Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report (2021)

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56 C H A P T E R 5 - H S M P R E D I C T I V E M O D E L HSM Predictive Model This chapter describes the findings obtained during the development of crash prediction models (CPMs) for freeways with part-time shoulder use (PTSU) operation. Specifically, it describes the fixed parameters crash prediction models that are recommended by the research team for inclusion in a future edition of the Highway Safety Manual (HSM) (AASHTO 2010). As discussed in Chapter 3, Modeling Approaches, the research team also developed CPMs using random parameters and latent class modeling techniques; these models are presented in Chapter 7, Advanced Crash Predictive Models. In addition to the documentation of the fixed parameters models presented in this chapter, the research team also prepared draft text for a future edition of the HSM. The draft text is available in Chapter 2 of the PTSU Safety Evaluation Guidelines, a separate document that was also prepared through National Cooperative Highway Research Program (NCHRP) Project 17-89. The CPMs in this chapter include variables that describe PTSU operational features and design elements. More precisely, these variables address the case where the shoulder is used by all vehicles and it is allowed on a static (i.e., fixed time schedule; static part-time shoulder use [S-PTSU]) or dynamic (i.e., traffic responsive; dynamic part-time shoulder use [D-PTSU]) basis during the day. This type of shoulder use is referred to herein as “PTSU operation.” Bus-on-shoulder (BOS) operation is implemented on some of the segments used to estimate the CPM coefficients; however, its presence was not found to have a significant effect on safety. Any reference in this chapter to “PTSU” is referring to shoulder use by all vehicle types; it is not referring to BOS operation. The CPMs in this chapter are used to predict the average crash frequency associated with one direction of travel on a freeway. Each CPM includes a safety performance function (SPF), one or more adjustment factors (AFs), a calibration factor, a severity distribution, and a crash type distribution. Note that AFs are called crash modification factors (CMFs) in the HSM (AASHTO 2010). The SPF, AFs, and calibration factor are multiplied together to form the predictive model equation. One predictive model is described herein for each of the following site types:  Freeway segment  Ramp entrance speed-change lane  Ramp exit speed-change lane These site types are defined in Chapter 3. The SPF is derived to estimate the crash frequency for a site with typical design and operating conditions. The AFs can be used to adjust the SPF estimate whenever one or more design or operational elements is atypical. All CPMs in the HSM include a crash type distribution and a crash severity distribution. These distributions are used with an associated CPM to estimate the average crash frequency for various combinations of crash type, severity, or both. Distributions were developed in Project 17-89 and are documented in Chapter 6, Crash Severity and Crash Type Distributions. To facilitate interpretation and implementation of the CPMs developed for this project, the variable names and definitions used in these CPMs are consistent with those used in Chapter 18 of the Highway Safety Manual Supplement (HSM Supplement) (AASHTO 2014).

57 The sections of this chapter provide a description of the predictive 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 and validation activities. Predictive Model Form The basic structure of the predictive model for freeways with PTSU was described previously in Chapter 3. This section describes the techniques used to derive analytic relationships for non-homogenous segments and for PTSU lane presence. These relationships were then used in the development of AFs in the predictive model. Analytic Relationship for Non-Homogeneous Segments As described in Chapter 3, the sites in the model estimation database were established using the segmentation criteria in Chapter 18 of the HSM Supplement (AASHTO 2014). In addition to these criteria, a site was also defined to begin at the start or end of a horizontal curve. These criteria ensured that each site was homogenous with respect to several variables (e.g., lane width, shoulder width, median width, speed-change lane presence, and horizontal curvature) but they were not necessarily homogenous in other variables with a known association with crash frequency. Some geometric elements (e.g., shoulder rumble strips, roadside barrier) were often found to start or stop at some point along the length of a site. When this occurred, the length of the site associated with the element’s initial condition and the length associated with its changed condition were recorded in the database. Geometric elements that were sometimes only partially located on a site are identified as follows:  Rumble strip presence on the inside shoulder.  Rumble strip presence on the outside shoulder.  Median barrier presence.  Outside (roadside) barrier presence.  Turnout presence. AFs are typically developed for application to homogeneous sites. Thus, they typically do not include variables that allow them to be modified for application to sites on which they only partially exist. For example, AFs (and CMFs) exist for freeway segments with shoulder rumble strips along their entire length, but AFs (or CMFs) are rarely, if ever, for the case where shoulder rumble strips are a present for only part of the segment length. The following equation was used to convert an AF for homogeneous sites into one that could be used for non-homogeneous sites. Equation 8 𝐴𝐹 , 1.0 𝑃 , 1.0 𝑃 , 𝐴𝐹 where AFi,agg = aggregated SPF adjustment factor for design or operational element i; PL,i = proportion of the site length with element i present; and AFi = SPF adjustment factor for design or operational element i. Analytic Relationship for Quantifying Association of PTSU Operation with Safety The previous section described the development of an AF form to address the situation where there is a spatial change in the site’s physical characteristics. This section describes the development of an AF form to address the situation where there is temporal change in the site’s operational characteristics. This type of AF form is needed to address the situation where the site has PTSU operation during some hours of the

58 day but not during the remaining hours of the day. The following equation was used to compute an equivalent AF for sites with a PTSU operating for a portion of the typical day. Equation 9 𝐴𝐹 , 1.0 𝑃 , 𝐴𝐹 , … 𝐴𝐹 , 𝑃 , 𝐴𝐹 , … 𝐴𝐹 , with, Equation 10 𝑃 , ∑ 5 𝑃 , , 2 𝑃 , , 7 24 where AFptsu,agg = aggregated SPF adjustment factor for sites with PTSU operation during some hours of the day; Pt,ptsu = proportion of time during the average day that PTSU operates; AFi,closed = SPF adjustment factor for design or operational element i present when the PTSU is not operating; AFi,open = SPF adjustment factor for design or operational element i present when the PTSU is operating; Popen, wkday,j = proportion of hour j that PTSU is operating during typical weekday of year; and Popen, wkend,j = proportion of hour j that PTSU is operating during typical weekend day of year. The model estimation database was structured to include facilities with PTSU operation (i.e., “treatment” facilities) and nearby “comparison” facilities that do not have PTSU operation. This paring of treatment and comparison facilities is intended to provide a more reliable estimate of incremental effect of PTSU operation on crash frequency. In the context of using regression analysis to quantify this effect, indicator variables were used in the regression model to facilitate a comparison of each treatment−comparison facility pair. This use of indicator variables is illustrated in the following equation for the simple case where there are two treatment−comparison facility pairs (one in State A and one in State B) and other unpaired facilities in other states. Equation 11 𝑁 exp 𝑏 𝑏 ln 𝐴𝐴𝐷𝑇 𝑏 , , 𝐼 , 𝑏 , , 𝑏 𝐼 , 𝑏 , , 𝐼 , 𝑏 , , 𝑏 𝐼 , where Nspf = predicted crash frequency for base conditions (crashes/yr); AADT = directional annual average daily traffic volume (veh/day); Ic A = indicator variable (= 1 if site on comparison facility in State A, 0 otherwise); Ic,B = indicator variable (= 1 if site on comparison facility in State B, 0 otherwise); It A = indicator variable (= 1 if site on treatment facility in State A, 0 otherwise); It,B = indicator variable (= 1 if site on treatment facility in State B, 0 otherwise); b2,c,A = regression coefficient for incremental effect of comparison facility in State A, relative to other states; b3,c,B = regression coefficient for incremental effect of comparison facility in State B, relative to other states; bptsu = regression coefficient quantifying incremental effect of PTSU presence; and bi = regression coefficient i. Equation 11 can be algebraically reduced to the following simpler form: Equation 12 𝑁 exp 𝑏 𝑏 ln 𝐴𝐴𝐷𝑇 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 𝐼 where IA = indicator variable (= 1 if site on treatment or comparison facility in State A, 0 otherwise); IB = indicator variable (= 1 if site on treatment or comparison facility in State B, 0 otherwise); It = indicator variable (= 1 if site on treatment facility in State A or B, 0 otherwise); and all other variables are as previously defined. The form shown in Equation 12 was used in the regression model, as described in a subsequent section. The PTSU lane is always preceded and succeeded by a transition zone. In the upstream transition zone, vehicles change lanes or adjust speed as they interact with vehicles preparing to enter the forthcoming

59 PTSU lane. Similarly, in the downstream zone, vehicles change lanes or adjust speed as they interact with vehicles that have just exited the PTSU lane. This transition zone is defined to be 800 feet (0.152 miles) in length based on the researchers’ experience with PTSU operations. To accurately quantify the influence of PTSU transition zone presence on freeway safety, a variable was defined to quantify the portion of a site’s length that included a transition zone. This variable is computed using the following equation. Equation 13 𝑃 𝐿transition, /𝐿 where Ptransition = proportion of site length with PTSU transition zone present upstream, downstream, or both; Ls = length of site (miles); and Ltransition,site = total length of PTSU transition zones within site (i.e., between site begin and end mileposts) (miles). The proportion computed using Equation 13 ranges between 0.0 (i.e., no transition zone present) and 1.0 (transition zone present for the length of the site). A transition zone can exist entirely within the length of one site or a portion of the zone can be located in two or more sites. In special cases, two separate transition zones can be located in the same site. These points are illustrated in Figure 5. a. Upstream transition zone. b. Downstream transition zone. c. Transition zones between PTSU lanes. Figure 5. Example calculation of the length of transition zone within a site. In Figure 5a, the site is shown as a freeway segment located upstream of a PTSU lane. The segment’s length is shown to exceed that of the transition zone (which always equals 0.152 miles). As a result, the proportion of the site length with PTSU transition zone Ptransition is computed to be less than 1.0. In Figure 5b, the site is shown as a freeway segment located downstream of a PTSU lane. Its length is shown to be less than that of the transition zone. As a result, the variable Ptransition is equal to 1.0. If the

60 next downstream segment of freeway is also evaluated, it too will have a non-zero value for Ptransition because it includes the remaining portion of the transition zone. In Figure 5c, the site is shown as a freeway segment between two PTSU lanes. Its length is shown to include two transition zones. As a result, the value of Ltransition,site is equal to the sum of the two lengths. Because this sum is less than the segment length, Ptransition is computed as a value less than 1.0. Modeling Approach Model Development Process The analytic form of the predictive model equation was based on the use of fatal-and-injury (FI) crash data (as opposed to “total” crashes, i.e., crashes of all severities). The reasons for following this process were discussed previously in Chapter 3. For this project, the FI prediction model was developed first, followed by the property-damage-only (PDO) model. The minimal influence of reporting threshold variation on the FI data ensured that it would provide the most insight regarding the model’s analytic form and the variables that truly have an effect on crash frequency. Once the FI model was developed and validated, all statistically valid AFs in the FI model were also considered during the development of the PDO model. New AFs that would be unique to the PDO model were not investigated during PDO model development because the issues associated with state-to-state variation in PDO reporting level tended to cloud the search for reliable associations. As a result of this process, the “FI model” and “PDO model” tended to have a similar analytic form. The retention of an AF in the FI and PDO models was based on consideration of its regression coefficients and overall model fit (i.e., magnitude, direction of effect, statistical significance, practical significance, and ability to reduce the Akaike information criterion [AIC] value). The analytic form of the FI prediction model was validated using a hold-out process where the data from one state were not used for FI model estimation. The data for this state were used to locally calibrate the estimated FI prediction model. Then, the calibrated FI model predictions were compared to the reported crash counts on a site-by-site basis and their differences evaluated using statistical tests. If these tests are satisfied, the valuation approach demonstrates the state-to-state transferability ability of the predictive model. It also provides evidence that the analytic form of the model provides reasonably sound relationships between crash frequency and the model’s independent variables. This latter benefit minimizes the need to validate the estimated PDO prediction model. Correlation of Variables with State A preliminary analysis of the highway safety database indicated that some of the site characteristic variables were correlated with the state in which the site was located. For example, sites for some states tended to have mostly two or three through lanes, while sites in other states tended to have four or more lanes. As a result of this correlation, the model development process required two stages. In the first stage, the regression model included only site characteristics variables (it did not include variables specific to the states). The regression coefficient for each site characteristic variable was examined for magnitude, direction, and statistical significance. If the magnitude, direction, and significance were acceptable, then the variable was retained in the model. At the conclusion of the first stage, the estimated regression model included only variables whose coefficients were considered acceptable. During the second stage of the model development process, the estimated regression model from the first stage was expanded to include one or more state-specific indicator 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 state that had significantly more or less crashes than the other states. One state- specific indicator variable was added to the model at a time. This process was repeated for all states

61 represented in the database. If the coefficient for a given state-specific indicator was found to be statistically 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 noted previously, the modeling approach was focused on the development of separate predictive models for each of the following site types: freeway segment, ramp entrance speed-change lane, and ramp exit speed-change lane. Additionally, for a given site type, one model would be developed to predict FI crash frequency and one model would be developed to predict PDO crash frequency. In summary, the model development plan was focused on the development of the six predictive model equations identified in the following list:  Freeway segment to predict FI crash frequency  Ramp entrance speed-change lane site to predict FI crash frequency  Ramp exit speed-change lane site to predict FI crash frequency  Freeway segment to predict PDO crash frequency  Ramp entrance speed-change lane site to predict PDO crash frequency  Ramp exit speed-change lane site to predict PDO crash frequency A preliminary regression analysis of the data indicated that the site sample size for speed-change lane equations was too small to develop a reliable, independent model for each of the six equations. To overcome this issue, it was decided that a combined modeling approach would be needed to develop semi-independent models for the three site types. Thus, combined modeling was used to develop semi- independent FI models for all three site types. It was also used to develop semi-independent PDO models for all three site types. With a combined modeling approach, some AFs are common to each of the site type models. Some AFs in the model for freeway segments are shown the same as the AFs in the model for the speed-change lane sites. That is, in some instances, the AF for traffic characteristic, geometric element, or traffic control feature i is the same in each equation. The regression coefficient associated with each AF is also the same. This approach asserts that the value of an AF common to both models is the same regardless of the site type, provided that the element or feature the AF represents (e.g., inside shoulder width) is the same at both sites. This approach assumes that some characteristics, elements, or features have a similar influence on crash frequency, regardless of whether the site is a freeway segment or a speed-change lane. The use of common AFs has the advantage of maximizing the sample size available to estimate the AF. In this manner, the data for all three site types are pooled to provide a more efficient estimate of the AF coefficient. The use of common AFs in multiple models requires the simultaneous regression analysis of all models. The total log-likelihood statistic for all three models combined was used to determine the best fit regression coefficients. A simulation analysis was undertaken to determine if this type of regression would bias the regression coefficients or their standard error. The results of this analysis indicated that (1) the coefficients were not biased and (2) that the standard error of those coefficients associated with a variable were not biased. The regression analysis is described in more detail in the Statistical Analysis Methods section of this chapter Overdispersion Parameter It was assumed that site crash frequency is Poisson distributed, and that the distribution of the mean crash frequency for a group of similar sites is gamma distributed. In this manner, the distribution of

62 crashes for a group of similar sites can be described by the negative binomial distribution. The variance of this distribution is computed using the following equation. Equation 14 𝑉 𝑋 𝑦𝑁 𝑦 𝑁𝐾 𝐿 where V[X] = crash frequency variance for a group of similar sites (crashes2); N = predicted average crash frequency (crashes/yr); X = reported crash count for y years (crashes); Ls = site length (miles); y = time interval for which X reported crashes were evaluated (i.e., study period) (year); and K = inverse dispersion parameter (miles-1). The overdispersion parameter, as defined and used in the HSM, is computed using the following equation: Equation 15 𝑘 1.0𝐾 𝐿 where k = overdispersion parameter and other variables are as previously defined. Cross-Sectional Database The database includes multiple years of data for each study site. However, study duration in “years” is represented as an offset variable in the regression model, and an average of each year’s annual average daily traffic (AADT) value for the study period is used in the regression model. For this reason, the database is described as cross sectional (as opposed to panel). One reason for using cross-sectional data for model estimation relates to the accuracy of the yearly AADT values in most highway safety databases. Examination of the AADT volume in the assembled database (and an examination of associated databases and state department of transportation [DOT] documentation) indicated that site AADT volume is frequently extrapolated by the state DOT from partial year counts taken at temporary count stations located several miles from the subject site. Thus, there are accuracy implications associated with this temporal and spatial extrapolation. Moreover, state DOT practice when a current count is not available for a site is sometimes to adjust the AADT volume from the last year it was counted (which could be several years previous); sometimes the practice is to leave the variable as missing. In fact, it is common for a site’s AADT volume to be missing for one or more years. In these cases, the researchers for this project have had to estimate a value using the AADT volume trends for adjacent years and adjacent sites. A second reason for using cross-sectional data for model estimation is to minimize the problems associated with over-representation of sites with zero crashes. Statistical methods have been developed to improve the fit of a model to this “zero-inflated” data. However, Lord et al. (2007) indicate that when these methods have been applied to highway crash data, they have (1) an inherent tendency to over-fit the data, (2) a theoretic explanation of dual-state highway safety that is problematic, and (3) the potential to obfuscate the interpretation of predictive model trend and coefficient meaning. Thus, summing each site’s crashes over a multiple-year study period minimizes the proportion of sites with zero crashes in the database and precludes the need for a dual-state distribution. In general, cross-sectional data can provide a more robust predictive model than panel data when the year-to-year variability in the independent variables (e.g., AADT) is largely random.

63 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 variations of the proposed predictive model were both nonlinear and discontinuous. The log-likelihood function for the negative binomial distribution was used to determine the best-fit model coefficients. Equation 14 was used to define the variance function for all models. The procedure 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 χ2 statistic, which is calculated using the following equation. Equation 16 𝜒 𝑋 𝑦 𝑁𝑉 𝑋 where χ2 = Pearson chi-square statistic; n = number of observations (i.e., sites 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 for which Xi reported crashes were evaluated (i.e., study period) (year). This statistic follows the χ2 distribution with n – p degrees of freedom, where n is the number of observations (i.e., sites) and p is the number of model variables (McCullagh and Nelder 1983). This statistic is asymptotic to the χ2 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 independent variable is at its mean value. This statistic can be computed using the following equation. Equation 17 𝑠 ∑ 𝑋 𝑦 𝑁𝑛 𝑝 𝑦 where sp = root mean square error of the model estimate (crashes/yr); all other variables are as previously defined. The scale parameter φ is used to assess the amount of variation in the observed data, relative to the specified distribution. This statistic is calculated by dividing Equation 16 by the quantity n – p. A scale parameter near 1.0 indicates that the assumed distribution of the dependent variable is approximately equivalent to that found in the data (i.e., negative binomial). Another measure of model fit is the coefficient of determination R2. This statistic is commonly used for normally distributed data. However, it has some useful interpretation when applied to data from other distributions when computed in the following manner (Kvalseth 1985). This coefficient is computed using the following equation. Equation 18 𝑅 1.0 ∑ 𝑋 𝑦 𝑁 ∑ 𝑋 𝑋 where 𝑋 = average crash frequency for all n observations (crashes/site); all other variables are as previously defined.

64 The last measure of model fit is the dispersion-parameter-based coefficient of determination Rk2. This statistic was developed by Miaou (1996) for use with data that exhibit a negative binomial distribution. It is computed using the following equation. Equation 19 𝑅 1.0 𝑘𝑘 where k = overdispersion parameter; and knull = overdispersion parameter based on the variance in the reported crash frequency. The null overdispersion parameter knull represents the dispersion 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 (i.e., a model with no independent variables but with the same error distribution, link function, and offset in years y). The overdispersion parameter k can be estimated using Equation 15 using the average site length for the variable L. Fatal-and-Injury Crash Frequency Prediction Model This section describes the development of predictive model equations based on FI crash data. One equation is applicable to freeway segments. A second equation is applicable to ramp entrance speed- change lane sites. A third equation is applicable to ramp exit speed-change lane sites. This section consists of six subsections. The first subsection describes the structure of the predictive equations as used in the regression analysis. The second subsection describes the regression statistics for each of the estimated equations. The third subsection describes a validation of the estimated equations. The fourth subsection describes the proposed safety prediction models. The fifth subsection describes the estimated AFs. The sixth subsection provides a sensitivity analysis of the predictive model equations over a range of traffic demands. Model Development This section describes the proposed predictive model equations and the methods used to develop them. The regression model is generalized to address three site types: freeway segments, ramp entrance speed- change lanes, and ramp exit speed-change lanes. The generalized form shows all the AFs in the model. Indicator variables are used to determine which AFs are applicable to each observation in the database based on its associated site type. The generalized form consists of AFs with indicator variables used to determine when the AF is applicable. For example, the generalized form of the outside clearance AF includes an indicator variable that is used to determine when the subject site is a freeway segment—in which case, the AF is applicable. If the site is speed-change lane, the AF is not applicable. All of the models described in this section are developed to predict crash frequency for all crash types combined (regardless of manner of collision or number of vehicles involved). For this reason, there is no subscript on the variables to denote the applicable crash type. The following generalized regression model is described using the following equations.

65 A. If the observation corresponds to a freeway segment, the following model is used. Equation 20 𝑁 , , 𝑦 𝐶 , , 𝑁 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , with, Equation 21 𝑁 , , 𝐿 , exp 𝑏 , , 𝑏 , , ln 𝐴𝐴𝐷𝑇 /1000 Equation 22 𝐴𝐹 , , 1.0 exp 𝑏 , , 5,730𝑅 Equation 23 𝐴𝐹 , , exp 𝑏 , , min 𝑊 , 13 12 Equation 24 𝐴𝐹 , , exp 𝑏 , , /𝑛 min 𝑊 , , 12 6 Equation 25 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 𝑏 , , /𝑛 Equation 26 𝐴𝐹 | , , 1.0 𝑃 exp 𝑏 , , /𝑛 𝑊 48 𝑃 exp 𝑏 , , /𝑛 min 𝑊 , 2 𝑊 48 Equation 27 𝑊 𝑊 𝑊 , 𝑊 , 𝑊 , 𝐼 , , 𝑊 , 𝐼 , , Equation 28 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 𝑏 , , 𝑛/𝑊 Equation 29 𝐴𝐹 | , , 1.0 𝑃 , exp 𝑓 , 𝑃 , exp 𝑓 , 𝑓 , , 𝑓 , Equation 30 𝑓 , 𝑏 , , /𝑛 min 𝑊 , , 12 𝐼 Equation 31 𝑓 , 𝑏 , , min 𝑊 , , 13 12 𝐼 Equation 32 𝑓 , 𝑏 , , 𝐼 Equation 33 𝑓 , , 𝑏 , , 1 𝐼 𝑃 , Equation 34 𝑃 , 𝐿 , /𝐿 , Equation 35 𝐴𝐹 , , 1.0 exp 𝑏 , , 𝑋 , 𝑏 , , ln 𝐴𝐴𝐷𝑇 , /1000𝑏 , , 𝐿 , 1.0 exp 𝑏 , , 𝐿 , 1.0 exp 𝑏 , , 𝑋 , 𝑏 , , ln 𝐴𝐴𝐷𝑇 , /1000𝑏 , , 𝐿 , 1.0 exp 𝑏 , , 𝐿 ,

66 Equation 36 𝐴𝐹 , , exp 𝑏 , , /𝑛 min 𝑊 , 12 10 Equation 37 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 𝑏 , , /𝑛 Equation 38 𝐴𝐹 | , , 1.0 𝑃 exp 𝑏 , , /𝑛 𝑊 𝑊 , 𝐼 , , 𝑊 20 𝑃 exp 𝑏 , , /𝑛 𝑊 20 Equation 39 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 𝑏 , , 𝑛/𝑊 Equation 40 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 𝑏 , , /𝑛 Equation 41 𝑃 𝐿 , /𝐿 , Equation 42 𝐶 , , exp 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 𝑏 , , 𝐼 B. If the observation corresponds to a ramp entrance speed-change lane, the following model is used. Equation 43 𝑁 , , 𝑦 𝐶 , , 𝑁 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 , , with, Equation 44 𝑁 , , 𝐿 , exp 𝑏 , , 𝑏 , , ln 𝐴𝐴𝐷𝑇 /1000 𝑏 , , 𝐴𝐴𝐷𝑇 /1000 Equation 45 𝐴𝐹 | , , 1.0 𝑃 , exp 𝑓 , 𝑃 , exp 𝑓 , 𝑓 , , 𝑓 , Equation 46 𝑓 , , 𝑏 , , 1 𝐼 𝑃 , Equation 47 𝑃 , 𝐿 , /𝐿 , Equation 48 𝐴𝐹 , , exp 𝑏 , , 1𝐿 1 0.142 C. If the observation corresponds to a ramp exit speed-change lane, the following model is used. Equation 49 𝑁 , , 𝑦 𝐶 , , 𝑁 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 | , , 𝐴𝐹 , , with, Equation 50 𝑁 , , 𝐿 , exp 𝑏 , , 𝑏 , , ln 𝐴𝐴𝐷𝑇 /1000 𝑏 , , 𝑛 2 .

67 Equation 51 𝐴𝐹 | , , 1.0 𝑃 , exp 𝑓 , 𝑃 , exp 𝑓 , 𝑓 , , 𝑓 , Equation 52 𝑓 , , 𝑏 , , 1 𝐼 𝑃 , Equation 53 𝑃 , 𝐿 , /𝐿 , Equation 54 𝐴𝐹 , , exp 𝑏 , , 1𝐿 1 0.071 where AADTfs = one-directional AADT volume of freeway segment (veh/day); AADTf = one-directional AADT volume of freeway in speed-change lane site (veh/day); AADTen = AADT volume of entrance ramp (veh/day); AADTb,ent = AADT volume of entrance ramp located at distance Xb,ent upstream of the subject segment (veh/day); AADTe,ext = AADT volume of exit ramp located at distance Xe,ext downstream of the subject segment (veh/day); bi = regression coefficient for condition i; AFhc,w,z = adjustment factor for horizontal curvature at site type w (w = fs: freeway segment, en: ramp entrance speed-change lane, ex: ramp exit speed-change lane, ast: all site types) and severity z (z = fi: fatal and injury, pdo: property damage only, as: all severities); AFisw,w,z = adjustment factor for inside shoulder width at site type w and severity z; AFirs|agg,w,z = aggregated adjustment factor for rumble strips on the inside shoulder at site type w and severity z; AFlc,w,z = adjustment factor for lane changes at site type w and severity z; AFlw,w,z = adjustment factor for lane width at site type w and severity z; AFmw|agg,w,z = aggregated adjustment factor for median width at site type w and severity z; AFmb|agg,w,z = aggregated adjustment factor for median barrier at site type w and severity z; AFosw,w,z = adjustment factor for outside shoulder width at site type w and severity z; AFptsu|agg,w,z = aggregated adjustment factor for PTSU operation at site type w and severity z; AFoc|agg,w,z = aggregated adjustment factor for outside clearance at site type w and severity z; AFors|agg,w,z = aggregated adjustment factor for rumble strips on the outside shoulder at site type w and severity z; AFob|agg,w,z = aggregated adjustment factor for outside (roadside) barrier at site type w and severity z; AFturnout|agg,w,z= aggregated adjustment factor for turnout presence at site type w and severity z; AFlen,w,z = adjustment factor for speed-change lane length at site type w and severity z; fw,closed = factor for width of PTSU lane (including tapered transitions) when shoulder use is not allowed; fw,open = factor for width of PTSU lane (including tapered transitions) when shoulder use is allowed; fptsu,open = factor for presence of PTSU lane (including tapered transitions) when shoulder use is allowed; fnear,open,w = factor for transition zone between, upstream or, or downstream of PTSU lane when shoulder use is allowed at site type w (w = fs: freeway segment, en: ramp entrance speed-change lane, ex: ramp exit speed-change lane, ast: all site types); IptsuLane = indicator variable for PTSU lane presence (= 1.0 if PTSU lane is present [Wptsu,s > 0], 0.0 otherwise); Iptsu,s,in = indicator variable for PTSU lane location in the subject travel direction (= 1.0 if inside shoulder is allocated to part-time vehicular traffic use at any time of the day; otherwise 0.0) (feet); Iptsu,o,in = indicator variable for PTSU lane location in the opposing travel direction (= 1.0 if inside shoulder is allocated to part-time vehicular traffic use at any time of the day; otherwise 0.0) (feet); Iptsu,s,o = indicator variable for PTSU lane location in the subject travel direction (= 1.0 if outside shoulder is allocated to part-time vehicular traffic use at any time of the day; otherwise 0.0) (feet); Iohio = indicator variable (= 1.0 if site is in Ohio, 0.0 otherwise); IMN35 = indicator variable (= 1.0 if site on I-35W treatment or comparison facility in Minnesota, 0.0 otherwise); IHI01 = indicator variable (= 1.0 if site on I-H1 treatment or comparison facility in Hawaii, 0.0 otherwise); IV264 = indicator variable (= 1.0 if site on I-264 treatment or comparison facility in Virginia, 0.0 otherwise); IV495 = indicator variable (= 1.0 if site on I-495 treatment or comparison facility in Virginia, 0.0 otherwise); IVA66 = indicator variable (= 1.0 if site on I-66 treatment or comparison facility in Virginia, 0.0 otherwise); IG400 = indicator variable (= 1.0 if site on GA 400 treatment or comparison facility in Georgia, 0.0 otherwise); IGA85 = indicator variable (= 1.0 if site on I-85 treatment or comparison facility in Georgia, 0.0 otherwise);

68 Ls,fs = length of freeway segment (miles); Ls,en = length of ramp entrance speed-change lane site being evaluated (≤ length of ramp entrance speed- change lane, as measured from gore point to taper point) (miles); Ls,ex = length of ramp exit speed-change lane site being evaluated (≤ length of ramp exit speed-change lane, as measured from gore point to taper point) (miles); Len = length of ramp entrance speed-change lane (miles); Lex = length of ramp exit speed-change lane (miles); Ltransition,site = total length of PTSU transition zones within site (i.e., between site begin and end mileposts) (miles); Lturnout,fs = length of turnout within segment (i.e., between segment begin and end mileposts) (miles); n = number of through lanes within site (including managed lanes but not including auxiliary lanes or PTSU lanes); Np,w,z = predicted average crash frequency of site type w (w = fs: freeway segment, en: ramp entrance speed- change lane, ex: ramp exit speed-change lane, ast: all site types) and severity z (z = fi: fatal and injury, pdo: property damage only, as: all severities) (crashes/yr); Nspf,w,z = predicted average crash frequency of a site with base conditions, for site type w and severity z (crashes/yr); Pib = proportion of site length with a barrier present in the median (i.e., inside); Pir = proportion of site length with a rumble strips present on the inside shoulder; Pob = proportion of site length with a barrier present on the outside (roadside); Por = proportion of site length with a rumble strips present on the outside shoulder; Pt,ptsu = proportion of time during the average day that PTSU operates (see Equation 10); Ptransition,w = proportion of site length with PTSU transition zone present upstream, downstream, or both for site type w (w = fs: freeway segment, en: ramp entrance speed-change lane, ex: ramp exit speed-change lane); Pturnout = proportion of segment length with a turnout present; R = radius of curve (feet); Whc = clear zone width (feet); Wicb = distance from edge of inside shoulder to barrier face (feet); Wis,s = paved inside shoulder width for the subject travel direction (does not include the portion of the shoulder used as a PTSU lane) (feet); Wis,o = paved inside shoulder width for the opposing travel direction (does not include the portion of the shoulder used as a PTSU lane) (feet); Wl = lane width (feet); Wm = median width (measured from near edges of traveled way in both directions) (feet); Wptsu,s = width of shoulder allocated to part-time vehicular traffic use in the subject travel direction (i.e., as an additional travel lane) (if PTSU is not provided at any time, this width equals 0.0) (feet); Wptsu,o = width of shoulder allocated to part-time vehicular traffic use in the opposing travel direction (i.e., as an additional travel lane) (if PTSU is not provided at any time, this width equals 0.0) (feet); Wocb = distance from edge of outside shoulder to barrier face (feet); Ws = paved outside shoulder width (does not include the portion of the shoulder used as a PTSU lane) (feet); Wum = non-shoulder part of median width (measured from near edges of shoulder in both directions) (feet); Xb,ent = distance from segment begin milepost to nearest upstream entrance ramp gore point (miles); Xe,ext = distance from segment end milepost to nearest downstream exit ramp gore point (miles); and y = time interval for which reported crashes were evaluated (i.e., study period) (year). The formulation of most of the AFs in the aforementioned regression equations is based on the formulations of similar CMFs in Chapter 18 of the HSM Supplement (AASHTO 2014). The base conditions established for the SPFs match those used in Chapter 18. These base conditions include:  Horizontal curve presence: not present  Lane width: 12 feet  Inside shoulder width (paved): 6 feet  Length of rumble strip on inside shoulder: 0.0 miles (i.e., not present)  Median width: 60 feet

69  Length of median barrier: 0.0 miles (i.e., not present)  Outside shoulder width: 10 feet  Length of rumble strip on outside shoulder: 0.0 miles (i.e., not present)  Clear zone width: 30 feet  Length of outside (roadside) barrier: 0.0 miles (i.e., not present)  PTSU operation: no PTSU operation during any hour of the day  PTSU lane width: 0 feet  Turnout length in segment: 0.0 miles (i.e., not present)  Ramp entrance speed-change lane length: 0.142 miles  Ramp exit speed-change lane length: 0.071 miles The variable names and definitions used in these equations are consistent with those used in Chapter 18 of the HSM Supplement (AASHTO 2014). Notably, the equations in Section 18.7.3 were used to compute the variables Pib, Wicb, Pob, and Wocb. The constant “48” in Equation 26 represents a base median width of 60 feet and a base inside shoulder width of 6 feet (i.e., 48 = 60 − 2×6). Similarly, the constant “20” in Equation 38 represents a base clear zone of 30 feet and a base outside shoulder width of 10 feet (i.e., 20 = 30 – 10). The final form of the regression models is described by the preceding equations. This form reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described herein 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. AFs for other variables were also examined but were not found to be helpful in explaining the variation in the reported crash frequency among sites. These variables included:  Proportion of directional AADT volume during hours where volume exceeds 1,000 vehicles/hour/lane  Directional AADT volume per lane  Presence of a managed lane (i.e., HOT or HOV lane)  Presence of a lane add  Presence of a lane drop  PTSU operation on the inside versus outside shoulder  Dynamic versus static PTSU operation  Presence of a weaving section  PTSU lane presence during hours that PTSU does not operate The influence of the variables in the preceding list could not be reliably quantified using regression analysis because they (a) have a relatively small effect on crash frequency, (b) are correlated with other variables in the data, or (c) both. This finding does not rule out the possibility that these factors have an influence on freeway safety. It is possible that they have some influence, but it will likely require the use of a before-after study that can isolate the individual effect of one factor on crash frequency. A preliminary analysis of the data indicated that there were 11 sites with a sufficiently high or low crash rate, relative to other sites, that they were considered possible outliers. This possibility was confirmed when their prediction errors were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explanation could be found to explain their outlier tendencies, so the sites were removed from the data used for model estimation. Model Estimation The predictive model estimation process was based on a combined regression modeling approach, as discussed previously in this chapter. With this approach, the generalized regression model (as represented

70 by Equation 20 to Equation 54) represents separate models for each of the three site types. The generalized model is estimated using a database that includes data for all three site types. This approach is needed because several AFs are common to two or three of the regression models. The models were estimated using data from the states of Hawaii, Minnesota, Ohio, and Virginia. The data from the state of Georgia were reserved for model validation. The findings from model validation are provided in the next section. The results of the generalized regression model estimation are presented in Table 32. Estimation of this model focused on FI crash frequency. The Pearson χ2 statistic for the model is 691, and the degrees of freedom are 632 (= n − p = 654 − 22). As this statistic is less than χ2 0.05, 632 (= 692), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.62. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk2, as developed by Miaou (1996). The Rk2 for the estimated model is 0.76. The t-statistics listed in the last column of Table 32 indicate 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 1.96 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 1.96, 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 AF or SPF predictions are documented in the subsequent sections. An indicator variable for the state of Ohio was included in the regression model. The coefficient for this variable bohio,ast,fi is shown in Table 32. It is statistically significant. Its value indicates that the sites in Ohio have about 107 percent more FI crashes than in the other states. This difference cannot be explained by state-to-state differences based on the variables used in the model. It is likely due to factors associated with Ohio freeways that are (a) different from the other states and (b) not represented in the database (e.g., signing, pavement condition, weather, speed limit, crash reporting threshold). The indicator variable for Ohio was removed, and an indicator variable was included in the regression model for the sites located in the state of Minnesota. The coefficient for this variable was very small and not statistically significant. This finding is evidence that the regression model form is able to explain differences in crash occurrence among the states of Minnesota, Georgia, Hawaii, and Virginia. Note that an indicator variable for Minnesota or for Ohio can be included in the model, but both indicator variables cannot be included in the model. This condition is present because all of the facilities in Georgia, Hawaii, and Virginia are associated with an Ixxxx indicator variable. As a result, the inclusion of variables for both Minnesota and Ohio result in all states being associated with an indicator variable, which would render the model indeterminate.

71 Table 32. Predictive model estimation statistics, FI crashes, all site types, four states. Model Statistics Value R2 (Rk2): 0.62 (0.76) Scale Parameter φ: 1.06 Pearson χ2: 691 (χ20.05, 632 = 692) Observations n: 654 sites (4,169 FI crashes) Standard Deviation sp: ±1.33 crashes/yr Estimated Coefficient Values Variable Description Value Std. Error t-statistic bhc,ast,fi Horizontal curvature −4.656 1.895 −2.46 bs,ast,fi Lane and shoulder width −0.02482 0.03421 −0.73 bmw,ast,fi Median width −0.00561 0.00722 −0.78 brs,ast,fi Shoulder rumble strip presence −0.4631 0.213 −2.17 bx,fs,fi Distance from ramp on lane changing 14.37 5.253 2.74 bv,fs,fi Ramp AADT on lane changing −1.365 0.6807 −2.01 blen,ast,fi Speed-change lane length 0.0252 0.02136 1.18 bob,ast,fi Barrier offset 0.02432 0.01995 1.22 b0,fs,fi Freeway segment −4.439 0.5314 −8.35 b1,ast,fi Directional AADT volume 1.376 0.1385 9.93 b0,en,fi Ramp entrance speed-change lane −4.169 0.5735 −7.27 b2,en,fi Entrance ramp AADT volume −0.05207 0.02815 −1.85 b0,ex,fi Ramp exit speed-change lane −5.936 0.7301 −8.13 b3,ex,fi Number of lanes adjacent to speed-change lane 1.284 0.4302 2.98 btout,fs,fi Turnout presence −0.8305 0.2651 −3.13 bptsuOpen,ast,fi PTSU lane presence and operation 1.372 0.2156 6.36 bnearOpen,ast,fi Transition zone presence up- or downstream of PTSU 1.56 0.5893 2.65 bohio,ast,fi Location in Ohio 0.7299 0.1452 5.03 bMN35,ast,fi Location on I-35W in Minnesota 0.1161 0.2395 0.48 bHI01,ast,fi Location on I-H1 in Hawaii −0.2146 0.1641 −1.31 bV264,ast,fi Location on I-264 in Virginia 0.5354 0.1624 3.30 bV495,ast,fi Location on I-495 in Virginia −0.107 0.2325 −0.46 bVA66,ast,fi Location on I-66 in Virginia 0.4176 0.1403 2.98 bG400,ast,fi Location on S.R. 400 in Georgia n.a. n.a. n.a. bGA85,ast,fi Location on I-85 in Georgia n.a. n.a. n.a. K Inverse dispersion parameter for all site types 10.41 0.910 11.4 n.a. = not applicable (sites in Georgia were reserved for model validation). The inverse dispersion parameter shown in Table 32 is relatively large when compared to that for other models reported in the literature. This trend implies that there is less unexplained site-to-site variability in the predicted mean crash frequency for groups of similar sites. Reasons for this trend are offered in the following list.  The data is cross sectional such that each independent variable value is averaged for each site over the analysis period. In contrast, with panel data, each independent variable value is measured for each site for each year in the analysis period. Thus, cross-sectional data inherently have less variability than panel data when the year-to-year variability in the independent variables is largely random (Lord and

72 Park 2008). Of particular note is the random variability in the AADT data found in most highway safety databases, as discussed previously in this chapter.  The manual collection of geometric variables using aerial photographs for this project was found to significantly reduce the variability of these variables, relative to that found in the equivalent variables in highway safety databases obtained from state agencies. For example, the standard deviation of lane width in one state database is 2.0 feet, while that in the manually assembled data for the same segments is 0.6 foot.  As described in Chapter 4, the researchers defined each site in the database using the segmentation criteria cited in Chapter 18 of the HSM Supplement (AASHTO 2014). This process resulted in less random variation in crashes on a site-by-site basis. An additional reason for the relatively large inverse dispersion parameter value stems from this project’s development of a “full” model (i.e., one with multiple variables). This type of model inherently explains more variability than a simple model (e.g., an AADT-to-a-power model), and results in a larger inverse dispersion parameter (Mitra and Washington 2007). Model Validation Model validation was a two-step process. The first step required using the estimated models to predict the crash frequency for sites from a fifth state (i.e., Georgia). The objective of this step was to demonstrate the robustness of the model’s analytic form and its transferability to another state. The second step required comparing the estimated AFs with similar CMFs reported in the literature, where such information was available. The objective of this step was to demonstrate that the estimated AFs were consistent with previous research findings. The findings from the first step of the validation process are described in this section. Those from the second step are described in the next section. The first step of the validation process consisted of several tasks. The first task was to quantify the local calibration factor for the regression model—which would be the first step for any agency using a safety prediction method in the HSM (AASHTO 2010). The calibration factor CGA was computed as a multiplicative adjustment to the regression model prediction, following the local calibration method described in Part C of the HSM. The inverse dispersion parameter was also computed for the Georgia data as 3.61. This value is much smaller than that computed for the sites in the four states used to estimate the regression model. It indicates that the average crash frequency for the Georgia sites is more varied than that for the sites in the four states used to estimate the regression model. For this reason, the value of K = 3.61 was used to assess model fit to the Georgia data. This task produced a calibrated model (i.e., the regression models with the coefficients from Table 32 plus the calibration factor). The calibration factor value for the Georgia data was computed as 0.74. This factor is based on all 61 sites in Georgia and represents a mix of freeway segments (47 sites), ramp entrance speed-change lanes (8 sites), and ramp exit speed-change lanes (6 sites). Separate calibration factors were not computed for each of the three site-type-specific regression models because there were fewer than 30 sites of each type in the Georgia data (30 sites is the minimum number recommended by the HSM for model calibration). The second task was to apply the calibrated models to the Georgia data to compute the predicted average crash frequency for each segment or speed-change lane. The predicted crash frequency was then compared to the reported crash frequency for each site. The third task was to compute the fit statistics and assess the robustness of the calibrated model. These fit statistics are listed in Table 33. The Pearson χ2 statistic for the combined model (= 42.2) is less than χ20.05 (= 79), so the hypothesis that the model fits the validation data cannot be rejected.

73 Table 33. Predictive model validation statistics, FI crashes, all site types, four states. Scale Parameter φ Pearson χ2 Observations n χ20.05, n-1 Standard Deviation sp 0.67 40.2 61 sites 79 ±2.1 crashes/yr The fit of the calibrated models is shown in Figure 6, which compares the predicted and reported crash frequency in the validation database. The trend line shown represents a “y = x” line. A data point would lie on this line if it’s predicted and reported crash frequencies were equal. The data points shown represent the predicted and reported FI crash frequency for the sites used to validate the regression model. Each data point shown in Figure 6 represents the average predicted and average reported FI crash frequency for a group of 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 segment observations were used for model calibration. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 20 FI crashes during the study period. Figure 6. Predicted versus reported FI crash frequency using Georgia validation data. The findings from this validation activity indicate that the trends in the validation data are not significantly different from those in the estimation data. These findings suggest that the model form is reasonably robust and transferable to other freeway sites (when locally calibrated) for the prediction of FI crash frequency. Based on these findings, the data for all states were combined and used to re-estimate the regression model. The larger sample size associated with the combined database reduced the standard error of several regression coefficients. Bared and Zhang (2007) also used this approach in their development of predictive models for urban freeways. The re-estimated model results are described in the next section. Combined Model The data from the five study states were combined, and the generalized regression model was estimated a second time using the combined data. The regression coefficients for the generalized regression model

74 are described in the next subsection. The subsequent three sections describe the model fit to each of the three site-type-specific regression models that comprise the generalized model. Combined Model Estimation The results of the generalized regression model estimation are presented in Table 34. Estimation of this model focused on FI crash frequency. The Pearson χ2 statistic for the model is 746, and the degrees of freedom are 692 (= n − p = 717 − 25). As this statistic is less than χ2 0.05, 692 (= 754), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.64. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk2, as developed by Miaou (1996). The Rk2 for the estimated model is 0.77. A comparison of the coefficient values in Table 32 and Table 34 for common variables indicates that the coefficient values did not change significantly with the addition of the data reserved for model validation (i.e., the Georgia data) to the estimation database. This result was expected based on the results of the model validation task. A comparison of the t-statistics in Table 32 and Table 34 for common variables indicates that most of the coefficients in Table 34 are more reliably known than those in Table 32. This improvement is a result of adding the Georgia data to the estimation database. A focused examination of the effect of speed-change lane length revealed that the coefficient value for ramp entrance speed-change lanes was different from that for ramp exit speed-change lanes. Their values were found to produce trends for the two associated AFs that are consistent with the corresponding CMFs for speed-change lane length in the HSM Supplement (AASHTO 2014). The validity of the two coefficients is also supported by their relatively large t-statistics (albeit neither t-statistic exceeded 1.96). This ability to estimate unique coefficient values for each speed-change lane type is a benefit of adding the Georgia data to the estimation database.

75 Table 34. Final predictive model estimation statistics, FI crashes, all site types, five states. Model Statistics Value R2 (Rk2): 0.64 (0.77) Scale Parameter φ: 1.04 Pearson χ2: 746 (χ20.05, 692 = 754) Observations n: 717 sites (4,647 FI crashes) Standard Deviation sp: ±1.34 crashes/yr Estimated Coefficient Values Variable Description Value Std. Error t-statistic bhc,ast,fi Horizontal curvature −4.888 2.145 −2.28 bs,ast,fi Lane and shoulder width −0.04106 0.03303 −1.24 bmw,ast,fi Median width −0.00601 0.00703 −0.85 brs,ast,fi Shoulder rumble strip presence −0.5159 0.2059 −2.51 bx,fs,fi Distance from ramp on lane changing 14.34 5.519 2.60 bv,fs,fi Ramp AADT on lane changing −1.305 0.7194 −1.81 blen,en,fi Ramp entrance speed-change lane length 0.06901 0.04172 1.65 blen,ex,fi Ramp exit speed-change lane length 0.03229 0.02181 1.48 bob,ast,fi Barrier offset 0.01664 0.01923 0.87 b0,fs,fi Freeway segment −4.556 0.5278 −8.63 b1,ast,fi Directional AADT volume 1.406 0.1373 10.24 b0,en,fi Ramp entrance speed-change lane −4.250 0.5646 −7.53 b2,en,fi Entrance ramp AADT volume −0.04994 0.02485 −2.01 b0,ex,fi Ramp exit speed-change lane −5.374 0.6587 −8.16 b3,ex,fi Number lanes adjacent to exit speed-change lane 0.9301 0.3438 2.70 btout,fs,fi Turnout presence −0.7873 0.3023 −2.60 bptsuOpen,ast,fi PTSU lane presence and operation 1.318 0.2061 6.40 bnearOpen,ast,fi Transition zone presence up- or downstream of PTSU 1.305 0.5662 2.31 bohio,ast,fi Location in Ohio 0.7923 0.1407 5.63 bMN35,ast,fi Location on I-35W in Minnesota 0.1833 0.2408 0.76 bHI01,ast,fi Location on I-H1 in Hawaii −0.1872 0.1647 −1.14 bV264,ast,fi Location on I-264 in Virginia 0.6090 0.1625 3.75 bG400,ast,fi Location on S.R. 400 in Georgia −0.7881 0.1808 −4.36 bV495,ast,fi Location on I-495 in Virginia −0.04034 0.2336 −0.17 bVA66,ast,fi Location on I-66 in Virginia 0.4551 0.1402 3.25 bGA85,ast,fi Location on I-85 in Georgia 0.3702 0.2677 1.38 K Inverse dispersion parameter for all site types 10.10 0.845 11.9 Model for Freeway Segments The results of the freeway segment model estimation are presented in Table 35. Estimation of this model focused on FI crash frequency. The Pearson χ2 statistic for the model is 553, and the degrees of freedom are 501 (= n − p = 510 − 9). As this statistic is less than χ2 0.05, 501 (= 554), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.63. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk2, as developed by Miaou (1996). The Rk2 for the estimated model is 0.78.

76 Table 35. Final predictive model estimation statistics, FI crashes, freeway segment, five states. Model Statistics Value R2 (Rk2): 0.63 (0.78) Scale Parameter φ: 1.09 Pearson χ2: 553 (χ20.05, 501 = 554) Inverse Dispersion Parameter K: 10.10 Observations n: 510 sites (4,032 FI crashes) Standard Deviation sp: ±1.53 crashes/yr The coefficients in Table 34 were combined with Equation 21 to obtain the estimated SPF for freeway segments. The form of the model is described by the following equation. Equation 55 𝑁 , , 𝐿 , exp 4.556 1.406 ln 𝐴𝐴𝐷𝑇 /1000 where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section. The fit of the estimated model is shown in Figure 7. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported FI crash frequency for a group of 10 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 58 FI crashes during the study period. Figure 7. Predicted versus reported FI crashes on freeway segments. Model for Ramp Entrance Speed-Change Lanes The results of the ramp entrance speed-change lane model estimation are presented in Table 36. Estimation of this model focused on FI crash frequency. The Pearson χ2 statistic for the model is 101, and the degrees of freedom are 99 (= n − p = 106 − 7). As this statistic is less than χ2 0.05, 99 (= 123), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.35. The Rk2 for the estimated model is 0.52.

77 Table 36. Final predictive model estimation statistics, FI crashes, ramp entrance speed-change lane, five states. Model Statistics Value R2 (Rk2): 0.35 (0.52) Scale Parameter φ: 0.96 Pearson χ2: 101 (χ20.05, 99 = 123) Inverse Dispersion Parameter K: 10.10 Observations n: 106 sites (319 FI crashes) Standard Deviation sp: ±0.66 crashes/yr The coefficients in Table 34 were combined with Equation 44 to obtain the estimated SPF for ramp entrance speed-change lanes. The form of the model is described by the following equation. Equation 56 𝑁 , , 𝐿 , exp 4.250 1.406 ln 𝐴𝐴𝐷𝑇 /1000 0.04994 𝐴𝐴𝐷𝑇 /1000 where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section. The fit of the estimated model is shown in Figure 8. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported FI crash frequency for a group of 10 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 14 FI crashes during the study period. Figure 8. Predicted versus reported FI crashes at ramp entrance speed-change lanes. Model for Ramp Exit Speed-Change Lanes The results of the ramp exit speed-change lane model estimation are presented in Table 37. Estimation of this model focused on FI crash frequency. The Pearson χ2 statistic for the model is 111, and the degrees of freedom are 94 (= n − p = 101 − 7). As this statistic is less than χ2 0.05, 94 (= 118), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.38. The Rk2 for the estimated model is 0.40.

78 Table 37. Final predictive model estimation statistics, FI crashes, ramp exit speed-change lane, five states. Model Statistics Value R2 (Rk2): 0.38 (0.40) Scale Parameter φ: 1.11 Pearson χ2: 111 (χ20.05, 94 = 118) Inverse Dispersion Parameter K: 10.10 Observations n: 101 sites (296 FI crashes) Standard Deviation sp: ±0.71 crashes/yr The coefficients in Table 34 were combined with Equation 50 to obtain the estimated SPF for ramp exit speed-change lanes. The form of the model is described by the following equation. Equation 57 𝑁 , , 𝐿 , exp 5.374 1.406 ln 𝐴𝐴𝐷𝑇 /1000 0.9301 𝑛 2 . where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section. The fit of the estimated model is shown in Figure 9. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported FI crash frequency for a group of 10 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 12 FI crashes during the study period. Figure 9. Predicted versus reported FI crashes at ramp exit speed-change lanes. Estimated Adjustment Factors Several AFs were estimated in conjunction with the SPFs. All of them were estimated using FI crash data. Collectively, they describe the relationship between various geometric factors and crash frequency. These AFs are described in this section and, where possible, compared with the findings from previous research as a means of model validation.

79 Horizontal Curve AF The estimated horizontal curve AF is described using the following equation. Equation 58 𝐴𝐹 , , 1.0 exp 4.888 5,730𝑅 The radii used to estimate this AF range from 1,430 to 24,170 feet. The base condition for this AF is an uncurved (i.e., tangent) segment. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The AF is shown in Figure 10 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The trend line based on Equation 58 is shown to produce smaller AF values than that obtained from the HSM supplement CMF. This result likely stems from the fact that the sites used to estimate Equation 58 are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed than the sites used to estimate the HSM CMF. Figure 10. Estimated horizontal curve AF for FI crashes. Lane Width AF The lane width AF is described using the following equation. Equation 59 𝐴𝐹 , , exp 0.04106 min 𝑊 , 13 12 The lane width used in this AF is an average for all through lanes on the segment. The AF is discontinuous, breaking at a lane width of 13 feet. The AF value does not change for lane width in excess of 13 feet. The widths used to estimate this AF range from 10.5 to 14.4 feet. The base condition for this AF is a 12-foot lane width. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The lane width AF is shown in Figure 11 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The two trend lines are in good agreement on the relationship between lane width and AF value.

80 Figure 11. Estimated lane width AF for FI crashes. Inside Shoulder Width AF The inside shoulder width AF is described using the following equation. Equation 60 𝐴𝐹 , , exp 0.04106/𝑛 min 𝑊 , , 12 6 The shoulder width used in this AF represents the paved width. The AF is discontinuous, breaking at a shoulder width of 12 feet. The AF value does not change for shoulder width in excess of 12 feet. This break-point was established based on trends in the outside shoulder width data. It is assumed that this trend also extends to inside shoulder width. The widths used to estimate this AF range from 0.7 to 11.0 feet. The number of through lanes range from 2 to 7. The base condition for this AF is a 6-foot inside shoulder width. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 60 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that shoulder width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the shoulder is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by shoulder presence. The inside shoulder width AF is shown in Figure 12 using two thick trend lines. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The three trend lines are in good agreement on the relationship between inside shoulder width and AF value.

81 Figure 12. Estimated inside shoulder width AF for FI crashes. Inside Shoulder Rumble Strip AF The inside shoulder rumble strip AF is described using the following equation. Equation 61 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.5159/𝑛 The proportion Pir represents the proportion of the site length with rumble strips present on the inside shoulder. It is computed by measuring the length of roadway with rumble strips on the inside shoulder and dividing by the site length. This AF is applicable to values of Pir that range from 0.0 to 1.0. The number of through lanes range from 2 to 7. The base condition for this AF is “no rumble strips present on the inside shoulder” (i.e., Pir = 0.0). This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed- change lanes. The variable for “number of through traffic lanes” n is included in Equation 61 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that rumble strip presence has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the rumble strips are laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by rumble strip presence. The inside shoulder rumble strip AF is shown in Figure 13 using two thick trend lines. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The three trend lines are in good agreement on the relationship between inside shoulder width and AF value.

82 Figure 13. Estimated inside shoulder rumble strip AF for FI crashes. Median Width AF The estimated median width AF is described using the following equation. Equation 62 𝐴𝐹 | , , 1.0 𝑃 exp 0.00601/𝑛 𝑊 48 𝑃 exp 0.00601/𝑛 min 𝑊 , 2 𝑊 48 with Equation 63 𝑊 𝑊 𝑊 , 𝑊 , 𝑊 , 𝐼 , , 𝑊 , 𝐼 , , The median width used in this AF is an average for the site. The AF is derived to be applicable to a site that has median barrier present along some portion of the site. The variable Wum represents width of the middle portion of the median that excludes the width of the inside shoulder in each travel direction. The variable Pib represents the proportion of the site length with a barrier present in the median. It is computed by measuring the length of roadway with median barrier present and dividing by the site length. The variable Wicb represents the distance the barrier is offset from the edge of the inside shoulder. More detailed guidance for computing the variables Pib and Wicb is provided in Section 18.7.3 of the HSM Supplement (AASHTO 2014). This AF is applicable to values of median width that range from 5 to 90 feet. The number of through lanes range from 2 to 7. The variable Wicb ranges from 0.75 to 20 feet. The variable Pib ranges from 0.0 to 1.0. The base condition for this AF is a 60-foot median width and an inside shoulder width of 6 feet (i.e., 48 = 60 – 6 – 6). This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 62 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that median width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the median is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by median width.

83 The median width AF is shown in Figure 14 using the thick, solid trend line labeled “proposed, 3 lanes; no barrier.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM, no barrier.” These two trend lines are in good agreement on the relationship between median width and AF value. The slope of the thick, solid trend line decreases with an increase in the number of lanes. The other two trend lines shown are discussed in the next section. Figure 14. Estimated median width AF for FI crashes. Median Barrier AF The estimated median barrier AF is described using the following equation. Equation 64 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.01664 𝑛/𝑊 The variables Pib and Wicb were described in the previous section. This AF is applicable to a number of through lanes ranging from 2 to 7. The variable Wicb ranges from 0.75 to 20 feet. The variable Pib ranges from 0.0 to 1.0. The base condition for this AF is “no barrier present in the median” (i.e., Pib = 0.0). This AF is applicable to freeway segments, ramp entrance speed- change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 64 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a multiplier) in the AF suggests that median barrier has more influence on crash frequency when there are more lanes. It is unclear whether this relationship is a result of (a) a correlation between the AF variables and other site characteristics or (b) an indication that non-recoverable lane departures are more prevalent on freeways with more lanes. This topic should be a subject of future research. The median barrier AF is shown in Figure 14 using the thick, solid trend line labeled “proposed, 3 lanes; barrier in center of median.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM Supplement (AASHTO 2014), barrier in center of median.” These two trend lines are in good agreement on the relationship between median width and AF value. The slope of the thick, solid trend line decreases with an increase in the number of lanes (albeit the trend line becomes more concave with an increase in lanes as a result of the aforementioned “lane” multiplier).

84 Lane Change AF The estimated lane change AF is described for freeway segments using the following equation. Equation 65 𝐴𝐹 , , 1.0 exp 14.34 𝑋 , 1.305 ln 𝐴𝐴𝐷𝑇 , /100014.34 𝐿 , 1.0 exp 14.34 𝐿 , 1.0 exp 14.34 𝑋 , 1.305 ln 𝐴𝐴𝐷𝑇 , /100014.34 𝐿 , 1.0 exp 14.34 𝐿 , This AF is based on five input variables. Two of these variables (i.e., Xb,ent and AADTb,ent) describe the distance to (and volume of) the entrance ramp just upstream of the subject segment. Two more of these variables (i.e., Xe,ext and AADTe,ext) describe the distance to (and volume of) the exit ramp just downstream of the subject segment. These four variables are described in more detail in Chapter 18, Figure 18-8, of the HSM Supplement (AASHTO 2014). This AF is applicable to ramp AADTs in the range of 450 to 30,680 veh/day. There is no upper limit on the distance variables Xb,ent and Xe,ext. The base condition for this AF is no upstream entrance ramp or downstream exit ramp (i.e., Xb,ent > 1.0 mile and Xe,ext > 1.0 mile). This AF is applicable to freeway segments. It is applicable to any segment in the vicinity of an upstream entrance ramp, a downstream exit ramp, or both. The AF is also applicable to a weaving section. To illustrate this AF, consider a 0.5-mile section of urban freeway with three through traffic lanes in the subject travel direction. The section consists of five segments that are each 0.1 miles in length. There is an interchange at one end of the section. The distance to the next interchange is sufficiently large that its ramp traffic has no influence on segment lane change activity. Under this scenario, Xe,ext > 1.0 mile such that the second term of Equation 65 equals 1.0. The first term of this equation is used to compute the corresponding AF value. The AF for each of the five segments is plotted in Figure 15. The example 0.5- mile freeway section is shown in plan-view in the upper left corner of this figure. a. Ramp AADT is 1,000 veh/day. b. Ramp AADT is 6,000 veh/day. Figure 15. Lane change AF as a function of distance from ramp gore – FI crashes. The AF values obtained from Equation 65 are shown in Figure 15 using a bar with a solid outline. Those obtained from the lane change CMF in the HSM Supplement (AASHTO 2014) are shown with a dashed outline. For the segment that starts at the ramp gore and extends 0.1 miles (i.e., segment center is x = 0.05 miles), the AF values are 1.33 and 1.03 for ramp AADTs of 1,000 and 6,000 veh/day, respectively. The AF values for the next segment are 1.08 and 1.01. AF values continue to decrease for each

85 subsequent segment. This decline in AF value reflects the decreasing number of lane changes with increasing distance from the ramp gore. The AF values from Equation 65 are shown in Figure 15 to be smaller than those from the HSM CMF. This result likely stems from the fact that the sites used to estimate Equation 65 are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed. Consider a 0.5-mile section of urban freeway with three through traffic lanes in the subject travel direction. The section consists of five segments that are each 0.1 miles in length. There is an interchange at each end of the section. Under this scenario, both terms of Equation 65 are used to compute the AF value. The AF for each of the five segments is plotted in Figure 16. The example 0.5-mile freeway section is shown in the upper left corner of this figure. a. Ramp AADT is 1,000 veh/day. b. Ramp AADT is 6,000 veh/day. Figure 16. Lane change AF for segments between a pair of interchanges – FI crashes. For the segment that starts at the ramp gore and extends 0.1 miles (i.e., segment center is x = 0.05 miles), the AF values are 1.33 and 1.03 for ramp AADTs of 1,000 and 6,000 veh/day, respectively. The AF values for the next segment are 1.08 and 1.01. The AF values continue to decrease until the middle segment is reached, and then they start to increase as the segments get closer to the next interchange. The regression coefficient associated with the ramp AADT term in Equation 65 is negative, which may seem counterintuitive at first glance. It indicates that the lane change AF is larger for segments associated with lower volume ramps. This trend may be explained by the fact that high-volume ramp flows tend to dominate the traffic stream such that a large portion of the traffic stream is changing lanes and all drivers are more aware of these maneuvers. Outside Shoulder Width AF The outside shoulder width AF is described for freeway segments using the following equation. Equation 66 𝐴𝐹 , , exp 0.04106/𝑛 min 𝑊 , 12 10 The shoulder width used in this AF represents the paved width. The AF is discontinuous, breaking at a shoulder width of 12 feet. The AF value does not change for shoulder width in excess of 12 feet. This breakpoint was established based on trends in the data. The widths used to estimate this AF range from 0.7 to 14.0 feet. The number of through lanes range from 2 to 7. The base condition for this AF is a 10- foot outside shoulder width. This AF is applicable to freeway segments.

86 The variable for “number of through traffic lanes” n is included in Equation 66 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that shoulder width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the shoulder is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by shoulder presence. The outside shoulder width AF is shown in Figure 17 using two thick trend lines. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The three trend lines are in reasonable agreement on the relationship between inside shoulder width and AF value. Figure 17. Estimated outside shoulder width AF for FI crashes. Outside Shoulder Rumble Strip AF The inside shoulder rumble strip AF is described for freeway segments using the following equation. Equation 67 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.5159/𝑛 The proportion Por represents the proportion of the site length with rumble strips present on the outside shoulder. It is computed by measuring the length of roadway with rumble strips on the outside shoulder and dividing by the site length. This AF is applicable to values of Por that range from 0.0 to 1.0. The number of through lanes range from 2 to 7. The base condition for this AF is “no rumble strips present on the outside shoulder” (i.e., Por = 0.0). This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 67 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that rumble strip presence has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the rumble strips are laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by rumble strip presence. The outside shoulder rumble strip AF is effectively the same as the inside shoulder rumble strip AF. Hence, the trends shown in Figure 13 are the same for the outside shoulder rumble strip AF.

87 Outside Clearance AF The estimated outside clearance AF is described for freeway segments using the following equation. Equation 68 𝐴𝐹 | , , 1.0 𝑃 exp 0.00601/𝑛 𝑊 𝑊 , 𝐼 , , 𝑊 20 𝑃 exp 0.00601/𝑛 𝑊 20 The clear zone width Whc used in this AF is an average for the site. The AF is derived to be applicable to a segment that has outside (roadside) barrier present along some portion of the site. The variable Whc represents the width of the clear zone, as measured from the edge of traveled way. The variable Pob represents the proportion of the site length with a barrier present on the outside (roadside); it is computed by measuring the length of roadway with outside (roadside) barrier present and dividing by the segment length. The variable Wocb represents the distance the barrier is offset from the edge of the inside shoulder. More detailed guidance for computing the variables Pob and Wocb is provided in Section 18.7.3 of the HSM Supplement (AASHTO 2014). This AF is applicable to values of clear zone width that range from 10 to 30 feet. The number of through lanes range from 2 to 7. The variable Wocb ranges from 1.0 to 20 feet. The variable Pob ranges from 0.0 to 1.0. The base condition for this AF is a 30-foot clear zone width and an outside shoulder width of 10 feet (i.e., 20 = 30 – 10). This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 68 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that clear zone width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the clear zone is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by clear zone width. The outside clearance AF is shown in Figure 18 using the thick, solid trend line labeled “proposed, 3 lanes; no barrier.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM Supplement (AASHTO 2014), urban; no barrier.” These two trend lines are in good agreement on the relationship between clear zone width and AF value. The slope of the thick, solid trend line decreases with an increase in the number of lanes. The other trend lines shown are discussed in the next section. Figure 18. Estimated outside clearance AF for FI crashes.

88 Outside Barrier AF The estimated median barrier AF is described for freeway segments using the following equation. Equation 69 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.01664 𝑛/𝑊 The variables Pob and Wocb were described in the previous section. This AF is applicable to a number of through lanes ranging from 2 to 7. The variable Wocb ranges from 1.0 to 20 feet. The variable Pob ranges from 0.0 to 1.0. The base condition for this AF is “no barrier present on the outside (roadside)” (i.e., Pob = 0.0). This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 69 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a multiplier) in the AF suggests that outside (roadside) barrier has more influence on crash frequency when there are more lanes. It is unclear whether this relationship is a result of (a) a correlation between the AF variables and other site characteristics or (b) an indication that non-recoverable lane departures are more prevalent on freeways with more lanes. This topic should be a subject of future research. The outside barrier AF is shown in Figure 18 using the thick, solid trend line labeled “proposed, 3 lanes; roadside has barrier 2-ft from shoulder edge for 100% of segment.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM Supplement (AASHTO 2014), roadside has barrier for 2-ft from shoulder edge for 100% of segment.” The AF from Equation 69 is 1.065, while the CMF from the HSM is 1.08—a difference of 1.5 percent. This difference is relatively small considering the uncertainty associated the coefficients and other model variables. The thick, solid trend line decreases (i.e., drops) slightly with an increase in the number of lanes. Ramp Entrance AF The estimated ramp entrance AF is described using the following equation. Equation 70 𝐴𝐹 , , exp 0.06901 1𝐿 1 0.142 The ramp entrance length Len is measured from the gore to the taper point of the speed-change lane, as defined in Chapter 18 (Figure 18-3) of the HSM Supplement (AASHTO 2014). The length values used to estimate this AF range from 0.06 to 0.32 miles. The base condition for this AF is a ramp entrance length of 0.142 miles. This value represents the median ramp entrance length in the estimation database. This AF is applicable to ramp entrance speed-change lanes. The ramp entrance AF is shown in Figure 19 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. Also shown is a CMF from Chapter 15 of the HSM (AASHTO 2010). To facilitate this comparison, the HSM CMFs were converted mathematically to a base length of 0.142 miles (i.e., 750 feet). The three trend lines are in reasonable agreement on the relationship between ramp entrance length and AF value. The proposed AF is typically between the two CMF values for all but the shortest speed-change lanes.

89 Figure 19. Estimated ramp entrance AF for FI crashes. Ramp Exit AF The estimated ramp exit AF is described for freeway segments using the following equation. Equation 71 𝐴𝐹 , , exp 0.03229 1𝐿 1 0.071 The ramp exit length Lex is measured from the gore to the taper point of the speed-change lane, as defined in Chapter 18 (Figure 18-3) of the HSM Supplement (AASHTO 2014). The length values used to estimate this AF range from 0.02 to 0.27 miles. The base condition for this AF is a ramp exit length of 0.071 miles. This value represents the median ramp exit length in the estimation database. This AF is applicable to ramp exit speed-change lanes. The ramp exit AF is shown in Figure 20 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. To facilitate this comparison, the HSM CMF was converted mathematically to a base length of 0.071 miles (i.e., 375 feet). The two trend lines are in reasonable agreement on the relationship between ramp entrance length and AF value.

90 Figure 20. Estimated ramp exit AF for FI crashes. Turnout Presence AF The estimated turnout presence AF is described for freeway segments using the following equation. Equation 72 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.7873/𝑛 with, Equation 73 𝑃 𝐿 , /𝐿 , The proportion Pturnout represents the proportion of the segment length that is adjacent to a turnout. It is computed by measuring the length of roadway adjacent to a turnout and dividing by the segment length. If the turnout extends beyond one or both of the segment boundaries, then only that portion of the turnout that lies within the segment is measured. This AF is applicable to values of Pturnout that range from 0.0 to 1.0. The number of through lanes range from 2 to 7. The base condition for this AF is “no turnout present” (i.e., Pturnout = 0.0). This AF is applicable to freeway segments with PTSU operation during some portion of the typical day. The variable for “number of through traffic lanes” n is included in Equation 72 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that turnout presence has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the turnout (like a shoulder) is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by turnout presence. The turnout AF is shown in Figure 21 using three thick trend lines; one line is shown for each of two, four, and six through traffic lanes. A review of the literature did not reveal any publications wherein the safety benefit of turnout presence on a PTSU facility was quantified. However, the t-statistic associated with the regression coefficient (i.e., t = −2.60) indicates that the coefficient is significantly different from 0.0 and implies that turnout presence does reduce crash frequency.

91 Figure 21. Estimated turnout presence AF for FI crashes. As noted in the discussion of Table 12, Jenior et al. (2016) recommend that turnouts should be constructed at desirably one-half mile intervals. This spacing combined with a turnout length of 800 feet (200-foot body plus a 300-foot entry taper and a 300-foot exit taper) corresponds to an average “turnout length as a proportion of segment length” of 0.30. Based on this proportion, AF values of 0.90, 0.95, and 0.96 are obtained from Figure 21 for two, four, and six lanes, respectively. PTSU Operation AF The estimated PTSU operation AF is described by the following equations. Equation 74 𝐴𝐹 | , , 1.0 𝑃 , exp 𝑓 , 𝑃 , exp 𝑓 , 𝑓 , , 𝑓 , with Equation 75 𝑓 , 0.04106/𝑛 min 𝑊 , , 12 𝐼 Equation 76 𝑓 , 0.04106 min 𝑊 , , 13 12 𝐼 Equation 77 𝑓 , 1.318 𝐼 Equation 78 𝑓 , , 1.305 1 𝐼 𝑃 , The variable Wptsu represents the average width of the shoulder that is allocated to vehicular traffic use (i.e., as an additional travel lane) during one or more hours of the typical day. If this width varies along the length of the site (e.g., as in the case where it the PTSU lane is tapered to add or drop the PTSU lane), then an average width is used for the subject site. The proportion Pt,ptsu represents the proportion of time during the average day that PTSU operates in the vicinity of (or at) the subject site. It has a nonzero value if (a) the site has a full-width PTSU lane or the tapered portion of a PTSU lane, (b) the site does not have a PTSU lane but a portion of it is within 0.152 miles of a PTSU lane (i.e., just upstream or downstream), or (c) the site is a speed-change lane site

92 and the PTSU lane effectively continues through the site even though it may not be marked as an exclusive PTSU lane (because ramp traffic is permitted to cross this area to enter or exit the freeway through lanes). This variable is computed using Equation 10. Cases “a” and “c” are referred herein to as a PTSU lane and case “b” is referred to as a PTSU transition zone. Cases “a” and “c” are addressed by Equation 77. Case “b” is addressed by Equation 78. The variable IptsuLane is an indicator variable that has a value of 1.0 if the site has a PTSU lane (i.e., Wptsu > 0.0). This AF is applicable to values of Pt,ptsu that range from 0.0 to 0.45. The number of through lanes range from 2 to 7. The width of the PTSU lane Wptsu ranges from 3.5 to 16.8 feet. The base condition for this AF is “no PTSU operation during any hour of the day” (i.e., Pt,ptsu = 0.0) and PTSU lane width Wptsu equal to 0.0 feet. This AF is applicable to sites with a nonzero value of the variable Pt,ptsu. The PTSU presence AF is shown in Table 38. AF values are listed for various combinations of PTSU turnout spacing, lane width, proportion time PTSU operating, and number of lanes. The turnout spacing values in the table are computed using the product of the PTSU operation AF and the turnout presence AF with a turnout spacing of 0.5 miles. The AF values in the upper two portions of Table 38 are shown to increase with an increase in the proportion time the PTSU is operating, a reduction in PTSU lane width, and an increase in number of lanes. The AF values decrease when turnouts are provided. The AF value listed in the table for the case where the proportion time PTSU operating equals 0.0 corresponds to the case where the width of the PTSU lane is effectively serving as additional shoulder width. The sites in the estimation database indicate that typical values of the conditions identified in Table 38 include 0.5-mile turnout spacing, a proportion time PTSU operating of 0.2, an 11-foot PTSU lane width, and four lanes. For these values, the AF value is 1.41. This AF value suggests that PTSU presence can increase the annual FI crash frequency by 41 percent for typical conditions. Margiotta et al. (2014, Table 6) evaluated the change in safety associated with PTSU implementation on two freeways in the United States. Specifically, they computed crash rates before and after PTSU operation was implemented on I-35W in Minnesota and on US 2 in Washington. I-35W was reported by Margiotta et al. (2014) to provide a PTSU lane with a width between 17 and 19 feet, operating for 9 hours each weekday, and where the subject travel direction served four traffic lanes. The presence of turnouts was not indicated. The change in total crash rate for I-35W was computed as a ratio (after + before) having the value of 1.16. The corresponding AF value from Table 38 for these conditions is 1.61. This AF value is somewhat larger than the crash rate ratio. It is possible that I-35W had other conditions (e.g., frequent turnouts) or changes (e.g., add a general-purpose lane) during the after period that accounts for the relatively small crash rate ratio. US 2 was reported by Margiotta et al. (2014) to provide a PTSU lane with a width of 14 feet, operating for 4 hours each weekday, and where the subject travel direction served two traffic lanes. The presence of turnouts was not indicated. The change in total crash rate for US 2 was computed as a ratio having the value of 1.20. The corresponding AF value from Table 38 for these conditions is 1.12. This AF value is slightly smaller than the crash rate ratio but still reasonably close given that information about other site characteristics that influence safety are not considered in the computation of the AF value. A review of the literature did not reveal any additional publications wherein the safety benefit of PTSU operation was reliably quantified. However, the t-statistics associated with the two regression coefficients (i.e., t for bptsuOpen is 6.40 and t for bnearOpen is 2.31) indicate that both coefficients are significantly different from 0.0. This finding implies that PTSU operation increases FI crash frequency.

93 Table 38. Estimated PTSU operation AF for FI crashes. PTSU Type PTSU Lane Width (feet) Proportion Time PTSU Operatinga AF Value by Number of Lanes 2 4 6 PTSU lane (no turnouts) 11 0.0 0.80 0.89 0.93 0.1 1.11 1.19 1.22 0.2 1.42 1.49 1.52 0.3 1.73 1.79 1.82 0.4 2.04 2.09 2.11 12 0.0 0.78 0.88 0.92 0.1 1.08 1.17 1.20 0.2 1.37 1.45 1.48 0.3 1.67 1.74 1.77 0.4 1.96 2.03 2.05 PTSU lane (turn-out every 0.5 miles) 11 0.0 0.72 0.85 0.89 0.1 1.00 1.13 1.18 0.2 1.28 1.41 1.46 0.3 1.56 1.70 1.75 0.4 1.84 1.98 2.04 12 0.0 0.71 0.84 0.89 0.1 0.97 1.11 1.16 0.2 1.24 1.38 1.43 0.3 1.51 1.65 1.70 0.4 1.77 1.92 1.97 PTSU transition zoneb Any 0.0 1.00 1.00 1.00 0.1 1.11 1.11 1.11 0.2 1.22 1.22 1.22 0.3 1.33 1.33 1.33 0.4 1.43 1.43 1.43 a Proportion time PTSU operating = (weekday hours × 5/7 + weekend hours × 2/7)/24 b Segment length is 0.27 miles. Sensitivity Analysis The relationship between crash frequency and traffic demand for freeway segments is illustrated in Figure 22 for a 1-mile freeway segment. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 22a correspond to the proposed model. The trends in Figure 22b correspond to the freeway segment model in the HSM Supplement (AASHTO 2014). The HSM model results were computed using a two-way AADT that is twice that of the directional AADT. The predicted crash frequency from the HSM model was then divided by two to obtain a result that is comparable to the proposed models. A comparison of the trends in Figure 22a with those in Figure 22b indicate that the proposed model predicts about as many crashes as the HSM model for a common directional AADT.

94 a. Proposed models. b. HSM models. Figure 22. Estimated freeway segment model for FI crashes. The relationship between crash frequency and traffic demand for ramp entrance speed-change lanes is illustrated in Figure 23. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 23a correspond to the proposed model. The trends in Figure 23b correspond to the corresponding model in the HSM Supplement (AASHTO 2014). A comparison of the trends in Figure 23a with those in Figure 23b indicate that the proposed model predicts about as many crashes as the HSM model for a common directional AADT. The sensitivity to number-of-lanes was examined in the estimation database, but it was not found to provide an improved fit to the data. a. Proposed models. b. HSM models. Figure 23. Estimated ramp entrance model for FI crashes. The relationship between crash frequency and traffic demand for ramp exit speed-change lanes is illustrated in Figure 24. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 24a correspond to the proposed model. The trends in Figure 24b correspond to the corresponding model in the HSM Supplement (AASHTO 2014). A comparison of the trends in Figure 24a with those in Figure 24b indicate that the proposed model predicts notably more

95 crashes than the HSM model for volumes in excess of about 70,000 veh/day. This difference was investigated for explanation by other model variables and by issues with the reported crash data; however, nothing notable was found. The difference likely stems from the fact that the sites used to estimate the proposed models are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed than those in the HSM database. a. Proposed models. b. HSM models. Figure 24. Estimated ramp exit model for FI crashes. Property-Damage-Only Crash Frequency Prediction Model This section describes the development of predictive model equations based on PDO crash data. One equation is applicable to freeway segments. A second equation is applicable to ramp entrance speed- change lanes. A third equation is applicable to ramp exit speed-change lanes. This section consists of four subsections. The first subsection describes the structure of the predictive equations as used in the regression analysis. The second subsection describes the regression statistics for each of the estimated equations and the proposed safety prediction models. The third section describes the estimated AFs. The last subsection provides a sensitivity analysis of the predictive model equations over a range of traffic demands. Model Development The proposed predictive model equations and the methods used to develop them were described in the preceding section Fatal-and-Injury Crash Frequency Prediction Model. These equations were shown as Equation 20 to Equation 54. In this section, the subscript “fi” is changed to “pdo” in each equation. The base conditions in the FI models are also applicable to the PDO models. The regression model is generalized to address three site types: freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The generalized form shows all the AFs in the model. Indicator variables are used to determine which AFs are applicable to each observation in the database based on its associated site type. All of the models described in this section are developed to predict crash frequency for all crash types combined (regardless of manner of collision or number of vehicles involved). For this reason, there is no subscript on the variables to denote the applicable crash type. The coefficients in the generalized regression model were estimated using the PDO crash data. However, the coefficients for the lane change AF (i.e., bx,fs,pdo and bv,fx,pdo) were not valid. As a result,

96 these coefficients and the associated lane change AF were removed from the model. Similarly, the coefficient for the inside and outside shoulder rumble strip AFs was not statistically significant and very near to 0.0, so the coefficient and the AFs were removed from the model. The coefficient for median width (i.e., bmw,ast,pdo) became illogically large, while the coefficient for lane and shoulder width (bs,ast,pdo) became smaller than that for median width and outside clearance. This reversal of magnitude and trend implied that there was more safety benefit to widening the median (or increasing the clear zone width) than increasing the lane or shoulder width. The implication was contrary to the trend in the FI data and in the HSM Supplement (AASHTO 2014). After some additional analysis, it was found that a good and logical fit to the data could be achieved using the following form for the median width AF and the outside clearance AF. Equation 79 𝐴𝐹 | , , 1.0 𝑃 exp 𝑏 , , /6.7 /𝑛 𝑊 48 𝑃 exp 𝑏 , , /6.7 /𝑛 min 𝑊 , 2 𝑊 48 Equation 80 𝐴𝐹 | , , 1.0 𝑃 exp 𝑏 , , /6.7 /𝑛 𝑊 𝑊 , 𝐼 , , 𝑊 20 𝑃 exp 𝑏 , , /6.7 /𝑛 𝑊 20 The regression coefficient bmw,ast,pdo in the median width AF (and the outside clearance AF) was replaced with “bs,ast,pdo/6.7.” The constant “6.7” was based on the ratio of the two regression coefficients from the FI regression model (i.e., 6.7 = bs,ast,fi /bmw,ast,fi). This change in the form of the two AFs forced the regression routine to find a fit to the data that ensured a logical safety relationship between the median width, clear zone width, lane width, and shoulder width. A preliminary analysis of the data indicated that there were 20 sites with a sufficiently high or low reported crash rate, relative to other sites, that they were considered possible outliers. This possibility was confirmed when their prediction errors were found to be very large. The independent variables for these sites were checked to confirm that they were reasonable. Ultimately, no explanation could be found to explain their outlier tendencies, so the sites were removed from the data used for model estimation. Model Estimation The predictive model estimation process was based on a combined regression modeling approach, as discussed previously in this chapter. With this approach, the three regression models that comprise the generalized regression model (as represented by Equation 20 to Equation 54) were estimated using a database that includes data for all three site types. This approach is needed because several AFs are common to two or three of the regression models. The models were estimated using data from the states of Georgia, Hawaii, Minnesota, Ohio, and Virginia. The regression coefficients for the generalized regression model are described in the Combined Model Estimation subsection below. The subsequent three sections describe the model fit to each of the three site-type-specific regression models that comprise the generalized model. Combined Model Estimation The results of the generalized regression model estimation are presented in Table 39. Estimation of this model focused on PDO crash frequency. The Pearson χ2 statistic for the model is 737, and the degrees of freedom are 687 (= n − p = 708 − 21). As this statistic is less than χ2 0.05, 687 (= 749), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.69. The Rk2 for the estimated model is 0.75.

97 Table 39. Final predictive model estimation statistics, PDO crashes, all site types, five states. Model Statistics Value R2 (Rk2): 0.69 (0.75) Scale Parameter φ: 1.04 Pearson χ2: 737 (χ20.05, 687 = 749) Observations n: 708 sites (11,173 PDO crashes) Standard Deviation sp: ±2.98 crashes/yr Estimated Coefficient Values Variable Description Value Std. Error t-statistic bhc,ast,pdo Horizontal curvature −5.469 2.618 −2.09 bs,ast,pdo Lane, shoulder, and median width −0.02725 0.02512 −1.09 blen,en,pdo Ramp entrance speed-change lane length 0.09908 0.03657 2.71 blen,ex,pdo Ramp exit speed-change lane length 0.04327 0.02104 2.06 bob,ast,pdo Barrier offset 0.01618 0.01755 0.92 b0,fs,pdo Freeway segment −3.133 0.4466 −7.02 b1,ast,pdo Directional AADT volume 1.295 0.1212 10.68 b0,en,pdo Ramp entrance speed-change lane −3.043 0.4774 −6.38 b2,en,pdo Entrance ramp AADT volume −0.02018 0.02083 −0.97 b0,ex,pdo Ramp exit speed-change lane −3.413 0.5329 −6.41 b3,ex,pdo Number of lanes adjacent to speed-change lane 0.598 0.2765 2.16 btout,fs,pdo Turnout presence −1.091 0.1492 −7.31 bptsuOpen,ast,pdo PTSU lane presence and operation 1.567 0.1808 8.67 bnearOpen,ast,pdo Transition zone presence up- or downstream of PTSU 1.515 0.4905 3.09 bohio,ast,pdo Location in Ohio 0.6849 0.09521 7.19 bMN35,ast,pdo Location on I-35W in Minnesota 0.3911 0.2255 1.73 bHI01,ast,pdo Location on I-H1 in Hawaii −1.563 0.1661 −9.40 bV264,ast,pdo Location on I-264 in Virginia −0.01825 0.1431 −0.13 bG400,ast,pdo Location on GA 400 in Georgia −0.8725 0.1536 −5.68 bV495,ast,pdo Location on I-495 in Virginia 0.2476 0.2151 1.15 bVA66,ast,pdo Location on I-66 in Virginia 0.2161 0.1300 1.66 bGA85,ast,pdo Location on I-85 in Georgia 0.5901 0.2511 2.35 K Inverse dispersion parameter for all site types 9.57 0.627 15.3 The t-statistics listed in the last column of Table 39 indicate 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 1.96 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 1.96, it was decided that the variable was important to the model, and its trend was found to be logical and consistent with either (a) that for the FI regression model or (b) 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 AF or SPF predictions are documented in the subsequent sections. Model for Freeway Segments The results of the freeway segment model estimation are presented in Table 40. Estimation of this model focused on PDO crash frequency. The Pearson χ2 statistic for the model is 557, and the degrees of freedom are 499 (= n − p = 505 − 6). This statistic is not less than χ2 0.05, 499 (= 552); however, it is less

98 than χ2 0.03, 499 (= 560). Thus, there is some evidence that the data are not fitting the model or that they are not distributed in a manner consistent with the negative binomial distribution. A check of the model fit to the data (as discussed in the subsequent paragraphs) indicated a good fit, so the relatively large Pearson χ2 statistic is likely an indication that there are some anomalies in the PDO crash data. A review of these data indicated that there was some “clustering” of crashes at individual mileposts for some of the Minnesota sites. This cluster was indicated by a large number of crashes being assigned to a common milepost—larger than could be explained by site characteristics and inconsistent with the more typical distribution of crashes along the freeway. Some of the affected sites were removed as outliers (as noted previously). However, a few of these sites remained in the estimation database and, while they were not sufficiently outstanding to be considered outliers, they resulted in a value that is higher than expected for the Pearson χ2 statistic. In spite of this finding, the model fit to the data was found to be reasonably good and the model predictions were found to be unbiased, so these few sites were retained in the database. The R2 for the model is 0.68. The Rk2 for the estimated model is 0.77. Table 40. Final predictive model estimation statistics, PDO crashes, freeway segment, five states. Model Statistics Value R2 (Rk2): 0.68 (0.77) Scale Parameter φ: 1.10 Pearson χ2: 557 (χ20.05, 499 = 552) Inverse Dispersion Parameter K: 9.57 Observations n: 505 sites (9,707 PDO crashes) Standard Deviation sp: ±3.42 crashes/yr The coefficients in Table 39 were combined with Equation 21 to obtain the estimated SPF for freeway segments. The form of the model is described by the following equation. Equation 81 𝑁 , , 𝐿 , exp 3.133 1.295 ln 𝐴𝐴𝐷𝑇 /1000 where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section. The fit of the estimated model is shown in Figure 25. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported PDO crash frequency for a group of 10 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 120 PDO crashes during the study period.

99 Figure 25. Predicted vs. reported PDO crashes on freeway segments. Model for Ramp Entrance Speed-Change Lanes The results of the ramp entrance speed-change lane model estimation are presented in Table 41. Estimation of this model focused on PDO crash frequency. The Pearson χ2 statistic for the model is 107, and the degrees of freedom are 97 (= n − p = 103 − 6). As this statistic is less than χ2 0.05, 97 (= 121), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.44. The Rk2 for the estimated model is 0.51. Table 41. Final predictive model estimation statistics, PDO crashes, ramp entrance speed-change lane, five states. Model Statistics Value R2 (Rk2): 0.44 (0.51) Scale Parameter φ: 1.05 Pearson χ2: 107 (χ20.05, 97 = 121) Inverse Dispersion Parameter K: 9.57 Observations n: 103 sites (736 PDO crashes) Standard Deviation sp: ±1.46 crashes/yr The coefficients in Table 39 were combined with Equation 44 to obtain the estimated SPF for ramp entrance speed-change lanes. The form of the model is described by the following equation. Equation 82 𝑁 , , 𝐿 , exp 3.043 1.295 ln 𝐴𝐴𝐷𝑇 /1000 0.02018 𝐴𝐴𝐷𝑇 /1000 where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section. The fit of the estimated model is shown in Figure 26. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported PDO crash frequency for a group of 10 sites. In general, the data shown in the figure

100 indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 24 PDO crashes during the study period. Figure 26. Predicted versus reported PDO crashes at ramp entrance speed-change lanes. Model for Ramp Exit Speed-Change Lanes The results of the ramp exit speed-change lane model estimation are presented in Table 42. Estimation of this model focused on PDO crash frequency. The Pearson χ2 statistic for the model is 73.1, and the degrees of freedom are 94 (= n − p = 100 − 6). As this statistic is less than χ2 0.05, 94 (= 118), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.41. The Rk2 for the estimated model is 0.37. Table 42. Final predictive model estimation statistics, PDO crashes, ramp exit speed-change lane, five states. Model Statistics Value R2 (Rk2): 0.41 (0.37) Scale Parameter φ: 0.74 Pearson χ2: 73.1 (χ20.05, 94 = 118) Inverse Dispersion Parameter K: 9.57 Observations n: 100 sites (730 PDO crashes) Standard Deviation sp: ±1.55 crashes/yr The coefficients in Table 39 were combined with Equation 50 to obtain the estimated SPF for ramp exit speed-change lanes. The form of the model is described by the following equation. Equation 83 𝑁 , , 𝐿 , exp 3.413 1.295 ln 𝐴𝐴𝐷𝑇 /1000 0.598 𝑛 2 . where all variables are as previously defined. The estimated AFs used with this SPF are described in a subsequent section.

101 The fit of the estimated model is shown in Figure 27. This figure compares the predicted and reported crash frequency in the estimation database. Each data point shown represents the average predicted and average reported PDO crash frequency for a group of 10 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 32 PDO crashes during the study period. Figure 27. Predicted versus reported PDO crashes at ramp exit speed-change lanes. Estimated Adjustment Factors Several AFs were estimated in conjunction with the SPFs. All of them were estimated using PDO crash data. Collectively, they describe the relationship between various geometric factors and crash frequency. These AFs are described in this section and, where possible, compared with the findings from previous research as a means of model validation. Horizontal Curve AF The estimated horizontal curve AF is described using the following equation. Equation 84 𝐴𝐹 , , 1.0 exp 5.469 5,730𝑅 The radii used to estimate this AF range from 1,430 to 24,170 feet. The base condition for this AF is an uncurved (i.e., tangent) segment. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The AF is shown in Figure 28 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. The trend line based on Equation 84 is shown to produce smaller AF values than that obtained from the HSM CMF. This result likely stems from the fact that the sites used to estimate Equation 84 are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed than the sites used to estimate the HSM CMF.

102 Figure 28. Estimated horizontal curve AF for PDO crashes. Lane Width AF The lane width AF is described using the following equation. Equation 85 𝐴𝐹 , , exp 0.02725 min 𝑊 , 13 12 The lane width used in this AF is an average for all through lanes on the segment. The AF is discontinuous, breaking at a lane width of 13 feet. The AF value does not change for lane width in excess of 13 feet. The widths used to estimate this AF range from 10.5 to 14.4 feet. The base condition for this AF is a 12-foot lane width. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The lane width AF is shown in Figure 29 using a thick, solid trend line. The HSM Supplement (ASHTO 2014) does not include a CMF for lane width. Nevertheless, the trend line is in good agreement on the relationship between lane width and AF value for FI crashes, as shown in Figure 11. Figure 29. Estimated lane width AF for PDO crashes.

103 Inside Shoulder Width AF The inside shoulder width AF is described using the following equation. Equation 86 𝐴𝐹 , , exp 0.02725/𝑛 min 𝑊 , , 12 6 The shoulder width used in this AF represents the paved width. The AF is discontinuous, breaking at a shoulder width of 12 feet. The AF value does not change for shoulder width in excess of 12 feet. This breakpoint was established based on trends in the outside shoulder width data. It is assumed that this trend also extends to inside shoulder width. The widths used to estimate this AF range from 0.7 to 11.0 feet. The number of through lanes range from 2 to 7. The base condition for this AF is a 6-foot inside shoulder width. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 86 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that shoulder width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the shoulder is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by shoulder presence. The inside shoulder width AF is shown in Figure 30 using two thick trend lines. The equivalent CMF from the HSM Supplement (ASHTO 2014) is shown using a thin, dashed trend line. The three trend lines are in good agreement on the relationship between inside shoulder width and AF value. Figure 30. Estimated inside shoulder width AF for PDO crashes. Median Width AF The estimated median width AF is described using the following equation. Equation 87 𝐴𝐹 | , , 1.0 𝑃 exp 0.00407/𝑛 𝑊 48 𝑃 exp 0.00407/𝑛 min 𝑊 , 2 𝑊 48 with Equation 88 𝑊 𝑊 𝑊 , 𝑊 , 𝑊 , 𝐼 , , 𝑊 , 𝐼 , ,

104 The median width used in this AF is an average for the site. The AF is derived to be applicable to a site that has median barrier present along some portion of the site. The variable Wum represents width of the middle portion of the median that excludes the width of the inside shoulder in each travel direction. The variable Pib represents the proportion of the site length with a barrier present in the median. It is computed by measuring the length of roadway with median barrier present and dividing by the site length. The variable Wicb represents the distance the barrier is offset from the edge of the inside shoulder. More detailed guidance for computing the variables Pib and Wicb is provided in Section 18.7.3 of the HSM Supplement (AASHTO 2014). This AF is applicable to values of median width that range from 5 to 90 feet. The number of through lanes range from 2 to 7. The variable Wicb ranges from 0.75 to 20 feet. The variable Pib ranges from 0.0 to 1.0. The base condition for this AF is a 60-foot median width and an inside shoulder width of 6 feet (i.e., 48 = 60 – 6 – 6). The coefficient “−0.00407” is equal to the value of the parameter bs,ast,pdo divided by 6.7, as described in the text associated with Equation 79. This AF is applicable to freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 87 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that median width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the median is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by median width. The median width AF is shown in Figure 31 using the thick, solid trend line labeled “proposed, 3 lanes; no barrier.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM, no barrier.” These two trend lines are in reasonable agreement on the relationship between median width and AF value, with the proposed AF showing more sensitivity to a change in width. The slope of the thick, solid trend line decreases with an increase in the number of lanes. The other two trend lines shown are discussed in the Median Barrier AF section. Figure 31. Estimated median width AF for PDO crashes.

105 Median Barrier AF The estimated median barrier AF is described using the following equation. Equation 89 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.01618 𝑛/𝑊 The variables Pib and Wicb were described in the previous section. This AF is applicable to a number of through lanes ranging from 2 to 7. The variable Wicb ranges from 0.75 to 20 feet. The variable Pib ranges from 0.0 to 1.0. The base condition for this AF is “no barrier present in the median” (i.e., Pib = 0.0). This AF is applicable to freeway segments, ramp entrance speed- change lanes, and ramp exit speed-change lanes. The variable for “number of through traffic lanes” n is included in Equation 89 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a multiplier) in the AF suggests that median barrier has more influence on crash frequency when there are more lanes. It is unclear whether this relationship is a result of (a) a correlation between the AF variables and other site characteristics or (b) an indication that non-recoverable lane departures are more prevalent on freeways with more lanes. This topic should be a subject of future research. The median barrier AF is shown in Figure 31 using the thick, solid trend line labeled “proposed, 3 lanes; barrier in center of median.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM Supplement (AASHTO 2014), barrier in center of median.” These two trend lines are in agreement that the AF value decreases with increasing median width. However, the slope of the AF trend line is not as large as that for the HSM CMF. This result likely stems from the fact that the sites used to estimate Equation 89 are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed than the sites used to estimate the HSM CMF. Outside Shoulder Width AF The outside shoulder width AF is described for freeway segments using the following equation. Equation 90 𝐴𝐹 , , exp 0.02725/𝑛 min 𝑊 , 12 10 The shoulder width used in this AF represents the paved width. The AF is discontinuous, breaking at a shoulder width of 12 feet. The AF value does not change for shoulder width in excess of 12 feet. This breakpoint was established based on trends in the data. The widths used to estimate this AF range from 0.7 to 14.0 feet. The number of through lanes range from 2 to 7. The base condition for this AF is a 10- foot outside shoulder width. This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 90 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that shoulder width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the shoulder is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by shoulder presence. The outside shoulder width AF is shown in Figure 32 using two thick trend lines. The HSM Supplement (AASHTO 2014) does not include a CMF for outside shoulder width on tangent (i.e., uncurved) sections, so a direct comparison with the proposed AF is not readily available. Nevertheless, the trend line is in good agreement on the relationship between outside shoulder width and AF value for FI crashes, as shown in Figure 17.

106 Figure 32. Estimated outside shoulder width AF for PDO crashes. Outside Clearance AF The estimated outside clearance AF is described for freeway segments using the following equation. Equation 91 𝐴𝐹 | , , 1.0 𝑃 exp 0.00407/𝑛 𝑊 𝑊 , 𝐼 , , 𝑊 20 𝑃 exp 0.00407/𝑛 𝑊 20 The clear zone width Whc used in this AF is an average for the site. The AF is derived to be applicable to a segment that has outside (roadside) barrier present along some portion of the site. The variable Whc represents the width of the clear zone, as measured from the edge of traveled way. The variable Pob represents the proportion of the site length with a barrier present on the outside (roadside). It is computed by measuring the length of roadway with outside (roadside) barrier present and dividing by the segment length. The variable Wocb represents the distance the barrier is offset from the edge of the inside shoulder. More detailed guidance for computing the variables Pob and Wocb is provided in Section 18.7.3 of the HSM Supplement (AASHTO 2014). This AF is applicable to values of clear zone width that range from 10 to 30 feet. The number of through lanes range from 2 to 7. The variable Wocb ranges from 1.0 to 20 feet. The variable Pob ranges from 0.0 to 1.0. The base condition for this AF is a 30-foot clear zone width and an outside shoulder width of 10 feet (i.e., 20 = 30 – 10). The coefficient “−0.00407” is equal to the value of the parameter bs,ast,pdo divided by 6.7, as described in the text associated with Equation 80. This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 91 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that clear zone width has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the clear zone is laterally more distant from the center of the traveled way such that the crashes occurring in the middle lanes are less influenced by clear zone width. The outside clearance AF is shown in Figure 33 using the thick, solid trend line labeled “proposed, 3 lanes; no barrier.” The HSM Supplement (AASHTO 2014) does not include a CMF for outside shoulder width on tangent (i.e., uncurved) sections, so a direct comparison with the proposed AF is not readily

107 available. Nevertheless, the trend line is in good agreement on the relationship between clear zone width and AF value for FI crashes, as shown in Figure 18. The other trend lines shown are discussed in the next section. Figure 33. Estimated outside clearance AF for PDO crashes. Outside Barrier AF The estimated median barrier AF is described for freeway segments using the following equation. Equation 92 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 0.01618 𝑛/𝑊 The variables Pob and Wocb were described in the previous section. This AF is applicable to a number of through lanes ranging from 2 to 7. The variable Wocb ranges from 1.0 to 20 feet. The variable Pob ranges from 0.0 to 1.0. The base condition for this AF is “no barrier present on the outside (roadside)” (i.e., Pob = 0.0). This AF is applicable to freeway segments. The variable for “number of through traffic lanes” n is included in Equation 92 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a multiplier) in the AF suggests that outside (roadside) barrier has more influence on crash frequency when there are more lanes. It is unclear whether this relationship is a result of (a) a correlation between the AF variables and other site characteristics or (b) an indication that non-recoverable lane departures are more prevalent on freeways with more lanes. This topic should be a subject of future research. The outside barrier AF is shown in Figure 33 using the thick, solid trend line labeled “proposed, 3 lanes; roadside has barrier 2-ft from shoulder edge for 100% of segment.” The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line labeled “HSM Supplement (AASHTO 2014), roadside has barrier for 2-ft from shoulder edge for 100% of segment.” The AF from Equation 92 is 1.05, while the CMF from the HSM is 1.09—a difference of 4 percent. This difference is relatively small considering the uncertainty associated the coefficients and other model variables. The thick, solid trend line decreases (i.e., drops) slightly with an increase in the number of lanes.

108 Ramp Entrance AF The estimated ramp entrance AF is described using the following equation. Equation 93 𝐴𝐹 , , exp 0.09908 1𝐿 1 0.142 The ramp entrance length Len is measured from the gore to the taper point of the speed-change lane, as defined in Chapter 18 (Figure 18-3) of the HSM Supplement (AASHTO 2014). The length values used to estimate this AF range from 0.06 to 0.32 miles. The base condition for this AF is a ramp entrance length of 0.142 miles. This value represents the median ramp entrance length in the estimation database. This AF is applicable to ramp entrance speed-change lanes. The ramp entrance AF is shown in Figure 34 using a thick, solid trend line. The equivalent CMF from the HSM Supplement (AASHTO 2014) is shown using a thin, dashed trend line. Also shown is a CMF from Chapter 15 of the HSM (AASHTO 2010). To facilitate this comparison, the HSM CMFs were converted mathematically to a base length of 0.142 miles (i.e., 750 feet). The HSM 2010 CMF and the proposed AF are in good agreement on the relationship between ramp entrance length and AF value for lengths in excess of 700 feet. For shorter lengths, the proposed AF value is higher than the HSM CMF values and more consistent with the proposed AF for FI crashes, as shown in Figure 19. This deviation should be a subject of future research. Figure 34. Estimated ramp entrance AF for PDO crashes. Ramp Exit AF The estimated ramp exit AF is described using the following equation. Equation 94 𝐴𝐹 , , exp 0.04327 1𝐿 1 0.071 The ramp exit length Lex is measured from the gore to the taper point of the speed-change lane, as defined in Chapter 18 (Figure 18-3) of the HSM Supplement (AASHTO 2014). The length values used to estimate this AF range from 0.02 to 0.27 miles. The base condition for this AF is a ramp exit length of

109 0.071 miles. This value represents the median ramp exit length in the estimation database. This AF is applicable to ramp exit speed-change lanes. The ramp entrance AF is shown in Figure 35 using a thick, solid trend line. The HSM Supplement (AASHTO 2014) has a CMF for ramp exit speed-change lanes, but it is not sensitive to ramp exit length. A CMF from Chapter 15 of the HSM (AASHTO 2010) is shown using a thin, dashed trend line. To facilitate this comparison, the HSM CMF was converted mathematically to a base length of 0.071 miles (i.e., 375 feet). The two trend lines are in reasonable agreement on the relationship between ramp entrance length and AF value. Figure 35. Estimated ramp exit AF for PDO crashes. Turnout Presence AF The estimated turnout presence AF is described for freeway segments using the following equation. Equation 95 𝐴𝐹 | , , 1.0 𝑃 1.0 𝑃 exp 1.091/𝑛 with, Equation 96 𝑃 𝐿 , /𝐿 , The proportion Pturnout represents the proportion of the segment length that is adjacent to a turnout. It is computed by measuring the length of roadway adjacent to a turnout and dividing by the segment length. If the turnout extends beyond one or both of the segment boundaries, then only that portion of the turnout that lies within the segment is measured. This AF is applicable to values of Pturnout that range from 0.0 to 1.0. The number of through lanes range from 2 to 7. The base condition for this AF is “no turnout present” (i.e., Pturnout = 0.0). This AF is applicable to freeway segments with PTSU operation during some portion of the typical day. The variable for “number of through traffic lanes” n is included in Equation 95 because it was found to improve the overall model fit to the data. The manner in which it is used (i.e., as a divisor) in the AF suggests that turnout presence has less influence on crash frequency when there are more lanes. In other words, when there are many lanes, the turnout (like a shoulder) is laterally more distant from the center of

110 the traveled way such that the crashes occurring in the middle lanes are less influenced by turnout presence. The turnout AF is shown in Figure 36 using three thick trend lines; one line is shown for each of two, four, and six through traffic lanes. A review of the literature did not reveal any publications wherein the safety benefit of turnout presence on a PTSU facility was quantified. However, the t-statistic associated with the regression coefficient (i.e., t = −7.31) indicates that the coefficient is significantly different from 0.0 and implies that turnout presence does reduce crash frequency. Figure 36. Estimated turnout presence AF for PDO crashes. As noted in the discussion of Table 12, Jenior et al. (2016) recommend that turnouts should be constructed at desirably one-half mile intervals. This spacing combined with a turnout length of 800 feet (200-foot body plus a 300-foot entry taper and a 300-foot exit taper) corresponds to an average “turnout length as a proportion of segment length” of 0.30. Based on this proportion, AF values of 0.87, 0.93, and 0.95 are obtained from Figure 36 for two, four, and six lanes, respectively. PTSU Operation AF The estimated PTSU operation AF is described by the following equation. Equation 97 𝐴𝐹 | , , 1.0 𝑃 , exp 𝑓 , 𝑃 , exp 𝑓 , 𝑓 , , 𝑓 , with Equation 98 𝑓 , 0.02725/𝑛 min 𝑊 , , 12 𝐼 Equation 99 𝑓 , 0.02725 min 𝑊 , , 13 12 𝐼 Equation 100 𝑓 , 1.567 𝐼 Equation 101 𝑓 , , 1.515 1 𝐼 𝑃 ,

111 The variable Wptsu represents the average width of the shoulder that is allocated to vehicular traffic use (i.e., as an additional travel lane) during one or more hours of the typical day. If this width varies along the length of the site (e.g., as in the case where it the PTSU lane is tapered to add or drop the PTSU lane), then an average width is used for the subject site. The proportion Pt,ptsu represents the proportion of time during the average day that PTSU operates in the vicinity of (or at) the subject site. It has a nonzero value if (a) the site has a full-width PTSU lane or the tapered portion of a PTSU lane, (b) the site does not have a PTSU lane but a portion of it is within 0.152 miles of a PTSU lane (i.e., just upstream or downstream), or (c) the site is a speed-change lane site and the PTSU lane effectively continues through the site even though it may not be marked as an exclusive PTSU lane (because ramp traffic is permitted to cross this area to enter or exit the freeway through lanes). This variable is computed using Equation 10. Cases “a” and “c” are referred herein to as a PTSU lane and case “b” is referred to as a PTSU transition zone. Cases “a” and “c” are addressed by Equation 100. Case “b” is addressed by Equation 101. The variable IptsuLane is an indicator variable that has a value of 1.0 if the site has a PTSU lane (i.e., Wptsu > 0.0). This AF is applicable to values of Pt,ptsu that range from 0.0 to 0.45. The number of through lanes range from 2 to 7. The width of the PTSU lane Wptsu ranges from 3.5 to 16.8 feet. The base condition for this AF is “no PTSU operation during any hour of the day” (i.e., Pt,ptsu = 0.0) and PTSU lane width Wptsu equal to 0.0 feet. This AF is applicable to sites with a nonzero value of the variable Pt,ptsu. The PTSU presence AF is shown in Table 43. AF values are listed for various combinations of PTSU turnout spacing, lane width, proportion time PTSU operating, and number of lanes. The turnout spacing values in the table are computed using the product of the PTSU operation AF and the turnout presence AF with a turnout spacing of 0.5 miles. The AF values in the upper two portions of Table 43 are shown to increase with an increase in the proportion time of the PTSU is operating, a reduction in PTSU lane width, and an increase in number of lanes. The AF values decrease when turnouts are provided. The AF value listed in the table for the case where the proportion time PTSU operating equals 0.0 corresponds to the case where the width of the PTSU lane is effectively serving as additional shoulder width. The sites in the estimation database indicate that typical values of the conditions identified in Table 43 include 0.5-mile turnout spacing, a proportion time of PTSU operating of 0.2, an 11-foot PTSU lane width, and four lanes. For these values, the AF value is 1.60; this value suggests that PTSU presence can increase the annual PDO crash frequency by 60 percent for typical conditions. A review of the literature on the safety effect of PTSU operation was provided in the text associated with Table 38. This review did not reveal any additional publications wherein the safety benefit of PTSU operation on PDO crash frequency was reliably quantified. However, the t-statistics associated with the two regression coefficients (i.e., t for bptsuOpen is 8.67 and t for bnearOpen = 3.09) indicate that both coefficients are significantly different from 0.0 and implies that PTSU operation increases PDO crash frequency.

112 Table 43. Estimated PTSU operation AF for PDO crashes. PTSU Type PTSU Lane Width (feet) Proportion Time PTSU Operatinga AF Value by Number of Lanes 2 4 6 PTSU lane (no turnouts) 11 0.0 0.86 0.93 0.95 0.1 1.27 1.33 1.35 0.2 1.67 1.73 1.75 0.3 2.08 2.13 2.14 0.4 2.49 2.53 2.54 12 0.0 0.85 0.92 0.95 0.1 1.24 1.31 1.33 0.2 1.64 1.70 1.72 0.3 2.03 2.08 2.10 0.4 2.43 2.47 2.49 PTSU lane (turn-out every 0.5 miles) 11 0.0 0.75 0.86 0.90 0.1 1.11 1.23 1.28 0.2 1.46 1.60 1.66 0.3 1.82 1.97 2.04 0.4 2.17 2.35 2.41 12 0.0 0.74 0.86 0.90 0.1 1.09 1.21 1.26 0.2 1.43 1.57 1.63 0.3 1.78 1.93 2.00 0.4 2.12 2.29 2.36 PTSU transition zoneb Any 0.0 1.00 1.00 1.00 0.1 1.13 1.13 1.13 0.2 1.27 1.27 1.27 0.3 1.40 1.40 1.40 0.4 1.54 1.54 1.54 a Proportion time PTSU operating = (weekday hours × 5/7 + weekend hours × 2/7)/24 b Segment length is 0.27 miles. Sensitivity Analysis The relationship between crash frequency and traffic demand for freeway segments is illustrated in Figure 37 for a 1-mile freeway segment. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 37a correspond to the proposed model. The trends in Figure 37b correspond to the freeway segment model in the HSM Supplement (AASHTO 2014). The HSM model results were computed using a two-way AADT that is twice that of the directional AADT. The predicted crash frequency from the HSM model was then divided by two to obtain a result that is comparable to the proposed models. A comparison of the trends in Figure 37a with those in Figure 37b indicate that the proposed model predicts about as many crashes as the HSM model for a common directional AADT.

113 a. Proposed models. b. HSM models. Figure 37. Estimated freeway segment model for PDO crashes. The relationship between crash frequency and traffic demand for ramp entrance speed-change lanes is illustrated in Figure 38. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 38a correspond to the proposed model. The trends in Figure 38b correspond to the corresponding model in the HSM Supplement (AASHTO 2014). A comparison of the trends in Figure 38a with those in Figure 38b indicate that the proposed model predicts about as many crashes as the HSM model for a common directional AADT. The sensitivity to number-of-lanes was examined in the estimation database but it was not found to provide an improved model fit to the data. a. Proposed models. b. HSM models. Figure 38. Estimated ramp entrance model for PDO crashes. The relationship between crash frequency and traffic demand for ramp entrance speed-change lanes is illustrated in Figure 39. The length of each trend line in this figure reflects the range of directional AADT volume in the data. The trends in Figure 39a correspond to the proposed model. The trends in Figure 39b correspond to the corresponding model in the HSM Supplement (AASHTO 2014). A comparison of the trends in Figure 39a with those in Figure 39b indicate that the proposed model predicts notably more

114 crashes than the HSM model for volumes in excess of about 70,000 veh/day. A similar trend was noted for FI crashes in the discussion associated with Figure 24. This difference was investigated for explanation by other model variables and by issues with the reported crash data. However, nothing notable was found from this investigation. The difference likely stems from the fact that the sites used to estimate the proposed models are collectively operating at a much higher volume-to-capacity ratio and a lower operating speed than those in the HSM database. a. Proposed models. b. HSM models. Figure 39. Estimated ramp exit model for PDO crashes.