CHAPTER 6

Case B CMF Development and Results

This chapter describes the methodology used to develop the case B CMFs and presents key findings from the analysis of the ATSPM data and crash data. It also describes the developed case B CMFs and their associated safety performance functions (SPFs). The specific CMFs developed are identified in the following list:

• Platoon Ratio CMF
• Split Failure CMF
• Yellow Actuation CMF
• Left-Turn Gap Availability CMF

In previous chapters reference was made to a CMF for “percent arrivals on green” based on the Purdue Coordination Report. This CMF was subsequently renamed as the “Platoon Ratio CMF” to more accurately describe the traffic measure upon which this CMF was developed. The platoon ratio measure represents the ratio of “percent arrivals on green” to “green to cycle length ratio.”

The research team also collected data to develop CMFs for red actuations. However, the research team was not able to generate a meaningful association between red actuations and crash frequency. This finding is discussed in a subsequent section of this document. As a result, no CMF is developed for the red actuation ATSPM report.

Case B CMF – Analysis Methodology and Findings

This section presents a set of CMFs that describe an inferred change in safety associated with the change in a specified ATSPM report. The presentation includes a description of the study design, a description of the intersections used to estimate the CMFs, a brief review of the statistical analysis methods, and a summary of the estimated CMFs.

Case B CMF – Study Design

This section provides a brief overview of the study design that was used to develop the case B CMFs. The following topics are addressed in separate subsections:

• Study Objective and Method
• Terminology
• CMF Development Approach
• Study Sites
• Database Structure

Study Objective and Method

The objective of the case B CMF study was to develop a set of CMFs that collectively describe the association between the change in a specified ATSPM report and the change in traffic safety at the intersection. Four ATSPM reports were considered for CMF development, they include:

• Percent arrivals on green (using the Purdue Coordination Report)
• Yellow and red actuations
• Split failure frequency

• Left-turn gap availability

For each ATSPM report, the researchers investigated the development of CMFs specific to two crash severity categories (i.e., fatal-and-injury crashes [FI], property-damage-only crashes [PDO]), two crash types (i.e., left-turn-with-opposing-through-vehicle crashes, non-left-turn-with-opposing-through-vehicle crashes), and two traffic periods (i.e., crashes during peak hours, crashes during off-peak hours).

These CMFs can be used by an ATSPM system manager to (1) identify intersections having a crash severity or crash type that is over-represented relative to other intersections, and (2) predict the change in average crash severity or crash type associated with a proposed change in ATSPM report value.

A cross-sectional study method was used to develop the proposed set of CMFs. Data was collected for several signalized intersections for which ATSPMs were used to manage the signals’ operation as part of a coordinated signal system. The database has a panel structure with one observation for each of the 24 hours of a typical weekday for each of three recent years at each intersection study site.

The FI crash severity category discussed in this chapter includes fatal (K), incapacitating injury (A), and 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 (e.g., models for predicting the frequency of the K, A, and B categories combined).

Terminology

A study site is defined as a single leg of a signalized intersection. The leg serves two-way traffic flow, with one or more lanes approaching the intersection and one or more lanes departing the intersection. For analysis purposes, the site length is defined to begin at the center of the intersection and extend back 400 ft along the leg. If the distance to the next signalized intersection is less than 800 feet, then the site length is reduced to one-half of the distance to the next signalized intersection.

Two traffic periods were evaluated to provide a sensitivity to traffic demand levels and the likelihood that progression quality may vary between peak and non-peak traffic periods. One traffic period consists of the collective set of peak hours of the typical week. The second analysis period consists of the collective set of non-peak hours of the typical week. A preliminary analysis of the volume data at the study sites indicated that the peak hours Monday through Friday are 7:00-9:00 am and 4:00-6:00 pm.

The evaluation time period is the number of years of data used to estimate the predictive model coefficients. For the case B CMF development, the evaluation time period includes the years 2021, 2022, and 2023. Each year of data represents a full calendar year. Aerial imagery (e.g., Google Earth) was used to confirm that there were no changes in roadway geometry at any site during the evaluation time period.

The developers of the next edition of the Highway Safety Manual (HSM) have determined that CMFs inferred from regression-based crash prediction models should be called “SPF Adjustment Factors” (or simply, Adjustment Factors [AFs]). This change in nomenclature is intended to distinguish AFs from CMFs in terms of their primary application (AFs are used in crash prediction models) and method of development (AFs from a cross-sectional study may have less reliability than CMFs from a before-after study). Henceforth, each case B CMF is referred to herein as an SPF Adjustment Factor.

A crash prediction model (CPM) is defined to include an SPF that predicts the average crash frequency for a specified set of base conditions, one or more adjustment factors (AFs) that are used to adjust the SPF prediction to account for non-base conditions at a site of interest, and a local calibration factor.

CMF Development Approach

The approach used to develop the case B AFs was to infer them from the independent variables in a regression model that relates crash frequency to various site characteristics (e.g., volume, speed limit, ATSPM reports) that are found to have a logical and statistically valid association with crash frequency.

This approach produces a crash prediction model (CPM) that can be used by analysts to (a) compute the average crash frequency for a site with specified characteristics, or (b) predict the effect of a change in site character on average crash frequency. If there are several AFs in the CPM, then the CPM can be used to predict the safety effect of logical combinations of site character changes.

The remaining paragraphs describe two forms of CPM framework that were used to develop the SPFs and AFs described in this document (i.e., the multiple-AF model and the one-AF model). For the reasons cited in these paragraphs, the multiple-AF model framework is the preferred form. It was used when supported by the available data. However, as described in subsequent sections, data constraints for one ATSPM report required use of the one-AF model framework.

where,

One advantage of this model framework is that the AF coefficients are jointly estimated such that the analyst can use the model to estimate the safety effect of concurrent changes to two or more ATSPM reports. The framework and its regression-based estimation indirectly can account for any interaction or overlap in effect within the set of AFs with reasonable accuracy (provided that the AFs are not strongly correlated).

To illustrate the utility of the multiple-AF model framework, consider the case where Equation 21 has AFs for percent-arrivals-on-green, yellow actuations, red actuations, split failure, and left-turn gap availability. The analyst has selected an intersection leg of interest and has used its ATSPM reports to quantify the variables needed to compute each of the five AFs. The AFs are then used in Equation 21 to compute the predicted average crash frequency for the leg of interest. The predicted value can be reliably compared with that for other legs at other intersections to determine if the subject leg has potential for safety improvement. The analyst can explore the effect of alternative signal operation strategies (via changes to one or more AFs) to predict the change in average crash frequency for the subject leg.

Continuing the example, the analyst could use Equation 21 to investigate the change in safety associated with a proposed change in the yellow change interval that would reduce yellow actuations. Equation 21 would be used directly by adjusting the AF associated with yellow actuations. If this change to the yellow change interval is expected to induce a change in percent-arrivals-on-green, red actuations, split failure, or left-turn gap availability and these secondary changes are quantified, then the analyst can use Equation 21 to compute the predicted average crash frequency for the subject site as a result of the expected change in all five AFs.

A disadvantage of this model framework is that the database assembled for model estimation must include the variable(s) needed for each AF at each site. If an AF for each of several ATSPM reports are to be included in a multiple-AF model, then each site used to estimate the model must have data for each of these reports. Note that this is a limitation for model development. It does not restrict multiple-AF model

application (e.g., by use of default values, the estimated model can still be used to evaluate sites that do not have all of the reports needed for the multiple-AF model).

where all variables are previously defined.

With this approach, a unique one-AF model is developed to evaluate the safety effect for each site characteristic of interest. If four ATSPM reports are of interest, then four one-AF models would be developed (i.e., N_p,1 = C × N_spf,1 × AF₁, N_p,2 = C × N_spf,2 × AF₂, N_p,3 = C × N_spf,3 × AF₃, N_p,4 = C × N_spf,4 × AF₄). One disadvantage of this approach is that regression-based “one-AF models” are inclined to predict the safety effect of a planned change in ATSPM report value along with the effect of any changes to report values that are correlated with the planned change (the correlation in reference here is that which may occur between data elements in the database, such as between yellow actuation and split failure frequency).

For example, consider the possibility that in the assembled database, a change in yellow actuations is correlated with a small change in split failure frequency. If a “one-AF model” for yellow actuations is used by an analyst to estimate the change in safety associated with a change in yellow actuations, the predicted average crash frequency will also reflect the small change in split failure frequency that occurs (to the extent that it occurred in the assembled database). If the site to which this one-AF model is being applied does not also experience the same small change in split failure frequency found in the database, then the predicted crash frequency from the one-AF model could be biased by this correlation. The multiple-AF model framework is intended to mitigate this bias whenever a correlation between independent variables is present, but not strong (when the correlations between independent variables are strong, then only one of the two correlated variables can be included in Equation 21).

A second disadvantage of the “one-AF model” form is that it does not provide a unique estimate of the predicted average crash frequency of a leg when two or more of the variables associated with the AFs are available (i.e., it provides one estimate for each model/AF used). For example, if an analyst desires to evaluate the effect of percent-arrivals-on-green and yellow actuations, then two one-AF models would be needed. Each model would be used to compute an estimate of the predicted average crash frequency for the leg. Theoretically the two estimates would be the same for a given site; however, they are unlikely to be equal due to uncertainty in the model. The analyst will require guidance as to how to interpret the disparity in these two estimates of predicted crash frequency for the site.

Study Sites

A total of 122 study sites (i.e., intersection legs) were identified at 42 intersections in three states (N. Carolina, Utah, and Georgia). ATSPM report data were obtained for each intersection for the three-year evaluation time period (except for North Carolina where two years of ATSPM data were obtained). In addition, crash history data were obtained for each intersection for the same time period. The crash data was requested from the corresponding state DOTs. The crash data request asked for all reported crashes on all intersection legs, extending back 500 ft from the intersection. The 500-ft zone was used to ensure that all intersection-related crashes could be subsequently identified by the researchers. This identification process is described in a subsequent section.

A preliminary review of the crash data revealed that some intersection legs were not represented in the crash data. These legs were typically on very minor public roads or driveways. A total of 13 legs were identified in this manner and removed from the database; leaving 109 candidate study sites (i.e., legs).

Some sites in some states did not have a detection configuration that would provide reliable data for some ATSPM reports (e.g., yellow actuations). Notably, only the Georgia sites had a detection configuration that could provide reliable data for the left-turn gap analysis. Similarly, only the Utah sites had a detection configuration that could provide reliable reports for the yellow-and-red-actuation analysis. ATSPM data were not available from North Carolina for Year 2021. Finally, a review of the split failure reports revealed that the detection at several sites in each state was unable to provide reliable split failure data in various years. In a collective manner, these issues resulted in there being only a subset of the 109 sites available for the development of the desired CPMs and associated SPF Adjustment Factors.

The number of sites available for each combination of state and year are listed in Table 42. The number of available sites is shown to vary by the various ATSPM reports used in the development of one or more AFs.

Table 42. Study-site count by state, year, and ATSPM report for case B CMF development.

^a – This variable is often referred to as the green-to-cycle-length (g/C) ratio.

n.a. – data not available.

Database Structure

The database assembled to develop the four AFs includes data describing the observed crash characteristics, geometric design, traffic volume, traffic control, and ATSPM data associated with each study site. Table 43 lists the key database variables collected for AF development.

Discussions with each of the operating agencies indicated that a common signal timing strategy was maintained for each weekday (i.e., Monday through Friday), but that this strategy was changed for weekend days. These strategy changes were difficult to describe in a quantitative manner such that the weekday vs. weekend safety effect could be isolated in the regression analysis. As a result, it was determined that the analysis should focus on weekday conditions to minimize the unquantifiable effect of strategy change on safety.

Table 43. Database variables for case B CMF development.

^a – Key variables listed; typically, a larger number of crash, roadway, driver, and vehicle variables were acquired.

^b – Data obtained from high-resolution signal controller and detector data provided by the operating agency; and summarized for each of the 24 hours of the typical weekday.

The database is structured as a flat file (using an Excel workbook). Each observation in the database describes one hour of the 24 hours during a typical weekday for each of three years at each study site. For example, one row in the database includes the geometric design, traffic volume, traffic control, and ATSPM data listed in Table 43 for one intersection leg. The traffic volume and ATSPM data in this row are listed

in three groups; with one group for each of the three years in the evaluation time period. There are 2,616 rows in the database (= 24 annual weekday hours/site × 109 sites).

Each of the 2,616 rows the database also includes the count of weekday crashes that occurred in the associated hour of day during one specific year of the evaluation period. The counts are grouped by severity (i.e., fatal-and-injury [FI], property-damage-only [PDO]) and by crash type (i.e., left-turn-with-opposing-through-vehicle crashes, non-left-turn-with-opposing-through-vehicle crashes); with one group included in the database for each of the three years in the evaluation time period.

The crashes included in the database are those that occurred within the boundaries of the associated site during the corresponding evaluation time period. The crash assignment process (as described in a subsequent section) was such that each intersection-related crash was uniquely assigned to one leg of the intersection; no crashes were unassigned, and no crashes were “double-counted”.

Case B CMF ‒ Database Assembly

This section describes the activities undertaken to assemble and review the model development database. The following topics are addressed in this section:

• Site Characteristics Summary
• Crash Data Verification
• Crash Data Screening
• Crash Assignment
• Crash Data Summary
• Exploratory Data Analysis

Site Characteristics Summary

As indicated by the discussion associated with Equation 21, the preferred approach to developing the desired AFs is to infer them from the variables in a multiple-variable regression model in which all ATSPM-based variables are represented. To accomplish this goal, sites were identified in the database that had all of the ATSPM report data listed in Table 43. Unfortunately, there are no sites for which all ATSPM report data are available. However, based on further investigation, it was determined that there were 23 sites suitable for developing a version of Equation 21 that includes one adjustment factor for each of the following three ATSPM reports: percent arrivals on green, yellow and red actuations, and split failure frequency. The investigation also determined that there are 23 different sites suitable for developing a version of Equation 22 that includes one adjustment factor for left-turn gap availability. The geometric design, traffic control, and traffic volume characteristics of the two site groups are described in Table 44 and Table 45, respectively.

Discussions with each of the operating agencies indicated that a common signal timing strategy was maintained for each weekday (i.e., Monday through Friday), but that this strategy was changed for weekend days. These strategy changes were difficult to describe in a quantitative manner such that their incremental safety effect could be isolated in the regression analysis. As a result, only those crashes occurring on a weekday were retained in the database.

Crash Assignment

This section describes the techniques used to identify crashes associated with a specific intersection leg. Initially, the crashes were categorized by “facility type” (i.e., signalized intersection and segment). The two types are mutually exclusive and were used to categorize all crashes in the database. In other words, every crash in the database was assigned to either a signalized intersection or a segment.

Then, the signalized intersections crashes were assigned to a specific intersection leg. The leg assignment process was such that each signalized-intersection-related crash was uniquely assigned to one leg of the intersection; no crashes were unassigned, and no crashes were “double-counted”.

The crash assignment technique was based on consideration of the crash report variables listed in Table 43. The specific variables used for the assignment varied among states because each operating agency’s crash data included slightly different variables (or similar variables but with different values). The assignment technique for each database source is described in the following subsections.

Crash Assignment for North Carolina Data

If one or more of the conditions in the preceding list applied, then the crash was categorized as “Intersection.” It was then defined as “Intersection – unsignalized” unless one or more of the following conditions applied:

• Distance to the nearest signalized intersection is 250 feet or less; or
• Vehicle 1 Contributing Circumstance variable indicates “Disregarded Traffic Signal”, “Right turn on red”; or
• Vehicle 2 Contributing Circumstance variable indicates “Disregarded Traffic Signal”, or “Right turn on red”.

If one or more of the conditions in the preceding list applied, then the crash was defined as “Intersection-signalized”.

center was visually determined to be at the centroid of the crashes found within the intersection conflict area. The milepost of this centroid was then used to define the center of the intersection.

Next, each crash was assigned to a leg based on matching its roadway ID with that of the leg and the distance between its milepost and that of the center of the intersection. A crash with a milepost that is smaller than that of the center of the intersection was assigned to the “upstream” leg of the intersection, relative to the direction of increasing milepost. Conversely, a crash with a milepost that is larger than that of the center of the intersection was assigned to the “downstream” leg of the intersection, relative to the direction of increasing milepost.

Finally, for those crashes whose milepost equaled that of the center of the intersection, the travel direction of vehicle 1 was used to assign the crash to a leg. For example, if the vehicle 1 travel direction is indicated to be “Eastbound”, then the crash was assigned to the west leg. Similarly, if the vehicle 1 travel direction is indicated to be “Westbound”, then the crash was assigned to the east leg. The same concept was applied to the north and south legs. The process was refined to accommodate travel directions identified as “northwest bound”, “northeast bound”, “southwest bound”, and so on; such that all possible travel direction combinations in the crash data were logically considered in the assignment.

Crash Assignment for Utah Data

If one or more of the conditions in the preceding list applied, then the crash was categorized as “Intersection.” It was then defined as “Intersection – unsignalized” unless one or more of the following conditions applied:

• Distance to the nearest signalized intersection on the subject arterial street is 250 feet or less, or
• Traffic Control Device variable indicates “traffic control signal”, or
• Traffic Control variable indicates “signal”.

If one or more of the conditions in the preceding list applied, then the crash was defined as “Intersection-signalized”.

Crash Assignment for Georgia Data

• Location at Impact variable indicates “on roadway – roadway intersection” andVehicle 2 Maneuver variable indicates “turning left,” or “turning right”; or
• Traffic Control Type variable indicates “traffic signal”, “stop sign”, or “yield sign”.

If one or more of the conditions in the preceding list applied, then the crash was categorized as “Intersection.” It was then defined as “Intersection – unsignalized” unless one or more of the following conditions applied:

• Traffic Control Type variable indicates “traffic signal”, or
• Distance to the nearest signalized intersection on the subject arterial street is 250 feet or less.

If one or more of the conditions in the preceding list applied, then the crash was defined as “Intersection-signalized”.

A variable named Intersection-Related is included in the Georgia crash data. Its value is “true” (indicating the crash is related to the intersection operation) or “false”. As discussed earlier, this variable was considered for use in the facility type assignment. However, this use was abandoned when it was discovered that the variable value was “true” for every crash in 2022 and 2023 at all four Georgia study sites. Many of the “true” values were associated with crashes whose other variable values indicated a non-signal-related crash. In contrast, for the years 2017 to 2020, the Intersection-Related variable had values that varied from “true” to “false”, with each group having logical values for other variables having “intersection-relationship” information.

Crash Data Summary

This section summarizes the crash counts at the sites used for model estimation. The counts used to estimate the AFs for three ATSPM reports (i.e., percent arrivals on green, yellow and red actuations, and split failure frequency) are described in Table 48. The counts used to estimate the AF for the left-turn-gap-availability report are described in Table 49. The counts in both tables correspond to weekday (Monday to Friday) crashes during the evaluation time period (i.e., years 2021, 2022, and 2023). One exception in Table 48 is for the North Carolina site for which ATSPM reports were available only for years 2022 and 2023. One exception in Table 49 is Signal Number 8162 for which the detector used to collect key ATSPM reports was functional only for years 2022 and 2023.

The development of an AF based on the left-turn-gap-availability ATSPM report was based on two crash type categories. The first category represents the crash between a left-turning vehicle and a vehicle on the opposing approach going straight through the intersection. Crashes in this category were considered “target” crashes. It was rationalized that these crashes were highly correlated with the “long” gaps quantified by the left-turn-gap-availability report. The vehicle-1-maneuver and vehicle-2-maneuver variables in the crash report were used to identify these crashes. A target crash was identified as one vehicle (either vehicle 1 or vehicle 2) having a “left turn” maneuver and the other vehicle having a “going straight ahead” maneuver. The second category represented all non-target crashes and are referred to herein as “NON-left-turn with opposing through vehicle crashes.” The crash counts in both categories include crashes of all severity levels (i.e., fatal, injury, and property-damage-only).

The trend shown in Figure 16 may reflect an unknown number of right-turn-on-red maneuvers that were counted by the ATSPM system (discussions with the UDOT staff also highlighted potential issues with the red actuation metric due to the detection issues related to both latency and the speed restriction feature that is typically used to eliminate right-turn-on-red maneuvers). These maneuvers are frequent in number and tend to be associated with fewer crashes than are vehicles that cross straight through the intersection during the red interval. Further investigation revealed that PVR is larger during periods with a lower traffic volume, which corresponds to larger gaps for the right-turn driver and less crash risk.

The two trend lines in Figure 17 indicate that the crash rate is higher for legs with lower volume. This trend may reflect the fact that high-volume legs are more likely to have favorable progression. As indicated in Figure 15, legs with favorable progression have a lower crash rate.

When comparing the trends in Figure 19a and Figure 19b, it appears that the trend in Figure 19b provides a better fit to the data. This finding suggests that an AF based on permissive capacity may provide more reliable predictions than an AF based on PGG.

Case B CMF ‒ Cross-Sectional Study Statistical Analysis Methods

This section summarizes the statistical analysis methods used to estimate the regression model coefficients and the associated SPFs and AFs. The following topics are addressed in this section:

• Modeling Approach
• Statistical Analysis Methods
• Analysis of Percent-Arrivals-on-Green, Split Failures, and Yellow Actuations
• Analysis of Left-Turn Gaps

Modeling Approach

The SPFs and AFs for both models were estimated using regression analysis of the data assembled for 109 sites in three states. The basic unit of analysis (i.e., the “observation”) is one hour of the typical weekday over the duration of one year at one site. The number of observations associated with each model are indicated at the bottom of Table 46 and Table 47. Thus, the models are developed to predict the annual average crash frequency for a specified hour of a typical weekday at one site.

where,

The turn movement proportions needed for Equation 25 were not available in the database, so default values of P_lt = 0.1 and P_th = 0.9 were used for all sites.

regression analysis, the resulting standard error estimates are more precise, and the associated statistical tests are more efficient.

Overdispersion Parameter

It was assumed that site crash frequency is Poisson distributed, and that the distribution of the mean crash frequency for a group of similar sites is gamma distributed. In this manner, the distribution of crashes for a group of similar sites can be described by the negative binomial distribution (also known as a Poisson-Gamma mixture distribution). The variance of this distribution is computed using the following equation.

where,

The overdispersion parameter k, as defined and used in the HSM, is computed using the following equation:

where k = overdispersion parameter and φ = inverse dispersion parameter.

Model Fit Statistics

Three statistics were used to assess CPM fit to the data. One measure of fit is the Pearson χ² statistic. This statistic is calculated using the following equation.

where,

This statistic follows the χ² distribution with n – p degrees of freedom, where n is the number of observations and p is the number of model variables (McCullagh and Nelder, 1983). This statistic is asymptotic to the χ² distribution for larger sample sizes.

The second measure of model fit is the scale parameter φ. It is used to assess the amount of variation in the observed data, relative to the specified distribution. This statistic is calculated by dividing Equation 18 by the quantity n – p. A scale parameter near 1.0 indicates that the assumed distribution of the dependent variable is approximately equivalent to that found in the data. In this instance, a value near 1.0 indicates that negative binomial distribution adequately describes the crash distribution among the study sites.

The third measure of model fit is the pseudo R² developed by McFadden (1974). This statistic provides a useful measure of model fit for maximum-likelihood regression. The pseudo R² compares the log likelihood value for the estimated model with that for the “null” model. The null model includes only an intercept term (i.e., no predictor variables). The pseudo R² is computed using the following equation.

where

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).

Analysis of Percent-Arrivals-on-Green, Split Failures, and Yellow Actuations

This section describes a set of CPMs that relate three ATSPM reports to crash frequency. It also presents the results from the model estimation process.

Model Development

Two CPMs were developed to describe the safety effect of percent-arrivals-on-green, split failures, and yellow actuations. One CPM predicts the frequency of fatal-and-injury (FI) crashes. The other CPM predicts the frequency of property-damage-only (PDO) crashes.

The functional form of (and independent variables in) the CPM for predicting FI crash frequency is the same as that for predicting PDO crash frequency. However, the model’s regression coefficients vary depending on whether the model is applied to FI or PDO crashes.

As noted in a previous section, split failure frequency is strongly correlated with yellow actuations such that both variables cannot be included in the same regression model. To overcome this issue, two regression analyses were undertaken. In the first analysis, the researchers estimated the coefficients for a model that included an AF for platoon ratio and an AF for the probability-of-a-vehicle-experiencing-a-split-failure psf. In the second analysis, the SPF and AF for platoon ratio where then used in a second regression analysis to estimate an AF for the proportion-of-approach-volume-entering-the-intersection-during-the-through-yellow-interval (PVY). These two models are described in the following paragraphs.

with

The final form of the regression model is described by the preceding equations. This form reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described herein represents that which provided the best fit to the data.

with

where

and all other variables are as previously defined.

All of the variables associated with Equation 37 describe the traffic and signal timing characteristics for a specified leg at a signalized intersection of interest. The underlined variable was obtained from the ATSPM yellow-and-red actuations report for the study sites represented in the estimation database.

The base conditions for this model are identified in the following list:

• Platoon ratio: 1.0
• Proportion of approach volume entering during the through yellow interval: 0.0
• Traffic period: off-peak hour (peak hours: 7:00 to 9:00 am and 4:00 to 6:00 pm)

The final form of the regression model is described by the preceding equations. This form reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described herein represents that which provided the best fit to the data.

Model Estimation

This section summarizes the results of the regression model estimation for both regression analysis 1 and 2, as described in the previous section.

The results for the FI-crash model are presented in Table 50. The Pearson χ² statistic for the model is 1695, and the degrees of freedom are 1603 (= n − p = 1608 − 5). Only one intercept variable is counted towards the value of p for this calculation given the intercept’s use in the negative multinomial error distribution. As this statistic is less than χ²_{0.05, 1603} (= 1697), the hypothesis that the model fits the data cannot be rejected. The R_p² for the model is 0.11. As described in a previous section, 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).

Table 51. Predictive model estimation statistics ‒ PDO crashes, platoon ratio and split failures.

n.s. – coefficient is not statistically significant (and is excluded from the model).

The t-statistics listed in the last column of Table 51 indicate a test of the null hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 1.96 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. As noted previously, a coefficient where the absolute value of the t-statistic was between 1.4 and 1.96, that was important to the model, and whose trend was found to be logical and consistent with intuition or previous research findings was also retained in the CPM.

The coefficient for the peak hour indicator variable was not statistically significant. As a result, it was removed from the model. The CPM for predicting PDO crash frequency does not include the adjustment factor for peak traffic hour AF_pk.

The fit of the estimated model is shown in Figure 21. The model was applied to each of the 23 sites (i.e., intersection legs) in the database. Each data point shown represents the average predicted and average reported PDO crash frequency for a group of 3 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 6.0 PDO crashes/year.

The results for the FI-crash model are presented in Table 52. The Pearson χ² statistic for the model is 1483, and the degrees of freedom are 1511 (= n − p = 1512 − 1). As this statistic is less than χ²_{0.05, 1511} (= 1602), the hypothesis that the model fits the data cannot be rejected. The R_p² for the model is 0.11.

Table 52. Predictive model estimation statistics ‒ FI crashes, yellow actuations.

The t-statistics listed in the last column of Table 52 indicate a test of the null hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 1.96 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05.

The fit of the estimated model is shown in Figure 22. The model was applied to each of the 21 sites in the database. Each data point shown represents the average predicted and average reported FI crash frequency for a group of 3 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 2.5 FI crashes/year.

The results for the PDO-crash model are presented in Table 53. The Pearson χ² statistic for the model is 1600, and the degrees of freedom are 1511 (= n − p = 1512 − 1). As this statistic is less than χ²_{0.05, 1511} (= 1602), the hypothesis that the model fits the data cannot be rejected. The R_p² for the model is 0.13.

Table 53. Predictive model estimation statistics ‒ PDO crashes, yellow actuations.

The t-statistics listed in the last column of Table 53 indicate a test of the null hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 1.96 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05.

The coefficient for the peak hour indicator variable was not statistically significant in the PDO model estimated in regression analysis 1. As a result, the CPM for predicting PDO crash frequency in regression analysis 2 does not include the adjustment factor for peak traffic hour AF_pk.

The fit of the estimated model is shown in Figure 23. The model was applied to each of the 21 sites in the database. Each data point shown represents the average predicted and average reported PDO crash frequency for a group of 3 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 6.0 PDO crashes/year.

Analysis of Left-Turn Gaps

This section describes a CPM that relates left-turn-gap-availability to crash frequency. It also presents the results from the model estimation process. The models are applicable only intersection legs that have an opposing left-turn movement that is served by permissive or protected/permissive left-turn operation.

Model Development

Two CPMs were developed to describe the safety effect of left-turn gap availability. One CPM predicts the frequency of left-turn-with-opposing-through-vehicle (LT) crashes (all severities). The other CPM predicts the frequency of all other (nonLT) crashes (all severities). Both models are developed using 23 sites for which the left-turn movement opposing the subject leg is served by permissive or protected/permissive left-turn operation.

The model for predicting LT crash frequency is the primary product of this development. It is intended to be used by analysts when evaluating the safety effect of alternative signal timing strategies that change left-turn gap availability. In contrast, the model for predicting nonLT crash frequency was primarily developed to confirm whether changes to left-turn gap-availability have an adverse effect on intersection safety. As a matter of good practice, before an analyst implements a treatment to reduce target crashes, he or she should first confirm that the treatment will not inadvertently increase non-target crashes. The nonLT CPM was developed to support this practice.

The functional form of (and independent variables in) the CPM for predicting LT crash frequency is the same as that for predicting nonLT crash frequency. However, the model’s regression coefficients vary

depending on whether the model is applied to LT or nonLT crashes. These two models are described in the following equations.

with

where,

The fraction “5/7” in Equation 39 is an offset variable that recognizes the dependent variable in the estimation data represents weekday crash counts (i.e., Monday through Friday).

All of the variables associated with Equation 39 describe the traffic and signal timing characteristics for a specified leg at a signalized intersection of interest. The underlined variables were obtained from the ATSPM reports for the study sites represented in the estimation database.

The model variables describe the leg of interest. Notably, the proportions P_lt,g,i and (g/C)_i describe the through traffic stream for the subject leg (not the opposing leg).

The base conditions for this model are identified in the following list:

• Approach speed limit: 40 mph
• Opposing left-turn permissive capacity: 1000 veh/h

The final form of the regression model is described by the preceding equations. This form reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described herein represents that which provided the best fit to the data.

Model Estimation

This section describes the results of the regression model estimation for the CPMs that include the left-turn gap availability AF. A total of 23 sites had the left-turn gap availability data needed to estimate this model. However, one site was identified as an outlier during a preliminary regression analysis. It was the

east leg of signal 7540, which was first noted in the discussion associated with Table 49. This site was removed from the data, leaving 22 sites in the estimation database.

The results for the LT-crash model are presented in Table 54. The Pearson χ² statistic for the model is 1508, and the degrees of freedom are 1556 (= n − p = 1560 − 4). Only one intercept variable is counted towards the value of p for this calculation given the intercept’s use in the negative multinomial error distribution. As this statistic is less than χ²_{0.05, 1556} (= 1649), the hypothesis that the model fits the data cannot be rejected. The R_p² for the model is 0.09. As described in a previous section, 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).

Table 54. Predictive model estimation statistics ‒ Left-turn-opposed crashes, gap availability.

The t-statistics listed in the last column of Table 54 indicate a test of the null hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 1.96 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05.

A second t-statistic threshold was established for those AF variables that were important to the model and whose AF trend was found to be logical and consistent with intuition or previous research findings. Coefficients associated with variables of this nature were tested against the t-statistic value of 1.4, which corresponds to a probability of error in rejecting the null hypothesis of 0.16. A coefficient where the absolute value of the t-statistic was between 1.4 and 1.96, that was important to the model, and whose trend was found to be logical and consistent with intuition or previous research findings was also retained in the CPM.

The left-turn capacity coefficient in Table 54, when used Equation 41, indicates that crash risk is higher at sites with a lower left-turn capacity. The speed limit coefficient indicates that the change in crash risk with a change in capacity is higher for sites with a higher speed limit.

The fit of the estimated model is shown in Figure 24. The model was applied to each of the 22 sites (i.e., intersection legs) in the database. To create this figure, the predicted crash frequency was computed for each hour of the typical day at each site and the results added for all 24 hours. Each data point shown represents the average predicted and average reported LT crash frequency for a group of 3 sites. In general, the data shown in the figure indicate that the model provides an unbiased estimate of predicted crash frequency for sites experiencing up to 1.7 LT crashes/year.

The results for the nonLT-crash model are presented in Table 55. The Pearson χ² statistic for the model is 1640, and the degrees of freedom are 1556 (= n − p = 1560 − 4). As this statistic is less than χ²_{0.05, 1556} (= 1649), the hypothesis that the model fits the data cannot be rejected. The R_p² for the model is 0.13.

Table 55. Predictive model estimation statistics ‒ Non-left-turn opposed crashes, gap availability.

n.s. – coefficient is not statistically significant (and is excluded from the model).

The t-statistics listed in the last column of Table 55 indicate a test of the null hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 1.96 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. As noted previously, a coefficient where the absolute value of the t-statistic was between 1.4 and 1.96, that

was important to the model, and whose trend was found to be logical and consistent with intuition or previous research findings was also retained in the CPM.

The coefficient for the speed limit variable was not statistically significant. As a result, it was removed from the model. The AF for left-turn permissive capacity does not include this variable.

The fit of the estimated model is shown in Figure 25. The model was applied to each of the 22 sites (i.e., intersection legs) in the database. Each data point shown represents the average predicted and average reported nonLT crash frequency for a group of 3 sites. The trend in the data suggests that an unexplained systematic effect may remain in the data. A subsequent investigation did not reveal the source of the observed systematic trend. Nevertheless, the primary purpose of this CPM is to confirm that the correlation between capacity and LT crash frequency is the same as that for nonLT crash frequency (i.e., that an analyst that increases left-turn permissive capacity to reduce LT crash frequency does not inadvertently increase nonLT crash frequency). The negative value for the left-turn capacity coefficient b_cap in Table 55 provides this confirmation.

Case B CMF – Findings from Cross-Sectional Study

This section presents the estimated SPFs and AFs that comprise the set of CPMs developed for the project. The presentation of each CPM component includes the parametrized equations and a figure illustrating the sensitivity of the prediction to various independent variables.

Analysis of Percent-Arrivals-on-Green, Split Failures, and Yellow Actuations

This section presents the estimated CPMs for predicting the safety effect of percent-arrivals-on-green (as embodied in the platoon ratio variable), peak hour traffic conditions, split failure, and yellow actuations. One subsection describes the CPM for predicting FI crash frequency. The second subsection describes the CPM for predicting PDO crash frequency. The SPF and AF components of each CPM are described in the associated subsection.

• CPM for FI Crash Frequency
• CPM for PDO Crash Frequency

CPM for FI Crash Frequency

This section describes the CPM for predicting FI crash frequency. For each SPF and AF component of the CPM, the description includes one subsection that (a) presents the parameterized equation and (b) includes a figure illustrating equation trends. The CPM (and its SPF and AF components) is described by the following equation.

where,

This equation is used to evaluate one specified hour for a given signalized intersection leg. The predicted value from the equation represents the annual average FI crash frequency for the specified hour. Multiple hours of the typical day can be evaluated by repeated application of the equation. The predicted average FI crash frequency for each hour can be added to estimate the average crash frequency for the combined set of hours.

There are two AFs listed in the parentheses of Equation 43. Only one of these two AF can be used when the CPM is applied to a site. This restriction reflects the fact that the two AF are highly correlated and tend to predict the risk associated with the same crash events.

If the operating conditions being evaluated exist only for a specific subset of the days of the typical week, then the predicted average crash frequency is multiplied by the number of applicable days per week and then divided by 7. For example, if the signal timing plan being evaluated applies only to weekdays (i.e., Monday to Friday), then the analyst would multiply the predicted average crash frequency by 5/7 (= 5 days per weekday/7 days per week).

Where the AAHT_a,i volume variable represents the annual average hourly volume during a specified hour i for the subject intersection leg. The coefficient “-2.016” corresponds to the Year 2023. The calibration factor C_f (to be provided by the analyst) will provide the necessary adjustments to ensure that the model predictions reflect the desired year of application and local conditions.

The SPF relationship between predicted FI crash frequency and hourly traffic volume is illustrated in Figure 26a. It is applicable to annual average hourly volumes as large as 1,200 veh/h but may be used with caution to evaluate volumes slightly larger than this value. The trend line indicates that intersection legs with higher volume have more frequent crashes.

The solid trend line in Figure 26a illustrates the values obtained from Equation 44. The dashed trend line illustrates the values obtained from Equation 44 multiplied by the adjustment factor for the peak traffic hour AF_pk. This adjustment factor is described in a subsequent paragraph.

The SPF relationship between predicted PDO crash frequency and hourly traffic volume is illustrated in Figure 26b. It is shown here to facilitate a comparison between the predicted FI and PDO crash frequencies. The SPF for predicting PDO crash frequency is described in the next section.

Where R_p is the platoon ratio for hour i. The base condition for this adjustment factor is a platoon ratio of 1.0. It is applicable to platoon ratios in the range of 0.8 to 1.6.

The estimated AF trend for FI crash frequency is shown in Figure 27a. The trend line in this figure illustrates the values obtained from Equation 45. The trend indicates that intersection approaches with favorable progression have a lower crash risk than those approaches with unfavorable progression.

The estimated AF trend for PDO crash frequency is shown in Figure 27b. It is shown here to facilitate a comparison of the AF values for FI and PDO crashes. The AF for PDO crash frequency is described in the next section.

Where the indicator variable for hour i I_pk,j has a value of 1 if the subject hour is in the range 7:00 to 9:00 am or 4:00 to 6:00 pm. If it is not in this range, then the indicator variable value is 0. The value of this AF is 0.70 for peak traffic hours and 1.0 for off-peak hours.

where,

The probability of a vehicle experiencing a split failure p_sf,i is computed using Equation 36 (it is described in the discussion associated with Equation 25). The base condition for this AF is no split failures for the left-turn phase or the through phase serving the subject approach lanes. It is applicable to approaches where the probability of vehicle experiencing split failure ranges from 0.0 to 0.30.

The estimated AF trend for FI crash frequency is shown in Figure 28a. The trend line in this figure illustrates the values obtained from Equation 47. The trend indicates that intersection approaches with a higher probability of split failure have a higher crash risk. The trend shown in Figure 28b is discussed in the next section. For a given probability of split failure, the trends indicate that crash risk is lower for those legs with a larger cycle length.

This equation is used to evaluate one specified hour for a given signalized intersection leg. The predicted value represents the annual average PDO crash frequency for the specified hour. Multiple hours of the typical day can be evaluated by repeated application of the equation. The predicted average PDO crash frequency for each hour can be added to estimate the average crash frequency for the combined set of hours.

There are two AFs listed in the parentheses of Equation 49. Only one of these two AF can be used when the CPM is applied to a site. This restriction reflects the fact that the two AF are highly correlated and tend to predict the risk associated with the same crash events.

If the operating conditions being evaluated exist only for a specific subset of the days of the typical week, then the predicted average crash frequency is multiplied by the number of applicable days per week and then divided by 7. For example, if the signal timing plan being evaluated applies only to weekdays (i.e., Monday to Friday), then the analyst would multiply the predicted average crash frequency by 5/7 (= 5 days per weekday/7 days per week).

Where the AAHT_a,i volume variable represents the annual average hourly volume during a specified hour i for the subject intersection leg. The coefficient “-1.307” corresponds to the Year 2023. The calibration factor C_f (to be provided by the analyst) will provide the necessary adjustments to ensure that the model predictions reflect the desired year of application and local conditions.

The SPF relationship between predicted PDO crash frequency and hourly traffic volume is illustrated in Figure 26b. It is applicable to annual average hourly volumes as large as 1,200 veh/h but may be used with caution to evaluate volumes slightly larger than this value. The trend line indicates that intersection legs with higher volume have more frequent crashes.

Where R_p is the platoon ratio for hour i. The base condition for this adjustment factor is a platoon ratio of 1.0. It is applicable to platoon ratios in the range of 0.8 to 1.6.

The estimated AF trend for PDO crash frequency is shown in Figure 27b. The trend line in this figure illustrates the values obtained from Equation 51. The trend indicates that intersection approaches with favorable progression have a lower crash risk than those approaches with unfavorable progression.

where,

The probability of a vehicle experiencing a split failure p_sf,i is computed using Equation 36. The base condition for this AF is no split failures for the left-turn phase or the through phase serving the subject approach lanes. It is applicable to approaches where the probability of vehicle experiencing split failure ranges from 0.0 to 0.30.

The estimated AF trend for PDO crash frequency is shown in Figure 28b. The trend line in this figure illustrates the values obtained from Equation 52. The trend indicates that intersection approaches with a higher probability of split failure have a higher crash risk. For a given probability of split failure, the trends indicate that crash risk is lower for those legs with a larger cycle length.

where,

The base condition for this AF is no vehicles entering during the through yellow interval (i.e., PVY_i = 0). It is applicable to approaches where the value of PVY_i is less than 0.1.

The estimated AF trend for PDO crash frequency is shown in Figure 29b. The trend line in this figure illustrates the values obtained from Equation 53. The trend indicates that, for a given approach volume, intersection approaches with a higher probability of intersection entry during yellow have a higher crash risk. For a given probability of entry, crash risk is shown to be higher for those approaches with higher volume.

Analysis of Left-Turn Gaps

This section presents the estimated CPMs for predicting the safety effect of left-turn gap availability. One subsection describes the CPM for predicting the frequency of left-turn-with-opposing-through-vehicle (LT) crashes. The second subsection describes the CPM for predicting the frequency of all other (nonLT) crashes. The SPF and AF components of each CPM are described in the associated subsection.

• CPM for Left-Turn-Opposed Crash Frequency
• CPM for NON-Left-Turn-Opposed Crash Frequency

CPM for Left-Turn-Opposed Crash Frequency

This section describes the CPM for predicting LT crash frequency (all severities). For each SPF and AF component of the CPM, the description includes one subsection that (a) presents the parameterized equation and (b) includes a figure illustrating equation trends. The CPM (and its SPF and AF components) is described by the following equation.

where,

This equation is used to evaluate one specified hour for a given signalized intersection leg. The intersection leg of interest must have an opposing left-turn movement that is served by permissive or protected/permissive left-turn operation.

The predicted value from the equation represents the annual average LT crash frequency for the specified hour. Multiple hours of the typical day can be evaluated by repeated application of the equation. The predicted average LT crash frequency for each hour can be added to estimate the average crash frequency for the combined set of hours.

If the operating conditions being evaluated exist only for a specific subset of the days of the typical week, then the predicted average crash frequency is multiplied by the number of applicable days per week and then divided by 7. For example, if the signal timing plan being evaluated applies only to weekdays (i.e., Monday to Friday), then the analyst would multiply the predicted average crash frequency by 5/7 (= 5 days per weekday/7 days per week).

Where the AAHT_a volume variable represents the annual average hourly volume during a specified hour i for the subject intersection leg. The coefficient “-3.298” corresponds to the Year 2023. The calibration factor C_f (to be provided by the analyst) will provide the necessary adjustments to ensure that the model predictions reflect the desired year of application and local conditions.

The SPF relationship between predicted LT crash frequency and hourly traffic volume is illustrated in Figure 30a. It is applicable to annual average hourly volumes as large as 1,200 veh/h but may be used with caution to evaluate volumes slightly larger than this value. The trend line indicates that intersection legs with higher volume have more frequent crashes.

The SPF relationship between predicted nonLT crash frequency and hourly traffic volume is illustrated in Figure 30b. It is shown here to facilitate a comparison between the predicted LT and nonLT crash frequencies. The SPF for predicting nonLT crash frequency is described in the next section.

where,

This equation is used to evaluate one specified hour for a given signalized intersection leg. The intersection leg of interest must have an opposing left-turn movement that is served by permissive or protected/permissive left-turn operation.

The predicted value from the equation represents the annual average nonLT crash frequency for the specified hour. Multiple hours of the typical day can be evaluated by repeated application of the equation. The predicted average nonLT crash frequency for each hour can be added to estimate the average crash frequency for the combined set of hours.

If the operating conditions being evaluated exist only for a specific subset of the days of the typical week, then the predicted average crash frequency is multiplied by the number of applicable days per week and then divided by 7. For example, if the signal timing plan being evaluated applies only to weekdays (i.e., Monday to Friday), then the analyst would multiply the predicted average crash frequency by 5/7 (= 5 days per weekday/7 days per week).