Safety Evaluation of Multiple Strategies at Stop-Controlled Intersections

Chapter 7. Development of SPFs

This section presents the SPFs developed for each crash type. The SPFs support the use of the EB methodology to estimate the safety effectiveness of this strategy.⁽¹⁵⁾ The research team developed negative binomial regression models to predict the number of crashes. In specifying a negative binomial error structure, the dispersion parameter, k, was estimated iteratively from the model and the data. For a given dataset, smaller values of k indicate relatively better models. The research team developed one SPF for each of the following intersection configurations:

• 3 x 22: Three-legged intersections with two lanes on the mainline and two lanes on the cross street.
  
  
• 4 x 22: Four-legged intersections with two lanes on the mainline and two lanes on the cross street.
  
  
• 3 x 42: Three-legged intersections with four lanes on the mainline and two lanes on the cross street.
  
  
• 4 x 42: Four-legged intersections with four lanes on the mainline and two lanes on the cross street.

The research team developed correlation matrices for variables and used them as guide for the SPF development process. This helped the research team avoid highly correlated variables in the models. The model development followed a process of forward selection for selecting variables with the best fit. The research team started with mainline and cross street traffic volumes and their variants (e.g., natural logarithm, ratio of cross street AADT, and mainline AADT). Other candidate explanatory variables were then added, one by one, to the model. The model was re-estimated and the goodness of fit was reevaluated with each variable addition.

The research team initially included annual adjustment variables (i.e., indicators for years 2005 to 2014) in the SPFs during the first iteration of model development. However, most of these variables did not result in statistically significant parameters or help improve the fit of the SPFs. The inclusion of annual adjustment variables also led to heavily under-predicted crashes for some years (i.e., small coefficients on the negative side and far from being well fit), especially for the later years that cover the after period. The team eventually decided to drop these annual adjustment variables from the models and considered another approach to account for the annual trend (discussed later in this chapter).

In some cases, the research team could not develop an adequate model for a specific crash type. In these cases, the team used the SPF for total crashes and adjusted by the proportion of the number of crashes for the given crash type in total crashes.

The definition of variables included in the final SPFs are as follows:

• Total_axbc = the predicted number of total crashes (all types and severity levels) for intersection with “a” legs, “b” lanes on the mainline, and “c” lanes on the cross street (e.g., 3 x 42 for three-legged intersections with four lanes on the mainline and two lanes on the cross street).
  
  
• FI_axbc = the predicted number of fatal and injury crashes for intersection with “a” legs, “b” lanes on the mainline, and “c” lanes on the cross street.
  
  
• Rear-End_axbc = the predicted number of rear-end crashes for intersection with “a” legs, “b” lanes on the mainline, and “c” lanes on the cross street.
  
  
• Right-Angle_axbc = the predicted number of right-angle crashes for intersection with “a” legs, “b” lanes on the mainline, and “c” lanes on the cross street.
  
  
• Night_axbc = the predicted number of nighttime crashes for intersection with “a” legs, “b” lanes on the mainline, and “c” lanes on the cross street.
  
  
• ml_aadt = AADT on the mainline (vehicles/day).
  
  
• xst_aadt = AADT on the cross street (vehicles/day).
  
  
• aadt = ml_aadt + xst_aadt, total traffic of intersection (vehicles/day).
  
  
• ratio1 = ln(xst_aadt)/ln(ml_aadt), with ln(xst_aadt) being the natural logarithm of AADT on cross street and ln(ml_aadt) being the natural logarithm of AADT on mainline.
  
  
• ratio2 = xst_aadt/ml_aadt, with xst_aadt and ml_aadt being the AADT on cross street and mainline, respectively.
  
  
• ratio3 = xst_aadt/(xst_aadt + ml_aadt), with xst_aadt and ml_aadt being the AADT on cross street and mainline, respectively.
  
  
• ratio4 = ln(xst_aadt)/ln(xst_aadt + ml_aadt), with ln(xst_aadt) being the natural logarithm of AADT on cross street and ln(xst_aadt + ml_aadt) being the natural logarithm of total traffic at intersection.
  
  
• urban = urban/rural indicator for the intersection (= 1 for urban, = 0 otherwise).
  
  
• β₁, β₂, β₃, β₄ = parameters estimated in the SPF development process using maximum likelihood method.
  
  
• k = overdispersion parameter.

SPFs for 3 x 22 Intersections

The SPF for total crashes at three-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 11.

Figure 11. Equation. Total crash SPF for 3 x 22 intersections.

Table 17 presents the total crash SPF parameters for three-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for fatal and injury crashes at three-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 12.

Figure 12. Equation. Fatal and injury crash SPF for 3 x 22 intersections.

Table 18 presents the fatal and injury crash SPF parameters for three-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for rear-end crashes at three-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 13.

Figure 13. Equation. Rear-end crash SPF for 3 x 22 intersections.

Table 19 presents the rear-end crash SPF parameters for three-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for right-angle crashes at three-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 14.

Figure 14. Equation. Right-angle crash SPF for 3 x 22 intersections.

Table 20 presents the right-angle crash SPF parameters for three-legged intersections with two lanes on the mainline and two lanes on the cross street.

The research team could not develop a statistically significant model for nighttime crashes. The SPF for total crashes was used with an adjustment factor to predict nighttime crashes.

SPFs for 4 x 22 Intersections

The SPF for total crashes at four-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 15.

Figure 15. Equation. Total crash SPF for 4 x 22 intersections.

Table 21 presents the total crash SPF parameters for four-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for fatal and injury crashes at four-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 16.

Figure 16. Equation. Fatal and injury crash SPF for 4 x 22 intersections.

Table 22 presents the fatal and injury crash SPF parameters for four-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for rear-end crashes at four-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 17.

Figure 17. Equation. Rear-end crash SPF for 4 x 22 intersections.

Table 23 presents the rear-end crash SPF parameters for four-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for right-angle crashes at four-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 18.

Figure 18. Equation. Right-angle crash SPF for 4 x 22 intersections.

Table 24 presents the right-angle crash SPF parameters for four-legged intersections with two lanes on the mainline and two lanes on the cross street.

The SPF for nighttime crashes at four-legged intersections with two lanes on the mainline and two lanes on the cross street is shown in figure 19.

Figure 19. Equation. Nighttime crash SPF for 4 x 22 intersections.

Table 25 presents the nighttime crash SPF parameters for four-legged intersections with two lanes on the mainline and two lanes on the cross street.

SPFs for 3 x 42 Intersections

The SPF for total crashes at three-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 20.

Figure 20. Equation. Total crash SPF for 3 x 42 intersections.

Table 26 presents the total crash SPF parameters for three-legged intersections with four lanes on the mainline and two lanes on the cross street.

The research team could not develop a statistically significant model for fatal and injury crash SPF at three-legged intersections with four lanes on the mainline and two lanes on the cross street. The SPF for total crashes was used with an adjustment factor to predict the number of fatal and injury crashes.

The SPF for rear-end crashes at three-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 21.

Figure 21. Equation. Rear-end crash SPF for 3 x 42 intersections.

Table 27 presents the rear-end crash SPF parameters for three-legged intersections with four lanes on the mainline and two lanes on the cross street.

The SPF for right-angle crashes at three-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 22.

Figure 22. Equation. Right-angle crash SPF for 3 x 42 intersections.

Table 28 presents the right-angle crash SPF parameters for three-legged intersections with four lanes on the mainline and two lanes on the cross street.

The research team could not develop a statistically significant model for nighttime crashes. The SPF for total crashes was used with an adjustment factor to predict nighttime crashes.

SPFs for 4 x 42 Intersections

The SPF for total crashes at four-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 23.

Figure 23. Equation. Total crash SPF for 4 x 42 intersections.

Table 29 presents the total crash SPF parameters for four-legged intersections with four lanes on the mainline and two lanes on the cross street.

The SPF for fatal and injury crashes at four-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 24.

Figure 24. Equation. Fatal and injury crash SPF for 4 x 42 intersections.

Table 30 presents the fatal and injury SPF parameters for four-legged intersections with four lanes on the mainline and two lanes on the cross street.

The SPF for rear-end crashes at four-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 25.

Figure 25. Equation. Rear-end crash SPF for 4 x 42 intersections.

Table 31 presents the rear-end SPF parameters for four-legged intersections with four lanes on the mainline and two lanes on the cross street.

The SPF for right-angle crashes at four-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 26.

Figure 26. Equation. Right-angle crash SPF for 4 x 42 intersections.

Table 32 presents the right-angle SPF parameters for four-legged intersections with four lanes on the mainline and two lanes on the cross street.

The SPF for nighttime crashes at four-legged intersections with four lanes on the mainline and two lanes on the cross street is shown in figure 27.

Figure 27. Equation. Nighttime crash SPF for 4 x 42 intersections.

Table 33 presents the nighttime SPF parameters for four-legged intersections with four lanes on the mainline and two lanes on the cross street.

Before–After Adjustment Factors

SPFs may include annual factors to account for potential time trends, as discussed in the first section of chapter 5. In this study, however, the SPFs did not include yearly indicator variables because after numerous attempts, the research team could not achieve a reasonable level of statistical significance for these individual variables. The research team decided to account for the time trend by using an aggregate before-to-after adjustment factor. Instead of using annual adjustment factors (i.e., one for each year), the research team used a single adjustment factor to account for the difference (i.e., crash trend) between the before and after periods. Because SCDOT did not install the treatment at all sites in the same year, the installation periods varied. For this reason, the team calculated one adjustment factor for each installation period (i.e., all intersections for which treatments were implemented in 2009–2010 have the same adjustment factor). Using these adjustment factors, the assumption is that all safety effects of unknown or immeasurable factors (e.g., weather) do not differ among reference and treatment sites or across intersection configurations. These factors were calculated based on the observed and predicted crashes at all reference sites. Figure 28 shows the equation used to calculate the before-after adjustment factors.

Figure 28. Equation. Before–after adjustment factor calculation.

Where:

Adj_Factor = the factor for adjusting the difference between the before and after period.

Obs_{_before} = observed number of crashes at reference sites during the before period.

Pred_before = predicted number of crashes at reference sites during the before period (calculated by SPF).

Obs_{_after} = observed number of crashes at reference sites during the after period.

Pred_after = predicted number of crashes at reference sites during the after period (calculated by SPF).

Table 34 to table 38 present the before–after adjustment factors for each installation time frame and crash type.