National Academies Press: OpenBook

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

Chapter: Chapter 6: Crash Severity and Crash Type Distributions

« Previous: Chapter 5: HSM Predictive Model
Page 113
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 113
Page 114
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 114
Page 115
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 115
Page 116
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 116
Page 117
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 117
Page 118
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 118
Page 119
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 119
Page 120
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 120
Page 121
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 121
Page 122
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 122
Page 123
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 123
Page 124
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 124
Page 125
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 125
Page 126
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 126
Page 127
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 127
Page 128
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 128
Page 129
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 129
Page 130
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 130
Page 131
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 131
Page 132
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 132
Page 133
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 133
Page 134
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 134
Page 135
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 135
Page 136
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 136
Page 137
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 137
Page 138
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 138
Page 139
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 139
Page 140
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 140
Page 141
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 141
Page 142
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 142
Page 143
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 143
Page 144
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 144
Page 145
Suggested Citation:"Chapter 6: Crash Severity and Crash Type Distributions." National Academies of Sciences, Engineering, and Medicine. 2021. Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report. Washington, DC: The National Academies Press. doi: 10.17226/26393.
×
Page 145

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

115 C H A P T E R 6 - CRASH SEVERITYAND CRASH TYPE DISTRIBUTIONS Crash Severity and Crash Type Distributions This chapter describes the findings obtained during the development of crash severity and crash type distributions for freeways with part-time shoulder use (PTSU) operation. A predictive model was developed for computing the crash severity distribution. A table of proportions was developed for computing the crash type distribution. The distributions are intended to be used with the predictive model equations (described in Chapter 5) to predict the average crash frequency associated with one direction of travel on a freeway. They could also be used with the advanced crash prediction models described in Chapter 7, Advanced Crash Predictive Models, or future models developed by others. Each predictive model equation includes a safety performance function (SPF), one or more adjustment factors (AFs), and a calibration factor. A crash prediction model (CPM) is comprised of a predictive model equation, severity distribution, and crash type distribution. The crash severity and crash type distribution described herein address each of the following site types:  Freeway segment  Ramp entrance speed-change lane  Ramp exit speed-change lane A separate document prepared through NCHRP Project 17-89, the PTSU Safety Evaluation Guidelines, documents the CPMs in the form of a safety predictive method for freeways with PTSU operation. The method describes how to use the CPMs to evaluate freeway safety, as may be influenced by roadway geometry, roadside features, traffic volume, and lane-change-related traffic maneuvers. This documentation is provided in a form suitable for inclusion in a future edition of the Highway Safety Manual (HSM) (AASHTO 2010). All CPMs in the HSM include procedures for estimating the average crash frequency by crash type severity. The corresponding distributions are used with a CPM to estimate the average crash frequency for various combinations of crash type, severity, or both. These procedures are described in this chapter. The distributions 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. 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).

116 This chapter consists of two sections. The first section describes the development of the proposed severity distribution model. The second section describes the development of the proposed crash type distribution tables. Severity Distribution Background Three crash severity distribution predictive models were developed—one model for each of the three site types identified previously. Each predictive model can be used to compute the proportion of crashes by severity level. Specifically, each model can be used to predict the proportion of K (fatal), A (incapacitating injury), B (non-incapacitating injury), and C (possible injury) crashes. In application, these proportions can be used with a fatal-and-injury (FI) crash frequency prediction model to predict the frequency of each crash severity level. In this regard, the predicted FI crash frequency is multiplied by the predicted proportion of crashes of a specific severity to obtain an estimate of the average frequency of crashes associated with the specified severity. The crash severity distribution prediction model is referred to in the literature as a severity distribution function (SDF). A typical SDF is a logit regression model that includes variables for various geometric design elements that are correlated with the severity of a crash. Models of this type were developed by Bonneson et al. (2012a) for the freeway safety prediction model in Chapter 18 of the HSM Supplement (AASHTO 2014). The development of the crash frequency prediction models is documented in Chapter 5, HSM Predictive Model. The development of crash severity distribution prediction models (i.e., SDFs) is described in this section. Together, these two sets of models can be used to compute the frequency of crashes for any desired severity level. This approach [i.e., separately developing (a) a model for predicting crash frequency and (b) an SDF for the predicting the severity distribution] is intended to minimize the frequency-severity indeterminacy problem described by Hauer (2006). Database Summary The purpose of this summary is to provide information about the range of data included in the database and to provide some insight to guide the development of the predictive model form. The discussion in this section is not intended to indicate conclusive results or recommendations. The proposed predictive models (and associated trends) are documented in later sections of this chapter. The database consists of data describing sites on three types of facilities. Facilities with PTSU operation were considered treatment facilities. Facilities not having PTSU operation but located near the treatment facilities and often having the same route designation were considered comparison facilities. Comparison facilities were used to specifically address the research question of whether PTSU operation has an effect on crash frequency or severity. Finally, facilities not having PTSU operation and located some distance away from treatment facilities were considered supplemental facilities. These facilities were added to the database to increase the sample size and the range of values for key independent variables. Data for supplemental facilities were initially collected to investigate the possible influence of BOS operation on safety. Crash data were obtained for the states of Georgia, Hawaii, Minnesota, Ohio, and Virginia. However, data for Hawaii were not included in the database used to develop the SDFs. These data were excluded because they do not distinguish between the A, B, and C severity levels. As a result, they could not be used to develop the severity distribution prediction model. The database for the SDFs includes data for 676 study sites, which collectively represent 13 freeway facilities in Georgia, Minnesota, Ohio, and Virginia. Treatment and comparison facilities are located in

117 Georgia, Minnesota, and Virginia. Supplemental facilities are located in Ohio and Minnesota. The 676 sites have a total length of 154.6 miles. About 22 percent of the total mileage consists of highway facilities with PTSU operation during 1 or more hours of the day. Additional information about the study sites is provided in Chapter 4, in the Database Summary section. Geometric Characteristics Table 44 summarizes the geometric and traffic characteristics of the sites in the highway safety database. The sites are located in urban areas and have lane counts that range from 2 to 7 in the subject direction of travel. The alignments of 250 sites (34 percent) are curved. Table 44. Summary geometric and traffic characteristics. Variablea Average Minimum Maximum Freeway Segment Volume Variables Directional AADT volume of segment, AADTfs (veh/day) 60,010 15,370 147,210 Proportion of AADT during hours where volume > 1,000 veh/h/lane 0.17 0.0 0.42 Horizontal Curve Variables Number of sites with horizontal curve 250 n.a. n.a. Horizontal curve radius, R (feet) 6,210 1,430 24,170 Cross Section Variables Number of lanes (in subject travel direction), n 3.3 2 7 Median width, Wm (feet) 35.0 5.1 90.9 Paved inside shoulder width, Wis (feet) 7.6 0.7 11.0 Inside PTSU lane width (left side), Wil,ptsu (feet) 11.7 3.5b 14.0 Lane width (excluding PTSU lane), Wl (feet) 11.9 10.5 14.4 Outside PTSU lane width (right side), Wl,ptsu (feet) 11.2 5.0b 16.8 Paved outside shoulder width, Ws (feet) 9.8 0.7 14.0 Rumble Strip Variables Proportion of site with rumble strips on the inside shoulder, Pir c 0.35 0.0 1.0 Proportion of site with rumble strips on the outside shoulder, Por c 0.37 0.0 1.0 Barrier Variables Proportion of site with barrier present in the median, Pib 0.88 0.0 1.0 Distance from inside shoulder to barrier face, Wicb (feet) 2.5 0.7 20.0 Distance from outside shoulder to barrier face, Wocb (feet) 1.7 1.0 20.0 Proportion of site with barrier present on the outside (roadside), Pob 0.43 0.0 1.0 Turnout Variables Number of sites with turnout beyond the shoulder 37 n.a. n.a. Length of turnout (from start of taper to end of taper), Ltout (feet) 520 75 1,500 Speed-Change Lane Variables AADT volume of entrance ramp, AADTen (veh/day) 6,890 450 25,210 AADT volume of exit ramp, AADTex (veh/day) 7,750 450 30,680 Length of entrance ramp, Len (miles) 0.16 0.06 0.32 Length of exit ramp, Lex (miles) 0.09 0.02 0.27 AADT = annual average daily traffic; n.a. = not applicable; veh/day = vehicles per day; veh/h/lane = vehicles per hour per lane a Variable names and definitions are consistent with those in Chapter 18 of the HSM Supplement (AASHTO 2014). b PTSU tapers from full width to zero width over segment; table value is an average width over length of segment. c Proportion of site with rumble strips is reported for sites that do not have PTSU on the associated side.

118 Table 44 was previously presented as Table 11 in Chapter 4. However, one additional row is added to Table 44. This row describes the proportion of AADT during high volume hours. This proportion is used to describe the level of congestion experienced during the average day. It can range in value from 0.0 to 1.0. It increases with an increase in the number of hours during which volumes exceed the “1,000 vehicles/h/lane” threshold value. If the volume during each hour of the day exceeds the threshold value, then the proportion equals 1.0. In general, this proportion is large when hourly volumes are continuously high or when there are a few peak hours with an exceptionally large volume. The “proportion of annual average daily traffic (AADT) during high volume hours” is used in the CPM in Chapter 18 of the HSM Supplement (AASHTO 2014). Guidance for computing its value is provided in Section 18.4 of the HSM. This proportion is used in the Chapter 18 predictive model equation to predict crash frequency. It is also used in the Chapter 18 SDF to estimate the crash severity distribution. With regard to the latter model, the proportion of K, A, and B severity crashes is predicted to decrease with an increase in the proportion of AADT during high-volume hours. Logically, operating speed decreases as the freeway becomes more congested (i.e., the proportion of AADT during high volume hours increases). The HSM, in Chapter 3, Appendix E, (AASHTO 2010) indicates that a reduction in speed corresponds to a reduction in fatal crashes. The “proportion of AADT during high volume hours” was computed using the annual average hourly volume distribution from the continuous traffic counting station nearest to each site in the database. This distribution was used with the site’s directional AADT to estimate the volume during each hour of the average day. The total volume during those hours for which the lane volume exceeded 1,000 vehicles per hour per lane was added together and divided by the AADT. This division produced the desired proportion. One variable not listed in Table 44 is the “proportion of time PTSU operates.” This value represents the proportion of time during the average day that PTSU is operating. It varied from 0.057 to 0.455 at the collective set of facilities with PTSU operation during some portion of the day. Additional discussion of this variable is provided in the text associated with Table 30 of Chapter 4 and Equation 10 of Chapter 5. Crash Characteristics Crash data were acquired from the states for a specified study period. The study period was defined to represent the range of recent years during which crash data were available. Additional detail describing the criteria used to define the study period for each facility is provided in Chapter 4. The dates associated with each study period are shown in Table 7 of Chapter 4. The study period at the collective set of facilities ranged from 1 to 5 years in duration. For some facilities, the study period varied along the length of the facility because PTSU operation was implemented at different dates along the facility or long-term construction was present. Multiple study period dates are shown for these facilities in Table 7. The distribution of reported crashes by state, PTSU operation, and severity category is shown in Table 45. There are a total of 676 study sites collectively representing four states and 3,233 site-years. The average study period duration is 4.8 years/site (= 3,233 site-years/676 sites). These sites collectively experienced a total of 16,296 crashes during the study period. The last row of Table 45 summarizes the crash data for the database assembled for NCHRP Project 17- 45, as reported by Bonneson et al. (2012a). The data shown are for urban freeway segments in the states of California, Maine, and Washington. The data include crashes located on freeway segments as well as at speed-change lane sites.

119 Table 45. Crash count distribution by state, PTSU operation, and severity. State PTSU Operation Number of Sites Total Years Crash Count by Severity (crashes) Total (crashes) Proportion FI FI PDO GA No 25 103 172 519 691 0.25 Yes1 38 153 318 928 1,246 0.26 Total: 63 256 490 1,447 1,937 0.25 MN No 251 1,255 717 2,103 2,820 0.25 Yes1 12 60 161 519 680 0.24 Total: 263 1,315 878 2,622 3,500 0.25 OH No 170 824 1,055 3,146 4,201 0.25 Yesa n.a. n.a. n.a. n.a. n.a. n.a. Total: 170 824 1,055 3,146 4,201 0.25 VA No 88 410 727 1,490 2,217 0.33 Yesa 92 427 1,383 3,058 4,441 0.31 Total: 180 837 2,110 4,548 6,658 0.32 All sites combined: 676 3,233 4,533 11,763 16,296 0.28 NCHRP 17-45 sites: 1,045 3,135 5,832 12,713 18,545 0.31 PDO = property-damage-only a Includes transition zone located between, just upstream of, or just downstream of PTSU segments. The last column of Table 45 lists the proportion of FI crashes for each row of the table. The proportion of 0.25 is shown for Georgia, Minnesota, and Ohio. In contrast, the Virginia data have a slightly larger proportion of FI crashes. This larger value reflects the fact that Virginia uses a larger property-damage- only (PDO) reporting threshold than the other three states (i.e., $1,500 for Virginia, $500 for Georgia, $1,000 for Minnesota, and $1,000 for Ohio after 9/7/2011). It is a reminder that the development of crash frequency prediction models using multi-state crash data should be based on the separate estimation of models for FI and PDO crashes. Table 46 provides the distribution of reported FI crashes by state, PTSU operation, and severity level. The column labeled “Proportion KAB” lists the proportion of crashes that have a K, A, or B severity. The proportions are shown to vary widely from state to state, with Minnesota having a proportion of 0.26 and Virginia having a proportion of 0.78. The proportion for Virginia is three times larger than that for Minnesota. Most of this variation likely stems from differences among states in either (a) the level of C severity crash reporting or (b) the criteria used to identify a C severity crash. It is a reminder of the need to locally calibrate the distribution of crash severity. Section B.1.4 of the HSM Supplement (AASHTO 2014) describes a procedure for locally calibrating an SDF.

120 Table 46. FI crash count distribution by state, PTSU operation, and severity. State PTSU Operation Crash Count by Severity (crashes) Total FI (crashes) Proportion KAB Odds KABa Odds Ratiob K A B C GA No 0 9 76 87 172 0.49 Yesc 0 15 119 184 318 0.42 Total: 0 24 195 271 490 0.45 0.81 0.63 MN No 5 9 170 533 717 0.26 Yesc 0 0 42 119 161 0.26 Total: 5 9 212 652 878 0.26 0.35 1.47 OH No 9 59 511 476 1,055 0.55 Yesc n.a. n.a. n.a. n.a. n.a. n.a. Total: 9 59 511 476 1,055 0.55 1.22 0.42 VA No 3 87 474 163 727 0.78 Yesc 7 115 958 303 1,383 0.78 Total: 10 202 1,432 466 2,110 0.78 3.53 0.14 All sites combined: 24 294 2,350 1,865 4,533 0.59 1.43 0.36 NCHRP 17-45 sites: 93 247 1,625 3,867 5,832 0.34 0.51 n.a. = not applicable. a Odds KAB = Proportion KAB /(1−Proportion KAB) b Odds ratio with respect to NCHRP 17-45 sites = Odds KAB17-45 / Odds KABi where i = state c Includes transition zone located between, just upstream of, or just downstream of PTSU segments. The last row of Table 46 lists the proportion of KAB crashes data for the database assembled for NCHRP Project 17-45; this proportion is “0.34.” It is in the range of proportions found in the four states identified in Table 46. The second-to-last column of Table 46 lists the odds that an FI crash is reported as a KAB crash severity (i.e., reported as either K, A, or B but not C). For example, the odds of an FI crash in the NCHRP 17-45 states being reported as either K, A, or B is 0.51 to 1. Conversely, the odds of an FI crash in the NCHRP 17-45 states being reported as C is 1.96 to 1 (= 1/0.51). The odds that an FI crash in Virginia is reported as either K, A, or B is 3.53 to 1. In other words, for every nine FI crashes in Virginia, about seven are reported as either K, A, or B. The odds in the second-to-last column of Table 46 can be used to compute an odds ratio, where the odds value for one state is related to the odds of another state. The odds ratio is an indication of the relative change in the reported KAB proportions when comparing two states. The odds ratios listed in the last column of the table relate the odds of a KAB crash in the NCHRP 17-45 data to that for each of the four states. For example, the odds ratio for the Minnesota data is 1.47. This value indicates that the odds of an FI crash being reported as KAB in the NCHRP 17-45 data is 47 percent larger than it is in Minnesota. The odds ratio for the Virginia data indicates that the odds of an FI crash being reported as KAB in the NCHRP 17-45 data is 86 percent (= [1 – 0.14] * 100) smaller than in Virginia. In fact, the odds ratios in Table 46 are equivalent to the local calibration factor “Csdf” that is computed using the procedure in Section B.1.4 of the HSM Supplement (AASHTO 2014). The wide range of odds ratios (i.e., calibration factors) shown in the table is further reminder of the need to use local values for the severity distribution (or to locally calibrate the HSM SDF). Table 47 lists the FI crash count distribution by PTSU operation, site type, and severity level. The data in this table are examined in the next section to determine whether the proportion of crashes for a given severity level varies based on PTSU operation.

121 Table 47. FI crash count distribution by PTSU operation, site type, and severity. PTSU Operation Site Type Number of Sites Total Years Crash Count by Severity (crashes) Total FI (crashes) K A B C No Freeway segment 368 1,778 12 146 1,079 1,076 2,313 Ramp ent. speed-change lane 87 422 3 7 86 104 200 Ramp exit speed-change lane 79 392 2 11 66 79 158 Total: 534 2,592 17 164 1,231 1,259 2,671 Yesa Freeway segment 113 508 6 117 979 533 1,635 Ramp ent. speed-change lane 13 61 1 6 63 30 100 Ramp exit speed-change lane 16 72 0 7 77 43 127 Total: 142 641 7 130 1,119 606 1,862 All sites combined: 676 3,233 24 294 2,350 1,865 4,533 a Includes transition zone located between, just upstream of, or just downstream of PTSU segments. Exploratory Data Analysis The database was examined using simple crash proportions to identify the possible association between specific site characteristics and the FI crash severity distribution. The insights obtained from this examination were used to (1) determine which characteristics are likely candidates for representation in the SDF as an adjustment factor and (2) guide the functional form development for the SDF. The discussion in this chapter is not intended to indicate conclusive results or recommendations. The proposed predictive models (and associated trends) are documented in a subsequent section. The following paragraphs describe the findings from the examination of crash severity proportions as a function of key site characteristics. Many of the characteristics in the database were evaluated in this manner. Those characteristics for which some trend was found are discussed in the following subsections. To facilitate the examination, the crash counts by severity level at each site were converted into an annual average crash frequency (i.e., crashes per year). This approach was used because of the wide range in study period duration among the study sites. Those sites having a long study period tend to have more crashes reported. When the counts from sites with a long study period are added to the counts from sites with a short study period, the resulting distribution proportions are biased to emphasize the sites having a long study period. To avoid this bias and ensure that each site was given equal weight in the severity distribution, the evaluation was based on annual crash frequency for each site. This issue is discussed further in the subsequent section of this chapter titled Study Period Duration. PTSU Operation The crash counts in Table 47 are shown as proportions in Table 48. The proportion of K crashes is based on relatively few crashes, so it is less reliably known. In contrast, the proportion of B and C crashes is based on many crashes, so they are more reliably known. The state-to-state variation in the proportion of KAB crashes, as shown in Table 46, represents a systematic variability that adds to the uncertainty associated with the examination of proportions aggregated for all states combined (such as those proportions shown in Table 48). As a result, minor trends (i.e., small changes in proportion for a change in character) that are observed in the proportions shown in this section may not be true indicators of systematic trend. Rather, they may be a result of random variation or an indirect result of correlations with state-to-state variation.

122 Table 48. FI crash proportions by PTSU operation, site type, and severity. PTSU Operation Site Type Crash Proportion by Severity Proportion KAB K A B C No Freeway segment 0.005 0.063 0.470 0.462 0.538 Ramp entrance speed-change lane 0.014 0.038 0.427 0.521 0.479 Ramp exit speed-change lane 0.012 0.070 0.418 0.500 0.500 Total: 0.006 0.062 0.464 0.468 0.532 Yesa Freeway segment 0.003 0.070 0.605 0.321 0.679 Ramp entrance speed-change lane 0.010 0.062 0.614 0.314 0.686 Ramp exit speed-change lane 0.000 0.058 0.621 0.321 0.679 Total: 0.003 0.069 0.607 0.321 0.679 All sites combined: 0.005 0.065 0.524 0.406 0.594 NCHRP 17-45 sites: 0.016 0.042 0.279 0.663 0.337 a Includes transition zone located between, just upstream of, or just downstream of PTSU segments. In spite of the aforementioned cautions, some general trends can be discerned from the proportions in Table 48. For example, the “proportion KAB” values shown in the last column of the table offer some insight about the differences among site types and PTSU operation. For example, considering the sites where PTSU is not operating, the proportion KAB values for speed-change lanes are smaller than those for freeway segments. This trend suggests that speed-change lanes are associated with a smaller proportion of severe crashes than is a freeway segment. This trend does not hold for sites where PTSU is operating. With regard to a comparison of sites with and without PTSU operation, the two rows labeled “Total” can be used to determine whether PTSU operation alters the crash severity distribution. Notably, at the sites with PTSU operation, the proportion of K crashes is smaller by about 50 percent (i.e., the proportion is 0.003 at sites with PTSU operation versus 0.006 at sites without PTSU operation). Similarly, the proportion of C crashes at sites with PTSU operation is about 31 percent smaller than at sites without PTSU operation. In contrast, the proportion of B crashes at sites with PTSU operation is about 30 percent larger than at sites without PTSU operation. These proportions suggest that PTSU operation may increase the proportion of B severity crashes, while reducing the K and C crashes. The proportion of A severity crashes is not much different between the two categories. Proportion of AADT during Hours with High Volume The findings from an examination of the “proportion of AADT during high volume hours” are shown in Figure 40. This proportion was computed for sites grouped by state and proportion intervals. Each group was defined to include a sufficient number of sites such that the computed proportion of KAB crashes for the group has statistical validity. Data for all four states are shown in the figure and a best-fit trend line is shown for the data for each state. The trend lines for three states show that there is a decrease in the proportion KAB crashes with an increase in the proportion of AADT during high volume hours. The data for Ohio do not show this sensitivity. This lack of sensitivity may be an artifact of other variables in the Ohio data that are associated with the level of congestion and crash severity. The “proportion of AADT during high volume hours” is included as a variable in the SDF for freeways provided in Chapter 18 of the HSM Supplement (AASHTO 2014). The regression coefficient associated with this variable has a negative sign, which indicates that the predicted proportion of K, A, and B crashes decreases with an increase in the proportion of AADT during high-volume hours. This trend is consistent with that shown in Figure 40 for Georgia, Minnesota, and Virginia.

123 Figure 40. Examination of proportion of AADT during hours with high volume. Proportion of Site Adjacent to Barrier The findings from an examination of the “proportion of the site adjacent to a barrier” are shown in Figure 41. The proportion was computed for each site by averaging (1) the proportion of the site length adjacent to a median barrier and (b) the proportion of the site length adjacent to barrier on the outside (roadside). The sites were grouped by state and by proportion intervals. Each group was defined to include a sufficient number of sites such that the computed proportion of KAB crashes for the group has statistical validity. Data for all four states are shown in the figure, and a best-fit trend line is shown for the data from each state. The trend lines for all four states show that there is a decrease in the proportion KAB crashes with an increase in the proportion of site adjacent to barrier. The “proportion of the site adjacent to a barrier” is included as a variable in the SDF for freeways provided in Chapter 18 of the HSM Supplement (AASHTO 2014). The regression coefficient associated with this variable has a negative sign, which indicates that the predicted proportion of K, A, and B crashes decreases with an increase in the proportion of the site adjacent to barrier. This trend is consistent with that shown in Figure 41 for the four states.

124 Figure 41. Examination of proportion of site adjacent to barrier. Severity Distribution Prediction Model Development This section describes the activities undertaken to develop the models for predicting the crash severity distribution. The following subsections provide a description of the basic model form, an overview of the modeling approach, an overview of the statistical analysis methods, and a discussion of the findings from the model estimation activities. Predictive Model Form The multinomial logit (MNL) model was used as the basic framework for the severity distribution model. The database assembled for model calibration included crash severity level as the dependent variable, with each reported crash serving as one observation. Geometric design features, traffic control features, and traffic characteristics were included as independent variables. The MNL model used in the SDF presented in the HSM Supplement (AASHTO 2014) has a structure that can be described using the following equations. Equation 102 𝑃 𝑆1/𝐶 𝑆 𝑆 𝑆 and Equation 103 𝑃 1 𝑃 𝑃 𝑃 with Equation 104 𝑆 𝑆 , 𝑓 , … 𝑓 , Equation 105 𝑆 𝑆 , 𝑓 , … 𝑓 , Equation 106 𝑆 𝑆 , 𝑓 , … 𝑓 ,

125 where Pl = proportion of FI crashes being described as severity l (l = K, A, B); PC = proportion of FI crashes being described as severity C; Sl = distribution score for severity l; Csdf = local calibration factor; Sl,b = base distribution score for severity l; fl,i = severity adjustment factor for the relationship between severity l of traffic characteristic, geometric element, or traffic control feature i (i = 1 to n); and n = total number of severity adjustment factors. The distribution score is a dimensionless number that indicates the relative frequency of crashes associated with a specific severity level (i.e., K, A, B, or C), given that a fatal or severe crash has occurred. Severity level C is assigned a score of “1/Csdf”. Smaller scores indicate a severity category that is less frequent. Larger scores indicate a severity level that is more frequent. If a severity level has a score smaller than 1/Csdf, then it is less frequent than level C. If a level has a score larger than 1/Csdf, then it is more frequent than level C. Modeling Approach Correlation of Variables with State. A preliminary analysis of the highway safety database indicated that some of the site characteristic variables are correlated with the state in which the site was located. For example, sites for some states tend to use median barrier extensively, while sites in other states tended to use median barrier less frequently. 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 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 described previously, one SDF was planned for each of the site types identified in the following list:  Freeway segment to predict FI crash severity  Ramp entrance speed-change lane to predict FI crash severity  Ramp exit speed-change lane to predict FI crash severity A preliminary regression analysis of the data indicated that the site sample size for speed-change lane equations was too small to develop reliable, multiple-variable site-type-specific models. To overcome this issue, it was decided that a combined modeling approach would be needed to develop one semi- independent model for the three site types combined. Thus, combined modeling was used to estimate one regression model using the data for all three site types. The estimated coefficients would then be used to

126 develop one SDF for each site type (i.e., one set of Equation 102 to Equation 106 for each site type) but with some adjustment factors being common to all SDFs and some unique to a site type. With a combined modeling approach, some severity adjustment factors are common to each of the site- type-specific models. That is, the adjustment factor for traffic characteristic, geometric element, or traffic control feature i is the same in each equation. The regression coefficient associated with each adjustment factor is also the same. Therefore, if the adjustment factor for a given characteristic, element, or feature is a function of variables (e.g., inside shoulder width) and the coefficient for these variables have the same value for a freeway segment and for a ramp entrance speed-change lane, then the adjustment factor was defined in the model to be the same for both site types. This approach recognizes that some characteristics, elements, or features have a similar influence on crash severity, regardless of whether the site is a freeway segment or a speed-change lane. The use of common adjustment factors has the advantage of maximizing the sample size available to estimate the adjustment factor regression coefficient. In this manner, the data for all three site types are pooled to provide a more efficient estimate of the coefficient value for some adjustment factors. The use of common adjustment factors in multiple models requires the simultaneous regression analysis of all models simultaneously. The total log-likelihood statistic for all three models combined was used to determine the best fit regression coefficients. Indicator variables were used to distinguish between the site types in the regression model. Model Development Process. The model development process was based on consideration of the p values for each regression coefficient and the model’s Akaike information criterion (AIC) value. Experience using the AIC for logistic regression modeling revealed that sole reliance on the AIC could lead to models that over-fit the data such that they included an unrealistically large number of independent variables. As a result, the Bayesian information criterion (BIC) was also given consideration during model development because it mitigated the potential for over-fitting the data. Independent variables were added to the model one at a time. When a variable was added (or removed), it was added to (or removed from) all three of the severity distribution score equations (i.e., Equation 104 to Equation 106). The process for evaluating each variable and the criteria for determining whether and how to retain it in the model is outlined in Table 49. The information in the second column of Table 49 describes the process used to develop the predictive model using the base distribution score equations and the independent variable q as an example. The process is equally applicable to any independent variable. The variable could be used to estimate the base distribution score or it could be used to estimate a severity adjustment factor. The first step is to evaluate the base model. This model includes only the intercept coefficient (i.e., bK,0, bA,0, bB,0) in each of the base distribution score equations; there are no adjustment factors. The next step is to evaluate model form 1. This model form includes the variable of interest q in each equation and a unique regression coefficient for each severity level (i.e., bK,q, bA,q, bB,q). Based on consideration of the AIC, BIC, and p values for each coefficient, one of three options is selected. With option 1, the variable is removed from all equations. With option 2, model form 1 is retained. With option 3, one of models 2, 3, or 4 is selected. If option 3 is selected, then two or three of the severity categories are combined and a common regression coefficient (e.g., bKA,q) is used for the combined categories.

127 Table 49. Model development process and decision criteria. Model Form Process and Decision Criteria Example Base Distribution Score Equations Fatal, SK Incapacitating Injury, SA Non- incapacitating Injury, SB Base Initial model form SK,b = bK,0 SA,b = bA,0 SB,b = bB,0 1 Add variable q and an associated regression coefficient to the model for each severity. 1. If AIC does not decrease (relative to base model) or if coefficients are not logical, then remove this variable and move on to the next variable. 2. If all coefficients have a p value < 0.15, then keep this form and go to next variable. 3. If one or more coefficients has a p value > 0.15, then consider one of the model forms below. SK,b = bK,0 + bK,q×q SA,b = bA,0 + bA,q×q SB,b = bB,0 + bB,q×q 2 Combine K and A terms if (1) bB,q has p value < 0.15 and (2) bK,q and bA,q have overlapping confidence intervals and same sign. Keep bB,q in the model. SK,b = bK,0 + bKA,q×q SA,b = bA,0 + bKA,q×q SB,b = bB,0 + bB,q×q 3 Combine A and B terms if (1) bK,q has p value < 0.15 and (2) bA,q and bB,q have overlapping confidence intervals and same sign. Keep bK,q in the model. SK,b = bK,0 + bK,q×q SA,b = bA,0 + bAB,q×q SB,b = bB,0 + bAB,q×q 4 Combine K, A, and B terms if bK,q, bA,q, and bB,q have overlapping confidence intervals and same sign. SK,b = bK,0 + bKAB,q×q SA,b = bA,0 + bKAB,q×q SB,b = bB,0 + bKAB,q×q Note: The p value of 0.15 is used as the threshold value (as opposed to a smaller value) to indicate an acceptable level of coefficient reliability. It is used in conjunction with other criteria that are intended to collectively provide logical, useful, and robust models. Statistical Analysis Methods The nonlinear regression procedure (NLMIXED) in the SAS software was used to estimate the regression model coefficients. This procedure was used because some variations of the regression model were both nonlinear and discontinuous. The log-likelihood function for the MNL logistic distribution was used to determine the best-fit model coefficients. The procedure was set up to estimate model coefficients based on maximum-likelihood methods. Weighted regression was used to quantify the regression coefficients. The log-likelihood of each observation was weighted by the reciprocal of its “number of years in the evaluation period.” The individual weight values wi were normalized by multiplying them by the constant W 𝑛/∑ 𝑤 , where n is the sample size. The normalized weights add up to the actual sample size and result in the covariance matrix of the coefficients being invariant to the scale of the weight variable. Weighted regression was used because of the wide range in study period duration among the study sites. Those sites having a long study period tend to have more crashes reported. When the counts from sites with a long study period are compared to the counts from sites with a short study period, the resulting regression coefficients are biased to emphasize the sites having a long study period. To avoid this bias and ensure that each site was given equal weight in the severity distribution, each observation was weighted using the aforementioned procedure. This issue is discussed further in the subsequent section of this chapter titled Study Period Duration. The measure of model fit is the pseudo R2 developed by McFadden (1974). This statistic provides a useful measure of model fit for multinomial logistic regression because it is based on likelihood maximization. The pseudo R2 compares the log likelihood value for the estimated model with that for the

128 “null” model. The null model includes only an intercept term (i.e., no predictor variables). The pseudo R2 is computed using the following equation. Equation 107 𝑅 1.0 𝐿𝐿𝐿𝐿 where R2 = pseudo R2; LLmodel = log-likelihood for the estimated model; and LLnull = log-likelihood for the null model. Pseudo R2 values can range from 0.0 to 1.0, with larger values indicating a better fit to the data. This interpretation is similar to that for the traditional R2 metric based on ordinary least squares. However, unlike the traditional R2, typical values for the pseudo R2 range from 0.0 to 0.4, with values in the range of 0.2 to 0.4 indicating a very good fit (McFadden 1977). Severity Distribution Prediction Model This section describes the model development, model estimation for each site type, and provides a sensitivity analysis of the predictive models over a range of site characteristics. Model Development. This subsection describes the regression model and the methods used to estimate its coefficients. The regression model is generalized to accommodate freeway segments, ramp entrance speed-change lanes, and ramp exit speed-change lanes. The generalized form shows the variables included in the model. For some variables, indicator variables are used to determine when the corresponding adjustment factor is applicable. The following equations describe the regression model that was calibrated using the severity data. A. If the observation corresponds to a freeway segment, the following model is used: Equation 108 𝑃 , 𝑆 ,1 𝑆 . 𝑆 , 𝑆 , and Equation 109 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 110 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 111 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 112 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 113 𝑆 , , exp 𝑏 , , Equation 114 𝑆 , , exp 𝑏 , , Equation 115 𝑆 , , exp 𝑏 , , Equation 116 𝑓 , , exp 𝑏 , , 0.5 𝑃 𝑃

129 Equation 117 𝑓 , , exp 𝑏 , , 𝑃 Equation 118 𝑓 , , exp 𝑏 , , 𝑃 , Equation 119 𝑓 , , exp 𝑏 , , 𝑃 , Equation 120 𝑓 , , exp 𝑏 , , 𝑃 , Equation 121 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 122 𝑃 , 𝑆 ,1 𝑆 . 𝑆 , 𝑆 , and Equation 123 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 124 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 125 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 126 𝑆 , , exp 𝑏 , , Equation 127 𝑆 , , exp 𝑏 , , Equation 128 𝑓 , , exp 𝑏 , , 𝑃 C. If the observation corresponds to a ramp exit speed-change lane, the following model is used: Equation 129 𝑃 , 𝑆 ,1 𝑆 . 𝑆 , 𝑆 , and Equation 130 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 131 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 132 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , 𝐶 , , Equation 133 𝑆 , , exp 𝑏 , , Equation 134 𝑆 , , exp 𝑏 , ,

130 Equation 135 𝑓 , , exp 𝑏 , , 𝑃 where bi = regression coefficient for condition i; Cstate,w,l = adjustment factor for crashes at a comparison site having severity l (l = K, A, B) at site type w (w = fs: freeway segment, en: ramp entrance speed-change lane, ex: ramp exit speed-change lane, ast: all site types); fbar,w,l = severity adjustment factor for the relationship between barrier presence and severity l (l = K, A, B) at site type w (w = fs: freeway segment, en: ramp entrance speed-change lane, ex: ramp exit speed- change lane, ast: all site types); fphv,w,l = severity adjustment factor for the relationship between “proportion of AADT during hours with high volume” and severity l at site type w; fptsu,w,l = severity adjustment factor for the relationship between PTSU operation and severity l at site type w; 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 S.R. 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); Pw,l = proporton of FI crashes being described as severity l (l = K, A, B) at site type w; Phv = proportion of AADT during hours where volume exceeds 1,000 veh/h/ln; Pib = proportion of site length with a barrier present in the median (i.e., inside); Pob = proportion of site length with a barrier present on the outside (roadside); Pt,ptsu = proportion of time during the average day that PTSU operates; Sb,w,l = base distribution score for severity l at site type w; and Sw,l = distribution score for severity l at site type w. 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 and Pob. The final form of the regression model reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described in the previous paragraphs represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defensible and properly bounded. AFs for other variables were also examined but were not found to be helpful in explaining the variation in crash severity among sites. These variables included:  Directional AADT volume per lane  Dynamic versus static PTSU operation  Horizontal curve presence  Lane width  Shoulder rumble strip presence The influence of the aforementioned variables could not be reliably quantified using regression analysis because they have a relatively small effect on crash severity, are correlated with other variables in the data, or both. This finding does not rule out the possibility that these factors have an influence on crash

131 severity. It is possible that they have some influence, but it will likely require the use of a larger database with more sites having PTSU operation. Model Estimation. The results of the regression model estimation are presented in Table 50. The pseudo R2 for the model is 0.39, which is considered to be a very good fit for a logistic model. Table 50. Predictive model estimation statistics – severity distribution. Model Statistics Value −2 Log-Likelihood for Null Model: 12,037 −2 Log-Likelihood for Full Model: 7,282.4 Pseudo R2: 0.39 AIC for Full Model: 7,320.4 Observations n: 4,533 fatal or injury crashes Estimated Coefficient Values Variable Description Value Std. Error t-statistic b0,ast,K Intercept for K severity at all site types −5.166 0.2811 −18.37 b0,fs,A Intercept for A severity at freeway segments −2.801 0.1701 −16.46 b0,fs,B Intercept for B severity at freeway segments −0.7990 0.1554 −5.14 b0,en,A Intercept for A severity at ramp ent. speed-change lanes −3.248 0.3337 −9.73 b0,en,B Intercept for B severity at ramp ent. speed-change lanes −0.9430 0.2085 −4.52 b0,ex,A Intercept for A severity at ramp exit speed-change lanes −2.853 0.3085 −9.25 b0,ex,B Intercept for B severity at ramp exit speed-change lanes −0.8767 0.2129 −4.12 bbar,ast,KAB Proportion of site adjacent to barrier; for KAB −0.4597 0.1883 −2.44 bphv,ast,KAB Proportion of AADT during high volume hours; for KAB −0.9931 0.4432 −2.24 bptsu,ast,K Proportion of time that PTSU operates; for K severity −4.313 2.341 −1.84 bptsu,ast,A Proportion of time that PTSU operates; for A severity −0.7180 0.6047 −1.19 bptsu,ast,B Proportion of time that PTSU operates; for B severity 0.1013 0.4080 0.25 bohio,ast,KAB Location in Ohio; for KAB severity 1.317 0.1177 11.19 bMN35,ast,KAB Location on I-35W in Minnesota; for KAB severity 0.2119 0.2282 0.93 bV264,ast,KAB Location on I-264 in Virginia; for KAB severity 2.894 0.1629 17.76 bG400,ast,KAB Location on I-495 in Virginia; for KAB severity 0.1204 0.1936 0.62 bV495,ast,KAB Location on I-66 in Virginia; for KAB severity 3.975 0.3180 12.5 bVA66,ast,KAB Location on S.R. 400 in Georgia; for KAB severity 2.307 0.1472 15.67 bGA85,ast,KAB Location on I-85 in Georgia; for KAB severity 1.598 0.1650 9.69 The t-statistics listed in the far right column of Table 50 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 2.0 indicate that the hypothesis can be rejected, with the probability of error in this conclusion being less than 0.05. For those variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the variable was important to the model, and its trend was found to be logical (even if the specific value was not known with a great deal of certainty as applied to this database). An indicator variable for the state of Ohio was included in the regression model. The coefficient for this variable bohio,ast,KAB is shown in Table 50; It is statistically significant. Its value indicates that the sites in Ohio have a larger proportion of crashes with K, A, or B severity than in the other states. This difference cannot be explained by state-to-state differences in value for the variables used in the model. It is likely due to differences between Ohio and the other states in either (a) the level of reporting for severity C crashes or (b) the criteria used to identify a C severity crash.

132 The indicator variable for Ohio was removed, and an indicator variable was included in the regression model for the sites located in Minnesota. The coefficient for this variable was approximately the same as that for the Ohio indicator variable but opposite in sign. The coefficients for all other variables were unchanged. It was statistically significant. Its value indicates that the sites in Minnesota have a smaller proportion of crashes with K, A, or B severity than in the other states. This difference cannot be explained by state-to-state differences in value for the variables used in the model. It is likely due to differences between Minnesota and the other states in either (a) the level of reporting for severity C crashes or (b) the criteria used to identify a C severity crash. Note that an indicator variable for Minnesota or 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 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. Model for Freeway Segments. The coefficients in Table 50 were combined with Equation 108 to Equation 120 to obtain the estimated SDF for freeway segments. The form of this model is described by the following equations: Equation 136 𝑃 , 𝑆 ,1/𝐶 , 𝑆 . 𝑆 , 𝑆 , and Equation 137 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 138 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 139 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 140 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 141 𝑆 , , exp 5.166 Equation 142 𝑆 , , exp 2.801 Equation 143 𝑆 , , exp 0.7990 Equation 144 𝑓 , , exp 0.4597 0.5 𝑃 𝑃 Equation 145 𝑓 , , exp 0.9931 𝑃 Equation 146 𝑓 , , exp 4.313 𝑃 , Equation 147 𝑓 , , exp 0.7180 𝑃 , Equation 148 𝑓 , , exp 0.1013 𝑃 , All variables are previously defined.

133 The coefficient in the adjustment factor for the “proportion of AADT during hours with high volume” (i.e., Equation 145) has a value of −0.9931. The negative value of the coefficient indicates that the probability of an FI crash having a K, A, or B severity decreases with an increase in the proportion of AADT during hours with high volume. This relationship was previously noted during the exploratory analysis. The trend lines shown in Figure 40 indicate that it exists in the data for three of the four states. The coefficient value in Equation 145 compares favorably with the coefficient values used in the freeway SDF provided in Chapter 18 of the HSM Supplement (AASHTO 2014). Specifically, the coefficient values for the Phv variable in the HSM SDF range from −0.853 to −0.924, depending on the severity level. This sensitivity to severity level could not be reliably quantified in the data because the variable Phv was found to be correlated with the variable describing the “proportion of time during the average day that PTSU operates” Pt,ptsu. By fixing the value of the Phv coefficient at the same value for all three severity levels, some of this correlation was mitigated such that the fixed value of −0.9931 could be reliably quantified. The coefficient in the adjustment factor for the “proportion of site adjacent to a barrier” (i.e., Equation 144) has a value of −0.4597. The negative value of the coefficient indicates that the probability of an FI crash having a K, A, or B severity decreases with an increase in the proportion of site adjacent to a barrier. This relationship was previously noted during the exploratory analysis. The trend lines shown in Figure 41 indicate that it exists in the data for all four states. The coefficient value in Equation 144 compares favorably with the coefficient values used in the freeway SDF provided in Chapter 18 of the HSM Supplement (AASHTO 2014). Specifically, the coefficient values for the “0.5 × (Pib + Pob)” variable construct in the HSM SDF range from −0.250 to −0.388, depending on the severity level. This sensitivity to severity level could not be reliably quantified in the data because the construct “0.5 × (Pib + Pob)” was found to be correlated with the facility/state indicator variables. This correlation can be seen in Figure 41, where the midpoint of each trend line and its associated proportion KAB crashes varies from state to state. By fixing the value of the “0.5 × (Pib + Pob)” coefficient at the same value for all three severity levels, some of this correlation was mitigated such that the fixed value of −0.4597 could be reliably quantified. Model for Ramp Entrance Speed-Change Lanes. The coefficients in Table 50 were combined with Equation 122 to Equation 128 to obtain the estimated SDF for ramp entrance speed-change lanes. The form of this model is described by the following equations. Equation 149 𝑃 , 𝑆 ,1/𝐶 , 𝑆 . 𝑆 , 𝑆 , and Equation 150 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 151 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 152 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 153 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 154 𝑆 , , exp 5.166 Equation 155 𝑆 , , exp 3.248 Equation 156 𝑆 , , exp 0.9430

134 Equation 157 𝑓 , , exp 0.4597 𝑃 All variables are previously defined. The adjustment factors fphv and fptsu in the preceding equations are provided in Equation 145 to Equation 148. Model for Ramp Exit Speed-Change Lanes. The coefficients in Table 50 were combined with Equation 129 through Equation 135 to obtain the estimated SDF for ramp exit speed-change lanes. The form of this model is described by the following equations. Equation 158 𝑃 , 𝑆 ,1/𝐶 , 𝑆 . 𝑆 , 𝑆 , and Equation 159 𝑃 , 1 𝑃 , 𝑃 , 𝑃 , with, Equation 160 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 161 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 162 𝑆 , 𝑆 , , 𝑓 , , 𝑓 , , 𝑓 , , Equation 163 𝑆 , , exp 5.166 Equation 164 𝑆 , , exp 2.853 Equation 165 𝑆 , , exp 0.8767 Equation 166 𝑓 , , exp 0.4597 𝑃 All variables are previously defined. The adjustment factors fphv and fptsu in the preceding equations are provided in Equation 145 through Equation 148. Sensitivity Analysis. This section examines the relationship between the predicted severity distribution and the variables included in the site-type SDFs described in the previous sections. Each site-type SDF includes a base distribution score equation for each of the three severity levels addressed in the model (i.e., Sb,K, Sb,A, and Sb,B). Based on the state and facility indicator variables in the model, the coefficients in each base distribution score equation correspond to Minnesota freeways. As a result, the three SDFs predict the severity distribution for Minnesota. As suggested by the discussion associated with Table 46, the severity distribution varies widely among the four states in the database. The SDFs could be “calibrated” (i.e., the coefficients in each base distribution score equation increased or decreased by a constant amount) to provide a good fit to any one of the four states. However, the SDF prediction would be notably biased when applied to the other three states. This issue was noted in the discussion of associated with Table 46 and underscores the importance of calibrating the SDF to local conditions. For the purpose of this sensitivity analysis, the three SDFs were calibrated to provide a best fit to the severity distribution predicted by the freeway SDF in Chapter 18 of the HSM Supplement (AASHTO 2014). This approach would allow for some comparison of the predicted values from the proposed SDFs with that from the HSM.

135 The procedure in Section B.1.4 of the HSM Supplement (AASHTO 2014) was used to compute one calibration factor for the three proposed SDFs. The distribution predicted by the HSM SDF was used in the procedure as the “observed” values, and that predicted by the proposed SDFs was used as the “predicted” values. Unique calibration factors for each site-type SDF were not estimated because the HSM SDF is offered as being equally applicable to freeway segments and speed-change lanes. A set of hypothetical site scenarios were developed for purpose of calibrating the three proposed SDFs and then comparing them to the HSM SDF. All scenarios were based on urban conditions with no PTSU operation. Nine scenarios were developed for each of the three site types. Each set of nine scenarios represents three values for the “proportion of AADT during high volume hours” and three values for the “proportion of site adjacent to barrier.” The HSM SDF was applied to each of the 27 scenarios (= 3 site types × 9 variable combinations). The HSM SDF variables for proportion of site with rumble strips, proportion of site with horizontal curve, and lane width were set at average values for the sites in the database used to estimate the proposed SDFs. The HSM SDF and the proposed SDFs (each with a calibration factor of 1.0) were applied to each of the 27 scenarios. The calibration procedure in Section B.1.4 of the HSM supplement was used to compute the one calibration factor that adjusts the proposed SDFs to provide a best fit to the HSM SDF. The calibration factor Csdf,fs was computed as 1.96 (as noted in a previous paragraph, the same value is used for Csdf,en and Csdf,ex). The calibrated SDFs were used to predict the proportion of FI crashes that are K, A, B, or C severity. The sum of the predicted K, A, and B proportions are compared in Figure 42 with those from the HSM SDF. The trend line shown in this figure is an “x = y” line, such that if the two models were in perfect agreement, the data points would lie on this line. The data points are shown to lie very near the line, which suggests the proposed SDFs (when calibrated) are in good agreement with the HSM SDF. This finding confirms that the proposed SDFs are consistent with the HSM SDF in representing the influence of “proportion of AADT during high volume hours” and “proportion of site adjacent to barrier” on severity. If the proposed SDFs were not calibrated, then the group of data points would maintain the same spatial arrangement, but they would shift to the left as a group, away from the “x = y” line. Figure 42. Comparison of HSM SDF with proposed SDFs.

136 Based on this examination, the proposed SDFs have a sensitivity to the input variables (proportion of AADT in high volume hours and proportion of site adjacent to barrier) that is consistent with that of the HSM SDF. The wide disparity in severity distributions among states is noted here again, as it was in the discussion associated with Table 46. It emphasizes the need for local calibration of the SDF. State-to-state differences in crash reporting and other factors also result in a wide disparity in crash frequency and an equivalent need for local calibration of crash frequency prediction models. This disparity is present in the data used to develop the crash frequency prediction models documented in Chapter 5, as identified by the indicator variables for specific states that are included in the models. In fact, this disparity is typically present in any multi-state database used to prediction models. The developers of the HSM (AASHTO 2010) recognized the aforementioned state-to-state disparity in crash frequency. They also anticipated that some agencies may not undertake local calibration of the HSM CPMs before using them for safety evaluation. In light of these concerns, the HSM developers adjusted the HSM model coefficients prior to HSM publication such that all models likely to be compared were calibrated to a common state. This approach mitigated some of the issues associated with the comparison of results from uncalibrated models developed using data for different states (e.g., comparing the results from the CPM for signalized intersections developed with data from State A with those for the CPM for unsignalized intersections developed with data from State B). Fortunately, in recent years this issue is less of a concern because agencies are now more inclined to locally calibrate the HSM crash frequency prediction models. However, there is also some anecdotal evidence that the HSM SDFs are still not being locally calibrated. For the same reasons that the HSM developers undertook a “common state” calibration of the crash frequency prediction models, the researchers adjusted the base coefficient values in the proposed SDFs such that the proposed SDF predictions provide a best-fit to the HSM SDF predictions. This adjustment facilitates more equitable comparisons between the two SDFs, should analysts use both methods to evaluate a freeway without first locally calibrating the SDFs. To seamlessly implement this adjustment in the proposed SDF, the natural log of the aforementioned calibration factor (i.e., 1.96) was added to the coefficient in the base distribution score equations for each site-type SDF prior to presenting them in the proposed HSM text (contained in the PTSU Safety Evaluation Guidelines document). The remainder of this section lists the predicted severity distribution obtained from each of the site-type SDFs. The three SDFs were calibrated using the aforementioned calibration factor (i.e., 1.96) such that they are comparable to the distribution values obtained from the HSM SDF (i.e., the distribution representing the states of California, Maine, and Washington). As noted previously, if the proposed SDFs are left uncalibrated in this manner, the predicted severity distribution would be most applicable to Minnesota freeways. Table 51 lists the predicted severity distribution for freeway segments as a function of PTSU operating time duration, proportion AADT during high volume hours, and proportion of site adjacent to barrier. The proportions shown indicate that the proportion of K, A, and B crashes decrease (while the proportion of C crashes increase) with an increase in the proportion of site adjacent to barrier. This trend is logical and consistent with the HSM SDF. The proportions shown in the table also indicate that an increase in the proportion AADT during high-volume hours corresponds to a decrease in the proportion of K, A, and B crashes. Finally, the proportions shown indicate that an increase in the proportion of time that PTSU operates corresponds to a decrease in K and A crashes (while the proportion of B crashes increases and the proportion of C crashes stays about the same). These latter changes in the K and B severity levels were previously noted in the examination of simple crash proportions associated with Table 48.

137 Table 51. Comparison of predicted severity distribution for freeway segments. Proportion Time PTSU Operates Proportion AADT in High Volume Hours Proportion Seg. Adjacent to Barrier Crash Proportion by Severity K A B C 0.0 0.05 0.1 0.005 0.056 0.417 0.521 0.5 0.005 0.051 0.378 0.567 0.9 0.004 0.046 0.339 0.611 0.25 0.1 0.005 0.051 0.375 0.570 0.5 0.004 0.045 0.336 0.615 0.9 0.004 0.040 0.299 0.657 0.45 0.1 0.004 0.045 0.333 0.618 0.5 0.004 0.040 0.296 0.660 0.9 0.003 0.035 0.261 0.700 0.5 0.05 0.1 0.001 0.039 0.439 0.521 0.5 0.001 0.036 0.397 0.567 0.9 0.000 0.032 0.357 0.611 0.25 0.1 0.001 0.035 0.394 0.570 0.5 0.000 0.032 0.353 0.615 0.9 0.000 0.028 0.314 0.657 0.45 0.1 0.000 0.031 0.350 0.618 0.5 0.000 0.028 0.311 0.660 0.9 0.000 0.025 0.275 0.700 Note: Results shown are based on using a calibration factor Csdf,fs = 1.96. Table 52 lists the predicted severity distribution for ramp entrance speed-change lanes. With regard to the variables in the second and third columns of the table, the proportions shown indicate that the trends noted in Table 51 for freeway segments are the same for ramp entrance speed-change lanes. The proportions shown also indicate that an increase in the proportion of time that PTSU operates corresponds to a decrease in K, A, and C crashes (while the proportion of B crashes increases). These changes in the K and B severity levels were previously noted in the examination of simple crash proportions associated with Table 48.

138 Table 52. Comparison of predicted severity distribution for ramp entrance speed-change lanes. Proportion Time PTSU Operates Proportion AADT in High Volume Hours Proportion Seg. Adjacent to Barrier Crash Proportion by Severity K A B C 0.0 0.05 0.1 0.006 0.039 0.391 0.564 0.5 0.005 0.035 0.351 0.609 0.9 0.005 0.031 0.313 0.651 0.25 0.1 0.005 0.035 0.348 0.612 0.5 0.005 0.031 0.310 0.655 0.9 0.004 0.027 0.274 0.695 0.45 0.1 0.004 0.031 0.307 0.658 0.5 0.004 0.027 0.271 0.698 0.9 0.003 0.024 0.237 0.735 0.5 0.05 0.1 0.001 0.027 0.410 0.562 0.5 0.001 0.024 0.368 0.607 0.9 0.001 0.022 0.328 0.650 0.25 0.1 0.001 0.024 0.365 0.610 0.5 0.001 0.022 0.325 0.653 0.9 0.000 0.019 0.287 0.693 0.45 0.1 0.001 0.021 0.322 0.656 0.5 0.000 0.019 0.284 0.696 0.9 0.000 0.017 0.249 0.734 Note: Results shown are based on using a calibration factor Csdf,en = 1.96. Table 53 lists the predicted severity distribution for ramp exit speed-change lanes. With regard to the variables in the first three columns of the table, the proportions shown indicate that the trends noted in Table 51 for freeway segments are the same for ramp exit speed-change lanes. By comparing across the proportions in Table 51, Table 52, and Table 53, the proportions listed suggest that freeway segments tend to have a smaller proportion of K and C crashes than speed-change lanes. Ramp entrance speed-change lanes tend to have the smallest proportion of A and B crashes, and the largest proportion of C crashes. Of particular note in these findings is that PTSU operation tends to be associated with a reduction in the most severe (i.e., K and A) crashes and an increase in B crashes. These trends will remain the same in the SDFs, regardless of the calibration factor value, because the calibration factor adjusts the KAB levels as a group in relation to the C severity level.

139 Table 53. Comparison of predicted severity distribution for ramp exit speed-change lanes. Proportion Time PTSU Operates Proportion AADT in High Volume Hours Proportion Seg. Adjacent to Barrier Crash Proportion by Severity K A B C 0.0 0.05 0.1 0.005 0.055 0.400 0.539 0.5 0.005 0.050 0.361 0.585 0.9 0.004 0.045 0.322 0.628 0.25 0.1 0.005 0.050 0.357 0.588 0.5 0.004 0.044 0.320 0.632 0.9 0.004 0.039 0.283 0.673 0.45 0.1 0.004 0.044 0.317 0.635 0.5 0.004 0.039 0.281 0.677 0.9 0.003 0.034 0.247 0.716 0.5 0.05 0.1 0.001 0.039 0.421 0.540 0.5 0.001 0.035 0.380 0.585 0.9 0.001 0.031 0.339 0.629 0.25 0.1 0.001 0.035 0.376 0.589 0.5 0.001 0.031 0.336 0.632 0.9 0.000 0.027 0.298 0.674 0.45 0.1 0.001 0.031 0.333 0.636 0.5 0.000 0.027 0.295 0.677 0.9 0.000 0.024 0.260 0.716 Note: Results shown are based on using a calibration factor Csdf,ex = 1.96. Crash Type Distribution This section describes the crash type distribution that was derived from the database assembled for the project. This distribution is described using a table of distribution values computed from the database. This type of table presentation is consistent with the other chapters in Part C of the HSM. The development of a distribution function using regression analysis was considered but not undertaken because the crash sample size is relatively small for most of the 10 crash types of interest and insufficient to develop statistically valid regression model relationships. Database Summary This section summarizes the crash data assembled for the purpose of computing the crash type distribution. Separate distributions are computed for FI crashes and PDO crashes. The remaining subsections describe the crash characteristics of the sites used to compute the crash type distribution. The purpose of this summary is to provide information about the range of data included in the database and to provide some insight to guide the development of the proposed crash distribution table. The discussion in this section is not intended to indicate conclusive results or recommendations. The proposed crash distribution tables are documented below. Data for the state of Hawaii were included in the database used to develop the crash type distributions because the Hawaii crash data include sufficient information to determine crash type. The Hawaii data could not be used to develop the SDFs (described in the first part of this chapter) because they do not distinguish between the A, B, and C severity levels. The database includes data for 728 study sites, collectively representing 14 freeway facilities in five states, with a total length of 164.8 miles. About 25 percent of the total mileage consists of highway facilities with PTSU operation during 1 or more hours of the day. Additional information about the study sites is provided in Chapter 4, in the Database Summary section.

140 FI Crash Type Distribution The crash type distribution of reported FI crashes is shown in Table 54. The crashes are categorized by state, site type, and PTSU operation. There are a total of 728 study sites collectively representing 3,493 site-years. The average study period duration is 4.8 years/site (= 3,493 site-years /728 sites); it ranged from 1 to 5 years/site. These sites collectively experienced a total of 4,807 FI crashes during the study period. Table 54. FI crash type distribution by state, site type, and PTSU operation. State Site Type PTSU Operation Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type (Crashes) Single-Vehicle Crash Count by Crash Type (Crashes) Total Crashes Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other GA FS No 18 74 0 7 114 12 0 0 3 0 1 1 138 Yes 30 121 0 15 199 41 3 1 11 3 2 13 288 ENSCL No 4 17 0 1 16 1 0 0 0 0 0 0 18 Yes 5 21 0 2 7 2 0 0 1 0 0 0 12 EXSCL No 3 12 0 1 10 3 0 0 1 1 0 0 16 Yes 3 12 0 0 16 2 0 0 0 0 0 0 18 HI FS No 12 60 2 4 13 3 1 0 7 0 1 5 36 Yes 27 135 0 8 131 13 3 0 23 8 1 11 198 ENSCL No 2 10 0 0 1 0 0 0 0 1 0 1 3 Yes 5 25 0 2 13 3 2 0 3 1 0 1 25 EXSCL No 3 15 0 0 5 2 0 0 0 0 0 1 8 Yes 3 15 0 0 2 1 0 0 0 0 0 0 3 MN FS No 164 820 3 8 364 47 31 2 91 2 7 60 615 Yes 11 55 2 3 125 11 3 0 6 0 0 1 151 ENSCL No 44 220 1 1 36 7 4 0 11 1 2 9 72 EXSCL No 43 215 0 0 20 4 0 0 4 0 1 2 31 Yes 1 5 0 0 8 2 0 0 0 0 0 0 10 OH FS No 123 589 0 8 452 188 9 9 207 11 13 34 931 ENSCL No 25 125 0 1 26 9 0 0 9 1 1 2 49 EXSCL No 22 110 0 1 23 21 2 0 22 1 1 4 75 VA FS No 63 295 0 48 447 34 11 2 68 1 3 16 630 Yes 72 332 0 89 847 72 18 0 148 2 3 17 1196 ENSCL No 14 60 2 5 42 3 0 0 5 0 1 3 61 Yes 8 40 0 8 57 7 1 0 11 0 0 4 88 EXSCL No 11 55 1 1 29 1 0 0 3 0 0 1 36 Yes 12 55 0 4 79 7 2 0 7 0 0 0 99 Grand Total 728 3493 11 217 3082 496 90 14 641 33 37 186 4807 Notes: Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments. The ten crash types listed in the column headings of Table 54 are consistent with those crash types listed in each chapter of HSM Part C (AASHTO 2010) and Chapter 18 of the HSM Supplement (AASHTO 2014). Five of the types are categorized as multiple-vehicle crashes. The other five are categorized as single-vehicle crashes. Multiple-vehicle FI crash types accounted for about 81 percent of the 4,807 FI crashes. Rear-end FI crashes represent the most frequently occurring crash type (79 percent) among multiple-vehicle FI crashes. In the crash data collected for NCHRP Project 17-45, multiple-vehicle FI crash types accounted for 71 percent of the crashes on urban freeways and rear-end FI crashes accounted for 75 percent of multiple- vehicle FI crashes (Bonneson et al. 2012a). These trends suggest that the sites represented in Table 54 may collectively experience more congestion than the sites included in the NCHRP Project 17-45 database because an increase in congestion level is often associated with an increase in rear-end crashes.

141 The FI crash type distribution categorized by site type and PTSU operation is shown in Table 55. About 87 percent of the reported FI crashes are on freeway segments, with roughly an even split of the remaining 13 percent among the two speed-change lane types. About 40 percent of the crashes are located on sites at which PTSU operates during some portion of the day. Table 55. FI crash type distribution by site type and PTSU operation. Site Type PTSU Operation Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type (Crashes) Single-Vehicle Crash Count by Crash Type (Crashes) Total Crashes Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 380 1838 5 75 1390 284 52 13 376 14 25 116 2350 Yes 140 643 2 115 1302 137 27 1 188 13 6 42 1833 ENSCL No 89 432 3 8 121 20 4 0 25 3 4 15 203 Yes 18 86 0 12 77 12 3 0 15 1 0 5 125 EXSCL No 82 407 1 3 87 31 2 0 30 2 2 8 166 Yes 19 87 0 4 105 12 2 0 7 0 0 0 130 Grand Total 728 3493 11 217 3082 496 90 14 641 33 37 186 4807 Notes: Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments. PDO Crash Type Distribution The crash type distribution of reported PDO crashes is shown in Table 56. The crashes are categorized by state, site type, and PTSU operation. There are a total of 728 study sites that collectively experienced a total of 11,937 PDO crashes during the study period. Multiple-vehicle PDO crash types accounted for about 82 percent of the 11,937 PDO crashes. Rear-end PDO crashes represent the most frequently occurring crash type (73 percent) among multiple-vehicle PDO crashes. In the crash data collected for NCHRP Project 17-45, rear-end PDO crashes accounted for 69 percent of multiple-vehicle PDO crashes (Bonneson et al. 2012a).

142 Table 56. PDO crash type distribution by state, site type, and PTSU operation. State Site Type PTSU Operation Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type (Crashes) Single-Vehicle Crash Count by Crash Type (Crashes) Total Crashes Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other GA FS No 18 74 1 20 258 112 4 0 11 10 3 5 424 Yes 30 121 3 47 535 161 13 1 35 18 3 24 840 ENSCL No 4 17 0 6 26 11 0 0 3 1 0 2 49 Yes 5 21 0 2 28 4 0 0 3 1 0 0 38 EXSCL No 3 12 0 6 21 11 2 0 1 1 0 4 46 Yes 3 12 0 3 34 10 1 0 2 0 0 0 50 HI FS No 12 60 0 2 14 7 1 0 7 2 1 2 36 Yes 27 135 0 5 63 16 6 0 12 3 0 3 108 ENSCL No 2 10 0 1 1 0 0 0 0 1 0 0 3 Yes 5 25 0 0 7 1 0 0 2 0 1 0 11 EXSCL No 3 15 0 0 2 0 1 0 2 0 0 0 5 Yes 3 15 0 2 3 3 2 0 1 0 0 0 11 MN FS No 164 820 8 33 948 285 66 10 367 8 10 50 1785 Yes 11 55 0 1 381 79 7 0 18 0 1 3 490 ENSCL No 44 220 2 6 106 44 10 0 53 1 0 6 228 EXSCL No 43 215 0 2 56 16 2 1 13 0 0 0 90 Yes 1 5 1 1 22 4 0 0 1 0 0 0 29 OH FS No 123 589 0 17 1272 660 72 102 457 86 24 55 2745 ENSCL No 25 125 0 1 76 41 5 5 48 6 1 5 188 EXSCL No 22 110 0 0 83 58 7 9 41 5 3 7 213 VA FS No 63 295 3 90 878 117 7 21 156 3 3 8 1286 Yes 72 332 1 164 1856 269 12 18 253 6 7 6 2592 ENSCL No 14 60 0 15 66 24 0 4 13 0 0 0 122 Yes 8 40 0 18 148 23 1 2 18 0 0 0 210 EXSCL No 11 55 0 4 56 8 0 2 10 1 0 1 82 Yes 12 55 0 15 187 24 1 4 23 0 0 2 256 Grand Total 728 3493 19 461 7127 1988 220 179 1550 153 57 183 11,937 Notes: Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments. The PDO crash type distribution categorized by site type and PTSU operation is shown in Table 57. About 86 percent of the reported PDO crashes are located on freeway segments, with roughly an even split of the remaining 14 percent among the two speed-change lane types. About 39 percent of the crashes are located on sites at which PTSU operates during some portion of the day. Table 57. PDO crash type distribution by site type and PTSU operation. Site Type PTSU Operation Number of Sites Total Years Multiple-Vehicle Crash Count by Crash Type (Crashes) Single-Vehicle Crash Count by Crash Type (Crashes) Total Crashes Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 380 1838 12 162 3370 1181 150 133 998 109 41 120 6276 Yes 140 643 4 217 2835 525 38 19 318 27 11 36 4030 ENSCL No 89 432 2 29 275 120 15 9 117 9 1 13 590 Yes 18 86 0 20 183 28 1 2 23 1 1 0 259 EXSCL No 82 407 0 12 218 93 12 12 67 7 3 12 436 Yes 19 87 1 21 246 41 4 4 27 0 0 2 346 Grand Total 728 3493 19 461 7127 1988 220 179 1550 153 57 183 11,937 Notes: Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments.

143 Crash Type Distribution Table Development This section describes the activities undertaken to develop the proposed crash type distribution tables. The first subsection describes the technique used to minimize the bias in crash proportions due to unequal study period duration among sites. The second subsection describes the procedure used to compute the standard error of the proportion associated with each cell of the distribution table. Study Period Duration In many HSM-based freeway evaluations, the analyst desires an estimate of the predicted crash frequency for a given crash type or severity level. In this situation, the analyst may use a predictive model or other means to estimate the average total crash frequency. This average is then used with the appropriate crash distribution value to compute the predicted crash frequency for the desired crash category. This calculation is described with the following equation. Equation 167 𝑁 𝑃 𝑁 where Ni = predicted crash frequency for crash category i, crashes/yr; Pi = predicted proportion of crashes being described as crash category i; and Nt = average total crash frequency, crashes/yr. There are several methods available to estimate the average total crash frequency for a site. One method is to use a predictive model to compute the predicted total crash frequency and use this value as the average total crash frequency. Other methods are described in Section C.7 of the HSM (AASHTO 2010). The predicted crash distribution proportions are used as an estimate of the predicted probability of a crash being described as crash category i. These proportions must be applicable to the site of interest (if they are not, then there is a possibility that the estimates are biased). “Applicability” is achieved when the distribution estimates are developed using data for similar sites (e.g., same site type, PTSU operation). Applicability can also be achieved when the distribution is computed using a locally calibrated SDF that includes variables for influential site characteristics (e.g., same proportion of site adjacent to barrier). The data assembled to develop the predicted crash distribution proportions (in either table or SDF form) must include the crash history for a large number of sites to ensure that the distribution proportions are statistically valid. Tables can include categories, and SDFs can include variables to control for observed differences among sites. This approach allows the analyst to select the desired proportion from a table (or compute it using an SDF), with the understanding that the predicted distribution proportions are based on sites that are similar to the site of interest. An issue emerges when the data assembled to compute the crash distribution proportions has, on a site- by-site basis, a wide range in the number of years of crash history. Those sites having many years of crash history tend to have more crashes reported. When the counts from sites with many years of crash history are combined with the counts from sites with a few years of crash history, the resulting distribution proportions weigh more heavily the sites having many years. This uneven weight (i.e., number of years) can create bias in the distribution when the number-of-years variable is correlated with unobserved site characteristics that influence the crash distribution. For example, if sites with many years of crash history are also found to have a higher speed, then number-of-years would be correlated with the crash severity distribution since higher speeds are likely associated with more severe crashes. To illustrate the points raised in the previous paragraph, consider the crash histories shown in Table 58 for four sites. The number of years of crash data for each site is indicated in column 2. Consider Site 2, which has 3 years of crash history. Thirty crashes of crash type X (e.g., FI crashes) occurred during the 3-

144 year period, and a total of 60 crashes occurred during the 3 years. The proportion of crash type X at this site is 0.5 (= 30/60). Table 58. Crash distribution example. Site Years Crash Type X Total Crashes Proportion Crash Type X (crashes/year) Total Crashes (crashes/year) Proportion 1 10 120 200 0.6 12 20 0.6 2 3 30 60 0.5 10 20 0.5 3 9 90 180 0.5 10 20 0.5 4 2 16 40 0.4 8 20 0.4 Total: 256 480 0.53 40 80 0.5 If it is rationalized that each year of data has equal weight (such that sites with more years of data are more informative about the distribution), then the proportion of crash type X in the database is computed using the sum of the third and fourth columns, as shown in the last row of the table. The average proportion of crash type X is computed to be 0.53 (= 256/480). Further examination of the fifth column indicates that two sites have a proportion of 0.5; one site has a proportion of 0.6; and the other has a proportion of 0.4. The trend here indicates that half of the sites have a proportion of 0.5. Although the sample size is small, the distribution of proportions is suggestive of a bell shape (i.e., normal distribution) and indicates that the typical site has a proportion of 0.5 (not 0.53). If an analyst wants an estimate of the proportion of crash type X for a new site, the typical value of 0.5 is the best estimate. If it is rationalized that each site has equal weight, then the proportion of crash type X in the database is computed using the sum from columns 6 and 7, as shown in the last row of the table. These two columns show the computed crash frequency (in crashes per year) for each site. The average proportion of crash type X is computed to be 0.50 (= 40/80). By giving each site equal weight when computing the distribution proportions, we obtain proportions that are representative of a typical site of interest. It follows that the value of 0.53 is not representative of the typical site. Rather, it is biased by the unbalanced number of years among sites. If the variability in the number of years is increased (e.g., one site has many years and the others have one year), then the difference between the proportions obtained by the two approaches (i.e., approach 1: each year of data has equal weight versus approach 2: each site has equal weight) will also increase. In contrast, if each site has the same number of years of data, then the same proportion will be obtained by either approach. This example was based on the assumption that a single distribution estimate (based on four similar sites) was needed for the site of interest. However, the concepts presented can extend to the case where an SDF is used to obtain the desired estimate as a function of site characteristics. For example, it is possible that the variation in proportions among sites in Table 58 is due to systematic effects. To predict the distribution for the typical site (having characteristics like those of the site of interest), the calculation of model coefficients requires logistic regression using a model that includes the site characteristics of interest. If the study period duration varies among sites, then weighted regression should be used, where the weight of each observation is equal to the reciprocal of its study period duration (in years).

145 Standard Error of Proportions Equation 168 and Equation 169 were used to compute the proportions and standard error of the proportion, respectively. These equations incorporate the study period duration in the calculations to avoid possible bias when the study period duration varies among sites. Equation 168 𝑝 𝑁𝑁 and Equation 169 𝑠 , 𝑝 𝑉 𝑁𝑁 𝑉 𝑁 𝑁 2 𝑉 𝑁 𝑉 𝑁 𝑁 𝑁 . with Equation 170 𝑁 𝑁𝑜 , /𝑦 Equation 171 𝑁 𝑁𝑜 , /𝑦 Equation 172 𝑉 𝑁 𝑁𝑜 , /𝑦 Equation 173 𝑉 𝑁 𝑁𝑜 , /𝑦 where sp,ct = standard error of the proportion p for crash type ct (ct = head on, right-angle, or …); pct = proportion crash type ct; Nct = crash frequency for crash type ct (all sites), crashes/yr; NT = total crash frequency (all sites and crash types), crashes/yr; V[Nct] = variance of crash frequency for crash type ct; V[NT] = variance of total crash frequency; Noct,i = reported crash frequency for crash type ct at site i (i = 1 to n), crashes/yr; Nok,i = reported crash frequency for crash type k at site i (k = 1 to m; i = 1 to n), crashes/yr; yi = number of years for study period at site i, yr; n = number of sites; and m = number of crash type categories. The statistic z used to test the null hypothesis that the proportion is equal to zero is computed using the following equation. This statistic is asymptotic to the normal distribution as sample size increases. Equation 174 𝑧 𝑝𝑠 , Crash Type Distribution Table This section describes the proposed crash type distributions for the FI and PDO crash severity categories. To compute the distribution values, the crash counts at each site were converted into an annual

146 crash frequency (i.e., crashes per year). This approach was taken to avoid bias in the computed proportions due to variation in study period duration among sites. This issue is discussed further in the previous section titled Study Period Duration. The goal in developing the crash type distribution was to compute separate distributions for key combinations of crash type, site type, and other site characteristic categories. However, sample size (in terms of number of crashes) was a concern for some of the combinations. Those combinations having a small number of crashes also have small and relatively uncertain distribution proportions. As a result, the combinations for which the distributions were computed were determined based on consideration of the standard error for the individual proportions and a statistical test of each proportion’s difference from zero (i.e., the null hypothesis was that the proportion was equal to 0.0). The number of combinations selected was initially small, but the number was increased as categories were added (and the sample size per combination reduced) until the largest number of statistically significant proportions was obtained. The statistics used in the statistical test are described in the previous section titled Standard Error of Proportions. The proposed FI crash type distribution is shown in the top half of Table 59. It includes separate distribution values for nine combinations of site type and PTSU operation. The sample size is too small to further disaggregate the data for the purpose of adding other categories (e.g., PTSU transition zone presence). The standard error for each proportion is shown in the bottom half of the table. For a given combination, the standard error tends to be small when the proportion is small or when the total crash frequency is large. As a result, some relatively small proportions can be significantly different from 0.0 if the total crash frequency for the combination is large. Table 59. Proposed FI crash type distribution. Crash Type Proportion Site Type PTSU Operation Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 0.002 0.033 0.599 0.122 0.022 0.005 0.154 0.006 0.010 0.048 Yes 0.001 0.061 0.712 0.080 0.014 0.001 0.098 0.007 0.003 0.023 ENSCL No 0.019 0.037 0.606 0.094 0.019 0.000 0.122 0.014 0.019 0.070 Yes 0.000 0.100 0.616 0.097 0.023 0.000 0.117 0.008 0.000 0.039 EXSCL No 0.006 0.020 0.526 0.189 0.012 0.000 0.175 0.014 0.012 0.047 Yes 0.000 0.028 0.808 0.100 0.014 0.000 0.050 0.000 0.000 0.000 Standard Error of Crash Type Proportion Site Type PTSU Operation Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 0.001 0.003 0.004 0.005 0.003 0.001 0.005 0.001 0.002 0.003 Yes 0.001 0.004 0.003 0.005 0.002 0.001 0.005 0.002 0.001 0.003 ENSCL No 0.010 0.011 0.013 0.014 0.008 0.004 0.016 0.007 0.008 0.013 Yes 0.007 0.020 0.015 0.019 0.011 0.007 0.020 0.007 0.007 0.014 EXSCL No 0.005 0.010 0.016 0.019 0.007 0.005 0.018 0.009 0.007 0.013 Yes 0.006 0.012 0.008 0.021 0.009 0.006 0.014 0.006 0.006 0.006 Notes: For those cells with a proportion of 0.0, the standard error estimate is based on the assumption that there was one crash during a 5- year period (which is the average evaluation period duration of all sites in the database). Based on this assumption, the estimated standard error is considered to be conservatively large for proportions equal to 0.0. Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments.

147 The proportions shown in Table 59 indicate that PTSU operation is associated with an increase in the proportion of right-angle and rear-end FI crashes. Ramp entrance speed-change lanes tend to be associated with more right-angle and rear-end FI crashes that the other two site types. The proposed PDO crash type distribution is shown in the top half of Table 60. It includes separate distribution values for nine combinations of site type and PTSU operation. As with the FI proportions, the proportions shown in Table 60 indicate that PTSU operation is associated with an increase in the proportion of right-angle and rear-end PDO crashes. Ramp entrance speed-change lanes tend to be associated with more right-angle PDO crashes that the other two site types. Table 60. Proposed PDO crash type distribution. Crash Type Proportion Site Type PTSU Operation Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 0.002 0.027 0.538 0.190 0.023 0.022 0.156 0.017 0.006 0.019 Yes 0.001 0.053 0.699 0.139 0.010 0.004 0.075 0.007 0.003 0.009 ENSCL No 0.003 0.054 0.468 0.207 0.024 0.020 0.187 0.015 0.002 0.020 Yes 0.000 0.077 0.708 0.106 0.004 0.007 0.089 0.005 0.004 0.000 EXSCL No 0.000 0.030 0.499 0.214 0.028 0.027 0.150 0.016 0.007 0.029 Yes 0.003 0.060 0.718 0.118 0.010 0.010 0.076 0.000 0.000 0.005 Standard Error of Crash Type Proportion Site Type PTSU Operation Multiple-Vehicle Crash Type Single-Vehicle Crash Type Head On Right Angle Rear End Sideswipe Same Dir. Other Animal Fixed Object Other Object Parked Vehicle Other FS No 0.001 0.002 0.003 0.003 0.002 0.002 0.003 0.001 0.001 0.002 Yes 0.001 0.003 0.002 0.004 0.001 0.001 0.003 0.001 0.001 0.001 ENSCL No 0.002 0.008 0.009 0.011 0.005 0.006 0.010 0.004 0.002 0.005 Yes 0.003 0.012 0.008 0.014 0.003 0.005 0.013 0.005 0.003 0.003 EXSCL No 0.002 0.007 0.010 0.012 0.007 0.006 0.011 0.005 0.004 0.007 Yes 0.002 0.010 0.008 0.012 0.005 0.005 0.011 0.002 0.002 0.003 Notes: For those cells with a proportion of 0.0, the standard error estimate is based on the assumption that there was one crash during a 5- year period (which is the average evaluation period duration of all sites in the database). Based on this assumption, the estimated standard error is considered to be conservatively large for proportions equal to 0.0. Site Type: FS = freeway segment; ENSCL = ramp entrance speed-change lane; EXSCL = ramp exit speed-change lane. PTSU Operation: Yes = includes transition zone located between, just upstream of, or just downstream of PTSU segments.

Next: Chapter 7: Advanced Crash Prediction Models »
Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report Get This Book
×
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

Part-time shoulder use is a congestion relief strategy that allows use of the left or right shoulders as travel lanes during some, but not all, hours of the day.

The TRB National Cooperative Highway Research Program's NCHRP Web-Only Document 309: Safety Performance of Part-Time Shoulder Use on Freeways, Volume 2: Conduct of Research Report describes the development of crash prediction models for freeways with PTSU operation.

Supplemental to the document is a Freeway Analysis Tool, which includes BOS Data, S D PTSU Data, and a Prediction Tool, as well as NCHRP Web-Only Document 309: Safety Performance of Part-Time Shoulder Use on Freeways, Volume 1: Informational Guide and Safety Evaluation Guidelines.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!