persion parameter K in Section 5.2.4 denote the adjusted parameter from Equation 5-21 (K = Kt). Park and Lord (2008) subsequently developed a procedure for computing an adjusted inverse dispersion parameter for the purpose of minimizing bias due to small sample size. They used data to demonstrate the effect of sample size on the inverse dispersion parameter. Specifically, they used a data- base with data for 458 sites. Three subsets of this database were used to replicate cases where fewer sites were available (cases with only 50, 70, and 100 sites). The results from their analysis are shown in columns 1 to 4 of Table 5-34. They considered the adjusted parameter value in the last row of column 4 to be the best estimate of the true value (1.29). Circulating Lanes Number of Legs Crash Severity Original Inverse Dispersion Parameter Mean Crash Frequency, crashes/yr Number of Observations Number of Model Variables Adjusted Inverse Dispersion Parameter Percent Change 1 3 FI 5.71 1.33 61 8 3.20 -43.9 4 FI 3.47 2.50 151 8 3.03 -12.7 3 PDO 2.07 4.38 61 4 1.84 -11.3 4 PDO 1.27 9.92 151 4 1.25 -1.5 2 3 FI 4.62 2.68 34 8 2.75 -40.5 4 FI 2.36 6.93 81 8 2.20 -7.0 3 PDO 0.98 13.24 34 5 0.94 -4.0 4 PDO 1.28 35.91 81 5 1.27 -0.8 Table 5-33. Examination of inverse dispersion parameters.

85 Comparison of the values in columns 3 and 4 show that the inverse dispersion parameter values are biased to be much larger than the true value for the smaller sample sizes. The adjusted values are shown to be nearer to the true value than the origi- nal value for any given sample size. The adjusted inverse dispersion parameters computed using Equation 5-21 are shown in column 5 of Table 5-34. The values in this column are shown to be closer to the true value (1.29) than those in column 4. For this reason, Equa- tion 5-21 was used to compute the adjusted inverse dispersion values. These values are shown in column 8 of Table 5-33. The two parameter values for the three-leg FI models have been reduced by more than 40% (from 5.71 to 3.20 and from 4.62 to 2.75), relative to the original values in column 4. 5.2.4.3 Statistical Analysis Methods The nonlinear regression procedure (NLMIXED) in the SAS software was used to estimate the proposed model coefficients. This procedure was used because the proposed predictive model is both nonlinear and discontinuous. The log-likelihood function for the negative binomial distribution was used to determine the best-fit model coefficients. Equation 5-19 was used to define the variance function for all models. The pro- cedure was set up to estimate model coefficients based on maximum-likelihood methods. Several statistics were used to assess model fit to the data. One measure of model fit is the Pearson c2 statistic. This statistic is calculated using the following equation. Equation 5-222 2 1 X y N V X i i i ii n ∑ [ ] ( )χ = − = where c2 = Pearson chi-square statistic; n = number of observations (i.e., segments, inter- sections, or roundabouts in database); V[Xi] = crash frequency variance for a group of similar locations, crashes2; Ni = predicted average crash frequency for observation i, crashes/yr; Xi = reported crash count for yi years for observation i, crashes; and yi = time interval during which Xi crashes were reported for observation i, yr. This statistic follows the c2 distribution with n – p degrees of freedom, where n is the number of observations (i.e., roundabouts), and p is the number of model variables (McCullagh and Nelder, 1983). This statistic is asymptotic to the c2 distribution for larger sample sizes. The root mean square error, sp, is a useful statistic for describing the precision of the model estimate. It represents the standard deviation of the estimate when each indepen- dent variable is at its mean value. This statistic can be com- puted using the following equation. Equation 5-23 2 1 2 s X y N n p y p i i i i n i ∑ ( ) ( ) = − − = where sp = root mean square error of the model estimate, crashes/yr, and all other variables are as previously defined. The scale parameter j is used to assess the amount of vari- ation in the observed data, relative to the specified distribu- tion. This statistic is calculated by dividing Equation 5-22 by the quantity n – p. A scale parameter near 1.0 indicates that the assumed distribution of the dependent variable is approx- imately equivalent to that found in the data (i.e., negative binomial). Another measure of model fit is the coefficient of deter- mination R2. This statistic is commonly used for normally distributed data. However, it has some useful interpretation when applied to data from other distributions when com- puted in the following manner (Kvalseth, 1985). This coeffi- cient is computed using the following equation. 1.0 Equation 5-242 2 1 2 1 R X y N X X i i i i n ii n ∑ ∑ ( ) ( ) = − − − = = Number of Observations Mean Crash Frequency, crashes/yr Original Inverse Dispersion Parameter Adjusted Inverse Dispersion Parameter Park and Lord (2008) Equation 5-21 50 0.22 14.20 10.02 2.63 70 0.21 2.01 1.48 0.94 100 0.28 2.11 1.81 1.20 458 0.25 1.37 1.29 1.17 Table 5-34. Comparison of procedures for computing the adjusted inverse dispersion parameter.

86 where X – is the average crash frequency for all n observations, crashes, and all other variables are as previously defined. The last measure of model fit is the dispersion parameter– based coefficient of determination Rk 2. This statistic was developed by Miaou (1996) for use with data that exhibit a negative binomial distribution. It is computed using the fol- lowing equation. 1.0 Equation 5-252R k k k null = − where k = overdispersion parameter (= 1/K, where K = inverse dispersion parameter); and knull = overdispersion parameter based on the variance in the observed crash frequency. The null overdispersion parameter knull represents the dis- persion in the reported crash frequency, relative to the overall average crash frequency for all sites. This parameter can be obtained using a null model formulation (a model with no independent variables but with the same error distribution, link function, and offset in years y). 5.2.4.4 FI Crash Frequency Prediction Model–One Circulating Lane This section describes the development of predictive models for roundabouts with one circulating lane. Specifi- cally, these models are applicable to roundabouts having only one circulating lane conflicting with each leg. They are not applicable to roundabouts that have two circulating lanes conflicting with one or more legs. The focus of this section is on models developed to predict FI crash frequency. One variation of the model is applicable to three-leg roundabouts, and a second variation of the model is applicable to four-leg roundabouts. This section consists of six subsections. The first sub section describes the structure of the safety predictive models as used in the regression analysis. The second subsection describes the regression statistics for each of the calibrated models. The third subsection describes a validation of the calibrated models. The fourth, fifth, and sixth subsections describe the proposed safety prediction models. Model Development. This subsection describes the pro- posed prediction models and the methods used to calibrate them. The regression model is generalized to accommodate three- and four-leg roundabouts in urban and rural areas. The generalized form shows all the CMFs in the model. Indicator variables are used to determine which CMFs are applicable to each observation in the database based on the associated num- ber of legs and area type. The generalized form consists of leg-specific CMFs with indicator variables used to determine when the CMF is appli- cable. For example, the generalized form of the inscribed circle diameter CMF includes an indicator variable that is used to determine when the subject roundabout is located in an urban area—in which case, the CMF is applicable. If the roundabout is in a rural area, the CMF is not applicable. The generalized regression model is described using the following equations. A. If the observation corresponds to a three-leg round- about (m = 3), the following model is used. Equation 5-26 ,3 ,3N y C N CMF CMF CMFy state SPF legs outbd ICD= × × × × × with 1000 Equation 5-27 ,3 0,3 ,3 3N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-283 1 2 3 EntAADT AADT AADT AADT= + + Equation 5-29 1 1 2 2 3 3CMF p CMF p CMF p CMFlegs ( ) ( ) ( )= × + × + × Equation 5-30 1 2 3 p AADT AADT AADT AADT j j= + + Equation 5-31 _ ,1 _ ,2 _ ,3CMF exp b I I Ioutbd outbd outbd only outbd only outbd only[ ]( )= × + + where Ny,m = predicted average crash count for y years for roundabout with m legs (m = 3, 4), crashes; y = time interval for reported crashes (i.e., evalu- ation period), year; Cstate = calibration factor for specific jurisdictions represented in the database; NSPF,m = predicted average crash frequency for base conditions on all legs for roundabout with m legs, crashes/yr; pj = proportion of total leg traffic volume associ- ated with roundabout leg j (j = 1 to m); CMFlegs = aggregate crash modification factor for all legs; CMFj = combined crash modification factor for leg j (j = 1 to m); CMFoutbd = CMF for outbound-only leg presence;

87 CMFICD = CMF for inscribed circle diameter at urban roundabouts; EntAADTm = entering AADT for roundabout with m legs (m = 3, 4), veh/d; Irural = area type indicator variable (= 1.0 if area is rural, 0.0 otherwise); AADTj = AADT volume for roundabout leg j (j = 1 to m), veh/d; Ioutbd_only,j = indicator variable for travel direction on leg j ( j = 1 to m) (= 1.0 when the leg serves only one-way outbound traffic, 0.0 otherwise); and bi = calibration coefficient for condition i. B. If the observation corresponds to a four-leg roundabout (m = 4), the following model is used. Equation 5-32 ,4 ,4N y C N CMF CMF CMFy state SPF legs outbd ICD= × × × × × with 1000 Equation 5-33 ,4 0,4 ,4 4N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-34 4 1 2 3 4 EntAADT AADT AADT AADT AADT= + + + Equation 5-35 1 1 2 2 3 3 4 4 CMF p CMF p CMF p CMF p CMF legs ( ) ( ) ( ) ( ) = × + × + × + × Equation 5-36 1 2 3 4 p AADT AADT AADT AADT AADT j j= + + + Equation 5-37 _ ,1 _ ,2 _ ,3 _ ,4 CMF exp b I I I I outbd outbd outbd only outbd only outbd only outbd only [ ] ( ) = × + + + All variables are as previously defined. The following equations are used in the three- and four-leg models. These equations are applicable to the overall round- about (i.e., they are not specific to the design of a given leg). The CMF for outbound-only leg presence CMFoutbd also has this characteristic; however, it is shown previously because its formulation varies with the number of legs present. Equation 5-38C exp b Istate wi wi[ ]( )= × 1 125 Equation 5-39 CMF exp b I ICDICD ICD rural[ ]( ) ( )= × − × − where Iwi = indicator variable for Wisconsin (= 1.0 if site is in Wisconsin, 0.0 otherwise) and ICD = inscribed circle diameter, ft. All other variables are as previously defined. The value of 125 in Equation 5-39 is the base inscribed circle diameter. This value was computed as the average diam- eter of those roundabouts in the database that have one circulating lane. The following CMFs are used in the three- and four-leg models. These equations are specific to the design of a given leg. As a result, each CMF is computed once for each leg j of the roundabout (j = 1 to m). Equation 5-40, ,CMF CMF CMFj bypass j ap j= × Equation 5-41, ,CMF exp b Ibypass j bypass bypass j[ ]= × Equation 5-42, ,CMF exp b n Iap j nbrDwy ap j dwyCrashIncl[ ]= × × where CMFbypass,j = CMF for right-turn bypass lane on leg j (j = 1 to m); Ibypass,j = indicator variable for presence of a right-turn bypass lane on leg j (j = 1 to m) (= 1.0 when the leg includes a right-turn bypass lane, 0.0 otherwise); CMFap,j = CMF presence of driveways or unsignalized access points on leg j (j = 1 to m); nap,j = number of driveways or unsignalized access points on leg j (j = 1 to m) (within 250 ft of yield line); and IdwyCrashIncl = indicator variable for possible presence of access point–related crashes in the crash data (= 1.0 when access point–related crashes are included in the crash data, 0.0 otherwise). All other variables are as previously defined. The final form of the regression model reflects the findings from several preliminary regression analyses where alterna- tive model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defen- sible and properly bounded. CMFs for other variables were also examined but were not found to be helpful in explaining the variation in the observed crash frequency among sites. These variables include angle to next leg, luminaire presence, circulating width, ramp pres- ence, and driveway leg presence. This finding does not rule

88 out the possibility that these factors have an influence on roundabout safety. It is possible that their effect is sufficiently small that it would require either a larger database (with more observations and a wider range in variable values) to detect using regression or the use of a before–after study that iso- lates the individual effect of one factor on crash frequency. The inscribed circle diameter CMF (CMFICD) demonstrated different trends in the CMF value depending on whether the site was located in a rural or urban area. The trend for urban roundabouts was logical and statistically valid. However, the trend for rural roundabouts was highly uncertain so it was not included in the inscribed circle diameter CMF. As a result, this CMF applies only to urban roundabouts. As described in Section 5.2.3.2, four sites were removed from the database because they had a high FI crash rate. The prediction errors for these sites were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explana- tion could be found so they were considered to be outliers and were removed from the data used for model calibra- tion. As was also described in Section 5.2.3.2, 24 sites were removed from the database because they had unusually high year-to-year variation in the AADT volume of one or more legs. All total, 28 sites were removed; 23 of the sites had one circulating lane. The model calibration process was executed with and without the 23 suspect sites (using the same model form) to determine whether their omission had an effect on the model fit. The model fit was assessed by examination of the AIC (normalized for sample size), the t-statistic for each calibra- tion coefficient, and the value of the coefficient for the AADT variable. The findings from this investigation revealed that the overall model fit was degraded when the 23 sites were included in the database. Of note was the AADT coefficient that was unusually small (less than 0.3) when the 23 sites were included; however, it was more consistent with other research findings (between 0.5 and 1.3) when the 23 sites were excluded. Model Calibration. The predictive model calibration pro- cess was based on a combined regression modeling approach, as discussed in Section 5.2.4.2. With this approach, the com- ponent models and CMFs (represented by Equations 5-26 to 5-42) are calibrated using a common database. This approach is needed because the CMFs are common to both the three- and four-leg models. The models were calibrated using a randomly selected set of sites constituting 80% of sites in the database. The sites not selected were reserved for model validation. The discussion in this subsection focuses on the findings from the model calibration. The findings from model validation are provided in the next subsection. The results of the combined regression model calibration are presented in Table 5-35. Calibration of this model focused Model Statistics Value R2 (Rk2): 0.41 (0.63) Scale Parameter φ: 0.97 Pearson χ2: 163 (χ20.05, 159 = 189) Adjusted Inverse Dispersion Parameter K: Three-leg: 1.98; Four-leg: 3.22 Observations n: 169 roundabouts (381 FI crashes) Standard Deviation sp: ±0.31 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic bwi Location in Wisconsin 0.479 0.183 2.62 brural Location in rural area 0.236 0.155 1.52 bICD Inscribed circle diameter -0.00726 0.00370 -1.96 boutbd Outbound-only leg presence -0.912 0.571 -1.60 bbypass Right-turn bypass lane presence -1.359 1.214 -1.12 bnbrDwy Driveways or unsignalized public street approaches 0.0881 0.0516 1.71 b0,3 Three-leg roundabout -4.233 0.852 -4.97 bAADT,3 Entering AADT at three-leg roundabout 0.981 0.357 2.75 b0,4 Four-leg roundabout -3.591 0.459 -7.82 bAADT,4 Entering AADT at four-leg roundabout 0.961 0.175 5.50 Table 5-35. Predictive model calibration statistics, FI crashes, one circulating lane.

89 on FI crash frequency. The Pearson c2 statistic for the model is 163, and the degrees of freedom are 159 (= n − p = 169 −10). As this statistic is less than c20.05, 159 (= 189), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.41. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk 2, as developed by Miaou (1996). The Rk 2 for the calibrated model is 0.63. The t-statistics listed in the last column of Table 5-35 indi- cate a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statis- tic is smaller than 2.0, it was decided that the variable was important to the model, and its trend was found to be logical and consistent with previous research findings (even if the specific value was not known with a great deal of certainty as applied to this database). The findings from an examination of the coefficient values and the corresponding CMF or SPF predictions are documented in Section 6.1.2. An indicator variable for Wisconsin was included in the regression model. The coefficient for this variable is shown in the middle of Table 5-35. It is statistically significant. Its value indicates that the roundabouts in Wisconsin have about 61% more crashes than those in the other jurisdictions. This difference cannot be explained by differences in area type, inscribed circle diameter, presence of outbound-only lanes, bypass lane presence, driveway leg presence, or AADT among the jurisdictions. It is likely due to factors at Wisconsin round- abouts that are different from the other jurisdictions and that are not represented in the database (e.g., signing, pavement condition, weather, and speed limit). Model Validation. This subsection describes the find- ings from a model validation activity. The objective of this activity was to demonstrate the robustness of the model form and its transferability to roundabouts not represented in the calibration sites. The validation activity entailed the application of the cali- brated model to the sites not included in the calibration data- base. Twenty percent of the sites were held out of the database for this purpose. The calibrated model was used to compute the predicted average crash count for each of the sites in the validation database. The observed and predicted counts were then compared statistically to assess the validity of the calibrated model. These fit statistics are listed in Table 5-36. The Pearson c2 statistic for the combined model (= 34.0) is less than c20.05 (= 58.1) so the hypothesis that the model fits the validation data cannot be rejected. The findings from this validation activity indicate that the trends in the validation data are not significantly different from those in the calibration data. These findings suggest that the model form is reasonably robust and transferable to other roundabouts (when locally calibrated) for the prediction of FI crash frequency. Based on these findings, the calibration and validation data were combined and used in a second regression model cali- bration. The larger sample size associated with the combined database reduced the standard error of several calibration coefficients. Bared and Zhang (2007) also used this approach in their development of predictive models for urban freeways. Combined Model. The data from the calibration and validation databases were combined, and the predictive model was calibrated a second time using the combined data. The calibration coefficients for the combined models are described in this subsection. The fit statistics and inverse dispersion parameter for each component model are also described. The results of the combined regression model calibration are presented in Table 5-37. Calibration of this model focused on FI crash frequency. The Pearson c2 statistic for the model is 214, and the degrees of freedom are 202 (= n − p = 212 − 10). As this statistic is less than c20.05, 202 (= 236), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.35. The Rk 2 for the calibrated model is 0.57. Model for Roundabouts with Three Legs. The statistics describing the calibrated model for three-leg roundabouts are presented in Table 5-38. Calibration of this model focused on FI crash frequency. The Pearson c2 statistic for the model is 59.0, and the degrees of freedom are 53 (= n − p = 61 − 8). As this statistic is less than c20.05, 53 (= 71.0), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.38. The Rk 2 for the calibrated model is 0.65. The coefficients in Table 5-37 were combined with Equa- tion 5-27 to obtain the calibrated SPF for three-leg round- abouts. The form of the model is described by the following equation. 4.404 1.084 1000 0.206 Equation 5-43 ,3 3N exp LN EntAADT I SPF rural [ ] ( )= − + × + × Scale Parameter φ Pearson χ2 Observations n χ20.05, n-1 Standard Deviation sp 0.809 34.0 43 roundabouts 58.1 ±0.27 crashes/yr Table 5-36. Predictive model validation statistics, FI crashes, one circulating lane.

90 The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-11. This figure compares the predicted and reported crash frequency in the calibration database. The thick trend line shown repre- sents a y = x line. A data point would lie on this line if its pre- dicted and reported crash frequencies were equal. The two thin lines identify the 90th percentile confidence interval. The data points shown represent the reported crash frequency for the roundabouts used to calibrate the corresponding component model. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. Each data point shown in Figure 5-11 represents the aver- age predicted and average reported crash frequency for a group of 10 roundabouts (i.e., 10 sites). The data were sorted by predicted crash frequency to form groups of sites with similar crash frequency. The purpose of this grouping was to reduce the number of data points shown in the figure and, thereby, to facilitate an examination of trends in the Model Statistics Value R2 (Rk2): 0.35 (0.57) Scale Parameter φ: 1.01 Pearson χ2: 214 (χ20.05, 202 = 236) Adjusted Inverse Dispersion Parameter K: Three-leg: 3.20; Four-leg: 3.03 Observations n: 212 roundabouts (459 FI crashes) Standard Deviation sp: ±0.30 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic bwi Location in Wisconsin 0.419 0.170 2.47 brural Location in rural area 0.206 0.140 1.47 bICD Inscribed circle diameter -0.00621 0.00336 -1.85 boutbd Outbound-only leg presence -0.853 0.431 -1.98 bbypass Right-turn bypass lane presence -1.095 1.008 -1.09 bnbrDwy Driveways or unsignalized public street approaches 0.0659 0.0477 1.38 b0,3 Three-leg roundabout -4.404 0.737 -5.97 bAADT,3 Entering AADT at three-leg roundabout 1.084 0.302 3.59 b0,4 Four-leg roundabout -3.503 0.416 -8.42 bAADT,4 Entering AADT at four-leg roundabout 0.915 0.159 5.76 Table 5-37. Final predictive model calibration statistics, FI crashes, one circulating lane. Model Statistics Value R2 (Rk2): 0.38 (0.65) Scale Parameter φ: 0.98 Pearson χ2: 59.0 (χ20.05, 53 = 71.0) Adjusted Inverse Dispersion Parameter K: 3.20 Observations n: 61 roundabouts (81 FI crashes) Standard Deviation sp: ±0.18 crashes/yr NOTE: original inverse dispersion parameter = 5.71; original overdispersion parameter = 0.175; adjusted overdispersion parameter = 0.312. Table 5-38. Final predictive model calibration statistics, FI crashes, one circulating lane, three legs.

91 data. The individual site observations were used for model calibration. Model for Roundabouts with Four Legs. The statistics describing the calibrated model for four-leg roundabouts are presented in Table 5-39. Calibration of this model focused on fatal-plus-injury crash frequency. The Pearson c2 statistic for the model is 155, and the degrees of freedom are 143 (= n − p = 151 − 8). As this statistic is less than c20.05, 143 (= 172), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.32. The Rk 2 for the calibrated model is 0.49. The coefficients in Table 5-37 were combined with Equa- tion 5-33 to obtain the calibrated SPF for four-leg round- abouts. The form of the model is described by the following equation. 3.503 0.915 1000 0.206 Equation 5-44 ,4 4N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-12. This figure compares the predicted and reported crash frequency in the calibration database. The thick trend line shown rep- resents a y = x line. A data point would lie on this line if its predicted and reported crash frequencies were equal. The two thin lines identify the 90th percentile confidence inter- val. The data points shown represent the reported crash frequency for the roundabouts used to calibrate the corre- sponding component model. In general, the data shown in the figure indicate that the model provides an unbiased esti- mate of expected crash frequency. 5.2.4.5 FI Crash Frequency Prediction Model, Two Circulating Lanes This section describes the development of predictive models for roundabouts with two circulating lanes conflict- ing with one or more legs. The focus of this section is on models developed to predict FI crash frequency. One varia- tion of the model is applicable to three-leg roundabouts, and a second variation of the model is applicable to four-leg roundabouts. Figure 5-11. Predicted versus reported FI crashes at three-leg roundabouts with one circulating lane. Model Statistics Value R2 (Rk2): 0.32 (0.49) Scale Parameter φ: 1.03 Pearson χ2: 155 (χ20.05, 143 = 172) Adjusted Inverse Dispersion Parameter K: 3.03 Observations n: 151 roundabouts (378 FI crashes) Standard Deviation sp: ±0.34 crashes/yr NOTE: original inverse dispersion parameter = 3.47; original overdispersion parameter = 0.288; adjusted overdispersion parameter = 0.330. Table 5-39. Final predictive model calibration statistics, FI crashes, one circulating lane, four legs. Figure 5-12. Predicted versus reported FI crashes at four-leg roundabouts with one circulating lane.

92 This section consists of four subsections. The first sub- section describes the structure of the safety predictive models as used in the regression analysis. The second subsection describes the regression statistics for each of the calibrated models. The third subsection describes a validation of the calibrated models. The fourth subsection describes the pro- posed safety prediction models. Model Development. This subsection describes the pro- posed prediction models and the methods used to calibrate them. The regression model is generalized to accommo- date three- and four-leg roundabouts in urban and rural areas. The generalized form shows all the CMFs in the model. Indicator variables are used to determine which CMFs are applicable to each observation in the database based on the associated number of legs and area type. The generalized regression model is described using the following equations. A. If the observation corresponds to a three-leg round- about (m = 3), the following model is used. Equation 5-45,3 ,3N y C N CMF CMFy state SPF legs outbd= × × × × with 1000 Equation 5-46 ,3 0,3 ,3 3N exp b b LN EntAADT b I SPF AADT rural rural [ ( )= + × + × 2 Equation 5-473 1 2 3 EntAADT AADT AADT AADT= + + Equation 5-48 1 1 2 2 3 3CMF p CMF p CMF p CMFlegs ( ) ( ) ( )= × + × + × Equation 5-49 1 2 3 p AADT AADT AADT AADT j j= + + Equation 5-50 _ ,1 _ ,2 _ ,3CMF exp b I I Ioutbd outbd outbd only outbd only outbd only[ ]( )= × + + where Ny,m = predicted average crash count for y years for roundabout with m legs (m = 3, 4), crashes; y = time interval for reported crashes (evalua- tion period), yr; Cstate = calibration factor for specific jurisdictions represented in the database; NSPF,m = predicted average crash frequency for base conditions on all legs for roundabout with m legs, crashes/yr; pj = proportion of total leg traffic volume asso- ciated with roundabout leg j (j = 1 to m); CMFlegs = aggregate crash modification factor for all legs; CMFj = combined crash modification factor for leg j (j = 1 to m); CMFoutbd = crash modification factor for outbound-only leg presence; EntAADTm = entering AADT for roundabout with m legs (m = 3, 4), veh/d; Irural = area type indicator variable (= 1.0 if area is rural, 0.0 otherwise); AADTj = AADT volume for roundabout leg j (j = 1 to m), veh/d; Ioutbd_only,j = indicator variable for travel direction on leg j (j = 1 to m) (= 1.0 when the leg serves only one-way outbound traffic, 0.0 otherwise); and bi = calibration coefficient for condition i. B. If the observation corresponds to a four-leg roundabout (i.e., m = 4), the following model is used. Equation 5-51,4 ,4N y C N CMF CMFy state SPF legs outbd= × × × × with 1000 Equation 5-52 ,4 0,4 ,4 4N exp b b LN EntAADT b I SPF AADT rural rural [ ( )= + × + × 2 Equation 5-53 4 1 2 3 4 EntAADT AADT AADT AADT AADT= + + + Equation 5-54 1 1 2 2 3 3 4 4 CMF p CMF p CMF p CMF p CMF legs ( ) ( ) ( ) ( ) = × + × + × + × Equation 5-55 1 2 3 4 p AADT AADT AADT AADT AADT j j= + + + Equation 5-56 _ ,1 _ ,2 _ ,3 _ ,4 CMF exp b I I I I outbd outbd outbd only outbd only outbd only outbd only [ ] ( ) = × + + + All variables are as previously defined. The following adjustment factor is used in the three- and four-leg models. This factor is applicable to the overall roundabout (it is not specific to the design of a given leg). The CMF for outbound-only leg presence CMFoutbd also has this characteristic; however, it is shown previously because its formulation varies with the number of legs present. Equation 5-57C exp b Istate ca ca[ ]( )= ×

93 where Ica = indicator variable for California (= 1.0 if site is in California, 0.0 otherwise), and all other variables as previ- ously defined. The following CMFs are used in the three- and four-leg models. These equations are specific to the design of a given leg. As a result, each CMF is computed once for each leg j of the roundabout (j = 1 to m). Equation 5-58, , ,CMF CMF CMF CMFj bypass j ew j cl j= × × Equation 5-59, ,CMF exp b Ibypass j bypass bypass j[ ]= × Equation 5-60, , , ,CMF exp b W Wew j ew ew j ew b j[ ]( )= × − 4 Equation 5-61, , ,CMF exp b n ncl j cl cl j el j[ ]( )= × × − where CMFbypass,j = CMF for right-turn bypass lane on leg j (j = 1 to m); Ibypass,j = indicator variable for presence of a right-turn bypass lane on leg j (j = 1 to m) (= 1.0 when the leg includes a right-turn bypass lane, 0.0 otherwise); CMFew,j = CMF for entry width on leg j (j = 1 to m); Wew,j = entry width on leg j (includes the width of all entering lanes and the adjacent shoulder; excludes the width of the adjacent right-turn bypass lane, if present), ft; Wew,b,j = base entry width on leg j (= 20 if one entering lane, 29 if two entering lanes), ft; CMFcl,j = CMF for number of circulating lanes conflict- ing with leg j (j = 1 to m); ncl,j = number of circulating lanes conflicting with leg j (j = 1 to m), lanes; and nel,j = number of entering lanes on leg j (j = 1 to m) (exclude right-turn bypass lanes), lanes. All other variables are as previously defined. The base entry widths used in Equation 5-60 are computed as the average width of those roundabouts in the database that have two circulating lanes. The circulating lane CMF CMFcl accounts for the influence of conflict points adjacent to the subject leg. The number of conflict points is estimated as the product of the number of circulating lanes and the number of entering lanes. This CMF includes four conflict points as the base value, so it has a value of 1.00 when there are two circulating lanes and two entering lanes. The final form of the regression model reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defensible and properly bounded. CMFs for other variables were also examined but were not found to be helpful in explaining the variation in the observed crash frequency among sites. These variables include angle to next leg, luminaire presence, circulating width, ramp pres- ence, and driveway leg presence. This finding does not rule out the possibility that these factors have an influence on roundabout safety. It is possible that their effect is sufficiently small that it would require either a larger database (with more observations and a wider range in variable values) to detect using regression or the use of a before-after study that isolates the individual effect of one factor on crash frequency. As described in Section 5.2.3.2, four sites were removed from the database because they had a high FI crash rate. The prediction errors for these sites were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explana- tion could be found so they were considered to be outliers and were removed from the data used for model calibra- tion. As was also described in Section 5.2.3.2, 24 sites were removed from the database because they had unusually high year-to-year variation in the AADT volume of one or more legs. All total, 28 sites were removed; five of the sites had two circulating lanes. The model calibration process was executed with and with- out the five suspect sites (using the same model form) to deter- mine whether their omission had an effect on the model fit. The model fit was assessed by examination of the AIC (normal- ized for sample size), the t-statistic for each calibration coeffi- cient, and the value of the coefficient for the AADT variable. The findings from this investigation revealed that the overall model fit was degraded when the five sites were included in the database. Model Calibration. The predictive model calibration pro- cess was based on a combined regression modeling approach, as discussed in Section 5.2.4.2. With this approach, the com- ponent models and CMFs (represented by Equations 5-45 to 5-61) are calibrated using a common database. This approach is needed because the CMFs are common to both the three- and four-leg models. The models were calibrated using a randomly selected set of sites constituting 80% of sites in the database. The sites not selected were reserved for model validation. The discussion in this subsection focuses on the findings from the model calibration. The findings from model validation are provided in the next subsection. The results of the combined regression model calibration are presented in Table 5-40. Calibration of this model focused

94 on FI crash frequency. The Pearson c2 statistic for the model is 88.7, and the degrees of freedom are 82 (= n − p = 92 −10). As this statistic is less than c20.05, 82 (= 104), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.67. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk 2, as developed by Miaou (1996). The Rk 2 for the calibrated model is 0.70. The t-statistics listed in the last column of Table 5-40 indi- cate a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statis- tic is smaller than 2.0, it was decided that the variable was important to the model, and its trend was found to be logical and consistent with previous research findings (even if the specific value was not known with a great deal of certainty as applied to this database). The findings from an examination of the coefficient values and the corresponding CMF or SPF predictions are documented in Section 6.1.2. An indicator variable for the State of California was included in the regression model. The coefficient for this vari- able is shown in the middle of Table 5-40. It is statistically significant. Its value indicates that the roundabouts in Cali- fornia have about 200 percent more crashes than those in the other jurisdictions. This difference cannot be explained by dif- ferences in area type, presence of outbound-only lanes, pres- ence of bypass lanes, entry width, number of circulating lanes, or AADT among the jurisdictions. It is likely due to factors at the California roundabouts that are different from the other jurisdictions and that are not represented in the database (e.g., signing, pavement condition, weather, speed limit, and driver behavior). Model Validation. This subsection describes the find- ings from a model validation activity. The objective of this activity was to demonstrate the robustness of the model form and its transferability to roundabouts not represented in the calibration sites. The validation activity entailed the application of the cali- brated model to the sites not included in the calibration data- base. Twenty percent of the sites were held out of the database for this purpose. The calibrated model was used to com- pute the predicted average crash count for each of the sites in the validation database. The observed and predicted counts were then compared statistically to assess the validity of the calibrated model. Model Statistics Value R2 (Rk2): 0.67 (0.70) Scale Parameter φ: 0.97 Pearson χ2: 88.7 (χ20.05, 82 = 104) Adjusted Inverse Dispersion Parameter K: Three-leg: 3.21; Four-leg: 2.52 Observations n: 92 roundabouts (576 FI crashes) Standard Deviation sp: ±0.96 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic bca Location in California 1.107 0.435 2.54 brural Location in rural area 0.293 0.197 1.49 boutbd Outbound-only leg presence -0.574 0.407 -1.41 bbypass Right-turn bypass lane presence -1.158 0.833 -1.39 bew Width of entering lanes -0.0402 0.0335 -1.20 bcl Number of circulating lanes 0.224 0.102 2.20 b0,3 Three-leg roundabout -4.012 1.149 -3.49 bAADT,3 Entering AADT at three-leg roundabout 1.371 0.398 3.45 b0,4 Four-leg roundabout -3.593 0.655 -5.49 bAADT,4 Entering AADT at four-leg roundabout 1.310 0.211 6.21 Table 5-40. Predictive model calibration statistics, FI crashes, two circulating lanes.

95 These fit statistics are listed in the top row of Table 5-41. The Pearson c2 statistic for the combined model (= 39.0) is not less than c20.05 (= 33.9), so the hypothesis that the model fits the validation data is rejected. Closer examination of the data revealed that there was one low-volume site that experienced four crashes in a 5-year period; yet, the calibrated model predicted 0.64 crashes for the same time period. This site was removed from the valida- tion database to obtain the fit statistics in the bottom row of Table 5-41. Using the reduced dataset, the Pearson c2 statistic for the combined model (= 24.9) is less than c20.05 (= 32.7), so the hypothesis that the model fits the reduced validation dataset cannot be rejected. It is rationalized that the one site is unusual, and that the results from the reduced validation dataset provide a more reliable indication of model validity. Based on these findings, the calibration and validation data were combined and used in a second regression model cali- bration. The larger sample size associated with the combined database reduced the standard error of several calibration coefficients. Bared and Zhang (2007) also used this approach in their development of predictive models for urban freeways. Combined Model. The data from the calibration and validation databases were combined, and the predictive model was calibrated a second time using the combined data. The calibration coefficients for the combined models are described in this subsection. The fit statistics and inverse dispersion parameter for each component model are also described. The analysis was conducted with and without the unusual site discussed in the previous section. The regression coefficient values associated with each database variation were not found to be notably different. The predicted value for the unusual site increased slightly; however, its residual error was not so large as to indicate that it should be considered an outlier suitable for removal from the database. As a result, the unusual site was not removed from the database used to determine the final model coefficients. The results of the combined regression model calibration are presented in Table 5-42. Calibration of this model focused on FI crash frequency. The Pearson c2 statistic for the model is 114, and the degrees of freedom are 105 (= n − p = 115 − 10). As this statistic is less than c20.05, 105 (= 130), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.67. The Rk 2 for the calibrated model is 0.68. Model for Roundabouts with Three Legs. The statistics describing the calibrated model for three-leg roundabouts are presented in Table 5-43. Calibration of this model focused on FI crash frequency. The Pearson c2 statistic for the model is 38.8, and the degrees of freedom are 26 (= n − p = 34 − 8). As this statistic is less than c20.05, 26 (= 38.9), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.52. The Rk 2 for the calibrated model is 0.77. The coefficients in Table 5-42 were combined with Equa- tion 5-46 to obtain the calibrated SPF for three-leg round- abouts. The form of the model is described by the following equation. 3.887 1.306 1000 0.250 Equation 5-62 ,3 3N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-13. This figure compares the predicted and reported crash fre- quency in the calibration database. The thick trend line shown represents a y = x line. A data point would lie on this line if its predicted and reported crash frequencies were equal. The two thin lines identify the 90th percentile confi- dence interval. The data points shown represent the reported crash frequency for the roundabouts used to calibrate the corresponding component model. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. With one exception, each data point shown in Figure 5-13 represents the average predicted and average reported crash frequency for a group of 10 roundabouts. The data point associated with a predicted crash frequency of 6.7 crashes per period represents the average for a group of four round- abouts. The data were sorted by predicted crash frequency to form groups of sites with similar crash frequency. The purpose of this grouping was to reduce the number of data points shown in the figure and, thereby, to facilitate an exami- nation of trends in the data. The individual site observations were used for model calibration. Model for Roundabouts with Four Legs. The statistics describing the calibrated model for four-leg roundabouts are Scale Parameter φ Pearson χ2 Observations n χ20.05, n-1 Standard Deviation sp 1.77 39.0 23 roundabouts 33.9 ±0.63 crashes/yr Statistics with Unusual Site Removed 1.13 24.9 22 roundabouts 32.7 ±0.62 crashes/yr Table 5-41. Predictive model validation statistics, FI crashes, two circulating lanes.

96 presented in Table 5-44. Calibration of this model focused on FI crash frequency. The Pearson c2 statistic for the model is 74.9, and the degrees of freedom are 73 (= n − p = 81 − 8). As this statistic is less than c20.05, 73 (= 93.9), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.65. The Rk 2 for the calibrated model is 0.63. The coefficients in Table 5-42 were combined with Equa- tion 5-52 to obtain the calibrated SPF for four-leg round- abouts. The form of the model is described by the following equation. 3.535 1.276 1000 0.250 Equation 5-63 ,4 4N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-14. This figure compares the predicted and reported crash frequency in the calibration database. The thick trend line shown rep- resents a y = x line. A data point would lie on this line if its Model Statistics Value R2 (Rk2): 0.67 (0.68) Scale Parameter φ: 1.00 Pearson χ2: 114 (χ20.05, 105 = 130) Adjusted Inverse Dispersion Parameter K: Three-leg: 2.75; Four-leg: 2.20 Observations n: 115 roundabouts (652 FI crashes) Standard Deviation sp: ±0.91 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic bca Location in California 1.151 0.458 2.51 brural Location in rural area 0.250 0.192 1.31 boutbd Outbound-only leg presence -0.787 0.409 -1.92 bbypass Right-turn bypass lane presence -0.840 0.784 -1.07 bew Width of entering lanes -0.0300 0.0329 -0.91 bcl Number of circulating lanes 0.196 0.0988 1.98 b0,3 Three-leg roundabout -3.887 0.891 -4.36 bAADT,3 Entering AADT at three-leg roundabout 1.306 0.316 4.14 b0,4 Four-leg roundabout -3.535 0.629 -5.61 bAADT,4 Entering AADT at four-leg roundabout 1.276 0.202 6.31 Table 5-42. Final predictive model calibration statistics, FI crashes, two circulating lanes. Model Statistics Value R2 (Rk2): 0.52 (0.77) Scale Parameter φ: 1.18 Pearson χ2: 38.8 (χ20.05, 26 = 38.9) Adjusted Inverse Dispersion Parameter K: 2.75 Observations n: 34 roundabouts (91 FI crashes) Standard Deviation sp: ±0.38 crashes/yr NŠ‹Œ: original inverse dispersion parameter = 4.62; original overdispersion parameter = 0.216; adjusted overdispersion parameter = 0.363. Table 5-43. Final predictive model calibration statistics, FI crashes, two circulating lanes, three legs.

97 these models are applicable only to roundabouts having one circulating lane conflicting with each leg. They are not appli- cable to roundabouts that have two circulating lanes con- flicting with one or more legs. The focus of this section is on models developed to predict PDO crash frequency. One variation of the model is applicable to three-leg roundabouts, and a second variation of the model is applicable to four-leg roundabouts. This section consists of four subsections. The first sub- section describes the structure of the safety predictive models as used in the regression analysis. The second subsection describes the regression statistics for each of the calibrated models. The third subsection describes a validation of the calibrated models. The fourth subsection describes the pro- posed safety prediction models. Model Development. This subsection describes the pro- posed prediction models and the methods used to calibrate them. The regression model is generalized to accommodate three- and four-leg roundabouts in urban and rural areas. The generalized form shows all the CMFs in the model. Indicator variables are used to determine which CMFs are applicable to each observation in the database based on the associated number of legs and area type. The generalized regression model is described using the following equations. A. If the observation corresponds to a three-leg round- about (m = 3), the following model is used. Equation 5-64,3 ,3N y N CMFy SPF legs= × × with 1000 Equation 5-65 ,3 0,3 ,3 3N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-663 1 2 3 EntAADT AADT AADT AADT= + + Figure 5-13. Predicted versus reported FI crashes at three-leg roundabouts with two circulating lanes. Model Statistics Value R2 (Rk2): 0.65 (0.63) Scale Parameter φ: 0.94 Pearson χ2: 74.9 (χ20.05, 73 = 93.9) Adjusted Inverse Dispersion Parameter K: 2.20 Observations n: 81 roundabouts (561 FI crashes) Standard Deviation sp: ±1.05 crashes/yr NŠ‹Œ: original inverse dispersion parameter =2.36; original overdispersion parameter = 0.424; adjusted overdispersion parameter = 0.455. Table 5-44. Final predictive model calibration statistics, FI crashes, two circulating lanes, four legs. predicted and reported crash frequencies were equal. The two thin lines identify the 90th percentile confidence interval. The data points shown represent the reported crash frequency for the roundabouts used to calibrate the corresponding com- ponent model. In general, the data shown in Figure 5-14 indicate that the model provides an unbiased estimate of expected crash frequency. There were a few roundabouts with a very high reported PDO crash frequency that were underpredicted by the model. These sites are in the group of 10 sites with a pre- dicted crash frequency of 19 crashes per period. Nevertheless, the average reported crash frequency for this group of sites is still within the 90th percentile confidence interval for the predicted value. 5.2.4.6 PDO Crash Frequency Prediction Model, One Circulating Lane This section describes the development of predictive mod- els for roundabouts with one circulating lane. Specifically,

98 Equation 5-67 1 1 2 2 3 3CMF p CMF p CMF p CMFlegs ( ) ( ) ( )= × + × + × Equation 5-68 1 2 3 p AADT AADT AADT AADT j j= + + where Ny,m = predicted average crash count for y years for roundabout with m legs (m = 3, 4), crashes; y = time interval for reported crashes (i.e., evalu- ation period), yr; NSPF,m = predicted average crash frequency for base conditions on all legs for roundabout with m legs, crashes/yr; pj = proportion of total leg traffic volume asso- ciated with roundabout leg j (j = 1 to m); CMFlegs = aggregate CMF for all legs; CMFj = combined CMF for leg j (j = 1 to m); EntAADTm = entering AADT for roundabout with m legs (m = 3, 4), veh/d; Irural = area type indicator variable (= 1.0 if area is rural, 0.0 otherwise); AADTj = AADT volume for roundabout leg j ( j = 1 to m), veh/d; and bi = calibration coefficient for condition i. B. If the observation corresponds to a four-leg roundabout (m = 4), the following model is used. Equation 5-69,4 ,4N y N CMFy SPF legs= × × with 1000 Equation 5-70 ,4 0,4 ,4 4N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-71 4 1 2 3 4 EntAADT AADT AADT AADT AADT= + + + Equation 5-72 1 1 2 2 3 3 4 4 CMF p CMF p CMF p CMF p CMF legs ( ) ( ) ( ) ( ) = × + × + × + × Equation 5-73 1 2 3 4 p AADT AADT AADT AADT AADT j j= + + + All variables are as previously defined. The following CMFs are used in the three- and four-leg models. These equations are specific to the design of a given leg. As a result, each CMF is computed once for each leg j of the roundabout (j = 1 to m). Equation 5-74,CMF CMFj ap j= Equation 5-75, ,CMF exp b n Iap j nbrDwy ap j dwyCrashIncl[ ]= × × where CMFap,j = CMF presence of driveways or unsignalized access points on leg j (j = 1 to m); 63 Figure 5-14. Predicted versus reported FI crashes at four-leg roundabouts with two circulating lanes.

99 nap,j = number of driveways or unsignalized access points on leg j ( j = 1 to m) (within 250 ft of yield line); and IdwyCrashIncl = indicator variable for possible presence of access point–related crashes in the crash data (= 1.0 when access point–related crashes are included in the crash data, 0.0 otherwise). All other variables are as previously defined. The final form of the regression model reflects the findings from several preliminary regression analyses where alterna- tive model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defensible and properly bounded. CMFs for other variables were also examined but were not found to be helpful in explaining the variation in the observed crash frequency among sites. These variables include angle to next leg, luminaire presence, circulating width, ramp pres- ence, and driveway leg presence. This finding does not rule out the possibility that these factors have an influence on roundabout safety. It is possible that their effect is sufficiently small that it would require either a larger database (with more observations and a wider range in variable values) to detect using regression or the use of a before–after study that iso- lates the individual effect of one factor on crash frequency. Each of the CMFs in the FI regression model (Equa- tions 5-26 to 5-42) were evaluated in the PDO model. How- ever, the coefficients associated with several of these CMFs were not found to be statistically valid, logical, or both. As a result, they were removed from the PDO model. This pat- tern (where the PDO model supports fewer CMFs than the FI model) was noted by Bonneson et al. (2012) in the devel- opment of FI and PDO models for freeway segments. The pattern is likely attributable to the relatively large variation in PDO crash frequency that is partly a result of differences in reporting threshold between and within jurisdictions, as noted previously in the discussion associated with Table 5-29. As described in Section 5.2.3.2, four sites were removed from the database because they had a high crash rate. The prediction errors for these sites were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explana- tion could be found so they were considered to be outliers and were removed from the data used for model calibra- tion. As was also described in Section 5.2.3.2, 24 sites were removed from the database because they had unusually high year-to-year variation in the AADT volume of one or more legs. In total, 28 sites were removed; 23 of the sites had one circulating lane. The model calibration process was executed with and without the 23 suspect sites (using the same model form) to determine whether their omission had an effect on the model fit. The model fit was assessed by examination of the AIC (normalized for sample size), the t-statistic for each calibra- tion coefficient, and the value of the coefficient for the AADT variable. The findings from this investigation revealed that the overall model fit was degraded when the 23 sites were included in the database. Of note was the AADT coefficient that was unusually small (less than 0.3) when the 23 sites were included; however, it was more consistent with other research findings (those between 0.5 and 1.3) when the 23 sites were excluded. Model Calibration. The predictive model calibration pro- cess was based on a combined regression modeling approach, as discussed in Section 5.2.4.2. With this approach, the com- ponent models and CMFs (represented by Equations 5-64 to 5-75) are calibrated using a common database. This approach is needed because the CMFs are common to both the three- and four-leg models. The models were calibrated using a randomly selected set of sites constituting 80% of sites in the database. The sites not selected were reserved for model validation. The discussion in this subsection focuses on the findings from the model calibration. The findings from model validation are provided in the next subsection. The results of the combined regression model calibra- tion are presented in Table 5-45. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 178, and the degrees of freedom are 163 (= n − p = 169 −6). As this statistic is less than c20.05, 163 (= 194), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.27. The Rk 2 for the calibrated model is 0.29. The t-statistics listed in the last column of Table 5-45 indi- cate a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the variable was impor- tant to the model, and its trend was found to be logical and consistent with previous research findings (or with the pre- viously calibrated FI models). The findings from an exami- nation of the coefficient values and the corresponding CMF or SPF predictions are documented in Section 6.1.2. Model Validation. This subsection describes the find- ings from a model validation activity. The objective of this activity was to demonstrate the robustness of the model form

100 and its transferability to roundabouts not represented in the calibration sites. The validation activity entailed the application of the calibrated model to the sites not included in the calibration database. Twenty percent of the sites were held out of the database for this purpose. The calibrated model was used to compute the predicted average crash count for each of the sites in the validation database. The observed and predicted counts were then compared statistically to assess the validity of the calibrated model. These fit statistics are listed in Table 5-46. The Pearson c2 statistic for the combined model (= 48.4) is less than c20.05 (= 58.1) so the hypothesis that the model fits the validation data cannot be rejected. The findings from this validation activity indicate that the trends in the validation data are not significantly different from those in the calibration data. These findings suggest that the model form is reasonably robust and transferable to other roundabouts (when locally calibrated) for the predic- tion of PDO crash frequency. Based on these findings, the calibration and validation data were combined and used in a second regression model cali- bration. The larger sample size associated with the combined database reduced the standard error of several calibration coefficients. Combined Model. The data from the calibration and validation databases were combined, and the predictive model was calibrated a second time using the combined data. The calibration coefficients for the combined models are described in this subsection. The fit statistics and inverse dispersion parameter for each component model are also described. The results of the combined regression model calibration are presented in Table 5-47. Calibration of this model focused on PDO crash frequency. The Pearson c2 statis- tic for the model is 218, and the degrees of freedom are 206 (= n − p = 212 − 6). As this statistic is less than c20.05, 206 (= 240), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.23. The Rk 2 for the cali- brated model is 0.26. Model for Roundabouts with Three Legs. The statistics describing the calibrated model for three-leg roundabouts are presented in Table 5-48. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 71.4, and the degrees of freedom are 57 (= n − p = 61 − 4). Model Statistics Value R2 (Rk2): 0.27 (0.29) Scale Parameter φ: 1.06 Pearson χ2: 178 (χ20.05, 163 = 194) Adjusted Inverse Dispersion Parameter K: Three-leg: 1.79; Four-Leg: 1.23 Observations n: 169 roundabouts (1,389 PDO crashes) Standard Deviation sp: ±1.30 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic brural Location in rural area 0.149 0.157 0.95 bnbrDwy Driveways or unsignalized public street approaches 0.0892 0.0583 1.53 b0,3 Three-leg roundabout -1.713 0.508 -3.37 bAADT,3 Entering AADT at three-leg roundabout 0.484 0.217 2.23 b0,4 Four-leg roundabout -1.836 0.480 -3.83 bAADT,4 Entering AADT at four-leg roundabout 0.858 0.196 4.37 Table 5-45. Predictive model calibration statistics, PDO crashes, one circulating lane. Scale Parameter φ Pearson χ2 Observations n χ20.05, n-1 Standard Deviation sp 1.15 48.4 43 roundabouts 58.1 ±1.15 crashes/yr Table 5-46. Predictive model validation statistics, PDO crashes, one circulating lane.

101 As this statistic is less than c20.05, 57 (= 75.6), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.30. The Rk 2 for the calibrated model is 0.18. The coefficients in Table 5-47 were combined with Equa- tion 5-65 to obtain the calibrated SPF for three-leg round- abouts. The form of the model is described by the following equation. 1.720 0.486 1000 0.168 Equation 5-76 ,3 3N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-15. This figure compares the predicted and reported crash frequency in the calibration database. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. Each data point shown in Figure 5-15 represents the aver- age predicted and average reported crash frequency for a group of 10 roundabouts (i.e., 10 sites). The data were sorted by predicted crash frequency to form groups of sites with similar crash frequency. The purpose of this grouping was to reduce the number of data points shown in the figure and, thereby, to facilitate an examination of trends in the data. The individual site observations were used for model calibration. Model for Roundabouts with Four Legs. The statistics describing the calibrated model for four-leg roundabouts are presented in Table 5-49. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model Model Statistics Value R2 (Rk2): 0.23 (0.26) Scale Parameter φ: 1.04 Pearson χ2: 218 (χ20.05, 206 = 240) Adjusted Inverse Dispersion Parameter K: Three-leg: 1.84; Four-leg: 1.25 Observations n: 212 roundabouts (1,765 PDO crashes) Standard Deviation sp: ±1.26 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic brural Location in rural area 0.168 0.139 1.21 bnbrDwy Driveways or unsignalized public street approaches 0.0885 0.0528 1.68 b0,3 Three-leg roundabout -1.720 0.467 -3.68 bAADT,3 Entering AADT at three-leg roundabout 0.486 0.198 2.46 b0,4 Four-leg roundabout -1.475 0.399 -3.70 bAADT,4 Entering AADT at four-leg roundabout 0.702 0.162 4.33 Table 5-47. Final predictive model calibration statistics, PDO crashes, one circulating lane. Model Statistics Value R2 (Rk2): 0.30 (0.18) Scale Parameter φ: 1.19 Pearson χ2: 71.4 (χ20.05, 57 = 75.6) Adjusted Inverse Dispersion Parameter K: 1.84 Observations n: 61 roundabouts (267 PDO crashes) Standard Deviation sp: ±0.63 crashes/yr N‰Š‹: original inverse dispersion parameter = 2.07; original overdispersion parameter = 0.482; adjusted overdispersion parameter = 0.543. Table 5-48. Final predictive model calibration statistics, PDO crashes, one circulating lane, three legs.

102 is 148, and the degrees of freedom are 145 (= n − p = 151 − 6). As this statistic is less than c20.05, 145 (= 174), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.16. The Rk 2 for the calibrated model is 0.16. The coefficients in Table 5-47 were combined with Equa- tion 5-70 to obtain the calibrated SPF for four-leg round- abouts. The form of the model is described by the following equation. 1.475 0.702 1000 0.168 Equation 5-77 ,4 4N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-16. This figure compares the predicted and reported crash fre- quency in the calibration database. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. 5.2.4.7 PDO Crash Frequency Prediction Model, Two Circulating Lanes This section describes the development of predictive mod- els for roundabouts with two circulating lanes conflicting with one or more legs. The focus of this section is on models devel- oped to predict PDO crash frequency. One variation of the model is applicable to three-leg roundabouts, and a second variation of the model is applicable to four-leg roundabouts. This section consists of four subsections. The first sub- section describes the structure of the safety predictive mod- els as used in the regression analysis. The second subsection describes the regression statistics for each of the calibrated models. The third subsection describes a validation of the calibrated models. The fourth subsection describes the pro- posed safety prediction models. Model Development. This subsection describes the pro- posed prediction models and the methods used to calibrate them. The regression model is generalized to accommodate Figure 5-15. Predicted versus reported PDO crashes at three-leg roundabouts with one circulating lane. Model Statistics Value R2 (Rk2): 0.16 (0.16) Scale Parameter φ: 0.98 Pearson χ2: 148 (χ20.05, 145 = 174) Adjusted Inverse Dispersion Parameter K: 1.25 Observations n: 151 roundabouts (1,498 PDO crashes) Standard Deviation sp: ±1.44 crashes/yr Nˆ‰Š: original inverse dispersion parameter = 1.27; original overdispersion parameter = 0.787; adjusted overdispersion parameter = 0.799. Table 5-49. Final predictive model calibration statistics, PDO crashes, one circulating lane, four legs.

103 three- and four-leg roundabouts in urban and rural areas. The generalized form shows all the CMFs in the model. Indicator variables are used to determine which CMFs are applicable to each observation in the database based on the associated number of legs and area type. The generalized regression model is described using the following equations. A. If the observation corresponds to a three-leg round- about (m = 3), the following model is used. Equation 5-78,3 ,3N y N CMFy SPF legs= × × with 1000 Equation 5-79 ,3 0,3 ,3 3N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-803 1 2 3 EntAADT AADT AADT AADT= + + Equation 5-81 1 1 2 2 3 3CMF p CMF p CMF p CMFlegs ( ) ( ) ( )= × + × + × Equation 5-82 1 2 3 p AADT AADT AADT AADT j j= + + where Ny,m = predicted average crash count for y years for roundabout with m legs (m = 3, 4), crashes; y = time interval for reported crashes, yr; NSPF,m = predicted average crash frequency for base conditions on all legs for roundabout with m legs, crashes/yr; pj = proportion of total leg traffic volume asso- ciated with roundabout leg j (j = 1 to m); CMFlegs = aggregate CMF for all legs; CMFj = combined CMF for leg j (j = 1 to m); EntAADTm = entering AADT for roundabout with m legs (m = 3, 4), veh/d; Irural = area type indicator variable (= 1.0 if area is rural, 0.0 otherwise); AADTj = AADT volume for roundabout leg j ( j = 1 to m), veh/d; and bi = calibration coefficient for condition i. B. If the observation corresponds to a four-leg roundabout (m = 4), the following model is used. Equation 5-83,4 ,4N y N CMFy SPF legs= × × with 1000 Equation 5-84 ,4 0,4 ,4 4N exp b b LN EntAADT b I SPF AADT rural rural [ ] ( )= + × + × 2 Equation 5-85 4 1 2 3 4 EntAADT AADT AADT AADT AADT= + + + Equation 5-86 1 1 2 2 3 3 4 4 CMF p CMF p CMF p CMF p CMF legs ( ) ( ) ( ) ( ) = × + × + × + × Equation 5-87 1 2 3 4 p AADT AADT AADT AADT AADT j j= + + + All variables are as previously defined. The following CMFs are used in the three- and four-leg models. These equations are specific to the design of a given leg. As a result, each CMF is computed once for each leg j of the roundabout (j = 1 to m). Equation 5-88, ,CMF CMF CMFj ew j cl j= × Equation 5-89, , , ,CMF exp b W Wew j ew ew j ew b j[ ]( )= × − 4 Equation 5-90, , ,CMF exp b n ncl j cl cl j el j[ ]( )= × × − where CMFew,j = CMF for entry width on leg j (j = 1 to m); Wew,j = entry width on leg j (includes the width of all entering lanes and the adjacent shoulder; excludes the width of the adjacent right-turn bypass lane, if present), ft; Figure 5-16. Predicted versus reported PDO crashes at four-leg roundabouts with one circulating lane.

104 Wew,b,j = base entry width on leg j (= 20 if one entering lane, 29 if two entering lanes), ft; CMFcl,j = CMF for number of circulating lanes conflicting with leg j (j = 1 to m); ncl,j = number of circulating lanes conflicting with leg j ( j = 1 to m), lanes; and nel,j = number of entering lanes on leg j (j = 1 to m) (exclude right-turn bypass lanes), lanes. All other variables are as previously defined. The base entry widths used in Equation 5-89 are computed as the average width of those roundabouts in the database that have two circulating lanes. The circulating lane CMF CMFcl accounts for the influence of conflict points adjacent to the subject leg. The number of conflict points is estimated as the product of the number of circulating lanes and the number of entering lanes. This CMF includes four conflict points as the base value, so it has a value of 1.00 when there are two circulating lanes and two entering lanes. The final form of the regression model reflects the findings from several preliminary regression analyses where alterna- tive model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defensible and properly bounded. CMFs for other variables were also examined but were not found helpful in explaining the variation in the observed crash frequency among sites. These variables include angle to next leg, luminaire presence, circulating width, ramp pres- ence, and driveway leg presence. This finding does not rule out the possibility that these factors have an influence on roundabout safety. It is possible that their effect is sufficiently small that it would require either a larger database (with more observations and a wider range in variable values) to detect using regression or the use of a before–after study that iso- lates the individual effect of one factor on crash frequency. Each of the CMFs in the FI regression model (Equa- tions 5-45 to 5-61) were evaluated in the PDO model. How- ever, the coefficients associated with several of these CMFs were not found to be statistically valid, logical, or both. As a result, they were removed from the PDO model. This pat- tern (where the PDO model supports fewer CMFs than the FI model) was noted by Bonneson et al. (2012) in the devel- opment of FI and PDO models for freeway segments. The pattern is likely attributable to the relatively large variation in PDO crash frequency that is partly a result of differences in reporting threshold between and within jurisdictions, as noted previously in the discussion associated with Table 5-29. As described in Section 5.2.3.2, four sites were removed from the database because they had a high crash rate. The prediction errors for these sites were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explana- tion could be found, so they were considered to be outliers and were removed from the data used for model calibra- tion. As was also described in Section 5.2.3.2, 24 sites were removed from the database because they had unusually high year-to-year variation in the AADT volume of one or more legs. In total, 28 sites were removed; five of the sites had two circulating lanes. The model calibration process was executed with and without the five suspect sites (using the same model form) to determine whether their omission had an effect on the model fit. The model fit was assessed by examination of the AIC (normalized for sample size), the t-statistic for each calibra- tion coefficient, and the value of the coefficient for the AADT variable. The findings from this investigation revealed that the overall model fit was degraded when the five sites were included in the database. Model Calibration. The predictive model calibration pro- cess was based on a combined regression modeling approach, as discussed in Section 5.2.4.2. With this approach, the com- ponent models and CMFs (represented by Equations 5-78 to 5-90) are calibrated using a common database. This approach is needed because the CMFs are common to both the three- and four-leg models. The models were calibrated using a randomly selected set of sites constituting 80% of sites in the database. The sites not selected were reserved for model validation. The discussion in this subsection focuses on the findings from the model calibration. The findings from model validation are provided in the next subsection. The results of the combined regression model calibra- tion are presented in Table 5-50. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 82.1, and the degrees of freedom are 85 (= n − p = 92 − 7). As this statistic is less than c20.05, 85 (= 107), the hypoth- esis that the model fits the data cannot be rejected. The R2 for the model is 0.51. The Rk 2 for the calibrated model is 0.39. The t-statistics listed in the last column of Table 5-50 indi- cate a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected, with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the variable was impor- tant to the model, and its trend was found to be logical and consistent with previous research findings (or with the previ- ously calibrated FI models). The findings from an examina- tion of the coefficient values and the corresponding CMF or SPF predictions are documented in Section 6.1.2.

105 Model Validation. This subsection describes the find- ings from a model validation activity. The objective of this activity was to demonstrate the robustness of the model form and its transferability to roundabouts not represented in the calibration sites. The validation activity entailed the application of the calibrated model to the sites not included in the calibration database. Twenty percent of the sites were held out of the database for this purpose. The calibrated model was used to compute the predicted average crash count for each of the sites in the validation database. The observed and predicted counts were then compared statistically to assess the validity of the calibrated model. These fit statistics are listed in Table 5-51. The Pearson c2 statistic for the combined model (= 14.9) is less than c20.05 (= 26.3), so the hypothesis that the model fits the validation data cannot be rejected. Based on these findings, the calibration and validation data were combined and used in a second regression model cali- bration. The larger sample size associated with the combined database reduced the standard error of several calibration coefficients. Combined Model. The data from the calibration and validation databases were combined, and the predictive model was calibrated a second time using the combined data. The calibration coefficients for the combined models are described in this subsection. The fit statistics and inverse dispersion parameter for each component model are also described. The results of the combined regression model calibration are presented in Table 5-52. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 102, and the degrees of freedom are 108 (= n − p = 115 − 7). As this statistic is less than c20.05, 108 (= 133), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.53. The Rk 2 for the calibrated model is 0.42. Model for Roundabouts with Three Legs. The statistics describing the calibrated model for three-leg roundabouts are presented in Table 5-53. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 29.9, and the degrees of freedom are 29 (= n − p = 34 − 5). As this statistic is less than c20.05, 29 (= 42.6), the hypothesis that Model Statistics Value R2 (Rk2): 0.51 (0.39) Scale Parameter φ: 0.90 Pearson χ2: 82.1 (χ20.05, 85 = 107) Adjusted Inverse Dispersion Parameter K: Three-leg: 1.01; Four-leg: 1.28 Observations n: 92 roundabouts (2,931 PDO crashes) Standard Deviation sp: ±5.50 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic brural Location in rural area 0.427 0.245 1.74 bew Width of entering lanes -0.0498 0.0416 -1.20 bcl Number of circulating lanes 0.261 0.135 1.94 b0,3 Three-leg roundabout -0.996 0.831 -1.20 bAADT,3 Entering AADT at three-leg roundabout 0.938 0.273 3.43 b0,4 Four-leg roundabout -1.307 0.584 -2.24 bAADT,4 Entering AADT at four-leg roundabout 1.078 0.186 5.81 Table 5-50. Predictive model calibration statistics, PDO crashes, two circulating lanes. Scale Parameter φ Pearson χ2 Observations n χ20.05, n-1 Standard Deviation sp 0.68 14.9 23 roundabouts 26.3 ±4.5 crashes/yr Table 5-51. Predictive model validation statistics, PDO crashes, two circulating lanes.

106 the model fits the data cannot be rejected. The R2 for the model is 0.17. The Rk 2 for the calibrated model is 0.35. The coefficients in Table 5-52 were combined with Equa- tion 5-79 to obtain the calibrated SPF for three-leg round- abouts. The form of the model is described by the following equation. 1.565 1.055 1000 0.496 Equation 5-91 ,3 3N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-17. This figure compares the predicted and reported crash fre- quency in the calibration database. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. With one exception, each data point shown in Figure 5-17 represents the average predicted and average reported crash frequency for a group of 10 roundabouts. The data point associated with a predicted crash frequency of 37 crashes per period represents the average for a group of four round- abouts. The data were sorted by predicted crash frequency to form groups of sites with similar crash frequency. The purpose of this grouping was to reduce the number of data Model Statistics Value R2 (Rk2): 0.53 (0.42) Scale Parameter φ: 0.90 Pearson χ2: 102 (χ20.05, 108 = 133) Adjusted Inverse Dispersion Parameter K: Three-leg: 0.940; Four-leg: 1.27 Observations n: 115 roundabouts (3,359 PDO crashes) Standard Deviation sp: ±5.28 crashes/yr Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic brural Location in rural area 0.496 0.224 2.22 bew Width of entering lanes -0.0390 0.0392 -1.00 bcl Number of circulating lanes 0.219 0.121 1.81 b0,3 Three-leg roundabout -1.565 0.717 -2.18 bAADT,3 Entering AADT at three-leg roundabout 1.055 0.243 4.35 b0,4 Four-leg roundabout -1.536 0.524 -2.93 bAADT,4 Entering AADT at four-leg roundabout 1.131 0.169 6.69 Table 5-52. Final predictive model calibration statistics, PDO crashes, two circulating lanes. Model Statistics Value R2 (Rk2): 0.17 (0.35) Scale Parameter φ: 0.91 Pearson χ2: 29.9 (χ20.05, 29 = 42.6) Adjusted Inverse Dispersion Parameter K: 0.940 Observations n: 34 roundabouts (450 PDO crashes) Standard Deviation sp: ±2.50 crashes/yr NOTE: original inverse dispersion parameter = 0.979; original overdispersion parameter = 1.021; adjusted overdispersion parameter = 1.064. Table 5-53. Final predictive model calibration statistics, PDO crashes, two circulating lanes, three legs.

107 points shown in the figure and, thereby, to facilitate an exami- nation of trends in the data. The individual site observations were used for model calibration. Model for Roundabouts with Four Legs. The statistics describing the calibrated model for four-leg roundabouts are presented in Table 5-54. Calibration of this model focused on PDO crash frequency. The Pearson c2 statistic for the model is 72.7, and the degrees of freedom are 76 (= n − p = 81 − 5). As this statistic is less than c20.05, 76 (= 97.4), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.52. The Rk 2 for the calibrated model is 0.39. The coefficients in Table 5-52 were combined with Equa- tion 5-84 to obtain the calibrated SPF for four-leg round- abouts. The form of the model is described by the following equation. 1.536 1.131 1000 0.496 Equation 5-92 ,4 4N exp LN EntAADT I SPF rural [ ] ( )= − + × + × The calibrated CMFs used with this SPF are described in Section 6.1.2. The fit of the calibrated model is shown in Figure 5-18. This figure compares the predicted and reported crash frequency in the calibration database. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency. There were a few roundabouts with a very high reported PDO crash frequency that were underpredicted by the model. These sites are in the group of 10 sites with a predicted crash frequency of 90 crashes per period. Nevertheless, the average reported crash frequency for this group of sites is still within the 90th percentile confi- dence interval for the predicted value. 5.2.5 Crash Severity Prediction Database Summary This section summarizes the data used to calibrate the severity prediction models. The first subsection summarizes the geometric design and crash characteristics associated with Figure 5-17. Predicted versus reported PDO crashes at three-leg roundabouts with two circulating lanes. Model Statistics Value R2 (Rk2): 0.52 (0.39) Scale Parameter φ: 0.91 Pearson χ2: 72.7 (χ20.05, 76 = 97.4) Adjusted Inverse Dispersion Parameter K: 1.27 Observations n: 81 roundabouts (2,909 PDO crashes) Standard Deviation sp: ±6.08 crashes/yr N‰Š‹: original inverse dispersion parameter = 1.28; original overdispersion parameter = 0.784; adjusted overdispersion parameter = 0.790. Table 5-54. Final predictive model calibration statistics, PDO crashes, two circulating lanes, four legs. Figure 5-18. Predicted versus reported PDO crashes at four-leg roundabouts with two circulating lanes.

108 the sites in the database. The second subsection presents the findings from an exploratory analysis of trends in the data. 5.2.5.1 Database Summary This section summarizes the data assembled for the pur- pose of calibrating the predictive models. Initially, the geo- metric characteristics are summarized. Then the crash data are summarized. The variables included in the database are described in Section 5.2.3. The purpose of this summary is to provide information about the range of data included in the database and to pro- vide some insight to guide the development of the predictive model form. The discussion in this subsection is not intended to indicate conclusive results or recommendations. The rec- ommended predictive models (and associated trends) are documented in Section 6.1.2. The database assembled for the project includes 355 round- abouts. However, data for 34 roundabouts were removed from the database for various reasons. One roundabout was removed because it had an unusually high FI crash rate and an unusually high PDO crash rate. Twenty-six roundabouts were removed because their crash reports do not distinguish between severity categories (i.e., the reports indicate only FI or PDO crash severity). Several roundabouts in New York were included in the aforementioned 26 roundabouts. The remaining seven roundabouts in New York were excluded because they have a very unusual severity distribution: 0 K, 2 A, 57 B, and 0 C. A total of 34 roundabouts were removed from the database, leaving 321 roundabouts for model development. Geometric Characteristics. The database included data for 321 roundabout study sites. The distribution of these sites by number of circulating lanes, number of legs, and area type is provided in Table 5-55. The first column of Table 5-55 separates the database into roundabouts with one or two circulating lanes. The sites indicated to have one circulating lane have one circulating lane conflicting with each leg. In contrast, only 15 of the 100 roundabouts identified as having two circulating lanes actually have two circulating lanes conflicting with each leg. Most of the 100 roundabouts identified as two circulating lanes actually have some combination of one and two circu- lating lanes within the roundabout. The inscribed circle diameter describes the circle that best fits the outside edge of the circulating lanes. This diameter does not vary significantly among rural and urban roundabouts. It averages 123 ft for roundabouts with one circulating lane and 177 ft for roundabouts with two circu- lating lanes. There are 25 roundabouts in the database that have a drive- way providing direct access to the circulating roadway. The driveway is not considered to be a leg because it does not have a splitter island. For this reason, the driveway was not counted toward the total number of legs cited in column 2 of Table 5-55 (a roundabout with three public street legs and a driveway was recorded as a three-leg roundabout). Nineteen roundabouts have one leg that serves only out- bound traffic. These roundabouts were often (but not always) operating as a crossroad-ramp terminal at an interchange. In this case, the one-way outbound lane corresponded to an entrance ramp for the freeway. Circu- lating Lanes Number of Legs Area Type Number of Round- abouts Inscribed Circle Diameter, ft Number of Roundabouts with... Driveway on Cir. Roadway 1-Way Outbound Legs Right-Turn Bypass Lane Lighting on 1 or more LegsAverage Std. Dev. 1 3 R 16 120 23 4 0 0 15 U 48 118 21 10 1 3 44 4 R 53 135 28 2 3 5 48 U 104 119 26 3 5 15 97 2 3 R 6 186 23 1 0 4 6 U 23 180 59 5 0 5 22 4 R 19 167 22 0 2 2 19 U 52 179 54 0 8 8 50 Grand Total 321 140 43 25 19 42 301 N: Area type: R = rural; U = urban or suburban. Table 5-55. Database sample size and summary of roundabout characteristics.

109 A total of 42 roundabouts have a right-turn bypass lane on one or more legs. About 75% of these roundabouts are located in urban or suburban areas. These bypass lanes represent a mixture of control types (e.g., add-lane, merge, and yield). A total of 301 roundabouts have lighting on one or more legs. The 20 roundabouts without lighting are nearly evenly split between urban and rural areas. A description of the entry width, number of access points, circulating lane width, and AADT for the study sites is pro- vided in Section 5.2.3.2. Crash Characteristics. The distribution of reported FI crashes is shown in Table 5-56. Vehicle–pedestrian and vehicle–bicycle crashes are not included. The crashes are State Circu- lating Lanes Number of Legs Number of Sites Total Years Crash Count by Severity Total FI CrashesK A B C CA 1 3 1 7.0 0 0 0 2 2 2 4 3 21.0 0 2 11 36 49 FL 1 3 18 134.0 1 6 18 20 45 4 37 309.1 2 10 52 64 128 2 3 5 43.0 0 3 7 7 17 4 13 93.0 1 10 24 27 62 KS 2 3 1 7.0 0 0 0 0 0 4 1 7.0 0 0 0 0 0 MI 1 3 6 16.0 0 0 1 2 3 4 20 87.0 0 0 8 14 22 2 3 7 26.0 0 1 3 6 10 4 11 38.0 0 2 3 14 19 MN 1 3 6 45.0 1 1 1 1 4 4 19 140.0 0 1 5 23 29 2 3 5 28.0 0 0 3 2 5 4 6 39.0 0 0 2 19 21 NC 1 3 10 70.0 0 0 0 5 5 4 16 141.0 0 1 5 23 29 2 4 1 10.0 0 0 0 2 2 PA 1 3 3 21.0 0 0 0 0 0 4 6 38.0 0 0 3 8 11 WA 1 3 16 145.1 0 1 5 19 25 4 24 232.1 0 2 18 51 71 2 3 5 41.0 0 1 5 10 16 4 12 96.0 1 6 36 125 168 WI 1 3 4 23.0 0 3 2 1 6 4 35 205.1 1 2 42 49 94 2 3 6 34.0 0 3 2 5 10 4 24 118.0 0 7 27 67 101 Grand Total 321 2214.6 7 62 283 602 954 Table 5-56. Crash severity data summary by state, counts.

110 categorized by state, number of circulating lanes, and num- ber of legs. There are 321 roundabout study sites collectively representing nine states. These roundabouts experienced 954 FI crashes. The number of consecutive years for which crash data were obtained is referred to as the “evaluation period.” Collectively, the evaluation periods at the roundabouts in the database varied from 1 to 15 years, with a median duration of 7 years. A summary of the reported FI crashes in the database is shown in Table 5-57. The crashes are categorized by number of circulating lanes, number of legs, and area type. Separate summaries are provided for each crash severity category. It should be noted that there are only five and two fatal crashes in the database for roundabouts with one and two circulating lanes, respectively. As a result, the proportions computed for K crashes are relatively uncertain. 5.2.5.2 Exploratory Data Analysis As a precursor to model development, the database was examined graphically to identify the possible association between specific site characteristics and the FI crash sever- ity distribution. The insights obtained from this examination were used to determine which characteristics are likely can- didates for representation in the model as an adjustment fac- tor and guide the functional form development for the SDF. The discussion in this subsection is not intended to indicate conclusive results or recommendations. The recommended predictive models are documented in Section 6.1.2. The paragraphs to follow describe the findings from the graphical examination of key site characteristics. Many of the characteristics in the database were evaluated graphically. Those characteristics for which some trend was found are discussed in this section. To facilitate the examination, the crash counts at each site were converted into an average annual crash frequency (crashes per year). This approach was used because of the wide range in evaluation period duration among the study sites. Those sites having a long evaluation period tend to have more crashes reported. When the counts from sites with a long evaluation period are added to the counts from sites with a short evaluation period, the resulting distribution proportions are biased to emphasize the sites having a long evaluation period. To avoid this bias and ensure that each site was given equal weight in the severity distribution, the evalu- ation was based on crash frequency. This issue is discussed further in Section 5.2.6.2. The computed crash frequency for each combination of circulating lanes, number of legs, and area type is shown in Table 5-58. Columns 4 through 8 list the crash frequency for each combination. The crash frequency values shown represent the sum of the crash frequency for each roundabout for a given combination. For example, the row associated with round- abouts with one circulating lane, three legs, and rural area rep- resents the sum of the crash frequencies for 16 roundabouts. The values in columns 4 through 8 of Table 5-58 are used to compute the crash severity distribution proportions shown in the last four columns. The proportions shown in any one row add to 1.0. Area Type. The findings from the examination of area type are shown in Table 5-59. Columns 2 through 6 list the crash frequency for each area type category. The crash fre- quencies shown represent the sum of the crash frequency for each roundabout in a given category. Circu- lating Lanes Number of Legs Area Type Number of Sites Total Years Crash Count by Severity Total FI CrashesK A B C 1 3 R 16 110.0 1 4 5 11 21 U 48 351.1 1 7 22 39 69 4 R 53 316.1 1 4 42 60 107 U 104 836.3 2 12 91 172 277 2 3 R 6 23.0 0 1 5 3 9 U 23 156.0 0 7 15 27 49 4 R 19 91.0 1 4 34 75 114 U 52 331.1 1 23 69 215 308 Grand Total 321 2214.6 7 62 283 602 954 N: Area type: R = rural; U = urban or suburban. Table 5-57. Crash severity data summary, counts.

111 The values in columns 2 through 6 of Table 5-59 are used to compute the proportions shown in the last four columns. The proportion of K crashes is 0.014 in rural areas and 0.005 in urban areas. These values suggest that the probability of a crash being designated as “fatal” is about 2.8 (= 0.014/0.005) times larger at a roundabout in a rural area than is a crash at a roundabout in an urban area. A similar trend occurs for B crashes, with the probability of a crash being designated as “nonincapacitating injury” is 1.4 (= 0.393/0.288) times larger in a rural area (relative to an urban area). The trend in A crashes is opposite to that for the K and B crashes. That is, the probability of a crash being designated as an “incapacitating-injury” crash is 0.8 (= 0.066/0.081) times smaller at a roundabout in a rural area than is a crash at a roundabout in an urban area. However, examination of the proportions in Table 5-58 indicates that this trend is found only at roundabouts with two circulating lanes. At roundabouts with one circulating lane, the probability of an A crash is larger in rural areas. Circulating Lanes. The findings from the examina- tion of number of circulating lanes are shown in Table 5-60. Columns 2 through 6 list the crash frequency for each cir- culating lane category. The crash frequencies shown repre- sent the sum of the crash frequency for each roundabout in a given category. For example, the row associated with one circulating lane represents the sum of the crash frequencies for 221 roundabouts. The proportions shown in Table 5-60 indicate that round- abouts with one circulating lane tend to have a larger propor- tion of K and B crashes, relative to those with two circulating lanes. In contrast, roundabouts with one circulating lane tend to have a proportion of A crashes that is slightly smaller than that for roundabouts with two circulating lanes. When K and A crash frequencies are combined, the proportion of K + A crashes for two circulating lanes is slightly smaller than that for one circulating lane. Number of Legs. The findings from the examination of number of legs are shown in Table 5-61. Columns 2 through 6 list the crash frequency for each number of legs category. The crash frequencies shown represent the sum of the crash frequency for each roundabout in a given category. For Circu- lating Lanes Number of Legs Area Type Crash Frequency by Severity, cr/yr FI Crash Freq., cr/yr Crash Proportion by Severity K A B C K A B C 1 3 R 0.1 0.7 0.8 1.5 3.1 0.041 0.218 0.259 0.482 U 0.2 1.8 4.0 4.8 10.8 0.019 0.166 0.369 0.446 4 R 0.3 0.8 7.9 9.5 18.5 0.018 0.044 0.426 0.512 U 0.2 1.3 11.1 19.4 32.0 0.007 0.040 0.346 0.607 2 3 R 0.0 0.1 1.9 0.8 2.9 0.000 0.050 0.675 0.275 U 0.0 1.2 2.2 3.7 7.1 0.000 0.163 0.310 0.527 4 R 0.1 1.0 5.0 9.8 15.9 0.006 0.064 0.316 0.615 U 0.1 3.7 10.8 33.3 47.9 0.002 0.077 0.225 0.696 Grand Total 1.1 10.6 43.7 82.8 138.1 0.008 0.076 0.316 0.600 N: Area type: R = rural; U = urban or suburban. Table 5-58. Crash frequency summary. Area Type Crash Frequency by Severity, cr/yr FI Crash Freq., cr/yr Crash Proportion by Severity K A B C K A B C R 0.5 2.6 15.6 21.5 39.8 0.014 0.066 0.393 0.541 U 0.5 7.9 28.0 61.3 97.2 0.005 0.081 0.288 0.630 Grand Total 1.1 10.6 43.7 82.8 137.0 0.008 0.077 0.319 0.604 N: Area type: R = rural; U = urban or suburban. Table 5-59. Examination of area type.

112 example, the row associated with three legs represents the sum of the crash frequencies for 93 roundabouts. The proportions shown in Table 5-61 indicate that round- abouts with three legs tend to have a larger proportion of K, A, and B crashes, relative to those roundabouts with four legs. Inscribed Circle Diameter. The findings from the examination of inscribed circle diameter are shown in Fig- ure 5-19. This figure shows the proportions for roundabouts with one and two circulating lanes. The proportions are shown graphically as a function of the inscribed circle diam- eter. The data points in the middle of the figure correspond to B crashes. The data points in the lower third of the figure correspond to the proportion for K and A crashes combined. The crash frequencies for these two severity categories were combined because of the small number of K crashes, espe- cially when distributed over the range of diameter values found in the database. Each data point in Figure 5-19 represents the propor- tion of crashes with a specified severity level for several sites combined. This combining of sites was necessary to provide some reliability in the computed proportion. Sites were com- bined into groups corresponding to roundabouts with simi- lar diameters. Then, the total crash frequency was computed for each group (by severity category). Each group was sized to have a similar level of total crash frequency (all severi- ties combined). It was determined that each group should include at least 15 crashes per year to provide some stability to the computed proportions. This approach facilitated the calculation of reasonably stable estimates of crash proportion by group and, thereby, it facilitated an examination of trends in the data. The two trend lines shown in Figure 5-19 for the two- circulating-lane roundabouts are lines of best fit to the grouped data points (the fit is based on linear regression). These two trend lines suggest that the proportions of K + A crashes and B crashes increase with increasing inscribed circle diameter. The two trend lines shown in Figure 5-19 for the one- circulating-lane roundabouts indicate that the proportion of K + A combined and B crashes decrease with increasing diameter, up to about 125 ft. This trend reverses for diam- eters in excess of about 125 ft, with the proportion of K + A combined and B crashes increasing with increasing diameter. This latter trend is consistent with that for two-circulating- lane roundabouts. The trends shown in Figure 5-19 were rationalized to be the result of unobserved variables that have an influence on speed and are correlated with inscribed circle diameter (e.g., speed limit, exit width). No evidence was found in the lit- erature that described a causal connection between diameter and crash severity. As a result, inscribed circle diameter was excluded from consideration as a variable in the severity dis- tribution model. Speed Limit. The findings from the examination of inscribed circle diameter provided motivation to investi- gate the relationship between speed and crash severity. The observed speed limit for each roundabout leg was used for this purpose. For this examination, the only roundabouts that were used were those for which speed limit data was available for each leg. An initial examination of the database indicated only 181 (of the 321) sites have speed limit data for each leg. It was noted during this examination that 75% of these sites Circulating Lanes Crash Frequency by Severity, cr/yr FI Crash Freq., cr/yr Crash Proportion by Severity K A B C K A B C 1 0.9 4.5 23.7 35.2 64.3 0.014 0.071 0.369 0.547 2 0.2 6.0 19.9 47.6 73.8 0.003 0.081 0.270 0.646 Grand Total 1.1 10.6 43.7 82.8 138.1 0.008 0.076 0.316 0.600 Table 5-60. Examination of circulating lanes. Number of Legs Crash Frequency by Severity, cr/yr FI Crash Freq., cr/yr Crash Proportion by Severity K A B C K A B C 3 0.3 3.8 8.9 10.8 23.8 0.014 0.158 0.374 0.454 4 0.8 6.8 34.8 72.0 114.3 0.007 0.059 0.304 0.630 Grand Total 1.1 10.6 43.7 82.8 138.1 0.008 0.076 0.316 0.600 Table 5-61. Examination of number of legs.

113 had the same speed limit on the opposing legs (the north and south legs oppose; the east and west legs oppose). Addition- ally, 84% of the sites had speed limits on opposing legs that were within 5 mph of one another. Based on these findings, it was determined that if the speed limit was missing on one leg, the speed limit for the opposing leg could be used as a reasonable estimate of the missing value. This approach increased the sample size to 229 (of the 321) sites. To facilitate comparison of leg speed limit and crash sever- ity at the overall roundabout, the weighted average speed limit was computed using leg AADT as the weight. The weighted average speed limit is considered to be a single value that is representative of the overall roundabout and is suitable for comparison with the site’s overall crash severity distribution. The findings from the examination of speed limit are shown in Figure 5-20. This figure shows the combined proportions for roundabouts with urban or rural area type, one or two circulating lanes, and one or two legs. The data points in the upper part of the figure correspond to B crashes. The data points in the lower half of the figure correspond to K and A crashes combined. The crash frequencies for these two severity categories were combined because of the small num- ber of K crashes. Each data point shown in Figure 5-20 represents the pro- portion of crashes with a specified severity level for several sites combined. This combining of sites was necessary to pro- vide some reliability in the computed proportion. Sites were combined into groups corresponding to 5-mph ranges (e.g., sites with a weighted average speed limit between 22.5 and 27.5 mph, sites with a weighted average speed limit between 27.5 and 32.5, etc.). The data for each group is shown to have a speed limit corresponding to the midpoint of its speed range (e.g., the data point for 25 mph represents all sites with an average speed limit between 22.5 and 27.5). This approach facilitated the calculation of reasonably stable estimates of crash proportion by group, and thereby it facilitated an exami- nation of trends in the data. The two trend lines shown in Figure 5-20 are lines of best fit to the grouped data points (the fit is based on linear regres- sion). These trend lines suggest that the proportion of K + A crashes increases in an exponential manner with increasing speed. Similarly, the proportion of B crashes increases in an exponential manner with increasing speed. These findings are consistent with those of Elvik (2005). He found that the fre- quency of FI crashes increased in an exponential relationship with increase in speed, and the rate of increase is largest for K and A crashes, less so for B crashes, and least for C crashes. 5.2.6 Crash Severity Prediction Model Development The development of a FI crash severity distribution pre- diction model is described in this section. This model can be a. Fatal or incapacitating injury crashes. b. Nonincapacitating injury Figure 5-19. Examination of inscribed circle diameter. Figure 5-20. Examination of speed limit.

114 used to predict the proportion of K (fatal), A (incapacitating- injury), B (nonincapacitating-injury), and C (possible-injury) crashes. These proportions are used with the predicted FI crash frequency to estimate the frequency of K, A, B, and C crashes. This approach is intended to minimize the frequency–severity indeterminacy problem described by Hauer (2006). The subsections to follow provide a description of the basic model form, an overview of the modeling approach, an over- view of the statistical analysis methods, and a discussion of the findings from the model calibration activities. 5.2.6.1 Predictive Model Form The multinomial logit (MNL) model is used as the basic framework for the severity distribution model. The database assembled for model calibration included crash severity level as the dependent variable, with each reported crash serving as one observation. Geometric design features, traffic control features, and traffic characteristics were included as indepen- dent variables. The MNL form used in the HSM (AASHTO, 2014) is shown in the following equations. 1 Equation 5-93P S C S S S l l sdf K A B = + + + and 1 Equation 5-94P P P PC K A B( )= − + + with . . . Equation 5-95, ,1 ,S S f fK K b K K n= × × . . . Equation 5-96, ,1 ,S S f fA A b A A n= × × . . . Equation 5-97, ,1 ,S S f fB B b B B n= × × where Pl = probability of the occurrence of crash severity l (l = K, A, B); PC = probability of the occurrence of crash severity C; Sl = distribution score for severity l; Csdf = local calibration factor; Sl,b = base distribution score for severity l; fl,i = severity adjustment factor for the effect on severity l of traffic characteristic, geometric element, or traffic control feature i (i = 1 to n); and n = total number of severity adjustment factors. The distribution score is a dimensionless number that indicates the relative frequency of crashes associated with a specific severity category (K, A, B, or C), given that a fatal or severe crash has occurred. Severity category C is assigned a score of 1/Csdf. Smaller scores indicate a severity category that is less frequent. Larger scores indicate a severity category that is more frequent. If a severity category has a score smaller than 1/Csdf, then it is less frequent than category C. If a category has a score larger than 1/Csdf, then it is more frequent than category C. The model structure described by Equations 5-93 to 5-97 does not recognize the somewhat independent nature of each roundabout leg. In this regard, a change can be made to the geometry of one or more legs without directly affecting the crash severity on a different leg to any significant degree. Similarly, the traffic characteristics of one leg can be differ- ent from that of another leg, which can result in the two legs having a different level of contribution to the overall crash severity distribution of the roundabout. An alternative model form was formulated for computing the severity distribution for the overall roundabout (includ- ing all legs and the circulating roadway). The model was derived by spatially disaggregating the intersection into its individual legs (and their associated portion of the circulat- ing roadway). With this approach, Equations 5-95 to 5-97 were expanded to the following form to more accurately reflect differences in traffic characteristics, geometric ele- ments, and traffic control features among roundabout legs. . . . . . . . . . . . . Equation 5-98 , 1 ,1,1 ,1, 2 ,2,1 ,2, , ,1 , , S S p f f p f f p f f l l b l l n l l n m l m l m n [ ] ( ) ( ) ( ) = × × + × + + × where pj = proportion of total leg traffic volume associated with roundabout leg j (j = 1 to m); fl,j,i = severity adjustment factor for the effect on severity l of traffic characteristic, geometric element, or traffic control feature i on leg j (i = 1 to n; j = 1 to m); and m = total number of legs (3 or 4), legs. All other variables are as previously defined. The proportion of total leg traffic volume on leg j is com- puted using the following equation. . . . Equation 5-99 1 2 p AADT AADT AADT AADT j j m = + + + where AADTj = AADT volume for roundabout leg j ( j = 1 to m).

115 5.2.6.2 Modeling Approach This section describes several elements of the modeling approach. It consists of four subsections. The first subsec- tion describes the approach for obtaining unbiased estimates of the distribution proportions. The second subsection describes the techniques used to minimize the influence of correlated variables. The third subsection describes the tech- niques used to develop models specific to various combina- tions of number of circulating lanes and number of legs. The last subsection describes the process used to select variables for inclusion in the model. Estimated Distribution Proportions. The predicted crash distribution is to estimate the predicted crash frequency for a given crash type or severity category at a site of interest when an estimate of the average total crash frequency for this site is available. This relationship is shown in the following equation. Equation 5-100,N P Np i i t= × where Np,i = predicted average crash frequency for crash category i, crashes/yr; Pi = predicted probability of the occurrence of crash category i; and Nt = average total crash frequency, crashes/yr. There are several methods available to estimate the aver- age total crash frequency for a site. One method is to use a predictive model to compute the predicted total crash frequency and use this value as the average total crash fre- quency. Other methods are described in Section C.7 of the HSM (AASHTO, 2010). The predicted crash distribution proportions are used as an estimate of the predicted probability of the occurrence of crash category i. These proportions must be applicable to the site of interest (if they are not, then there is a possibility that the estimates are biased). Applicability is achieved when the distribution estimates are developed using data for similar sites (e.g., same number of legs, same area type). Applicability can also be achieved when the distribution is computed using a severity distribution function that includes variables for the influential site characteristics (e.g., same leg speed limit). The data assembled to develop the predicted crash distri- bution proportions (either table or function) must include the crash history for a large number of sites to ensure that the distribution represents the typical site and that the pro- portions are statistically valid. Statistical methods are used to control for observed differences among sites. This approach allows the analyst to select from a table (or compute using a function) the predicted distribution proportions for typi- cal site whose characteristics are similar to the site of interest (e.g., rural area type, three legs). An issue emerges when the data assembled to compute the crash distribution proportions has, on a site-by-site basis, a wide range in the number of years of crash history. Those sites having many years of crash history tend to have more crashes reported. When the counts from sites with many years of crash history are added to the counts from sites with a few years of crash history, the resulting distribution propor- tions weigh more heavily the sites having many years. This uneven weight can create bias in the distribution when the number-of-years variable is correlated with unobserved site characteristics that influence the crash distribution (e.g., if sites with many years of crash history are also found to have higher speeds, then number of years would be correlated with the crash severity distribution since higher speeds are likely associated with more severe crashes). To illustrate the points raised in the previous paragraph, consider the crash histories shown in Table 5-62 for four sites. The number of years of crash data for each site is indicated in column 2. Consider Site 2. It has 3 years of crash history. Thirty crashes of crash type X occurred during the 3-year period. A total of 60 crashes occurred during the 3 years. The proportion of crash type X at this site is 0.5 (= 30/60). Site Years Crash Type X, crashes Total Crashes, crashes Proportion Crash Type X, crashes/yr Total Crashes, crashes/yr Proportion 1 10 120 200 0.6 12 20 0.6 2 3 30 60 0.5 10 20 0.5 3 9 90 180 0.5 10 20 0.5 4 2 16 40 0.4 8 20 0.4 Total: 256 480 0.53 40 80 0.50 Table 5-62. Crash distribution example.

116 Option A. Each Year of Data Has Equal Weight. If it is rationalized that each year of data has equal weight (such that sites with more years of data are more informative about the distribution), then the proportion of crash type X in the data- base is computed using the sum of columns 3 and 4, as shown in the last row of the table. The average proportion of crash type X is computed to be 0.53 (= 256/480). Option B. Each Site Has Equal Weight. On the other hand, if it is rationalized that each site has equal weight, then the proportion of crash type X in the database is computed using the sum from columns 6 and 7, as shown in the last row of the table. These two columns show the computed aver- age crash frequency (in crashes per year) for each site. The average proportion of crash type X is computed to be 0.50 (= 40/80). Assessment of Options A and B. Further examination of column 5 indicates that two sites have a proportion of 0.5; one site has a proportion of 0.6 and the other 0.4. The trend here indicates that one-half of the sites have a proportion of 0.5. Although the sample size is small, the distribution of pro- portions is suggestive of a bell shape (normal distribution) and indicates that the typical site has a proportion of 0.5 (not 0.53). If an analyst wants an estimate of the proportion of crash type X for a new site, the typical value of 0.5 is the best estimate. The aforementioned best estimate is also obtained by giving each site equal weight. Option B yields proportions that are representative of a typical site. It follows that the value of 0.53 is not representative of the typical site. Rather, it is biased by the unbalanced number of years among sites. If the variability in the number of years is increased (e.g., one site has many years, and the others have one year), then the difference between the proportions obtained by the two approaches (i.e., each year of data has equal weight versus each site has equal weight) will also increase. In contrast, if each site has the same number of years of data, then the same proportion will be obtained by either approach. This example was based on the assumption that a single distribution estimate was needed for the site of interest. How- ever, the concepts presented can extend to the case where a severity distribution function is used to obtain the desired estimate as a function of site characteristics. For example, it is possible that the variation in proportions among sites in Table 5-62 is due to systematic effects. To predict an unbiased estimate of the distribution for a typical site, the calculation of model coefficients requires weighted logistic regression using a model that includes the site characteristics of interest, where the weight of each observation is equal to the recipro- cal of number of years of crash data for the associated site. Correlated Variables. A preliminary analysis of the data- base indicated that some of the site characteristic variables were correlated with roundabout location. For example, round abouts for some jurisdictions were mostly located in rural areas, while those in other jurisdictions were located in urban areas. As a result of this correlation, the model devel- opment process required two stages. In the first stage, the regression model included only site characteristics variables (it did not include variables specific to the jurisdictions). The regression coefficient for each site characteristic variable was examined for magnitude, direction, and statistical sig- nificance. If the magnitude, direction, and significance were acceptable, then the variable was retained in the model. At the conclusion of the first stage, the calibrated regression model included only variables whose coefficients were considered acceptable. During the second stage of the model development pro- cess, the calibrated regression model from the first stage was expanded to include one or more jurisdiction-specific indi- cator variables. The coefficient associated with this variable would serve to adjust the model prediction (similar to a local calibration factor) for sites in a jurisdiction that had signifi- cantly more or less severe crashes than the other jurisdictions. One jurisdiction-specific indicator variable was added to the model at a time. This process was repeated for all jurisdic- tions represented in the database. If the coefficient for a given jurisdiction-specific indicator was found to be statistically sig- nificant and if it did not notably alter the magnitude, direc- tion, or significance of any site characteristic variable, then it was retained in the model. Combined Regression Modeling. Section 5.2.4 described the development of two FI crash prediction models: one model for roundabouts with one circulating lane and one model for roundabouts with two circulating lanes. Each model com- bined the data for three- and four-leg roundabouts because a preliminary regression analysis indicated that the site sample size was too small to separately develop a reliable model for three-leg roundabouts and for four-leg roundabouts. Sample size considerations also challenged the develop- ment of separate severity distribution functions for various combinations of circulating lanes and legs. To overcome this challenge, it was decided that one severity distribution model would be developed to address all four combinations. Inde- pendent variables in the model would be used to account for the number of circulating lanes and number of legs. The general form of the regression model is shown in the following equations. 1 Equation 5-101, , , , , , , , , , P S S S S l q m l q m K q m A q m B q m = + + +

117 and 1 Equation 5-102, , , , , , , ,P P P PC q m K q m A q m B q m( )= − + + with A. Distribution scores for a three-leg roundabout (m = 3): . . . . . . . . . Equation 5-103 , ,3 , , ,3 1 ,1,1 ,1, 2 ,2,1 ,2, 3 ,3,1 ,3, S S p f f p f f p f f l q l b q l l n l l n l l n [ ] ( ) ( ) ( ) = × × + × + × B. Distribution scores for a four-leg roundabout (m = 4): . . . . . . . . . . . . Equation 5-104 , ,4 , , ,4 1 ,1,1 ,1, 2 ,2,1 ,2, 3 ,3,1 ,3, 4 ,4,1 ,4, S S p f f p f f p f f p f f l q l b q l l n l l n l l n l l n [ ] ( ) ( ) ( ) ( ) = × × + × + × + × where Pl,q,m = probability of the occurrence of crash severity l (l = K, A, B) at a roundabout with q lanes and m legs (q = 1 or 2; m = 3 or 4); q = total number of circulating lanes (= 1 if each leg is conflicted by one circulating lane, 2 if one or more legs are conflicted by two circulating lanes), lanes; PC,q,m = probability of the occurrence of crash severity C at a roundabout with q lanes and m legs; Sl,q,m = distribution score for severity l at a roundabout with q lanes and m legs; and Sl,b,q,m = base distribution score for severity l at a round- about with q lanes and m legs. All other variables are as previously defined. The base distribution scores associated with these models are defined as Equation 5-105, , , ,0 , ,S exp b b q b mK b q m K K cl K lg[ ]= + × + × Equation 5-106, , , ,0 , ,S exp b b q b mA b q m A A cl A lg[ ]= + × + × Equation 5-107, , , ,0 , ,S exp b b q b mB b q m B B cl B lg[ ]= + × + × where q = total number of circulating lanes (= 1 if each leg is conflicted by one circulating lane, 2 if one or more legs are conflicted by two circulating lanes), lanes; m = total number of legs (= 3 or 4), legs; bl,cl = calibration coefficient for severity l and number of circulating lanes; and bl,lg = calibration coefficient for severity l and number of legs. All other variables are as previously defined. The severity adjustment factors in Equations 5-103 and 5-104 can have a range of functional forms. The most general form for the factors applicable to a roundabout leg is shown in the following equations. Equation 5-108, , , ,f exp b XK j i K i j i[ ]= × Equation 5-109, , , ,f exp b XA j i A i j i[ ]= × Equation 5-110, , , ,f exp b XB j i B i j i[ ]= × where Xj,i describes the traffic characteristic, geometric ele- ment, or traffic control feature i on leg j (i = 1 to n; j = 1 to m), bl,i is the calibration coefficient for severity l and feature i, and all other variables are as previously defined. The severity adjustment factors in Equations 5-103 and 5-104 are the same. That is, the adjustment factor for a given traffic characteristic, geometric element, or traffic control feature i is the same in each equation. The calibration coef- ficient associated with each factor is also the same. Therefore, if the factor for a given characteristic, element, or feature is a function of variables (e.g., entry width), and the variables have the same value at a three-leg roundabout and at a four- leg roundabout, then the factor value is the same for both roundabouts. This approach recognizes that some character- istics, elements, or features have a similar influence on crash severity, regardless of the number of roundabout legs or cir- culating lanes. Model Development Process. The model development process was based on consideration of the p values for each calibration coefficient and the model’s AIC value. Indepen- dent variables were added to the model one at a time. This variable was added to either the base distribution score or a severity adjustment factor equation. When a variable was added (or removed), it was added to (or removed from) all three of the severity distribution score equations (i.e., those equations associated with severities K, A, and B). The process for evaluating each variable and the criteria for determining whether to retain it in the model are outlined in Table 5-63. The information in column 2 of Table 5-63 describes the process used to develop the predictive model using the base distribution score equations and the number of circulating lanes variable as an example. It is equally applicable to other

118 variables and other equations. The first step is to evaluate the null model. This model includes only the intercept coef- ficient (bK,0, bA,0, bB,0) in each of the base distribution score equations. The next step is to evaluate model form 1. This model form includes the variable of interest X in each equation and a unique regression coefficient for each severity level (bK,X, bA,X, bB,X). Based on consideration of the AIC and p values for each coefficient, one of three options is selected. With Option 1, the variable is removed from all equations. With Option 2, model form 1 is retained. With Option 3, one of models 2, 3, or 4 is selected. If Option 3 is selected, then two or three of the severity categories are combined, and a common regression coefficient (e.g., bKA,X) is used for the combined categories. 5.2.6.3 Statistical Analysis Methods The nonlinear regression procedure (NLMIXED) in the SAS software was used to estimate the proposed model coefficients. This procedure was used because the proposed predictive model is both nonlinear and discontinuous. The log-likelihood function for the MNL logistic distribution was used to deter- mine the best-fit model coefficients. The procedure was set up to estimate model coefficients based on maximum-likelihood methods. Weighted regression was used to quantify the regression coefficients. The log-likelihood of each observation was weighted by the reciprocal of its “number of years in the evaluation period.” The individual weight values wi were nor- malized by multiplying them by the constant W(= n/∑ni=1wi), where n is the sample size. The normalized weights add up to the actual sample size and result in the covariance matrix of the coefficients being invariant to the scale of the weight variable. Weighted regression was used because of the wide range in evaluation period duration among the study sites. Those sites having a long evaluation period tend to have more crashes reported. When the counts from sites with a long evaluation period are added to the counts from sites with a short evalua- tion period, the resulting distribution proportions are biased to emphasize the sites having a long evaluation period. To avoid this bias and ensure that each site was given equal weight in the severity distribution, each observation was weighted using the aforementioned procedure. This issue is discussed further in the previous subsection titled Estimated Distribu- tion Proportions. 5.2.6.4 Severity Distribution Prediction Model This section consists of two subsections. The first sub- section describes the structure of the safety predictive models as used in the regression analysis. The second subsection describes the regression statistics for each of the calibrated models and presents the proposed prediction model. Model Form Process and Decision Criteria Example Base Distribution Score Equations Fatal, SK Incapacitating Injury, SA Nonincapacitating Injury, SB Null Initial model form SK,b = bK,0 SA,b = bA,0 SB,b = bB,0 1 With ncl and an associated unique regression coeficient added to the model for each severity. 1. If AIC does not decrease (relative to null model) or if coeficients not logical, then remove this variable and move on to the next variable. 2. If all coeficients have a p value < 0.15, then keep this form and go to next variable. 3. If one or more coeficients has a p value > 0.15, then consider one of the model forms below. SK,b = bK,0 + bK,cl × q SA,b = bA,0 + bA,cl × q SB,b = bB,0 + bB,cl × q 2 Combine K and A terms if (1) bB,cl has p value < 0.15 and (2) bK,cl and bA,cl have overlapping conidence intervals and same sign. Keep bB,cl in the model. SK,b = bK,0 + bKA,cl × q SA,b = bA,0 + bKA,cl × q SB,b = bB,0 + bB,cl × q 3 Combine A and B terms if (1) bK,cl has p value < 0.15 and (2) bA,cl and bB,cl have overlapping conidence intervals and same sign. Keep bK,cl in the model. SK,b = bK,0 + bK,cl × q SA,b = bA,0 + bAB,cl × q SB,b = bB,0 + bAB,cl × q 4 Combine K, A, and B terms if bK,cl bA,cl and bB,cl have overlapping conidence intervals and same sign. SK,b = bK,0 + bKAB,cl × q SA,b = bA,0 + bKAB,cl × q SB,b = bB,0 + bKAB,cl × q NOTE: The p value of 0.15 is used as the threshold value (as opposed to a smaller value) to indicate an acceptable level of coeficient reliability. It is used in conjunction with other criteria that are intended to collectively provide logical, useful, and robust models. Table 5-63. Model development process and decision criteria.

119 Model Development. This subsection describes the pro- posed prediction model and the methods used to calibrate it. The regression model is generalized to accommodate three- and four-leg roundabouts with one or two circulating lanes in urban and rural areas. The generalized form shows the variables included in the model. For some variables, indica- tor variables are used to determine when the corresponding adjustment factor is applicable. The generalized form of the model was previously shown in Equations 5-101 to 5-110. It includes leg-specific adjustment factors. Several leg-specific factors were evaluated in a series of preliminary regression analyses including right-turn bypass lane presence, lighting presence, entry width, and number of access points. However, only speed limit was found to have a statistically significant effect on the crash severity distribution. Thus, the leg-specific factors include only speed limit in the final model form. The following equations describe the regression model that was calibrated using the severity data. A. If the observation corresponds to a three-leg round- about (m = 3), the following model is used. 1 Equation 5-111, ,3 , ,3 , ,3 , ,3 , ,3 P S S S S l q l q K q A q B q = + + + and 1 Equation 5-112, ,3 , ,3 , ,3 , ,3P P P PC q K q A q B q( )= − + + with Equation 5-113 , ,3 , , ,3 1 ,1, 2 ,2, 3 ,3,S S p f p f p fK q K b q K sl K sl K sl[ ]= × + × + × Equation 5-114 , ,3 , , ,3 1 ,1, 2 ,2, 3 ,3,S S p f p f p fA q A b q A sl A sl A sl[ ]= × + × + × Equation 5-115 , ,3 , , ,3 1 ,1, 2 ,2, 3 ,3,S S p f p f p fB q B b q B sl B sl B sl[ ]= × + × + × 3 Equation 5-116, , ,3 ,0 , ,S exp b b q bK b q K KA cl KA lg[ ]= + × + × 3 Equation 5-117, , ,3 ,0 , ,S exp b b q bA b q A KA cl KA lg[ ]= + × + × 3 Equation 5-118, , ,3 ,0 , ,S exp b b q bB b q B B cl B lg[ ]= + × + × 100 35 100 Equation 5-119 , , , 2 2f exp b SLK j sl KAB sl j{ }( ) ( )= × −  100 35 100 Equation 5-120 , , , 2 2f exp b SLA j sl KAB sl j{ }( ) ( )= × −  100 35 100 Equation 5-121 , , , 2 2f exp b SLB j sl KAB sl j{ }( ) ( )= × −  Equation 5-122 1 2 3 p AADT AADT AADT AADT j j= + + where Pl,q,m = probability of the occurrence of crash severity l (l = K, A, B) at a roundabout with q lanes and m legs (q = 1 or 2; m = 3 or 4); q = total number of circulating lanes (= 1 if each leg is conflicted by one circulating lane, 2 if one or more legs are conflicted by two circulating lanes), lanes; m = total number of legs (3 or 4), legs; PC,q,m = probability of the occurrence of crash severity C at a roundabout with q lanes and m legs; Sl,q,m = distribution score for severity l at a roundabout with q lanes and m legs; Sl,b,q,m = base distribution score for severity l at a round- about with q lanes and m legs; pj = proportion of total leg traffic volume associated with roundabout leg j ( j = 1 to m); fl,j,sl = severity adjustment factor for the effect of speed limit on severity l on leg j ( j = 1 to m); SLj = speed limit on leg j ( j = 1 to m); bl,i = calibration coefficient for severity l and condi- tion i; and AADTj = AADT volume for roundabout leg j ( j = 1 to m). B. If the observation corresponds to a four-leg roundabout (m = 4), the following model is used. 1 Equation 5-123, ,4 , ,4 , ,4 , ,4 , ,4 P S S S S l q l q K q A q B q = + + + and 1 Equation 5-124, ,4 , ,4 , ,4 , ,4P P P PC q K q A q B q( )= − + + with Equation 5-125 , ,4 , , ,4 1 ,1, 2 ,2, 3 ,3, 4 ,4,S S p f p f p f p fK q K b q K sl K sl K sl K sl[ ]= × + × + × + × Equation 5-126 , ,4 , , ,4 1 ,1, 2 ,2, 3 ,3, 4 ,4,S S p f p f p f p fA q A b q A sl A sl A sl A sl[ ]= × + × + × + × Equation 5-127 , ,4 , , ,4 1 ,1, 2 ,2, 3 ,3, 4 ,4,S S p f p f p f p fB q B b q B sl B sl B sl B sl[ ]= × + × + × + ×

120 4 Equation 5-128, , ,4 ,0 , ,S exp b b q bK b q K KA cl KA lg[ ]= + × + × 4 Equation 5-129, , ,4 ,0 , ,S exp b b q bA b q A KA cl KA lg[ ]= + × + × 4 Equation 5-130, , ,4 ,0 , ,S exp b b q bB b q B B cl B lg[ ]= + × + × Equation 5-131 1 2 3 4 p AADT AADT AADT AADT AADT j j= + + + All variables are as previously defined. The final form of the regression model reflects the findings from several preliminary regression analyses where alterna- tive model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defen- sible and properly bounded. The speed limit adjustment factor for three-leg roundabouts is shown in Equations 5-119 to 5-121 for severity levels K, A, and B, respectively. These equations are equally applicable to the four-leg model without modification. The adjustment fac- tor is specified to have a base condition of 35 mph, which is the average leg speed limit in the database. This base value pro- duces a factor value of 1.0 when the leg speed limit is 35 mph. Adjustment factors for several geometric elements were examined. Only speed limit was found to be helpful in explaining the variation in crash severity among sites. This finding does not rule out the possibility that other variables have an influence on crash severity. It is possible that the effect of the other variables is sufficiently small that it would require a larger database (with more observations and a wider range in variable values) to detect using regression. Calibrated Model. The predictive model calibration pro- cess was based on a combined regression modeling approach, as discussed in Section 5.2.4.2. With this approach, the com- ponent models (represented by Equations 5-111 to 5-131) are calibrated using a common database. This approach is needed because the adjustment factor for speed limit (and the geometric elements that were considered) is common to all models, regardless of the number of legs or circulating lanes. The results of the combined regression model calibration are presented in Table 5-64. The t-statistics listed in the last col- umn of this table indicate a test of the hypothesis that the coef- ficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the vari- able was important to the model, and its trend was found to be logical (even if the specific value was not known with a great deal of certainty as applied to this database). The findings from an examination of the coefficient values and the corresponding model predictions are documented in Section 6.1.2. The coefficients in Table 5-64 were combined with Equa- tions 5-116 to 5-118 to obtain the calibrated base distribution score equations for three-leg roundabouts. These equations (shown in Table 5-65) are used in Equations 5-113 to 5-115 Model Statistics Value -2 Log-Likelihood for Null Model: 1344.5 -2 Log-Likelihood for Full Model: 1318.8 AIC for Full Model: 1334.8 Observations n: 718 fatal or injury crashes Calibrated Coef icient Values Variable Inferred Effect of… Value Std. Error t-statistic bK,0 Intercept for fatal crash (K) -0.1853 1.25 -0.15 bA,0 Intercept for incapacitating crash (A) 2.1120 1.18 1.78 bB,0 Intercept for nonincapacitating crash (B) 1.5445 0.853 1.81 bKA,cl Number of circulating lanes on K or A crashes 0.1601 0.272 0.59 bB,cl Number of circulating lanes on B crashes -0.3224 0.164 -1.97 bKA,lg Number of legs on K or A crashes -1.1491 0.302 -3.80 bB,lg Number of legs on B crashes -0.4212 0.217 -1.94 bKAB,sl Posted speed limit on K, A, or B crashes 3.1187 1.143 2.73 Table 5-64. Predictive model calibration statistics, severity distribution.

121 to compute the corresponding distribution scores for each severity category. Similarly, the coefficients were combined with Equations 5-128 to 5-130 to obtain the calibrated base distribution score equations for four-leg roundabouts. These equations (shown in Table 5-65) are used in Equations 5-125 to 5-127 to compute the corresponding distribution scores for each severity category. The speed limit adjustment factor is computed using the following equations. 3.1187 100 35 100 Equation 5-132 , , 2 2f exp SLK j sl j{ }( ) ( )= × −  3.1187 100 35 100 Equation 5-133 , , 2 2f exp SLA j sl j{ }( ) ( )= × −  3.1187 100 35 100 Equation 5-134 , , 2 2f exp SLB j sl j{ }( ) ( )= × −  where fl,j,sl = severity adjustment factor for the effect of speed limit on severity l on leg j ( j = 1 to m) and SLj = speed limit on leg j ( j = 1 to m). The same coefficient value (3.1187) is used in each equa- tion. This usage is based on the findings of the regression analysis, which indicated that the difference in the coefficient values among severity categories was not significant. 5.2.7 Crash-Type Prediction Database Summary This section summarizes the crash data assembled for the purpose of computing the crash-type distribution. Sepa- rate distributions are computed for FI crashes and for PDO crashes. The geometric characteristics for the roundabouts represented in the FI data are summarized in Section 5.2.5.1. The geometric characteristics for the roundabouts repre- sented in the PDO data are summarized in Section 5.2.3. This section consists of two subsections. The first sub- section describes the characteristics of the FI crashes used to compute the FI crash-type distribution. The second sub- section describes the characteristics of the PDO crashes used to compute the FI crash-type distribution. 5.2.7.1 FI Crash Characteristics The database assembled for the project includes 355 round- abouts. However, data for 34 roundabouts were removed from the database for various reasons. The reasons for excluding these sites are provided in Section 5.2.5.1. The crash-type distribution of reported FI crashes is shown in Table 5-66. Vehicle–pedestrian and vehicle–bicycle crashes are not included. The crashes are categorized by state, number of circulating lanes, and number of legs. There are 321 round about study sites collectively representing nine states. These roundabouts experienced 954 FI crashes. There are 10 crash types identified in the database. Five of the types are categorized as multiple-vehicle crashes. The other five are categorized as single-vehicle crashes. Multiple- vehicle FI crash types accounted for about 67% of the 954 FI crashes. Rear-end FI crashes represent the most frequently (30%) occurring crash type among FI crashes. There are no animal-related FI crashes in the database. There were five crashes for which the crash report did not include crash type information. These crashes were recorded as “Unknown Type” and are shown in the second-to-last column of the table. The number of consecutive years for which crash data were obtained is referred to as the “evaluation period.” Collectively, the evaluation periods at the roundabouts in the database varied from 1 to 15 years, with a median duration of 7 years. A summary of the reported FI crashes in the database is shown in Table 5-67. The crashes are categorized by number of circulating lanes, number of legs, and area type. Separate summaries are provided for crash-type category. Circu- lating Lanesa Number of Legs Equation for Base Distribution Score (by Severity)b Fatal (K) Incapacitating Injury (A) Nonincapacitating Injury (B) 1 3 SK,b,1,3 = exp[-3.4725] SA,b,1,3 = exp[-1.1752] SB,b,1,3 = exp[-0.0415] 4 SK,b,1,4 = exp[-4.6216] SA,b,1,4 = exp[-2.3243] SB,b,1,4 = exp[-0.4627] 2 3 SK,b,2,3 = exp[-3.3124] SA,b,2,3 = exp[-1.0151] SB,b,2,3 = exp[-0.3639] 4 SK,b,2,4 = exp[-4.4615] SA,b,2,4 = exp[-2.1642] SB,b,2,4 = exp[-0.7851] N : a A roundabout is considered to have one circulating lane if each leg is con„licted by one circulating lane. A roundabout is considered to have two circulating lanes if one or more legs are con„licted by two circulating lanes. b Sl,b,q,m = base distribution score for severity l at a roundabout with q lanes and m legs. Table 5-65. Calibrated base distribution score equations.

122 State Circu- lating Lanes Number of Legs Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type Single-Vehicle Crash Count by Crash Type Unknown Type Total Crashes Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other CA 1 3 1 7.0 0 0 1 0 0 0 0 0 0 1 0 2 2 4 3 21.0 2 11 24 0 5 0 3 0 0 4 0 49 FL 1 3 18 134.0 0 5 16 3 8 0 4 0 0 8 1 45 4 37 309.1 4 22 25 5 14 0 31 0 1 22 4 128 2 3 5 43.0 0 1 0 4 3 0 4 0 0 5 0 17 4 13 93.0 0 8 10 7 8 0 11 0 0 18 0 62 KS 2 3 1 7.0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 7.0 0 0 0 0 0 0 0 0 0 0 0 0 MI 1 3 6 16.0 0 2 1 0 0 0 0 0 0 0 0 3 4 20 87.0 0 1 11 1 0 0 8 0 0 1 0 22 2 3 7 26.0 0 0 3 0 1 0 4 0 0 2 0 10 4 11 38.0 0 4 4 2 3 0 5 0 0 1 0 19 MN 1 3 6 45.0 1 1 1 0 0 0 1 0 0 0 0 4 4 19 140.0 0 1 12 2 4 0 7 0 0 3 0 29 2 3 5 28.0 0 1 0 1 0 0 2 0 0 1 0 5 4 6 39.0 0 9 2 7 1 0 1 0 0 1 0 21 NC 1 3 10 70.0 0 0 4 1 0 0 0 0 0 0 0 5 4 16 141.0 0 3 11 1 5 0 0 0 0 9 0 29 2 4 1 10.0 0 1 1 0 0 0 0 0 0 0 0 2 PA 1 3 3 21.0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 38.0 0 5 4 1 0 0 0 0 0 1 0 11 WA 1 3 16 145.1 0 0 12 0 8 0 5 0 0 0 0 25 4 24 232.1 1 0 28 2 8 0 25 0 0 7 0 71 2 3 5 41.0 0 0 4 0 5 0 5 0 0 2 0 16 4 12 96.0 1 0 48 21 60 0 29 0 0 9 0 168 WI 1 3 4 23.0 0 0 0 0 0 0 2 0 0 4 0 6 4 35 205.1 0 11 29 11 1 0 17 0 0 25 0 94 2 3 6 34.0 0 2 1 3 0 0 1 0 0 3 0 10 4 24 118.0 0 17 31 29 3 0 6 0 0 15 0 101 Grand Total 321 2214.6 9 105 283 101 137 0 171 0 1 142 5 954 Table 5-66. FI crash-type distribution by state.

123 5.2.7.2 PDO Crash Characteristics The database assembled for the project includes 355 round- abouts. However, data for nine roundabouts were removed from the database for various reasons. One roundabout was removed because it had high FI and PDO crash rates. Eight roundabouts from Ontario were removed because the asso- ciated crash reports do not distinguish between the desired 10 crash-type categories. The crash-type distribution of reported PDO crashes is shown in Table 5-68. Vehicle–pedestrian and vehicle–bicycle crashes are not included. The crashes are categorized by state, number of circulating lanes, and number of legs. There are 346 roundabout study sites collectively representing 10 states. These roundabouts experienced 4,895 PDO crashes. There are 10 crash types identified in the database. Multiple-vehicle PDO crash types accounted for about 77% of the 4,895 PDO crashes. Other multiple-vehicle PDO crashes represent the most frequently (25%) occurring crash type among PDO crashes. There are 24 animal-related PDO crashes (0.5%) in the database and 128 PDO crashes of unknown type. A summary of the reported PDO crashes in the database is shown in Table 5-69. The crashes are categorized by number of circulating lanes, number of legs, and area type. Separate summaries are provided for crash-type category. 5.2.8 Crash-Type Distribution Table Development The development of a crash-type distribution table is described in this section. This table can be used to predict the proportion of crashes associated with a specific crash type or manner of collision (e.g., right angle, fixed object). These proportions are used with the predicted total crash frequency to estimate the frequency of the specific crash type. The crash-type distribution is described using a table of distribution values computed from the database described in Section 5.2.7. The use of a table to describe the distribution proportions is consistent with the other chapters in Part C of HSM (AASHTO, 2010). The development of models for pre- dicting the crash-type distribution as a function of variables was beyond the project scope of work. There are three subsections in this section. The first sub- section describes the approach used to develop the crash-type distribution table. The second subsection describes the sta- tistics used to compute the standard error of the distribution proportions. The third subsection describes the calibrated distribution table. 5.2.8.1 Development Approach To compute the distribution values, the crash counts at each site were converted into an annual crash frequency. This approach was used because of the wide range in evaluation period duration among the study sites. Sites having a long evaluation period tend to have more crashes reported. When the counts from sites with a long evaluation period are added to the counts from sites with a short evaluation period, the resulting distribution proportions are biased to emphasize the sites having a long evaluation period. To avoid this bias and ensure that each site was given equal weight in the crash- type distribution, the evaluation was based on proportions Circu- lating Lanes Number of Legs Area Type Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type Single-Vehicle Crash Count by Crash Type Unknown Type Total Crashes Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 R 16 110.0 0 1 8 0 4 0 2 0 0 6 0 21 U 48 351.1 1 7 27 4 12 0 10 0 0 7 1 69 4 R 53 316.1 1 8 27 11 5 0 31 0 1 22 1 107 U 104 836.3 4 35 93 12 27 0 57 0 0 46 3 277 2 3 R 6 23.0 0 0 1 2 1 0 2 0 0 3 0 9 U 23 156.0 0 4 7 6 8 0 14 0 0 10 0 49 4 R 19 91.0 0 2 34 15 42 0 12 0 0 9 0 114 U 52 331.1 3 48 86 51 38 0 43 0 0 39 0 308 Grand Total 321 2214.6 9 105 283 101 137 0 171 0 1 142 5 954 N: Area type: R = rural; U = urban or suburban. Table 5-67. FI crash-type distribution.

124 State Circu- lating Lanes Number of Legs Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type Single-Vehicle Crash Count by Crash Type Unknown Type Total Crashes Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other CA 1 3 1 7.0 0 1 0 0 0 0 1 0 0 0 0 2 2 4 3 21.0 1 28 49 0 62 0 22 0 0 1 0 163 FL 1 3 18 134.0 1 16 19 4 14 2 11 0 0 10 6 83 4 38 317.1 14 74 74 3 42 3 28 2 7 37 42 326 2 3 5 43.0 0 1 3 5 3 0 5 0 0 3 0 20 4 14 95.0 11 32 33 28 15 0 9 0 0 14 17 159 KS 2 3 1 7.0 0 1 3 1 0 0 0 0 0 0 0 5 4 5 33.0 2 52 16 48 1 0 7 1 0 2 0 129 MI 1 3 6 16.0 0 3 8 1 0 1 3 0 0 1 0 17 4 20 87.0 1 33 92 15 16 8 33 0 2 5 0 205 2 3 7 26.0 0 14 26 18 13 2 15 0 0 4 0 92 4 11 38.0 0 35 50 38 13 7 21 0 0 1 0 165 MN 1 3 6 45.0 0 0 4 0 0 0 4 0 0 1 0 9 4 19 140.0 0 10 33 6 16 0 40 1 0 4 0 110 2 3 5 28.0 0 2 5 5 0 0 4 0 0 1 0 17 4 6 39.0 0 21 19 45 17 0 23 1 0 0 0 126 NC 1 3 10 70.0 1 5 5 1 2 0 1 1 0 4 0 20 4 16 141.0 0 20 24 10 41 1 15 0 0 10 0 121 2 4 1 10.0 0 13 1 5 1 0 0 0 0 1 0 21 PA 1 3 3 21.0 0 0 0 1 0 0 4 0 0 0 0 5 4 6 38.0 0 7 3 2 0 0 17 0 1 1 0 31 WA 1 3 16 145.1 0 0 24 18 51 0 22 0 0 0 0 115 4 24 232.1 0 0 57 29 146 0 78 0 1 4 0 315 2 3 5 41.0 1 0 6 13 30 0 10 0 0 0 0 60 4 12 96.0 0 0 140 134 525 0 155 2 0 5 0 961 WI 1 3 4 23.0 0 1 1 0 0 0 18 0 0 3 0 23 4 35 205.1 1 106 94 85 2 0 92 0 0 37 1 418 2 3 6 34.0 0 4 10 10 0 0 19 0 0 20 0 63 4 24 118.0 3 94 73 244 10 0 62 0 0 31 0 517 NY 1 3 5 37.0 0 0 3 0 12 0 4 0 0 2 6 27 4 8 62.0 0 18 32 0 16 0 12 0 0 4 3 85 2 3 1 7.0 0 3 16 0 18 0 7 0 0 0 2 46 4 5 41.0 0 95 114 2 162 0 4 0 0 11 51 439 Grand Total 346 2397.7 36 689 1037 771 1228 24 746 8 11 217 128 4895 Table 5-68. PDO crash-type distribution by state.

125 computed using the average crash frequency (in crashes per year) for each site. This approach is discussed further in Sec- tion 5.2.6.2 (in the subsection titled Estimated Distribution Proportions). The distribution values are based on the frequency of crashes for which the crash type was identified on the crash report. Crashes identified as having “unknown type” were not used to compute the distribution proportions. It is assumed that the crashes of unknown type are distributed in the same proportions as the known crash types. If the assumption is valid, the computed proportions should not be biased by the exclusion of the unknown crashes. The goal in developing the crash-type distribution was to compute separate distributions for each combination of crash type, circulating lanes, number of legs, and area type. However, sample size (in terms of number of crashes) was a concern for some of the combinations (e.g., single-vehicle crashes at roundabouts with one lane and three legs). Those combinations having a small number of crashes have rela- tively uncertain distribution values. As a result, the combi- nations for which the distributions were computed were determined based on consideration of the standard error for the individual proportions and a statistical test of each pro- portion’s difference from zero (i.e., the null hypothesis was that the proportion was equal to 0.0). The number of com- binations retained was initially small, but the number was increased (and the data disaggregated into more categories) until the largest number of statistically significant propor- tions was obtained. The statistics used in the statistical test are described in Section 5.2.8.2. 5.2.8.2 Standard Error of Proportions Equations 5-135 and 5-136 were used to compute the pro- portions and standard error of the proportion, respectively. These equations incorporate the evaluation period duration in the calculations to avoid possible bias when the evaluation period duration varies among sites. Equation 5-135p N N ct ct T = and 2 Equation 5-136 , 2 2 0.5 s p V N N V N N V N V N N N p ct ct ct ct T T ct T ct T ] ]] ][ [[ [= + −   with Equation 5-137,1N No yct ct i ii n∑= = Equation 5-138,11N No yT k i ii n k m ∑∑= == Equation 5-139, 21V N No yct ct i ii n∑][ = = Equation 5-140, 211V N No yT k i ii n k m ∑∑][ = == N: Area type: R = rural; U = urban or suburban. Circu- lating Lanes Number of Legs Area Type Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type Single-Vehicle Crash Count by Crash Type Unknown Type Total Crashes Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 R 19 132.0 0 8 22 10 17 1 20 0 0 4 8 90 U 50 366.1 2 18 42 15 62 2 48 1 0 17 4 211 4 R 58 351.1 3 62 126 46 34 9 105 0 2 40 5 432 U 108 871.3 13 206 283 104 245 3 210 3 9 62 41 1179 2 3 R 7 30.0 0 13 25 9 30 0 17 0 0 4 2 100 U 23 156.0 1 12 44 43 34 2 43 0 0 24 0 203 4 R 21 100.0 9 116 202 165 360 5 77 1 0 14 19 968 U 60 391.1 8 254 293 379 446 2 226 3 0 52 49 1712 Grand Total 346 2397.7 36 689 1037 771 1228 24 746 8 11 217 128 4895 Table 5-69. PDO crash-type distribution.

126 where sp,ct = standard error of the proportion p for crash type ct (ct = head on, right angle, or . . .); pct = proportion crash type ct; Nct = crash frequency for crash type ct (all sites), cr/yr; NT = total crash frequency (all sites and crash types), cr/yr; V[Nct] = variance of crash frequency for crash type ct; V[NT] = variance of total crash frequency; Noct,i = observed crash frequency for crash type ct at site i (i = 1 to n), cr/yr; Nok,i = observed crash frequency for crash type k at site i (k = 1 to m; i = 1 to n), cr/yr; yi = number of years for evaluation period at site i, yr; n = number of sites; and m = number of crash type categories. The statistic z used to test the null hypothesis that the proportion is equal to zero is computed using the following equation. This statistic is asymptotic to the normal distribu- tion as sample size increases. Equation 5-141 , z p s ct p ct = 5.2.8.3 Calibrated Distribution The recommended FI crash-type distribution is shown in the top half of Table 5-70. It includes separate distribution val- ues for four combinations of circulating lanes and number of legs. The sample size was too small to further disaggregate the data for the purpose of adding area type as an additional cat- egory. The standard error for each proportion is shown in the bottom half of the table. For a given combination, the stan- dard error tends to be small when the proportion is small or when the total crash frequency is large. As a result, some rela- tively small proportions can be significantly different from 0.0 when the total crash frequency for the combination is large. The recommended PDO crash-type distribution is shown in the top half of Table 5-71. It includes separate distribution values for eight combinations of circulating lanes, number of legs, and area type. 5.3 Leg-Level Crash Prediction Models for Design 5.3.1 Introduction This section describes the development of leg-level models for predicting crash frequency. The leg-level models are used Crash-Type Proportion Circu- lating Lanes Number of Legs Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 0.007 0.168 0.356 0.045 0.139 0.000 0.109 0.000 0.000 0.175 4 0.011 0.115 0.298 0.078 0.071 0.000 0.216 0.000 0.002 0.209 2 3 0.000 0.072 0.137 0.109 0.124 0.000 0.325 0.000 0.000 0.233 4 0.008 0.142 0.268 0.177 0.152 0.000 0.127 0.000 0.000 0.126 Standard Error of Crash-Type Proportion Circu- lating Lanes Number of Legs Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 0.006 0.059 0.038 0.017 0.015 0.009 0.018 0.009 0.009 0.028 4 0.004 0.013 0.012 0.014 0.009 0.003 0.013 0.003 0.002 0.017 2 3 0.012 0.024 0.038 0.020 0.026 0.012 0.065 0.012 0.012 0.037 4 0.004 0.015 0.012 0.015 0.010 0.002 0.013 0.002 0.002 0.015 NOTE: For those crash types in a given row having a proportion of 0.0, the standard error estimate is based on the assumption that there was one crash during a 7-year period (which is the average evaluation period duration of all sites in the database). Based on this assumption, the estimated standard error is considered to be conservatively large for proportions equal to 0.0. Table 5-70. Recommended FI crash-type distribution.

127 to predict the average frequency of crashes associated with a specific roundabout leg, disaggregated by specific crash type. The crash types studied include the following: • Entering-circulating, • Exiting-circulating, • Rear-end on approach, • Single-vehicle on approach, • Circulating-circulating, • Single-vehicle circulating, and • Total. These models are envisioned to be used for design-stage applications and, as such, include one or more CMFs that can be used to adjust the predicted crash frequency to reflect the safety influence of existing or proposed roundabout design elements. This chapter describes the data and modeling approach. There were insufficient vehicle–pedestrian and vehicle– bicycle crashes represented in the assembled database to enable the development of a model for predicting the fre- quency of these crashes. 5.3.2 Database Summary The selected variables in the database used for developing leg-level models are listed in Table 5-72. Crash-Type Proportion Circu- lating Lanes Number of Legs Area Type Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 R 0.000 0.070 0.411 0.099 0.151 0.017 0.183 0.000 0.000 0.069 U 0.008 0.121 0.226 0.053 0.241 0.008 0.225 0.002 0.000 0.117 4 R 0.004 0.149 0.248 0.136 0.070 0.014 0.261 0.000 0.003 0.116 U 0.010 0.192 0.263 0.093 0.187 0.002 0.188 0.002 0.009 0.054 2 3 R 0.000 0.147 0.215 0.131 0.262 0.000 0.186 0.000 0.000 0.060 U 0.002 0.072 0.227 0.256 0.131 0.005 0.178 0.000 0.000 0.128 4 R 0.025 0.164 0.216 0.230 0.258 0.005 0.076 0.001 0.000 0.025 U 0.005 0.174 0.178 0.265 0.199 0.003 0.138 0.002 0.000 0.037 Standard Error of Crash-Type Proportion Circu- lating Lanes Number of Legs Area Type Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe, Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other 1 3 R 0.008 0.016 0.065 0.017 0.014 0.014 0.018 0.008 0.008 0.026 U 0.005 0.024 0.032 0.009 0.020 0.005 0.014 0.002 0.004 0.023 4 R 0.002 0.013 0.011 0.014 0.009 0.004 0.013 0.002 0.002 0.014 U 0.002 0.009 0.010 0.007 0.008 0.001 0.009 0.001 0.003 0.006 2 3 R 0.005 0.029 0.030 0.036 0.024 0.005 0.034 0.005 0.005 0.025 U 0.002 0.024 0.028 0.029 0.012 0.003 0.013 0.003 0.003 0.015 4 R 0.008 0.011 0.010 0.014 0.004 0.002 0.008 0.001 0.001 0.006 U 0.002 0.007 0.007 0.007 0.005 0.002 0.006 0.001 0.001 0.005 N: For those crash types in a given row having a proportion of 0.0, the standard error estimate is based on the assumption that there was one crash during a 7-year period (which is the average evaluation period duration of all sites in the database). Based on this assumption, the estimated standard error is considered to be conservatively large for proportions equal to 0.0. Area Type: R = rural; U = urban or suburban. Table 5-71. Recommended PDO crash-type distribution.

128 The database assembled for the leg-level models included 150 roundabouts. From these, only legs with all of entering, exiting, and circulating AADTs available were kept. Legs not meeting these definitions were few in number and deemed to be fundamentally different in terms of operation and so were not included. The area type, traffic volume, and crash data were obtained from various public agencies responsible for each roundabout study site. All other data elements listed in Table 5-72 were obtained from aerial imagery for the years represented by the crash data. Crash reports were manually reviewed in order to define each crash as a specific roundabout crash type. The database used for developing the models included data for 534 roundabout legs drawing from the 150 roundabouts in the dataset. The distribution of these sites by number of circulating lanes, entering lanes, and area type is provided in Tables 5-73 to 5-75. Table 5-76 provides a summary of the AADT volume esti- mates for the entering, exiting, and circulating AADT for the leg. The distribution of reported vehicle–vehicle crashes is shown in Table 5-77. 5.3.3 Model Development This section describes the activities undertaken to develop the leg-level models. The subsections to follow provide a description of the basic model form, an overview of the modeling approach, an overview of the statistical analysis methods, and a discussion of the findings from the model estimation activities. The predicted average crash frequency for each site is computed using a prediction model. The general form of the generalized linear safety predictive model is shown below. Equation 5-1421 1 . . .N exp AADT expa b x nxn= ( )β + β where N = predicted average crash frequency, crashes/yr; AADT = AADT term; X1 . . . Xn = a series of predictor variables; a, b, c, β1 . . . βn = are estimated parameters; and k = the estimated overdispersion parameter. Category Variable Description Descriptive Name_Leg Route number or street name (by leg) State State in which roundabout is located (postal code) Area_Type Urban/suburban or rural RampTerm One or more roundabout legs is a ramp to or from a freeway StartDate Starting date of the evaluation period EndDate Ending date of the evaluation period Roadway NumberLegs Number of legs at roundabout (3 or 4) CirculatingLanes Number of circulating lanes in the roundabout (1 or 2) (by leg) EnteringLanes Number of entering lanes (by leg) ExitingLanes Number of exiting lanes (by leg) Bypass Presence of a right-turn bypass lane (yes/no) (by leg) EntryWidth Width of the approaching lanes (plus shoulders, if provided) (by leg) Angle Angle to next leg going in counterclockwise direction (by leg) ICD Inscribed circle diameter CirculatingWidth Width of circulating lanes (plus shoulders, if provided) (by leg) Trafic volume Entering AADT Average daily trafic entering the leg in the study period Exiting AADT Average daily trafic exiting the leg in the study period Circulating AADT Average daily trafic circulating past the leg in the study period Other Luminaires Number of luminaires within 250 ft of roundabout (by leg) NumberAccess Number of access points within 250 ft of roundabout (by leg) Table 5-72. Summary of data attributes and sources for leg-level models.

Circulating Lanes Entering Lanes Area Type Number of Legs Inscribed Circle Diameter, ft. Number of Legs with … Average Std. Dev. Right- Turn Bypass Lane Combination of 1 and 2 Circulating Lanes at Roundabout One Exiting Lane Two Exiting Lanes 1 1 R U 195 105 132.7 119.8 28.6 23.4 4 7 2 3 194 103 1 2 21 R U 22 68 159.0 172.3 23.1 32.4 1 2 20 68 3 10 19 58 2 1 R U 13 51 171.7 195.0 14.6 60.7 0 2 13 45 13 51 0 0 2 R U 19 61 166.7 236.8 25.5 88.3 0 3 10 22 7 20 12 41 N: Area type: R = rural; U = urban or suburban. 1 Instances where there are two entering lanes and only one circulating lane occur where there is a single circulating lane prior to the entry and two following the entry. This may occur where the major road is a four-lane road, and the minor road is a two-lane road. Table 5-73. Database sample size and summary of roundabout leg characteristics: ICD and leg-specific characteristics. Circulating Lanes Entering Lanes Area Type Number of Legs Entry Width per lane, ft. Angle, deg. Circulating Width, ft. Average Std. Dev. Average Std. Dev. Average Std. Dev. 1 1 R U 195 105 19.7 17.9 3.2 2.5 94.9 99.2 21.7 23.8 20.9 20.0 3.3 3.7 2 R U 22 68 13.8 14.6 1.3 1.4 96.7 100.2 20.6 29.5 25.5 24.6 5.4 6.4 2 1 R U 13 51 22.5 20.3 3.1 3.1 85.1 92.6 7.1 13.6 30.2 32.0 1.5 1.9 2 R U 19 61 14.5 1.9 14.3 1.6 95.5 96.1 22.7 22.5 30.7 32.1 2.0 3.7 N: Area type: R = rural; U = urban or suburban. Table 5-74. Database sample size and summary of roundabout leg characteristics: average entry widths, angles, and circulating widths. Circulating Lanes Entering Lanes Area Type Number of Legs Luminaires Number of Access Points Posted Speed, mph Average Std. Dev. Average Std. Dev. Average Std. Dev. 1 1 R U 195 105 2.1 2.0 1.4 1.2 0.8 1.3 1.1 1.6 41.2 32.1 11.1 7.6 2 R U 22 68 2.5 3.1 0.9 1.5 0.5 0.9 0.9 1.5 46.2 36.0 9.2 7.9 2 1 R U 13 51 1.8 3.1 1.3 1.4 1.0 1.7 1.2 1.9 40.6 30.3 12.1 5.2 2 R U 19 61 2.7 3.0 1.6 1.6 0.8 1.0 1.1 1.5 43.2 33.8 8.1 8.3 N: Area type: R = rural; U = urban or suburban. Table 5-75. Database sample size and summary of roundabout leg characteristics: average number of luminaires, access points, and posted speeds.

130 Circulating Lanes Entering Lanes Area Type Number of Legs Entering AADT Exiting AADT Circulating AADT Average Std. Dev. Average Std. Dev. Average Std. Dev. 1 1 R U 195 105 2,523 4,444 1,665 2,340 2,532 4,540 1,675 2,406 2,253 3,744 1,538 2,418 2 R U 22 68 5,534 5,325 2,123 3,188 5,524 5,342 2,138 3,212 2,725 2,409 2,588 2,197 2 1 R U 13 51 1,242 2,609 1,266 1,1917 1,242 2,609 1,266 1,917 6,076 5,629 2,135 3,236 2 R U 19 61 4,016 4,624 1,980 2,296 4,016 4,654 1,980 2,368 4,136 4,323 2,103 2,166 N: Area type: R = rural; U = urban or suburban. Table 5-76. Database summary of traffic volume for leg-level data. Circulating Lanes Entering Lanes Area Type Number of Legs Total Ent- Circ Ext- Circ REA SVA Circ- Circ SVCirc 1 1 R U 195 105 340 315 98 (29) 102 (32) 3 (1) 11 (3) 68 (20) 92 (29) 75 (22) 23 (7) 2 (1) 3 (1) 88 (26) 70 (22) 2 R U 22 68 260 450 54 (21) 161 (36) 78 (30) 57 (13) 31 (12) 61 (14) 21 (8) 46 (10) 42 (16) 43 (10) 16 (6) 62 (14) 2 1 R U 13 51 43 148 13 (30) 48 (32) 2 (5) 10 (7) 3 (7) 27 (18) 3 (7) 15 (10) 9 (21) 27 (18) 9 (21) 12 (8) 2 R U 19 61 173 674 56 (32) 190 (28) 40 (23) 238 (35) 8 (5) 90 (13) 11 (6) 29 (4) 27 (16) 56 (8) 18 (10) 46 (7) N: Area type: R = rural; U = urban or suburban. Total = all vehicle crashes not involving a pedestrian or bicycle. Ent-Circ = a crash between an entering and a circulating vehicle. Ext-Circ = a crash between an exiting and a circulating vehicle. REA = a rear-end crash on the approach. SVA = a single-vehicle crash on the approach. Circ-Circ = a crash between two circulating vehicles. SVCirc = a single-vehicle crash in the circulating roadway. Table 5-77. Number of crashes by crash type (percentages in brackets) for leg-level data.

131 Alternative model forms were considered with a focus on developing CMFs for roundabout geometry-related vari- ables. Those models were not successful and, therefore, were not adopted. Estimation of the models employed generalized estimat- ing equations to account for the correlation in crash counts among legs at the same roundabouts. The process of developing the models followed three steps: 1. Develop preliminary models using only AADT-related variables, 2. Develop preliminary models with additional predictor variables using Equation 1, and 3. Investigate alternate model forms for the models and over- dispersion parameters. Due to the limited data, a validation sample of data was not withheld when developing the models. Variable definitions for the predictor variables for all models follow: • EntAADT = entering AADT; • ExtAADT = exiting AADT; • ApprAADT = approach AADT (equal to EntAADT + ExtAADT); • CircAADT = circulating AADT; • ICD = inscribed circle diameter in ft; • Angle = angle in degrees; • COS(Angle) = equal to cosine of angle in radians; • CircWidth = circulating width in ft; • TwoEnteringLanes = equal to 1 if two entering lanes, 0 other- wise; • TwoExitingLanes = equal to 1 if two exiting lanes, 0 other- wise; • EntryWidAvg = average width of entering lanes in ft; • NumberAccess = number of access points within 250 ft of entry point on leg; • Luminaires = number of luminaires within 250 ft of entry point on leg; • PostedSpeed = posted speed on approach in mph; • AreaType = equal to 1 if rural, 0 if urban; and • State = equal to 0 if located in Wisconsin; 1 otherwise. 5.3.3.1 Entering-Circulating Crash Models This section describes the development of predictive models for entering-circulating crashes. Separate models were pursued by the number of circulating and entering lanes because many of the geometric variables are directly related to the number of lanes provided. Alternate models were attempted, including combining sites based on the number of entering lanes. Each alternate model is reported in Tables 5-78 through 5-94; each table is preceded by the model form. Equation 5-143 EntCirc exp EntAADT CircAADT expa b c d ICD e Bypass f Statevar= ( )× + × + × Variable Parameter Estimate (s.e.) Intercept a -7.8580 (1.5714) EntAADT b 0.6091 (0.1281) CircAADT c 0.3020 (0.1062) ICD d -0.0068 (0.0043) Bypass e -0.9982 (0.4845) Statevar f -0.9628 (0.3128) Overdispersion Parameter k 0.7470 (0.2187) Table 5-78. Entering-circulating SPFs, one circulating and one entering lanes. Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -9.5763 (2.6693) -11.4943 (2.4136) EntAADT b 0.9636 (0.2696) 0.9609 (0.2701) CircAADT c 0.3917 (0.1786) 0.3633 (0.1722) ICD d -0.0082 (0.0046) -0.0080 (0.0046) Angle e -0.0234 (0.0069) – COSAngle f – 1.6403 (0.4020) Overdispersion Parameter k 0.6232 (0.2151) 0.6018 (0.2131) Table 5-79. Entering-circulating SPFs, one circulating and two entering lanes. Equation 5-144 EntCirc exp EntAADT CircAADT exp a b c d ICD e Angle f COS Angle = ( )( )× + × + ×

132 Equation 5-145 EntCirc exp EntAADT CircAADT exp a b c d ICD e Angle g CircWidth h TwoEnteringLanes = ( )× + × + × + × Equation 5-147 EntCirc exp EntAADT CircAADT exp a b c e Angle f COS Angle g CircWidth = ( )( )× + × + × Variable Parameter Estimate (s.e.) Intercept a -9.6906 (1.2989) EntAADT b 0.7175 (0.1184) CircAADT c 0.3552 (0.0910) ICD d -0.0074 (0.0031) Angle e -0.0091 (0.0045) Circulating Width g 0.0315 (0.0205) Two Entering Lanes h 1.1092 (0.2487) Overdispersion Parameter k 0.9350 (0.1824) Table 5-80. Entering-circulating SPFs, one circulating and one or two entering lanes. Equation 5-146 EntCirc exp EntAADT CircAADT expa b c d ICD g CircWidth= ( )× + × Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -4.3332 (4.0105) -6.3627 (3.9088) EntAADT b 0.3608 (0.2096) 0.3344 (0.2210) CircAADT c 0.6711 (0.3422) 0.6282 (0.3483) ICD d -0.0148 (0.0061) -0.0185 (0.0066) Circulating Width g -0.1041 (0.0652) – Overdispersion Parameter k 1.0734 (0.5316) 1.2891 (0.5771) Table 5-81. Entering-circulating SPFs, two circulating and one entering lanes. Variable Parameter Estimate (s.e.) Option 1 Option 2 Option 3 Option 4 Intercept a -8.3493 (3.0338) -15.6743 (3.4922) -9.9541 (3.1785) -17.3375 (3.4243) EntAADT b 0.8054 (0.1783) 0.9868 (0.2033) 0.8221 (0.1798) 0.9987 (0.2049) CircAADT c 0.7398 (0.2216) 0.9455 (0.2684) 0.7623 (0.2222) 0.9673 (0.2705) Angle e -0.0134 (0.0053) -0.0153 (0.0052) – – COSAngle f – – 0.8997 (0.3992) 1.0277 (0.3833) Circulating Width g -0.1375 (0.0578) – -0.1353 (0.0590) – Overdispersion Parameter k 0.7759 (0.2202) 0.8850 (0.2366) 0.7854 (0.2222) 0.8926 (0.2381) Table 5-82. Entering-circulating SPFs, two circulating and two entering lanes.

133 Equation 5-148 EntCirc exp EntAADT CircAADT exp a b c d ICD e Angle f COS Angle g CircWidth h TwoEnteringLanes i EntrywidAvg = ( )( )× + × + × + × + × +  For KABC crashes there are only 127 crashes, which is not enough to develop separate models by number of circulating lanes and entering lanes. Therefore, all sites were combined. Variable Parameter Estimate (s.e.) Option 1 Option 2 Option 3 Option 4 Intercept a -5.1248 (2.9781) -10.0132 (2.2471) -6.1794 (3.0968) -10.9775 (2.2222) EntAADT b 0.5608 (0.1572) 0.6400 (0.1490) 0.5673 (0.1575) 0.6453 (0.1487) CircAADT c 0.5579 (0.1661) 0.6316 (0.1698) 0.5700 (0.1650) 0.6407 (0.1696) ICD d -0.0046 (0.0031) -0.0061 (0.0033) -0.0046 (0.0032) -0.0062 (0.0033) Angle e -0.0105 (0.0035) -0.0094 (0.0043) – – COSAngle f – – 0.7131 (0.2579) 0.6146 (0.3334) Circulating Width g -0.1756 (0.0631) – -0.1775 (0.0635) – Two Entering Lanes h 1.4058 (0.4520) 0.7107 (0.1995) 1.4150 (0.4586) 0.7070 (0.2006) Entry Width per Lane i 0.0722 (0.0449) – 0.0728 (0.0451) – Overdispersion Parameter k 0.8206 (0.2170) 0.9755 (0.2402) 0.8256 (0.2183) 0.9795 (0.2413) Table 5-83. Entering-circulating SPFs, two circulating and one or two entering lanes.

134 Equation 5-149 EntCirc exp EntAADT CircAADT exp a b c d ICD e Angle f COS Angle h TwoEnteringLanes = ( )( )× + × + × + × with more crashes for the one circulating lane legs and fewer crashes for the two circulating lane legs. Equation 5-150 ExtCirc exp ExtAADT CircAADT expa b c e CircWidth g State= ( )× + × Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -13.0182 (2.4290) -14.6408 (2.2413) EntAADT b 0.9374 (0.1946) 0.9380 (0.1937) CircAADT c 0.4749 (0.1487) 0.4764 (0.1493) ICD d -0.0049 (0.0025) -0.0048 (0.0025) Angle e -0.0177 (0.0082) – COSAngle f – 1.2227 (0.4922) 2 Entering Lanes h 1.1599 (0.2487) 1.1546 (0.2488) Overdispersion Parameter k 0.4337 (0.2874) 0.4201 (0.2841) Table 5-84. Entering-circulating KABC SPFs, one or two circulating and one or two entering lanes. 5.3.3.2 Exiting-Circulating Crash Models This section describes the development of predictive models for exiting-circulating crashes. Separate models were pursued by the number of circulating and exiting lanes because many of the geometric variables are directly related to the number of lanes provided. Alternate models were attempted, including combining sites based on the number of exiting lanes. Each alternate is reported in the tables below preceded by the model form. A model was not successfully estimated using sites with only one circulating lane and one exiting lane. This is not surprising given that there are only 18 such crashes in the database at 310 legs. For legs with two circulating and two exiting lanes, no model was successfully estimated with any non-AADT-related variables, thus providing no CMFs. The model forms and parameter estimates for the success- fully estimated models that produced CMFs are provided below. All models logically indicate that exiting-circulating crashes increase with increases in exiting and circulating AADT. A larger inscribed circle diameter is associated with fewer crashes in the models for one circulating and either one or two exiting lanes and in the model for two circulating and one exiting lane. An increased circulating width is associated Variable Parameter Estimate (s.e.) Intercept a -29.7315 (8.4312) ExtAADT b 2.0150 (0.7415) CircAADT c 0.5511 (0.3103) Circulating Width e 0.1808 (0.0368) State g 1.6362 (0.5156) Overdispersion Parameter k 1.8837 (0.5828) Table 5-85. Exiting-circulating crash model, one circulating and two exiting lanes. Equation 5-151 ExtCirc exp ExtAADT CircAADT exp a b c d ICD e CircWidth f TwoExitingLanes g State = ( )× + × + × + × Variable Parameter Estimate (s.e.) Intercept a -18.4950 (3.9340) ExtAADT b 1.3685 (0.3333) CircAADT c 0.2062 (0.2237) ICD d -0.0146 (0.0069) Circulating Width e 0.0823 (0.0291) Two Exiting Lanes f 2.6555 (0.6792) State g 1.1171 (0.5861) Overdispersion Parameter k 2.0345 (0.6795) Table 5-86. Exiting-circulating crash model, one circulating and one or two exiting lanes.

135 For legs with one circulating and either one or two exit- ing lanes, the parameter estimate for circulating AADT has a large standard error. For legs with two circulating and one exiting lanes, two models were estimated, one with and one without circulating width. The two models were estimated because Option 1 indi- cates that exiting-circulating crashes decrease with an increase in circulating width, which is opposite to the results for legs with one circulating lane. Equation 5-152 ExtCirc exp ExtAADT CircAADT expa b c d ICD e CircWidth g State= ( )× + × + × Equation 5-153 ExtCirc exp ExtAADT CircAADT exp a b c e CircWidth f TwoExitingLanes g State = ( )× + × + × Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -5.8705 (5.4794) -14.0278 (5.1071) ExtAADT b 0.4317 (0.2366) 0.4381 (0.2922) CircAADT c 1.0853 (0.5380) 1.2133 (0.5313) ICD d -0.0148 (0.0049) -0.0194 (0.0055) Circulating Width e -0.2582 (0.0862) – State g 0.9808 (0.5080) 0.8823 (0.4814) Overdispersion Parameter k 0.6159 (0.5465) 0.8564 (0.6767) Table 5-87. Exiting-circulating crash model, two circulating and one exiting lanes. For legs with two circulating and one or two exiting lanes, two models were again estimated because the results for cir- culating width indicated a decrease in exiting-circulating crashes with an increase in circulating width. Both models exhibit a high overdispersion parameter, indicating that com- bining the legs with one and two exiting lanes is not a good choice for the two circulating lane legs. Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -11.2279 (5.3399) -13.6570 (3.2058) ExtAADT b 0.6176 (0.2277) 0.6903 (0.2784) CircAADT c 0.7709 (0.5579) 0.5887 (0.2766) Circulating Width e -0.1124 (0.0612) – Two Exiting Lanes f 1.7930 (0.2722) 1.6157 (0.2822) State g 0.8814 (0.5734) 0.7586 (0.5385) Overdispersion Parameter k 2.7334 (0.5840) 2.8866 (0.6271) Table 5-88. Exiting-circulating crash model, two circulating and one or two exiting lanes. 5.3.3.3 Rear-End Approach Crash Models This section describes the development of predictive models for rear-end crashes occurring on the approach. All legs were modeled together. The model form and parameter estimates for the successfully estimated model are provided below. The model indicates that rear-end approach crashes increase with increasing approach and circulating AADT as well as with the number of access points on the approach. As the number of luminaires increase the number of crashes of this type are associated with a decrease.

136 Equation 5-154 RearEnd Approach exp ApprAADT CircAADT exp a b c d NumberAccess e Luminaires = ( )× + × Equation 5-155 SV Approach exp ApprAADT expa b c PostedSpeed d AreaType+e State= ( )× + × × Variable Parameter Estimate (s.e.) Intercept a –14.4195 (1.6284) ApprAADT b 1.0978 (0.1555) CircAADT c 0.3034 (0.0921) NumberAccess d 0.0894 (0.0557) Luminaires e –0.0652 (0.0421) Overdispersion Parameter k 1.0659 (0.1965) Table 5-89. Rear-end approach crash model. 5.3.3.4 Single-Vehicle Approach Crash Models This section describes the development of predictive mod- els for single-vehicle crashes occurring on the approach. All legs were modeled together. The model form and parameter estimates for the successfully estimated models are provided below. Two models were developed, one using the posted speed limit on the approach and the other considering the area type (urban or rural). Posted speed limit was not known for all legs, so the model including this variable is based on 420 of the 534 legs. The models indicate that single-vehicle approach crashes increase with increasing speed limit and are higher in rural than urban areas. This may reflect the fact that speed limits in rural areas are typically higher. Given the lower overdispersion parameter for Option 1, which includes posted speed limit, that model may be preferred for adoption of CMFs for single-vehicle approach crashes. Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -6.8027 (1.1780) -6.5618 (1.1977) ApprAADT b 0.3392 (0.1195) 0.5031 (0.1273) PostedSpeed c 0.0441 (0.0099) – Rural Area Type d – 0.3628 (0.2373) State e -0.7495 (0.2091) -0.9474 (0.2346) Overdispersion Parameter k 0.8153 (0.2657) 1.1070 (0.2807) Table 5-90. Single-vehicle approach crash model. 5.3.3.5 Circulating-Circulating Crash Models This section describes the development of predictive mod- els for circulating-circulating crashes. Separate models were pursued for leg with one and two circulating legs, but a model was only successfully estimated for two circulating lane leg sites. The model form and parameter estimates for the success- fully estimated model is provided below. The model indicates that circulating-circulating crashes decrease with an increas- ing circulating width when two circulating lanes are present. Equation 5-156 Circulating Circulating exp CircAADT expa b c CircWidth− = ( )× Variable Parameter Estimate (s.e.) Intercept a -2.8066 (2.5657) CircAADT b 0.3963 (0.2438) Circulating Width c -0.0870 (0.0534) Overdispersion Parameter k 1.4571 (0.4378) Table 5-91. Circulating- circulating crash model, two circulating lanes.

137 5.3.3.6 Single-Vehicle Circulating Plus Single-Vehicle Approach Crash Models No models were successfully estimated for single-vehicle circulating crashes. These were combined with single-vehicle approach crashes to estimate a new model. All legs were mod- eled together. The model form and parameter estimates for the successfully estimated models are provided below. Two models were developed, one using the posted speed limit on the approach and the other considering the area type (urban or rural). Posted speed limit was not known for all legs, so the model including this variable is based on 420 of the 534 legs. The models indicate that single-vehicle crashes increase with increasing speed limit and are higher in rural than urban areas. This may reflect the fact that speed limits in rural areas are typically higher. Single-vehicle crashes are higher at legs with two entering lanes and lower with wider circulating widths. Given the lower overdispersion param- eter for Option 1, which includes posted speed limit, that model may be preferred for adoption CMFs for single-vehicle approach plus single-vehicle circulating crashes. Equation 5-157 SV Approach SV Circulating exp ApprAADT exp a b c PostedSpeed d CircWidth e TwoEnteringLanes f AreaType g State + = ( )× + × + × + × + × 5.3.3.7 Total Crash Models Total crash models were also estimated to allow for the pre- diction of all other crash types for which a model is not pro- vided by subtracting the sum of predicted crash types from the total crash prediction. Because of the nature of the mod- els, it is possible that in certain ranges of predictive variables the sum of estimates of specific crash types could be more than the estimate for total. In this case the sum of estimates by crash type should be used as the estimate for total crashes. Models were estimated separately for single- and two- circulating-lane sites. Where successful, variables indicating the area type, number of entering, and number of exiting lanes were included. Equation 5-158 Total exp ApprAADT CircAADT expa b c d AreaType e TwoEnteringLanes= ( )× + × Variable Parameter Estimate (s.e.) Option 1 Option 2 Intercept a -6.0580 (0.8603) -4.6149 (0.9617) ApprAADT b 0.4055 (0.0925) 0.3694 (0.1078) PostedSpeed c 0.0350 (0.0075) – CircWidth d -0.0232 (0.0100) -0.0157 (0.0114) Two Entering Lanes e 0.5464 (0.1461) 0.6093 (0.1548) Rural AreaType f – 0.3348 (0.1763) State g -0.3624 (0.1712) -0.5768 (0.1959) Overdispersion Parameter k 0.9410 ()0.1574 1.1998 (0.1694) Table 5-92. Single-vehicle circulating plus single-vehicle approach crash model. Variable Parameter Estimate (s.e.) Intercept a -10.5458 (1.1847) ApprAADT b 0.8197 (0.0929) CircAADT c 0.2747 (0.0657) Rural Area Type d 0.3673 (0.1820) 2 Entering Lanes e 0.9827 (0.1536) Overdispersion Parameter k 0.6921 (0.0841) Table 5-93. Total crash model, one circulating lane.

138 Equation 5-159 Total exp ApprAADT CircAADT exp a b c d AreaType e TwoExitingLanes f TwoExitingLanes = ( )× + × + × • Benefit–cost analysis: Design exceptions and funding deci- sions are often made on the basis of a benefit–cost analysis, so being able to better quantify roundabout design features would help make more informed decisions during differ- ent stages of project planning and design. • Calibration: Attendees were generally aware of the impor- tance of calibration. The intersection control evaluation (ICE) case study showing that the NCHRP 17-70 models may predict the same number of PDO crashes as a sig- nal using the Part C models in the HSM was a point of discussion on the importance of calibration, since these models were developed using datasets from different states and time periods. • Pedestrian crashes: Some attendees had hoped the crash prediction models would be able to address pedestrian treatments and crashes. • High-speed locations: There was interest in the differ- ence between crashes at high-speed locations compared to lower-speed urban and suburban locations. Eight respondents described results that they found either counterintuitive or incorrect. These items included • A wider circulatory roadway width resulting in lower crashes (two responses), • Wider entry widths for single-lane roundabouts resulting in fewer crashes (two responses), and • A greater inscribed circle diameter resulting in fewer crashes (one response). Two individuals responded that they found some results either counterintuitive or incorrect, but they did not provide an explanation as to what specifically they believed fell into this category. Finally, based on feedback received from the workshop, it seems likely that the areas that will be of most interest to practitioners include • Using the planning-level models as part of ICE efforts, • Using the intersection-level design models to assess the differences in expected crashes when comparing single- lane versus multilane roundabouts or different types of multilane roundabout configurations, and • Using leg-level design models to evaluate existing round- abouts with concerns about a specific crash type. 5.5 Effect of Driver Learning Curve on Roundabout Safety Performance 5.5.1 Description of Methodology This section describes the investigation of a potential driver learning curve at newly constructed roundabouts. The Variable Parameter Estimate (s.e.) Intercept a -7.1029 (1.6107) ApprAADT b 0.4443 (0.1039) CircAADT c 0.3306 (0.1421) Rural Area Type d 0.4194 (0.2393) 2 Entering Lanes e 0.2950 (0.1381) 2 Exiting Lanes f 0.3805 (0.1139) Overdispersion Parameter k 0.9429 (0.1437) Table 5-94. Total crash model, two circulating lanes. 5.4 Practitioner Validation A Practitioner Workshop was held on June 26, 2017. It was hosted by the Louisiana Department of Transportation and Development (DOTD) at the Louisiana Transportation Research Center’s Transportation Training and Education Center in Baton Rouge, Louisiana. The purpose of the work- shop was to present the proposed crash prediction methods (the final of which are presented in Chapter 6 of this report) to a range of practitioners and obtain their feedback on the func- tionality of the models. The following subsections summa- rize feedback received during the workshop and the research team’s actions based on the outcome of the workshop. Twenty-five individuals attended the workshop, including five panel members (or representatives from their agency). Local attendees included Louisiana DOTD staff represent- ing a range of disciplines from planning through design and representatives from a local city and metropolitan planning organization. During the workshop, attendees provided input on what they wanted the results of this project to be able to help them with and feedback on what was presented. Key areas of inter- est included the following: • Multilane roundabouts: Attendees were interested in treatments that can be applied to reduce crashes at multi- lane roundabouts and to make better informed decisions regarding multilane roundabout design.

139 thesis tested is that at new roundabouts driver behavior may improve with familiarity. This would be seen in a reduction in the frequency and/or severity of crashes over time. To investigate a potential driver learning curve, those roundabouts where the opening year was known with cer- tainty were used. This included 109 roundabouts. Crash prediction models at the intersection-level were developed with one of the predictor variables being the number of years since opening. To avoid the confounding factor of crash data from different seasonal periods (e.g., if one location opened to traffic in February and another in July), only full years of crash data were considered, starting with the first full year after opening to traffic. In developing the models, each site-year was used as an observation, and generalized estimating equations model- ing was applied to account for the temporal correlation in crash counts for the same site. Due to the limited number of sites available, a subset could not be set aside to use as a validation sample. Several possible predictor variables were considered, including • Area type (urban versus rural), • Ramp terminal versus nonramp terminal, • Number of circulating lanes, • Number of legs, • Number of entering lanes per leg, • Number of exiting lanes per leg, • Posted speed limit, • Year of crash count, and • Major and minor road AADT. At the onset, a number of modeling challenges and goals were defined, as outlined below: Goals: • To see if evidence of a driver learning trend was consistent between states, • To investigate if evidence of a driver learning curve is dif- ferent for single- versus multilane roundabouts, • To investigate if evidence of a driver learning curve is dif- ferent for urban versus rural roundabouts, and • To separately analyze total, KABC, and PDO crashes. Challenges: • Florida (1) and New York (5) provided few sites, so they should possibly be excluded; however, they were included for discussion purposes. • Considering the appropriate number of after-period years so as to not inadvertently mask evidence of a driver learn- ing curve while not looking at too few as to miss evidence of a driver learning curve. • The need to consider that observed trends due to a driver learning curve may be mixed in with general trends over time in crash frequency by attempting to include the year of crash count in the model. • Because the opening year differs between sites, the mix of sites and sites per state will change by year postopening, and this may confound the identification of a driver learn- ing curve if all available site-years of data are used. • Because of the limited number of sites, data for all site types may need to be combined. The following describes the approach used to evaluate the potential effects of driver learning curve. The intersection-level models for investigating the driver learning curve follow the categories for planning or network screening models. These include two categories for area type, two categories for number of circulating lanes, and three severity categories. In combination, there are 12 (= 2 × 2 × 3) potential intersection-level models. These models are identi- fied in the following list. • Rural single-lane, total; • Rural single-lane, KABC; • Rural single-lane, PDO; • Rural multilane, total; • Rural multilane, KABC; • Rural multilane, PDO; • Urban single-lane, total; • Urban single-lane, KABC; • Urban single-lane, PDO; • Urban multilane, total; • Urban multilane, KABC; and • Urban multilane, PDO. The predicted average crash frequency for each site is com- puted using a prediction model. The general form of the safety predictive model for investigating a driver learning curve is shown below. Equation 5-1601 1 . . .N exp EXPOSURE expa c x nxn= ( )β + β where N = predicted average crash frequency, crashes/yr; EXPOSURE = the variable for traffic volumes; X1 . . . Xn = a series of predictor variables; a, b, c, β1 . . . βn = estimated parameters; and k = the estimated overdispersion parameter. Consideration of a driver learning curve is accomplished by using the number of years postconstruction as one of the predictor variables.

140 5.5.2 Data Summary for Driver Learning Curve Analysis This section identifies the data used for model estimation. Table 5-95 provides general summary statistics of the data available for investigating a driver learning curve. Table 5-96 provides the frequency of sites by state and the totals for the data used in Table 5-95, for each value of years in service (i.e., years postconstruction), grouped in three ranges. It is observed that after 5 years the number of sites available becomes much smaller and after 10 years there are very few sites providing data. Table 5-97 and Figure 5-21 provide the average crash rate per MEV grouped by years in service. The crash rate was determined for each site-year and the average over all sites is reported. Table 5-97 and Figure 5-21 indicate that there may be a slight decrease in crash rate over time for total and PDO crashes, with a few outliers at years of service equal to 13 and 14 for which only five sites remain. The trend for KABC crashes, which is based on relatively few crashes, appears quite flat until the outlier at Year 14. Although comparative analyses of these data can be some- what informative, caution should be exercised in interpreting the apparent trends since each year of service is a different mix of sites and sites per state, characteristics such as num- ber of legs etc., and a linear relationship between crashes and AADT needs to be assumed, which is also problematic in interpreting trends. To investigate the potential driver learn- ing curve multivariable regression models need to be devel- oped to account for confounding factors. 5.5.3 Model Development for Driver Learning Curve Analysis Generalized estimating equations modeling with a nega- tive binomial error distribution was applied to estimate all of the potential models. The first step taken was to investigate if separate models could be developed for each site category defined by area type (rural versus urban) and number of circulating lanes (one or Variable Frequency of Sites Sites by State FL – 1 MI – 45 NY – 5 WA – 41 WI – 17 Site-Years by State FL – 5 MI – 173 NY – 46 WA – 364 WI – 123 Site-Years by Area Type Rural – 175 Urban – 536 Site-Years by Ramp Terminal Yes – 68 No – 643 Site-Years by Number of Legs 3 – 223 4 – 488 Site-Years by Number of Circulating LanesA 1 – 489 2/hybrid – 222 A Hybrid sites are those roundabouts where there are two circulating lanes in front of one or more approaches and at least one approach with only one circulating lane at the point of entry. Table 5-95. Major characteristics of sites used for driver learning curve. State 1 to 5 Years 6 to 10 Years 11 to 14 Years FL 1 – 1 2 – 1 3 – 1 4 – 1 5 – 1 6 – 0 7 – 0 8 – 0 9 – 0 10 – 0 11 – 0 12 – 0 13 – 0 14 – 0 MI 1 – 45 2 – 35 3 – 29 4 – 19 5 – 13 6 – 12 7 – 7 8 – 5 9 – 4 10 –4 11 – 0 12 – 0 13 – 0 14 – 0 NY 1 – 5 2 – 5 3 – 5 4 – 5 5 – 5 6 – 5 7 – 4 8 – 3 9 – 3 10 – 2 11 – 1 12 – 1 13 – 1 14 – 1 WA 1 – 41 2 – 41 3 – 41 4 – 41 5 – 41 6 – 34 7 – 31 8 – 24 9 – 20 10 – 17 11 – 15 12 – 10 13 – 4 14 – 4 WI 1 – 17 2 – 17 3 – 17 4 – 17 5 – 17 6 – 9 7 – 8 8 – 6 9 – 5 10 – 4 11 – 4 12 – 2 13 – 0 14 – 0 Sites Used in Table 5-95 1 – 109 2 – 99 3 – 93 4 – 83 5 – 77 6 – 60 7 – 50 8 – 38 9 – 32 10 – 27 11 – 20 12 – 13 13 – 5 14 – 5 Table 5-96. Frequency of roundabouts by state and after-period years available for driver learning curve.

141 Years in Service Total Crashes per MEV KABC Crashes per MEV PDO Crashes per MEV 1 0.721 0.084 0.636 2 0.793 0.096 0.697 3 0.672 0.125 0.546 4 0.763 0.102 0.660 5 0.770 0.121 0.649 6 0.643 0.086 0.557 7 0.606 0.105 0.501 8 0.668 0.120 0.549 9 0.624 0.095 0.529 10 0.483 0.109 0.374 11 0.613 0.104 0.510 12 0.571 0.125 0.446 13 1.423 0.097 1.326 14 1.449 0.505 0.944 Table 5-97. Average crashes per MEV by years in service. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 Cr as h Ra te Years In Service total crashes per MEV KABC crashes per MEV PDO crashes per MEV Figure 5-21. Average crashes per MEV by years in service. two). Model variables included only major and minor road AADT and the number of roundabout legs. Although most parameter estimates were of an expected magnitude and sign, the standard errors were quite high, indicating that the sample sizes are too low to develop reliable models when separated by site type. The number of sites (not site-years of data) per site type was 21 for rural single lane, 12 for rural multilane, 50 for urban single lane, and 26 for urban multilane. Models were then attempted by combining sites by urban versus rural, by single versus multilane, and by combining all sites together. The models were consistent in these three groupings with the lowest standard errors obtained when all sites were grouped together. For this reason, all sites were combined for further model development, with site-type dependent variables, in addition to AADT. These site-type variables are correlated with AADT, meaning their coefficients should not be interpreted to infer safety effects of changing these variables for a given AADT. Several options were explored, including the following: • Considering the number of years postconstruction as either a continuous or factor variable, • Including a term for the state in which the roundabout is located, • Allowing the effect of year postconstruction to vary based on the number of circulating lanes, and • Considering a range of the number of years post- construction to consider.

142 Bagdade, J., B. Persaud, K. McIntosh, J. Yassin, C. Lyon, C. Redinger, J. Whitten, and W. Butch. 2011. Evaluating the Performance and Safety Effectiveness of Roundabouts. Report RC-1566. Opus International Consultants, Inc., West Bloomfield, Mich. Bared, J. G., and W. Zhang. 2007. Safety Assessment of Interchange Spacing on Urban Freeways. FHWA-HRT-07-031. Federal Highway Administration, Washington, D.C. Bonneson, J., D. Lord, K. Zimmerman, K. Fitzpatrick, and M. Pratt. 2006. Development of Tools for Evaluating the Safety Implications of Highway Design Decisions. FHWA/TX-07/0-4703-4. Texas Trans- portation Institute, College Station, Tex. Bonneson, J., S. Geedipally, M. Pratt, and D. Lord. 2012. Safety Predic- tion Methodology and Analysis Tool for Freeways and Interchanges. Final Report. NCHRP Project 17-45. Texas Transportation Insti- tute, College Station, Tex. Dixon, K., and J. Zheng. 2014. Safety Performance for Roundabout Application in Oregon. Paper No. 14-5636. Presented at the Annual Meeting of the Transportation Research Board, Washington, D.C. Elvik, R. 2005. Speed and Road Safety. Transportation Research Record: Journal of the Transportation Research Board, No. 1908. Transporta- tion Research Board, National Research Council, Washington, D.C., pp. 59–69. Elvik, R. 2011. Assessing Causality in Multivariate Accident Models. Accident Analysis & Prevention. Vol. 43, pp. 253–264. Hauer, E. 2006. The Frequency-Severity Indeterminacy. Accident Analysis & Prevention. Vol. 38, pp. 78–83. Isebrands, H. 2011. Quantifying Safety and Speed Data for Rural Round- abouts with High-Speed Approaches. Graduate Theses and Disserta- tions. Paper 10378. Iowa State University, Ames, Iowa, Kvalseth, T. O. 1985. Cautionary Note About R2. The American Stat- istician. American Statistical Association, Vol. 39, No. 4, (Part 1), November, pp. 279–285. Lord, D. 2006. Modeling Motor Vehicle Crashes Using Poisson-Gamma Models: Examining the Effects of Low Sample Mean Values and Small Sample Size on the Estimation of the Fixed Dispersion Parameter. Accident Analysis & Prevention, Vol. 38, No. 4, Elsevier Ltd., Oxford, Great Britain, pp. 751–766. McCullagh, P., and J. A. Nelder. 1983. Generalized Linear Models. Chapman and Hall, New York, N.Y. Miaou, S. P. 1996. Measuring the Goodness-of-Fit of Accident Predic- tion Models. FHWA-RD-96-040. Federal Highway Administration, Washington, D.C. Park, B.-J., and D. Lord. 2008. Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter. Transpor- tation Research Record: Journal of the Transportation Research Board, No. 2061. Transportation Research Board, Washington, D.C., pp. 9–19. Rodegerdts, L., M. Blogg, E. Wemple, E. Myers, M. Kyte, M. Dixon, G. List, A. Flannery, R. Troutbeck, W. Brilon, N. Wu, B. Persaud, C. Lyon, D. Harkey, and D. Carter. 2007. NCHRP Report 572: Roundabouts in the United States. National Cooperative Highway Research Program, Transportation Research Board, Washington, D.C. Rodegerdts, L., J. Bansen, C. Tielser, J. Knudsen, E. Myers, M. Johnson, M. Moule, B. Persaud, C. Lyon, S. Hallmark, H. Isebrands, R. B. Crown, B. Guichet, and A. O’Brien. 2010. NCHRP Report 672: Roundabouts: An Informational Guide. 2nd edition. National Cooperative High- way Research Program, Transportation Research Board, Wash- ington, D.C. Parameter Estimate Standard Error p-Value a -5.4370 2.1624 0.0119 STATE MI 0.5207 NY -0.3531 WA -0.0558 WI 0.0000 0.3328 0.3585 0.2799 0.0000 0.1177 0.3247 0.8420 n/a b 0.7424 0.2298 0.0012 c -0.6880 0.2302 0.0028 d -0.9573 0.2288 < 0.0001 e 0.0514 0.0333 0.1220 Overdispersion (k) 0.7560 0.0944 < 0.0001 Table 5-98. Driver learning curve analysis: total crash model two As an example of the models developed, Equation 5-161 and Table 5-98 report on the model form and results for a model for total crashes using 5 years of postconstruction data. In this example, the parameter associated with years postconstruction, e, indicates an increase in crash frequency with time, although this estimate is not statistically signifi- cant at the 90% confidence limit. Equation 5-161 N exp MAJAADT MINAADT exp a STATE b c NUMBERLEGS d CIRCLANES e POSTYR ( )= + ( ) + × + × + × where N = predicted average crash frequency, crashes/yr; STATE = an additive intercept term dependent on state; MAJAADT = AADT on the major road; MINAADT = AADT on the minor road; NUMBERLEGS = 1 if a 3-leg roundabout; 0 if 4 legs; CIRCLANES = 1 if a single-lane roundabout; 0 if more than 1 circulating lanes; and POSTYR = a parameter representing the year in service postconstruction. 5.6 References and Bibliography AASHTO. 2010. Highway Safety Manual, 1st edition. American Asso- ciation of State Highway and Transportation Officials, Washing- ton, D.C. AASHTO. 2014. Highway Safety Manual. 1st edition, 2014 Supple- ment. American Association of State Highway and Transportation Officials, Washington, D.C.