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

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17 C H A P T E R 3 - M O D E L I N G A P P R O A C H E S Modeling Approaches This chapter describes the site types studied in NCHRP Project 17-89 and the framework for the crash prediction model (CPM) associated with each site type. The CPMs themselves are described in later chapters of this report. This project developed “one-directional CPMs,” where the CPM is used once to evaluate a given freeway travel direction and it is used a second time to evaluate the opposing travel direction. In contrast, Chapter 18 of the Highway Safety Manual Supplement (HSM Supplement) (AASHTO 2014) has bi- directional CPMs, where the CPM is used to evaluate both travel directions together. Aside from this difference, the Project 17-89 site type definitions and the form of crash prediction models are similar to those in the HSM Supplement. Site Types For analysis purposes, a freeway facility is considered to consist of a contiguous set of freeway segments, ramp entrance speed-change lane site, and ramp exit speed-change lane sites. These components are generally referred to as “sites.” Figure 2 illustrates the three site types in the context of a short length of freeway near an interchange. The figure shows five sites (with grey shading) for the right-to-left direction of travel. There are three freeway segments, one ramp exit speed-change lane site, and one ramp entrance speed-change lane site. Although ramps are shown in the figure for the purpose of depicting the interchange, this project did not develop CPMs for ramps. As indicated in Chapter 18 of the HSM Supplement (AASHTO 2014), site boundaries are typically defined by speed-change-lane presence or by a change in the cross section. This guidance is equally applicable to a one-directional freeway facility evaluation. Specifically, HSM-based site boundaries are defined by the presence of a speed-change lane, a change in number of through lanes, or a relatively large change in the width of various cross section components. For the one-directional freeway CPMs developed in this project, site boundaries are also defined by the start or end of a horizontal curve and the start or end of a PTSU lane.

18 Figure 2. Illustrative sites for one-directional freeway facility evaluation. Predictive Model This section presents background information on HSM models, discusses the model framework used for this project, and discusses other modelling approaches that were initially considered but not used. Background Part C of the HSM provides one predictive method for each of the following facility types: rural two- lane roads, rural multi-lane highways, urban and suburban arterials, freeways, and ramps. The predictive method for each facility type is described in a separate chapter within Part C. A predictive method consists of (1) one or more CPMs and (2) guidelines for using these CPMs and interpreting the results. A CPM is used to estimate the predicted average crash frequency of a specific type of site (e.g., segment, signalized intersection, etc.) with specific geometric design elements and traffic control features. Each CPM includes a predictive model equation that has the following general form: Equation 1 𝑁 𝐶 𝑁 𝐴𝐹 , … ,𝐴𝐹 where Np = predicted average crash frequency, crashes per year (crashes/year); C = local calibration factor; NSPF = predicted crash frequency for site with base conditions (crashes/year); AFi = adjustment factors for geometric design element, or traffic control feature i (i = 1 to m); and m = total number of AFs.

19 Each predictive model equation includes a safety performance function (SPF), one or more adjustment factors (AFs), and a local calibration factor (C). The SPF is used to predict the crash frequency NSPF for a site having characteristics that match a specified set of “base conditions.” These conditions describe the typical site’s design elements and control features (e.g., 12 feet lane width). The set of AFs are used to adjust NSPF such that the CPM can provide reliable estimates of the predicted crash frequency Np for sites that do not match all base conditions (e.g., lanes less than or greater than 12 feet in width). The HSM currently uses the term “crash modification factor (CMF)” to refer to the model terms that are applied to NSPF when site conditions do not match base conditions. In the next edition of the HSM, the phrase “SPF adjustment factor” (AF) is expected to replace the phrase “crash modification factor” in Part C. As a result, the acronym AF is used throughout this report when describing models developed in this project. The HSM CPMs are bi-directional, meaning that they analyze both directions of the road simultaneously and there is no underlying model or distribution of crashes associated with each direction of travel. Most HSM chapters include tools for predicting the frequency of crashes by severity category (i.e., K, A, B, C, O). The nature of these tools varies among the chapters. In HSM Chapter 10, a table of proportions is used to describe the severity distribution. The proportion of interest is multiplied by the value obtained from Equation 1 to obtain an estimate of the predicted crash frequency for the associated severity category. HSM Chapters 18 and 19 provide severity distribution functions (SDFs) that predict the proportion of crashes associated with each of the crash SDF categories. Each SDF is represented as a multinomial logistic regression model that includes variables (and associated regression coefficients) for those traffic and geometric characteristics that influence severity. Most HSM chapters also include tools for predicting the frequency of crashes by type. (e.g., collision with animal, rear-end collision, etc.). The proportion of interest is multiplied by the value obtained from Equation 1 to obtain an estimate of the predicted crash frequency for the associated crash type. Modeling Approach The CPMs developed for this project follow the structure used in the CPMs in Part C of the HSM (AASHTO 2010). The predictive model equation in each CPM was developed as a regression model having multiple independent variables that relate crash frequency to various site characteristics. The independent variables were also used to establish representative base conditions for the SPF (following the guidance provided in Appendix A of HSM Part C). Additionally, the variables were also used to derive AFs for those specific site characteristics that were found to have a logical and statistically valid association with crash frequency. The CPMs are intended to be used to evaluate alternative freeway operational features or design elements, as may be developed during the preliminary design or final design stages of the project development process. Each CPM is developed to predict the average crash frequency of one direction of travel through one freeway site. Model Development based on Crash Severity There are two options for developing models that can be used to obtain an estimate of a site’s predicted fatal-and-injury (FI), property-damage-only (PDO), and total (i.e., all severities combined) crash frequency. One option is to develop one CPM for predicting FI crash frequency and a second CPM for predicting PDO crash frequency. Total crash frequency is then estimated by adding the predicted FI crash frequency and the predicted PDO crash frequency.

20 The second option is to develop one CPM for predicting FI crash frequency and a second CPM for predicting total crash frequency. With this option, PDO crash frequency is computed by subtracting the predicted FI crash frequency from the predicted total crash frequency. This option may produce SPFs and AFs whose individual predictions do not compare well to those obtained from their counterparts in FI models and PDO models. The reasons for this limitation are related to the challenges of developing a reliable predictive model form based on total crashes. These reasons are explained in the following paragraphs. One reason for the aforementioned limitation of developing models based on total crashes is that these models increase the potential for creating suboptimal formulations for the SPF and AF functions. A suboptimal formulation will bias in the model predictions at some factor levels. Elvik (2011) discusses the potential for biased estimates and misleading conclusions when the wrong model form is used. Recent research indicates that the underlying causal mechanisms for FI crashes are different from those for PDO crashes. The reason for the differences is that some roadway geometric elements can influence FI crash occurrence more than they influence PDO crash occurrence (and vice versa). An example of this influence is outside (roadside) barrier presence, which tends to reduce severe crashes and increase PDO crashes. As a result of the differing causal mechanisms, the functional relationship between a geometric element’s dimension (or presence) and crash frequency may have a different shape for FI crashes than it does for PDO crashes. Evidence of these differing relationships can be found by comparing the FI and PDO CMFs in Chapter 18 of the HSM Supplement (AASHTO 2014). These comparisons will reveal differences in curvature and slope among the AFs for the two severity categories. In the extreme, these differences can produce situations where the computed PDO crash frequency has a negative value. Another reason for the aforementioned limitation relates to the variation in crash reporting threshold between and within jurisdictions. An evaluation of PDO crash rates in the database indicated a wide variation in their representation in the crash distribution for each of the study jurisdictions. This variation is likely to be the result of (a) differences in the legal reporting threshold between jurisdictions and (b) differences in the level of adherence to this threshold within jurisdictions. This variation in PDO crash representation can cloud the search for association between database variables and crash frequency. In contrast, FI crashes are more consistently reported among jurisdictions and thus provide a more reliable basis for model structure development. In recognition of these issues, many of the more-recently developed models for Part C of the HSM have included separate models for predicting FI crash frequency and for predicting PDO crash frequency (e.g., NCHRP Project 17-58, NCHRP Project 17-62, and NCHRP Project 17-70). After considering the issues described in the previous paragraph, the researchers determined that the best approach for developing the freeway CPMs was to estimate a predictive model for each of two severity categories (i.e., FI and PDO). The predictions from these two models would be added to obtain an estimate of total crash frequency. The FI crash severity category discussed in this report includes fatal (K), incapacitating injury (A), non-incapacitating injury (B), and possible injury (C) crashes. This approach for modeling FI crash frequency is in contrast to that used by some researchers who have developed models for other severity combinations, such as models that predict just the frequency of the K, A, and B categories combined. Model Framework The predictive model framework for freeway segments developed in this project and recommended for inclusion in a future edition of the HSM is presented in Equation 2. This equation consists of two terms, where Equation 3 and Equation 4 each correspond to one term. Equation 2 𝑁 , , , 𝑁 , , , 𝑁 , , ,

21 Equation 3 𝑁 , , , 𝐶 , , 𝑁 , , , 𝐴𝐹 , , , … 𝐴𝐹 , , , Equation 4 𝑁 , , , 𝐶 , , 𝑁 , , , 𝐴𝐹 , , , … 𝐴𝐹 , , , where Np,fs,at,z = predicted average crash frequency of a freeway segment for all crash types at and severity z (z = fi: fatal-and-injury, pdo: property-damage-only, as: all severities) (crashes/year); Nspf,fs,at,z = predicted average crash frequency of a freeway segment with base conditions, for all crash types at and severity z (z = fi: fatal-and-injury, pdo: property-damage-only) (crashes/year); AFm,fs,at,z = adjustment factor associated with feature m in a freeway segment, all crash types at, and severity z (z = fi: fatal-and-injury, pdo: property-damage-only); and Cfs,ac,y,z = calibration factor for freeway segments for all crash types at and severity z (z = fi: fatal-and-injury, pdo: property-damage-only). Equation 2 shows that freeway segment crash frequency is estimated as the sum of components: FI crash frequency and PDO crash frequency. Equation 3 is used to estimate the FI crash frequency, and Equation 4 is used to estimate the PDO crash frequency. The predictive model framework for ramp entrance speed-change lanes is the same as for freeway segments except that the subscript “en” is substituted for “fs” in each variable. Similarly, the predictive model framework for ramp exit speed-change lanes is the same as for freeway segments except that the subscript “ex” is substituted for “fs” in each variable. The models are fully documented in Chapter 5, HSM Predictive Model, of this report. SDFs were developed to predict the percent of the K, A, B, and C severity-level crashes within the overall FI crash frequency predicted by the CPM. SDFs and crash type distributions are presented in Chapter 6. Advanced Model Types Models in Part C of the HSM are generally estimated using fixed parameters count regression modeling. The distribution of crashes is assumed to follow a negative binomial distribution. The models developed for this project also used count regression modeling with a negative binomial distribution of crashes. Three techniques were used to estimate the models for this project:  Fixed parameters modeling  Random parameters modeling  Latent class modeling Note that the term “parameter” is synonymous with the terms “coefficient” and “regression coefficient,” as used in this document. Chapter 5 describes the fixed parameters models that were developed. These models are proposed for inclusion in a future edition of the HSM. Random parameters and latent class models are presented in Chapter 7. Random parameter and latent class models are advantageous because they explicitly account for unobserved heterogeneity within a dataset and may reveal relationships between variables and crash frequency that are masked by unobserved heterogeneity fixed parameters models. Random parameter and latent class models are best suited for analysis of sites contained within the dataset used to estimate them. Models in the HSM, though, are routinely used by practitioners to analyze sites outside of the estimation dataset. The lack of an established method for applying random parameter and latent class models to sites outside of the estimate dataset was a key factor in not recommending them

22 for inclusion in a future edition of the HSM. The remainder of this section presents an overview of the three model types and explains the differences between them. The SPF portion of each of the three models contains variables, such as length and annual average daily traffic (AADT), and coefficients developed in the modeling process. Equation 5, Equation 6, and Equation 7 present the SPFs for FI crashes on freeway segments that were developed with fixed parameters, random parameters, and latent class modeling, respectively. The SPFs are fully documented in Chapters 5 and 7. They are presented here to concisely illustrate the differences associated with their application: Equation 5 𝑁 𝐿 exp 4.556 1.406 ln (from fixed parameters model) Equation 6 𝑁 𝐿 exp 𝑏 . (from random parameters model) Equation 7 𝑁 𝐿 exp 𝑏 (from latent class model) where L = length of freeway segment (miles); and AADT = one-directional AADT volume of the freeway (veh/day). For the random parameters model, the variable b1 has different values for each site in the database used to estimate the model. The mean of the values of b1 is −4.842, with a standard deviation of 0.215. For the latent class model, the variable b2 has different values for the “class 1” and “class 2” sites in the database. The values are −5.491 and −3.035 for class 1 and class 2, respectively. Similarly, the values of variable b3 are 1.259 and 1.288 for class 1 and class 2, respectively. The fixed parameters model SPF can readily be applied to any freeway segment (if AADT and length are known). Application of the random parameters model SPF requires an analyst to select a value for the b1 term; however, there is no established practice for making this selection. Similarly, application of the latent class model requires an analyst to assign a site to “class 1” or “class 2.” However, there is no established practice for making this selection. For these reasons, the random parameters models and latent class models are not well suited for analysis of sites not contained in the dataset from which the associated models were developed. AFs can be included with each of the three modeling techniques. These AFs are typically of the form “AF = ex,” where x is an equation containing regression coefficients and variables. The regression coefficients are constants with the fixed parameters model. The coefficients can be either fixed or random with the random parameters model. They can be one of two values with the latent class models. Additionally, the equation (i.e., the “x” portion of the term ex) may be linear or nonlinear with the fixed parameters model, but it is always linear in the random parameters and latent class models. This is a limitation within the current state of practice that is dictated by the available software packages used to estimate random parameters and latent class models. At the time this research project was undertaken, these packages did not support nonlinear exponential terms. Approaches Considered and Not Used In the early stages of this project, the research team considered two other approaches for estimating the safety performance of freeways with PTSU operation— (1) adapting the current HSM Part C Freeway CPM for PTSU operation and (2) developing PTSU CMFs based on a before-after study. This section describes these approaches and explains why they were not selected.

23 Adapt the Existing HSM Part C Freeway CPM This approach involved adapting the existing HSM freeway CPM such that it could be used to evaluate freeways with PTSU operation. The adaptation process would have led to the development of new PTSU- related AFs that would be used with the existing HSM method to evaluate freeways with PTSU operation. With this approach, the characteristics of the existing HSM freeway method would have been retained. Specifically, the evaluation of a freeway segment would be based on a bi-directional evaluation approach. In this regard, the CPM developed using this approach would be used to evaluate both segment travel directions together, where one or both travel directions support PTSU operation. A key reason that the research team did not select this approach is that it would not have produced a single-directional model. PTSU operation is often considered in a single direction on a freeway, and such a design would be more challenging for analysis with a bi-directional model. Develop Crash Modification Factors This approach would have developed one or more CMFs using retrospective before-after data. These CMFs would be used with the existing HSM freeway CPM. Retrospective before-after data are obtained from cross section databases, where the sites in the database are known to have had PTSU operation implemented (and no other changes) at a point in time several years prior to the most recent year of data availability. For each of these sites, data are available for several years before and after implementation. Other sites in the database that did not undergo any change in conditions for the same years are used as reference sites. This approach was not selected because some of the PTSU facilities studied had PTSU operation implemented many years ago and “before” crash data would not have been available for these sites. Additionally, this approach would have provided practitioners with an analysis technique that is less robust than an HSM Part C CPM.