Crash Modification Factors in the Highway Safety Manual: Resources for Evaluation (2023)

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Appendix K Calibration of Prediction Models of Rural Segments for the 2nd Edition of the HSM K-1 

Table of Contents 1. Introduction ....................................................................................................................................... K-4 2. Summary of Data............................................................................................................................... K-4 3. Methodology and Models .................................................................................................................. K-7 4. Calibration Results .......................................................................................................................... K-10 5. Validation of SPF Adjustment Factors ............................................................................................ K-13 5.1 Rural Two-Lane Undivided Roads ............................................................................................ K-14 Lighting ....................................................................................................................................... K-14 TWLTL ....................................................................................................................................... K-16 RHR ............................................................................................................................................. K-17 Horizontal Curvature ................................................................................................................... K-20 Lane Width .................................................................................................................................. K-22 Shoulder Width............................................................................................................................ K-24 5.2 Rural Four-Lane Undivided Roads ........................................................................................... K-27 Lighting ....................................................................................................................................... K-27 Shoulder Width............................................................................................................................ K-28 5.3 Rural Four-Lane Divided Roads ............................................................................................... K-31 Lighting ....................................................................................................................................... K-31 Shoulder Width............................................................................................................................ K-32 Median Width .............................................................................................................................. K-35 5.4 Summary on Validation of Adjustment Factors ........................................................................ K-37 Table 1. Summary of Rural Two-Lane Undivided Segment ..................................................................... K-5 Table 2. Summary of Rural Multi-Lane Undivided Segment ................................................................... K-5 Table 3. Summary of Rural Multi-Lane Divided Segment ....................................................................... K-6 Table 4. Site Summary .............................................................................................................................. K-6 Table 5. Summary of Crash Data .............................................................................................................. K-6 Table 6. Summary of Volume Data........................................................................................................... K-7 Table 7. Base Condition SPFs, Two-Lane Undivided Segments .............................................................. K-8 Table 8. Base Condition SPFs, Four-Lane Undivided Segments .............................................................. K-9 Table 9. Base Condition SPFs, Four-Lane Divided Segments ................................................................ K-10 Table 10. Calibration Results and Recommended Calibration Factors, Two-Lane Undivided Segments ...... K-11 Table 11. Calibration Results and Recommended Calibration Factors, Four-Lane Undivided Segments ...... K-12 Table 12. Calibration Results and Recommended Calibration Factors, Four-Lane Divided Segments ............................................................................................................................. K-12 Table 13. Illustration of Approach 1 ....................................................................................................... K-13 Table 14. Rural Two-Lane Undivided Lighting Approach 1 .................................................................. K-15 Table 15. Rural Two-Lane Undivided Lighting Approach 2, estimate (standard error) ......................... K-15 K-2 

Table 16. Rural Two-Lane Undivided TWLTL Adjustment Factors ...................................................... K-16 Table 17. Rural Two-Lane Undivided TWLTL Approach 1 .................................................................. K-17 Table 18. Rural Two-Lane Undivided TWLTL Approach 2, estimate (standard error) ......................... K-17 Table 19. Rural Two-Lane Undivided RHR Adjustment Factors Formulae ........................................... K-18 Table 20. Rural Two-Lane Undivided Implied RHR Adjustment Factors for Total Crashes ................. K-18 Table 21. Rural Two-Lane Undivided RHR Approach 1........................................................................ K-19 Table 22. Rural Two-Lane Undivided RHR Approach 2, estimate (standard error) .............................. K-19 Table 23. Rural Two-Lane Undivided Horizontal Curves Approach 3................................................... K-20 Table 24. Rural Two-Lane Undivided Lane Width Formulae ................................................................ K-23 Table 25. Rural Two-Lane Undivided Lane Width Approach 3 ............................................................. K-23 Table 26. Rural Two-Lane Undivided Lane Width Approach 3 By Lane Width ................................... K-24 Table 27. Rural Two-Lane Undivided Shoulder Width Formulae .......................................................... K-24 Table 28. Rural Two-Lane Undivided Lane Width Formulae ................................................................ K-25 Table 29. Rural Four-Lane Undivided Shoulder Width Parameters ....................................................... K-28 Table 30. Rural Two-Lane Undivided Lane Width Formulae ................................................................ K-29 Table 31. Rural Four-Lane Divided Lighting Approach 1 ...................................................................... K-32 Table 32. Rural Four-Lane Divided Shoulder Width Parameters ........................................................... K-32 Table 33. Rural Two-Lane Undivided Lane Width Formulae ................................................................ K-33 Table 34. Rural Four-Lane Divided Median Width Parameters ............................................................. K-36 Table 35. Rural Two-Lane Undivided Median Width Formulae ............................................................ K-36 Figure 1. Rural Two-Lane Undivided Horizontal Curves KABCO Predictions ..................................... K-21 Figure 2. Rural Two-Lane Undivided Horizontal Curves SD KABCO Predictions ............................... K-21 Figure 3. Rural Two-Lane Undivided Horizontal Curves OD KABCO Predictions .............................. K-22 Figure 4. Rural Two-Lane Undivided Horizontal Curves SV KABCO Predictions ............................... K-22 Figure 5. Rural Two-Lane Undivided Shoulder Width KABCO Predictions ......................................... K-26 Figure 6. Rural Two-Lane Undivided Shoulder Width SD KABCO Predictions ................................... K-26 Figure 7. Rural Two-Lane Undivided Shoulder Width OD KABCO Predictions .................................. K-27 Figure 8. Rural Two-Lane Undivided Shoulder Width SV KABCO Predictions ................................... K-27 Figure 9. Rural Four-Lane Undivided Shoulder Width KABCO Predictions ......................................... K-29 Figure 10. Rural Four-Lane Undivided Shoulder Width SD KABCO Predictions ................................. K-30 Figure 11. Rural Four-Lane Undivided Shoulder Width OD KABCO Predictions ................................ K-30 Figure 12. Rural Four-Lane Undivided Shoulder Width SV KABCO Predictions ................................. K-31 Figure 13. Rural Four-Lane Divided Shoulder Width KABCO Predictions ........................................... K-34 Figure 14. Rural Four-Lane Divided Shoulder Width SD KABCO Predictions..................................... K-34 Figure 15. Rural Four-Lane Divided Shoulder Width OD KABCO Predictions .................................... K-35 Figure 16. Rural Four-Lane Divided Shoulder Width SV KABCO Predictions..................................... K-35 Figure 17. Rural Four-Lane Divided Median Width KABCO Predictions ............................................. K-37 K-3 

1. INTRODUCTION This Appendix documents the production of common-jurisdiction calibration factors for the rural segment base condition crash prediction models developed in NCHRP Project 17-62 and that are expected to be included in the 2nd edition of the HSM. This paper also documents the validation of SPF adjustment factors recommended in NCHRP 17-72 for the same site types in Ohio. For the purposes of developing the calibration factors, data were obtained from Ohio for three segment types for the period from 2013 to 2017: 1. Rural two-lane undivided 2. Rural four-lane undivided 3. Rural four-lane divided To develop calibration factors for the base models, the data provided were reduced to only include segments meeting the relevant base conditions where possible. In some cases, the base condition criteria were relaxed (including sites outside of AADT ranges used for SPF estimation) in order to provide for a sufficient sample size. The calibrations were performed for the base models because a) SPF adjustment factors were not developed in the 17-62 project, and b) the calibration process is intended to account for jurisdictional and temporal differences in factors such as crash reporting practices, weather etc. that are unrelated to roadway design and the adjustment factors should be more universal for the average location. If adjustment factors were to be developed and applied to the model predictions the same calibration factors would be applied. 1 2. SUMMARY OF DATA For rural two-lane undivided segments Table 1 lists the base condition variables, the base conditions and the potential for using the data to also validate the adjustment factors recommended from NCHRP 17- 72. The data available did not include information on the number of driveways, vertical curvature, or grade within a segment. For these variables, the base condition has been assumed. Furthermore, if the base conditions for lane width and shoulder width were adopted very few segments would be available. In order to obtain a useable sample size, segments with a lane width between 10 and 12 feet and shoulder width between 4 and 7 feet were included in the calibration data. With regards to validating the adjustment factors, this is not possible for the assumed variables of driveway density, vertical curvature, and grade. It is also not possible for variables showing no variation from the base condition in the data, including, shoulder type, centerline rumble strips, passing lanes and automated speed enforcement. 1 Note that for HSM users calibrating the models to their own data, the calibration procedure developed in NCHRP 17-62 advises: “If calibration is being done for base models and appropriate skills are available or could be acquired, it is recommended to try to directly estimate a model with the final calibration dataset and adopt the model if successfully estimated and performs better than the calibration factor and calibration function.” K-4 

Table 1. Summary of Rural Two-Lane Undivided Segment Variables and Base Conditions Potential for Validating Adjustment Variable Base Condition Factor Lane width (LW) 12 feet Yes Shoulder width (SW) 6 feet Yes Shoulder type Paved No – no variation Roadside hazard rating (RHR) 3 Yes Driveway density (DD)1 5 driveways per mile No – assumed value Horizontal curvature None Yes Vertical curvature2 None No – assumed value Centerline rumble strips None No – no variation Passing lanes None No – no variation Two-way left-turn lanes None Yes Lighting None Yes Automated speed enforcement None No – no variation Grade Level3 0% (see note below) No – assumed value 1 assumed to be 5 driveways per mile 2 assumed to be no vertical curves 3 assumed to be 0% The base conditions for rural multi-lane undivided segments are shown in Table 2. If the base conditions for lane width and shoulder width were adopted very few segments would be available. In order to obtain a useable sample size, segments with a lane width between 10 and 12 feet and shoulder width greater or equal to 4 feet were included in the calibration data. Sideslope has been assumed to be 1V:7H for all segments. With regards to validating the adjustment factors this is not possible for the assumed variables of sideslope. It is also not possible for variables showing no variation from the base condition in the data, including, shoulder type and automated speed enforcement. Table 2. Summary of Rural Multi-Lane Undivided Segment Variables and Base Conditions Potential for Validating Adjustment Variable Base Condition Factor Lane width (LW) 12 feet Yes Shoulder width (SW) ≥ 6 feet Yes Shoulder type Paved No – no variation Sideslopes1 1V:7H or flatter No – assumed value Lighting None Yes Automated speed enforcement None No – no variation 1 assumed to be 1V:7H The base conditions for rural multi-lane divided segments are shown in Table 3. All base conditions could be met while ensuring an adequate sample size. With regards to validating the adjustment factors this is not possible for variables showing no variation from the base condition in the data, including, shoulder type and automated speed enforcement. K-5 

Table 3. Summary of Rural Multi-Lane Divided Segment Variables and Base Conditions Potential for Validating Adjustment Variable Base Condition Factor Lane width (LW) 12 feet Yes Right shoulder width (SW) ≥ 8 feet Yes Shoulder type Paved No – no variation Median width ≥ 30 feet Yes Lighting None Yes Automated speed enforcement None No – no variation Tables 4 and 5 show the number of segments, the sum of mileage and the sum of crashes by crash and severity type for the calibration of base condition SPFs. While the mileage of segments is adequate for calibration it is notable that there are very few intersecting direction and opposite direction crashes for the four-lane undivided and divided sites. For all crash types the numbers of crashes can be low for the higher severity categories. Table 4. Site Summary Site Type Rural 2 Lane Undivided Rural Multilane Undivided Rural Multilane Divided Number of Segments 990 86 346 Sum of Mileage 328 miles 30 miles 162 miles Table 5. Summary of Crash Data Severity Rural 2 Lane Rural Multilane Rural Multilane Crash Type Type Undivided Undivided Divided All KABCO 1,316 316 957 KABC 536 144 293 KAB 378 102 194 KA 113 26 50 Same Direction KABCO 338 124 290 KABC 141 51 102 KAB 80 26 57 KA n/a 8 n/a Opposite Direction KABCO 155 27 22 KABC 93 22 11 KAB 75 17 10 KA 37 6 5 Intersecting KABCO n/a 38 n/a Direction KABC n/a 27 n/a KAB n/a 20 n/a KA n/a 5 n/a Single Vehicle KABCO 726 105 611 KABC 262 41 171 KAB 201 36 120 KA 54 6 27 n/a indicates that a base condition model is not available for that crash/severity type K-6 

Table 6 shows the AADT ranges for the calibration dataset along with the AADT ranges used for base condition SPF estimation. Table 6. Summary of Volume Data Calibration Dataset SPF Estimation Dataset Min Max Mean Min Max Mean AADT AADT AADT AADT AADT AADT Rural 2 Lane Undivided 235 16125 4227 210 21622 4573 Rural Multilane Undivided 592 21589 10293 250 21667 7193 Rural Multilane Divided 472 41424 11937 2325 66504 16212 3. METHODOLOGY AND MODELS The methodology for calculating the calibration factors is documented in the HSM 1st edition. This equation is: 𝑛𝑛sites 𝑛𝑛𝑐𝑐𝑐𝑐 ∑𝑖𝑖=1 ∑𝑗𝑗=1 𝑁𝑁𝑜𝑜,𝑤𝑤(𝑖𝑖),𝑥𝑥(𝑖𝑖),𝑦𝑦,𝑧𝑧,𝑖𝑖,𝑗𝑗 𝐶𝐶𝑤𝑤,𝑥𝑥,𝑦𝑦,𝑧𝑧 = 𝑛𝑛sites 𝑛𝑛𝑐𝑐𝑐𝑐 ∑𝑖𝑖=1 ∑𝑗𝑗=1 𝑁𝑁𝑝𝑝𝑝𝑝,𝑤𝑤(𝑖𝑖),𝑥𝑥(𝑖𝑖),𝑦𝑦,𝑧𝑧,𝑖𝑖,𝑗𝑗 where Cw, x y, z = calibration factor to adjust SPF for local conditions for site type w, cross section or control type x, crash type y, and severity z; No, w(i), x(i), y, z, i, j = observed crash frequency for site i with site type w(i) and year j (includes cross section or control type x(i) for crash type y, and severity z) (crashes/yr); Npu, w(i), x(i), y, z, i = predicted average crash frequency unadjusted by calibration factor for site i with site type w(i) (includes cross section or control type x(i) for crash type y, and severity z) (crashes/yr); nsites = number of sites in the calibration database (site); and nca = number of years in the calibration period (yr). For all rural segment models the model form developed by Project 17-62 is: 𝑁𝑁𝑠𝑠𝑠𝑠𝑠𝑠,17−62 = 𝑒𝑒𝑒𝑒𝑒𝑒[𝑏𝑏0 + 𝑏𝑏1 × ln(𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴) + ln(𝐿𝐿)] and the overdispersion parameter is determined by 𝑘𝑘 = 1�𝑒𝑒𝑒𝑒𝑒𝑒[𝑐𝑐 + ln(𝐿𝐿)]. The calibration factors were estimated using the FHWA tool “The Calibrator” 2. This spreadsheet based tool can be used to estimate calibration factors and well as calculate several goodness-of-fit measures to assess the success of the calibration. 2 https://safety.fhwa.dot.gov/rsdp/toolbox-content.aspx?toolid=150 K-7 

Tables 7 to 9 present the parameter estimates (and standard errors in parentheses) for the rural segment base condition SPFs developed in NCHRP 17-62. Table 7. Base Condition SPFs, Two-Lane Undivided Segments Crash Type Severity b0 b1 c -7.463 0.927 1.999 KABCO (0.520) (0.062) (0.166) -9.006 0.977 1.479 KABC (0.798) (0.095) (0.255) Total -8.499 0.852 1.100 KAB (1.003) (0.120) (0.327) -9.853 0.872 KA 2.527 (2.703) (1.472) (0.172) -15.456 1.658 1.214 KABCO (1.168) (0.135) (0.292) -17.721 1.807 1.326 Same direction KABC (1.684) (0.190) (0.550) -16.183 1.526 1.355 KAB (2.313) (0.262) (1.339) -10.525 1.085 0.636 KABCO (1.230) (0.147) (0.254) -11.461 1.100 0.582 KABC (1.573) (0.185) (0.430) Opposite direction -10.972 0.999 0.228 KAB (1.842) (0.218) (0.517) -11.190 0.947 30.408 KA (2.021) (0.235) (0.014) -5.798 0.674 2.005 KABCO (0.572) (0.069) (0.223) -6.582 0.613 1.117 KABC (0.975) (0.117) (0.347) Single vehicle -6.919 0.592 0.809 KAB (1.227) (0.148) (0.460) -10.949 0.899 0.446 KA (2.381) (0.280) (1.254) K-8 

Table 8. Base Condition SPFs, Four-Lane Undivided Segments Crash Type Severity b0 b1 c -9.129 1.055 0.476 KABCO (1.001) (0.112) (0.130) -9.6520 1.0088 0.611 KABC (1.2192) (0.1350) (0.221) Total -9.704 0.950 0.783 KAB (1.447) (0.160) (0.390) -9.799 0.847 -0.2157 KA (2.335) (0.259) (0.5427) -13.541 1.431 0.0327 KABCO (1.616) (0.178) (0.183) Same direction -16.6504 1.654 0.365 KABC (2.2606) (0.245) (0.365) -10.209 1.000 -0.825 KABCO (2.145) (0.241) (0.211) -10.944 0.978 -1.199 Intersecting direction KABC (2.913) (0.325) (0.331) -11.340 0.955 -0.764 KAB (3.227) (0.356) (0.567) -15.344 1.495 -0.923 KABCO (2.912) (0.321) (0.304) -16.518 1.540 0.365 KABC (3.174) (0.343) (0.824) Opposite direction -18.421 1.711 13.203 KAB (3.572) (0.382) (224.650) -16.573 1.482 0.885 KA (3.998) (0.431) (2.254) -7.127 0.688 1.018 KABCO (1.196) (0.133) (0.379) -6.738 0.545 13.202 Single vehicle KABC (1.558) (0.173) (121.940) -6.941 0.518 0.476 KAB (2.044) (0.228) (0.879) K-9 

Table 9. Base Condition SPFs, Four-Lane Divided Segments Crash Type Severity b0 b1 c -9.644 1.050 0.669 KABCO (1.519) (0.156) (0.296) -10.817 1.064 1.023 KABC (1.999) (0.203) (0.851) Total -10.69 0.983 2.090 KAB (2.456) (0.248) (3.255) -7.690 0.508 11.238 KA (5.401) (0.554) (289.120) -14.701 1.479 -0.473 KABCO (2.920) (0.299) (0.210) -18.512 1.730 -1.620 Same direction KABC (6.115) (0.625) (0.521) -14.914 1.261 -2.190 KAB (9.572) (0.983) (0.883) -17.478 1.470 9.638 KABCO (5.829) (0.575) (438.060) -17.132 1.403 1.553 KABC Opposite (7.121) (0.707) (42.172) direction -20.211 1.656 9.871 KAB1 (8.927) (0.874) (396.870) -20.211 1.656 9.871 KA1 (8.927) (0.874) (396.870) -7.990 0.816 1.262 KABCO (1.580) (0.161) (0.715) -9.473 0.879 10.025 KABC (2.093) (0.212) (586.580) Single vehicle -10.952 0.973 1.422 KAB (2.925) (0.296) (2.264) -1.524 -0.176 9.978 KA (6.838) (0.719) (913.790) 1 For KA Opposite Direction crashes the estimated models are identical due to a lack of B severity crashes in the original dataset. 4. CALIBRATION RESULTS The results of the calibration are shown in Tables 10 to 12. For each base condition model calibrated the total number of observed and uncalibrated predicted crashes are provided as well as the calibration factor, its covariance and two statistics from the CURE plots for the predicted values. These two statistics are the maximum deviation from 0 of the cumulative residuals and the percentage of observations outside of the two standard deviation limits. The Notes column indicates which calibration factor is recommended in the event that the calculated calibration factor is based on few crashes. Generally, if the number of crashes was less than 100 it was judged insufficient. The coefficient of variation of the calibration factor, CV(C), was also considered with values less than or equal to 0.15 indicating a reliable calibration. This guidance is subjective and did not necessarily rule out a recommendation to use the estimated calibration factor in all cases. For example, in some cases it was considered acceptable when the number of crashes was close to 100 and it was judged K-10 

that the calibration factor was similar to the calibration factor for the same crash type with a severity category that included more crashes. Table 10. Calibration Results and Recommended Calibration Factors, Two-Lane Undivided Segments Observed Predicted Calibration Max % CURE SPF CV(C) Notes Crashes Crashes Factor CURE Dev Dev KABCO 1,316 2,204.30 0.60 0.06 124.93 91% KABC 536 723.04 0.74 0.08 66.06 85% KAB 378 412.00 0.92 0.09 44.85 26% KA 113 126.20 0.90 0.11 14.64 36% OD 155 400.17 0.39 0.16 24.01 46% KABCO OD 93 178.56 0.52 0.13 12.32 56% KABC Use OD OD KAB 75 122.26 0.61 0.16 7.65 45% KABC factor Use OD OD KA 37 62.95 0.59 0.17 3.89 52% KABC factor SD 338 415.48 0.81 0.24 33.61 50% KABCO SD 141 158.94 0.89 0.15 24.04 87% KABC Use SD SD KAB 80 63.51 1.26 0.09 9.66 35% KABC factor SV 726 1,348.76 0.54 0.09 67.56 70% KABCO SV 262 367.22 0.71 0.11 29.02 29% KABC SV KAB 201 219.48 0.92 0.13 26.73 28% Use SV SV KA 54 53.12 1.02 0.14 13.21 73% KAB factor K-11 

Table 11. Calibration Results and Recommended Calibration Factors, Four-Lane Undivided Segments Max % Observed Predicted Calibration SPF CV(C) CURE CURE Notes Crashes Crashes Factor Dev Dev KABCO 316 345.62 0.91 0.10 17.95 21% KABC 144 131.64 1.09 0.13 11.29 31% KAB 102 71.23 1.43 0.14 8.58 26% Use KABC factor KA 26 24.24 1.07 0.20 1.72 42% Use KABC factor OD KABCO 27 47.57 0.57 0.29 4.93 40% Use KABCO factor OD KABC 22 22.71 0.97 0.41 6.51 48% Use KABC factor OD KAB 17 17.71 0.96 0.27 4.36 41% Use KABC factor OD KA 6 12.28 0.49 0.41 1.26 65% Use KABC factor SD KABCO 124 155.65 0.80 0.16 9.30 23% Use SD KABCO SD KABC 51 59.91 0.85 0.24 9.40 26% factor SV KABCO 105 77.26 1.36 0.15 8.19 12% Use SV KABCO SV KABC 41 29.41 1.39 0.23 7.31 47% factor Use SV KABCO SV KAB 36 18.60 1.94 0.26 6.45 47% factor INT 38 69.33 0.55 0.28 6.02 31% Use KABCO factor KABCO INT KABC 27 26.94 1.00 0.25 6.08 30% Use KABC factor INT KAB 20 14.55 1.37 0.29 5.38 333% Use KABC factor Table 12. Calibration Results and Recommended Calibration Factors, Four-Lane Divided Segments Max % Observed Predicted Calibration SPF CV(C) CURE CURE Notes Crashes Crashes Factor Dev Dev KABCO 957 1,017.53 0.94 0.05 50.00 12% KABC 293 359.56 0.81 0.07 8.55 11% KAB 194 189.47 1.02 0.08 9.08 11% KA 50 43.04 1.16 0.14 6.42 16% Use KAB factor OD 22 21.89 1.01 0.43 4.05 16% Use KABCO factor KABCO OD KABC 11 16.33 0.67 0.41 2.37 29% Use KABC factor OD KAB 10 8.42 1.19 0.70 1.71 27% Use KAB factor OD KA 5 8.42 0.59 0.45 2.26 36% Use KAB factor SD KABCO 290 383.30 0.76 0.07 18.86 57% SD KABC 102 93.47 1.09 0.10 6.17 15% SD KAB 57 38.82 1.47 0.13 2.37 15% Use SD KABC factor SV KABCO 611 580.88 1.05 0.05 34.64 12% SV KABC 171 239.13 0.72 0.10 11.11 10% SV KAB 120 132.62 0.90 0.11 9.34 11% SV KA 27 34.48 0.78 0.28 4.40 14% Use SV KAB factor K-12 

5. VALIDATION OF SPF ADJUSTMENT FACTORS The validation of adjustment factors focused on the validation of each adjustment factor individually. The validation is a cross-sectional comparison so the optimum approach is to compare two sets of data where the only difference is in the variable of interest. The approach taken was to use sites that met all base conditions with the exception of the variable to which the adjustment factor applies. For example, where the adjustment factor for lighting is of interest, all the sites used meet the base conditions except for lighting where both sites with and without lighting are used. Three analysis approaches were explored depending on the nature of the adjustment factors. Approach 1 The first approach is to compare the observed to predicted number of crashes using the base condition model at each level of the variable, e.g., sites with lighting and those without. If that variable has an effect on safety, then the ratio of observed to predicted crashes will differ between the levels of the variable. The factor for each level, i, is the sum of observed crashes divided by the sum of predicted crashes as shown in the following equation: ∑𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝑒𝑒𝑒𝑒 𝐶𝐶𝑖𝑖 = ∑𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝑒𝑒𝑒𝑒 To do so, the following steps were undertaken: Step 1: Apply the base condition models to a dataset of sites meeting all base conditions with the exception of the variables of interest. For this variable site selection is not constrained. Step 2: Calculate the factor Ci for each level of the variable. Step 3: Adjustment factors can then be estimated for each level of the variable. Each factor is related to the base condition level by dividing the factor for level i by the factor for the base level. To illustrate, in the table below a variable has 3 levels and a factor has been estimated for each, ranging from 1.0 to 1.2. Level 1 is the base condition for the SPF. The adjustment factors for the other levels are estimated by dividing each factor Ci by the Ci for level 1. Note that in this example the Ci for level 1 is not 1.0. This may occur if the sites used did not exactly match those used for calibrating the base condition models or due to rounding the numbers of predicted crashes. Table 13. Illustration of Approach 1 Adjustment Level of Variable Ci Factors 1 0.98 1.00 2 1.20 1.22 3 1.50 1.53 These adjustment factors were then compared to the recommended adjustment factors. K-13 

Approach 2 The second approach made use of generalized linear regression modeling (GLM). In this approach the expected number of a crashes is modeled with the base condition model prediction as an offset. The variable of interest is then included in the model to estimate the adjustment factor. The equations below illustrate this approach. 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒𝑒𝑒 = (𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)(𝑆𝑆𝑆𝑆𝑆𝑆)𝑓𝑓(𝑉𝑉𝑉𝑉𝑉𝑉) ln(𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒𝑒𝑒) − ln [(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)(𝑆𝑆𝑆𝑆𝑆𝑆)] = 𝑙𝑙𝑙𝑙(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)ln (𝑓𝑓(𝑉𝑉𝑉𝑉𝑉𝑉)) where • SPF = base model prediction • intercept = a constant term • f(VAR) = function representing the relationship between crashes and the variable of interest Approach 3 For some variables, the recommended adjustment factors are not a single value or simple equation. In these cases, the approach to validation was to compare the sum of observed and predicted values for the base model and base model plus adjustment factor when applied to sites that did not meet the base conditions for the variable of interest. Additionally, where the variable is a continuous variable with sufficient variation plots of the cumulative residuals for the two predictions were compared to see if the use of the adjustment factors improves the predictions at the non-base condition sites. The remaining sections document the validation effort for each rural road segment type. 5.1 Rural Two-Lane Undivided Roads Lighting Presence of lighting was not in fact one of the recommended adjustment factors but was nevertheless explored. The base condition for lighting is the absence of roadway segment lighting. The HSM 1st edition adjustment factor applies to total crashes and is determined using the formula: AFlighting = 1.0 – [(1.0-0.72*pinr-0.83*pnr) x pnr] pinr = proportion of total nighttime crashes for unlit segments that are KABC severity (0.382 is default) ppnr = proportion of total nighttime crashes for unlit segments that are PDO (0.618 is default) pnr = proportion of total crashes for unlit segments that occur at night (0.370 is default) With the default values used AFlighting is equal to 0.92. Approaches 1 and 2 were both applied. Table 14 shows the number of sites, observed crashes and results of the approach 1 analysis. The only crash category with at least 100 crashes was total crashes and the implied adjustment factor of presence of lighting is 1.29. This does not agree with the HSM lighting adjustment factor of 0.92, although it may not be statistically different due to the small estimation sample. K-14 

Table 14. Rural Two-Lane Undivided Lighting Approach 1 No. Observed Crashes Lighting Sites KA KAV KABC KABCO SD KABCO OD KABCO SV KABCO Not Present 963 113 378 536 1316 338 155 726 Present 70 4 14 25 107 27 15 46 Calibration Factors No. Lighting C C C C C SD C OD C SV Sites KA KAB KABC KABCO KABCO KABCO KABCO Not Present 963 0.99 1.00 1.00 1.00 1.00 0.99 1.00 Present 70 0.56 0.58 0.75 1.29 1.37 1.55 0.99 Implied CMF 0.57 0.58 0.75 1.29 1.37 1.57 0.99 In approach 2 negative binomial models were calibrated for KABCO and single-vehicle KABCO (SVKABO) crashes. Although an adjustment factor does not exist for single-vehicle crashes it was of interest to attempt to calibrate a model. Other crash types had very few crashes in total so were not analyzed. The model form is: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒𝑒𝑒 = (𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)(𝑆𝑆𝑆𝑆𝑆𝑆)𝑒𝑒𝑒𝑒𝑒𝑒𝛽𝛽∗𝑙𝑙𝑙𝑙𝑙𝑙ℎ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 where SPF = the calibrated base condition SPF Lighting = 1 if lighting is present, 0 if not present Overdispersion = the overdispersion parameter of the negative binomial model Table 15 provides the parameter estimates and the implied adjustment factors. In contrast to Approach 1 the adjustment factor is less than 1 for KABCO crashes and close to the value of 0.92 when using the default values for the formula. For single-vehicle KABCO crashes the adjustment factor is lower at 0.79. The statistical significance of the parameters is however very poor, which would translate into a high variance for the estimated AF. Table 15. Rural Two-Lane Undivided Lighting Approach 2, estimate (standard error) Lighting Implied Crash Type Intercept Overdispersion Present AF 0.1422 -0.0157 KABCO 0.8747 0.98 (0.0502) (0.1803) 0.1149 -0.2378 SV KABCO 1.1635 0.79 (0.0612) (0.2350) Based on the results from Approach 1 the adjustment factor is not validated for KABCO crashes. Using Approach 2, there is some evidence that the direction of effect and magnitude of the adjustment factor is reasonable, although it has a relatively large standard error and is not statistically significantly different from a CMF value of 1.0. K-15 

TWLTL Table 16 shows the recommended adjustment factors for presence of a two-way-left-turn lane (TWLTL). Table 16. Rural Two-Lane Undivided TWLTL Adjustment Factors Crash Type Crash Severity AF All KABCO 0.64 KABC 0.64 KAB 0.64 KA 0.64 Single vehicle KABCO 1.00 KABC 1.00 KAB 1.00 KA 1.00 Same direction KABCO 0.53 KABC 0.53 KAB 0.53 KA 0.53 Opposing direction KABCO No recommended CMF KABC No recommended CMF KAB No recommended CMF KA No recommended CMF Approaches 1 and 2 were both applied. For TWLTL, if all other base conditions were kept, there are only 5 sites with 6 total crashes with a TWLTL. To increase the sample size segments with lighting were excluded and for the other base condition variables the recommended adjustment factors were applied. This allowed for 205 sites with 122 total crashes. Table 17 shows the number of sites, observed crashes and results of the approach 1 analysis. The only crash category with at least 100 crashes was total crashes and the implied adjustment factor of presence of lighting is 0.64 which happens to exactly match the recommended adjustment factor. For same direction KABCO crashes the implied adjustment factor of 0.61 is close the recommended value of 0.53 although there are only 35 same direction KABCO crashes at sites with TWLTL. The implied adjustment factor of 0.50 for SVKABCO, based on only 64 single vehicle crashes at sites with a TWLTL, disagrees with the recommended value of 1.0. K-16 

Table 17. Rural Two-Lane Undivided TWLTL Approach 1 No. Observed Crashes TWLTL Sites KA KAB KABC KABCO SD KABCO OD KABCO SV KABCO Not 34,177 2692 9488 13318 31120 5338 3128 20389 Present Present 205 8 23 42 122 35 13 64 Calibration Factors No. TWLTL C C C C C SD C OD C SV Sites KA KAB KABC KABCO KABCO KABCO KABCO Not 34,177 0.76 0.79 0.84 0.77 0.70 0.71 0.80 Present Present 205 0.38 0.32 0.41 0.49 0.45 0.43 0.48 Implied CMF 0.50 0.41 0.49 0.64 0.64 0.61 0.60 In approach 2, negative binomial models were calibrated for KABCO crashes. Models were not successfully developed for other crash types. The model form is: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒𝑒𝑒 = (𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)(𝑆𝑆𝑆𝑆𝑆𝑆)𝑒𝑒𝑒𝑒𝑒𝑒𝛽𝛽∗𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 where SPF = the calibrated base condition SPF TWLTL = 1 if a TWLTL is present, 0 if not present Overdispersion = the overdispersion parameter of the negative binomial model Table 18 provides the parameter estimates and the implied adjustment factor which at 0.69 is close to the estimate of 0.64 from Approach 1. Based on the relative consistency of the results from Approach 1 and Approach 2 the adjustment factors can be considered as validated. Although Approach 1 resulted in a different estimate for single vehicle KABCO crashes it was based on few crashes at sites with TWLTL. Table 18. Rural Two-Lane Undivided TWLTL Approach 2, estimate (standard error) TWLTL Implied Crash Type Intercept Overdispersion Present AF -0.1641 -0.3717 KABCO 1.224 0.69 (0.0098) (0.1479) RHR Table 19 shows the recommended adjustment factor formulae for roadside hazard rating (RHR), where 𝑒𝑒𝑒𝑒𝑒𝑒(−0.6869+0.0668×𝑅𝑅𝑅𝑅𝑅𝑅) All KABCO 𝐴𝐴𝐴𝐴 = exp(−0.4865) PSV = Proportion of crashes that are single-vehicle PNON-SV = Proportion of crashes that are not single-vehicle K-17 

Table 19. Rural Two-Lane Undivided RHR Adjustment Factors Formulae Crash Type Crash Severity AF All KABCO KABC KAB Use formula for all severities KA Single vehicle KABCO KABC (AFALL-PNON-SV)/PSV KAB KA Same direction KABCO 1.00 KABC 1.00 KAB 1.00 KA 1.00 Opposing direction KABCO 1.00 KABC 1.00 KAB 1.00 KA 1.00 The Ohio dataset has RHR values from 3 to 7. The implied AAFs for total crashes using the formulae in Table 19 are shown in Table 20. Approaches 1 and 2 were both applied. Table 20. Rural Two-Lane Undivided Implied RHR Adjustment Factors for Total Crashes RHR AF 3 1.00 4 1.07 5 1.14 6 1.22 7 1.31 Table 21 shows the number of sites, observed crashes and results of the Approach 1 analysis. The only crash type with at least 100 crashes in each RHR category was all KABCO crashes. For all KABCO crashes, the implied adjustment factors are reasonably consistent in trend with the recommended adjustment factors for RHR categories 4 and 7 but not categories 5 and 6. For other crash types it is difficult to make strong conclusions because the number of crashes in each category can be small. As a result there are no evident consistent trend, with fewer crashes often predicted for RHR levels higher than 3. K-18 

Table 21. Rural Two-Lane Undivided RHR Approach 1 No. Observed Crashes RHR Sites KA KAB KABC KABCO SD KABCO OD KABCO SV KABCO 3 963 113 378 536 1316 338 155 726 4 811 85 307 445 1004 262 101 542 5 512 59 227 326 719 204 75 386 6 214 21 66 120 275 79 28 141 7 96 11 41 65 127 40 8 65 Calibration Factors No. RHR C C C C C SD C OD C SV Sites KA KAB KABC KABCO KABCO KABCO KABCO 3 963 0.99 1.00 1.00 1.00 1.00 0.99 1.00 4 811 1.13 1.22 1.27 1.15 1.22 0.99 1.11 5 512 0.96 1.10 1.12 1.00 1.12 0.89 0.98 6 214 0.92 0.87 1.09 1.02 1.01 0.86 0.99 7 96 1.40 1.56 1.76 1.39 1.70 0.74 1.27 In approach 2 negative binomial models were calibrated for KABCO and SVKABCO crashes. The model form is: 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑒𝑒𝑒𝑒 = (𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦)(𝑆𝑆𝑆𝑆𝑆𝑆)𝑒𝑒𝑒𝑒𝑒𝑒𝛽𝛽∗𝑅𝑅𝑅𝑅𝑅𝑅 where SPF = the calibrated base condition SPF RHR = a categorical variable with a unique parameter estimate for each level Overdispersion = the overdispersion parameter of the negative binomial model Table 22 provides the parameter estimates and the implied adjustment factors. As with Approach 1, the results for KABCO crashes seem to validate the recommended adjustment factors for RHR 4 and 7 but not for RHR 5 and 6. The results for SVKABCO crashes follows the same pattern. Table 22. Rural Two-Lane Undivided RHR Approach 2, estimate (standard error) Crash Type Intercept RHR Overdispersion Implied AFs KABCO 0.1400 3 – 0 (0) 0.8404 3 – 1.00 (0.0495) 4 – 0.1179 (0.0731) 4 – 1.13 5 – 0.0403 (0.0835) 5 – 1.04 6 – -0.0816 (0.1218) 6 – 0.92 7 – 0.2445 (0.1619) 7 –1.28 SV KABCO 0.1139 3 – 0 (0) 1.1397 3 – 1.00 (0.0607) 4 – 0.1061 (0.0900) 4 – 1.11 5 – 0.0822 (0.1030) 5 – 1.09 6 – -0.0688 (0.1504) 6 – 0.93 7 – 0.2120 (0.2008) 7 – 1.24 K-19 

Given the relative consistency in the results for the two approaches for KABCO crashes and of these results with the trend in the HSM CMFs, it may be cautiously considered that the adjustment factors for RHR are validated for these crashes. Horizontal Curvature The recommended adjustment factor for all crash types is determined using the formula below: 80.2 1.55𝐿𝐿𝑐𝑐 + − 0.012𝑆𝑆 𝑅𝑅 𝐴𝐴𝐴𝐴 = 1.55𝐿𝐿𝑐𝑐 where Lc = length of horizontal curve in miles S = 1 is spiral transition present and 0 if not R = radius of curvature in feet In applying the equation, if the radius of curvature (R) is less than 100-ft, R is set to 100-ft. If the length of the horizontal curve (Lc) is less than 100 feet, Lc is set to 100ft. Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor applied to sites with horizontal curves. Table 23 shows the number of sites, observed and predicted crashes for each crash type. As expected, the application of the base models alone underpredicts the number of crashes at sites with curves, with the exception of SD KABCO crashes. When the adjustment factor is applied the predictions overpredict crashes for all crash types. Table 23. Rural Two-Lane Undivided Horizontal Curves Approach 3 Model No. Sites OBS KABCO PRED KABCO Base Model 651 523 397 Base Model + AAF 651 523 706 Model No. Sites OBS SV KABCO PRED SVKABCO Base Model 651 332 224 Base Model + AAF 651 332 404 Model No. Sites OBS OD KABCO PRED OD KABCO Base Model 651 58 46 Base Model + AAF 651 58 82 Model No. Sites OBS SD KABCO PRED SD KABCO Base Model 651 93 98 Base Model + AAF 651 93 169 K-20 

Figures 1 to 4 plot the cumulative residuals (observed-predicted) versus AADT on the x-axis without and with applying the adjustment factors. The purpose of the plots is to assess if the accuracy of the predictions is better when using the adjustment factor. For all crash types, the adjustment factor correctly increases the predictions but the increase is too large in all cases with the result that the cumulative residuals are actually closer to zero without using the adjustment factor. This issue is particularly bad at radii < 1,000 ft. In the figures the plot denoted pCMF includes the adjustment factor application. 200.0000 100.0000 0.0000 0 5000 10000 15000 -100.0000 cum res pKABCO cum res pCMFKABCO -200.0000 -300.0000 -400.0000 -500.0000 Figure 1. Rural Two-Lane Undivided Horizontal Curves KABCO Predictions 20.0000 0.0000 0 5000 10000 15000 -20.0000 cum res pSDKABCO -40.0000 cum res -60.0000 pCMFSDKABCO -80.0000 -100.0000 -120.0000 Figure 2. Rural Two-Lane Undivided Horizontal Curves SD KABCO Predictions K-21 

20.0000 10.0000 0.0000 0 5000 10000 15000 -10.0000 cum res pODKABCO -20.0000 cum res -30.0000 pCMFODKABCO -40.0000 -50.0000 -60.0000 Figure 3. Rural Two-Lane Undivided Horizontal Curves OD KABCO Predictions 150.0000 100.0000 50.0000 0.0000 cum res pSVKABCO 0 5000 10000 15000 -50.0000 cum res -100.0000 pCMFSVKABCO -150.0000 -200.0000 -250.0000 Figure 4. Rural Two-Lane Undivided Horizontal Curves SV KABCO Predictions Based on the results the adjustment factor for horizontal curvature is not performing well for this dataset. It should be noted in passing that the HSM AF does or consider the effect of deflection angle, more precisely how this angle impacts the amount of change in tangent length with change in radius. Subsequent research has provided a methodology for estimating an AF that considers this effect. Lane Width The recommended adjustment factor for total crashes is determined using the formula below: AF = (AFra -1) x pra + 1 K-22 

where AFra = formula calculated in Table 23 pra = proportion of crashes composed of single-vehicle run-off-road, head-on, opposite-direction- sideswipe and same-direction crashes For the specific crash types the formulae in Table 24 are applied directly. Table 24. Rural Two-Lane Undivided Lane Width Formulae AADT (veh/day) Lane Width < 400 400 to 2000 >2000 9-ft or less 1.05 1.05+2.81x10-4(AADT-400) 1.50 10-ft 1.02 1.02+1.75x10-4(AADT-400) 1.30 11-ft 1.01 1.01+2.50x10-5(AADT-400) 1.05 12-ft or more 1.00 1.00 1.00 Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor. Table 25 shows the number of sites, observed and predicted crashes for each crash type. For all crash types there is an underprediction prior to using the adjustment factor, sometimes quite small, and the use of the adjustment factor results in the sum of predictions being greater than the sum of crashes. Table 25. Rural Two-Lane Undivided Lane Width Approach 3 Model No. Sites OBS KABCO PRED KABCO Base Model 1049 1404 1370 Base Model + AAF 1049 1404 1474 Model No. Sites OBS SV KABCO PRED SV KABCO Base Model 1049 779 758 Base Model + AAF 1049 779 819 Model No. Sites OBS OD KABCO PRED OD KABCO Base Model 1049 164 161 Base Model + AAF 1049 164 173 Model No. Sites OBS SD KABCO PRED SD KABCO Base Model 1049 356 345 Base Model + AAF 1049 356 368 Table 26 shows the observed and predicted crashes with and without the adjustment factor applied by lane width. The bold cells indicate for each base model/base model plus adjustment factor pair which prediction is closer to the observed value. The data indicate that more often than not the adjustment factor gets the sum of predicted crashes closer to the observed value. K-23 

Table 26. Rural Two-Lane Undivided Lane Width Approach 3 By Lane Width Lane OBS PRED OBS SV PRED SV OBS OD PRED OD OB SD PRED SD Model Width KABCO KABCO KABCO KABCO KABCO KABCO KABCO KABCO 9’ Base Model 88 47 53 30 9 5 18 8 Base Model 9’ 88 70 53 43 9 8 18 12 + AAF 10’ Base Model 215 179 152 108 21 20 27 37 Base Model 10’ 215 229 152 137 21 26 27 48 + AAF 11’ Base Model 665 627 356 348 88 74 174 154 Base Model 11’ 665 658 356 365 88 77 174 162 + AAF 12’ Base Model 436 517 218 273 46 62 137 145 Base Model 12’ 436 517 218 273 46 62 137 145 + AAF Based on the results the adjustment factor for lane width generally improves the crash predictions although it can still result in over- or underprediction. Shoulder Width The recommended adjustment factor for all crash types is determined using the formulae in Table 26 and as shown in Table 27 where AFra = formula calculated in Table 27 pra = proportion of crashes composed of single-vehicle run-off-road, head-on, opposite-direction- sideswipe and same-direction crashes Table 27. Rural Two-Lane Undivided Shoulder Width Formulae AADT (veh/day) Shoulder Width < 400 400 to 2000 >2000 0-ft 1.10 1.10+2.50x10-4(AADT-400) 1.50 2-ft 1.07 1.07+1.43x10-4(AADT-400) 1.30 4-ft 1.02 1.02+8.125x10-5(AADT-400) 1.15 6-ft 1.00 1.00 1.00 8-ft or more 0.98 0.98+6.875x10-5(AADT-400) 0.87 K-24 

Table 28. Rural Two-Lane Undivided Lane Width Formulae Crash Type Crash Severity AF (std. err.) All KABCO (AFra-1.0)xpra+1.0 KABC KAB KA Single vehicle KABCO Use AFra KABC KAB KA Same direction KABCO Use AFra KABC KAB KA Opposing direction KABCO Use AFra KABC KAB KA Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor. Figures 5 to 8 plot the cumulative residuals (observed-predicted) versus shoulder width on the x-axis without and with applying the adjustment factors. In the figures the plot denoted CMF includes the adjustment factor application. For total crashes the predictions are improved but still show fairly large residuals from should widths 0 to 3 feet. For same direction KABCO crashes the results with adjustment factor greatly improve the cumulative residuals plot and do not show significant amounts of bias. For opposite direction KABCO crashes the cumulative residuals are improved when the adjustment factor is applied but still showing some bias between widths of 0 to 4 feet. For single-vehicle KABCO crashes use of the adjustment factor improves the cumulative residuals but still shows some bias from 0 to 3 feet. K-25 

1400.0000 1200.0000 1000.0000 800.0000 cumres KABCO 600.0000 cumres CMF KABCO 400.0000 200.0000 0.0000 -2 0 2 4 6 8 10 -200.0000 Figure 5. Rural Two-Lane Undivided Shoulder Width KABCO Predictions 300.0000 250.0000 200.0000 150.0000 cumres SDKABCO 100.0000 cumres CMF 50.0000 SDKABCO 0.0000 -2 0 2 4 6 8 10 -50.0000 -100.0000 Figure 6. Rural Two-Lane Undivided Shoulder Width SD KABCO Predictions K-26 

100.0000 80.0000 60.0000 40.0000 cumres ODKABCO 20.0000 0.0000 cumres CMF -2 0 2 4 6 8 10 ODKABCO -20.0000 -40.0000 -60.0000 -80.0000 Figure 7. Rural Two-Lane Undivided Shoulder Width OD KABCO Predictions 900.0000 800.0000 700.0000 600.0000 500.0000 cumres SVKABCO 400.0000 300.0000 cumres CMF SVKABCO 200.0000 100.0000 0.0000 -100.0000 0 -2 2 4 6 8 10 -200.0000 Figure 8. Rural Two-Lane Undivided Shoulder Width SV KABCO Predictions Based on the results the use of the adjustment factors improve the predictions but do still show bias at low values of shoulder width, roughly up to 3 or 4 feet. The exception is same direction crashes where little bias is seen. 5.2 Rural Four-Lane Undivided Roads Lighting For rural four-lane undivided roads there are not enough sites meeting base conditions, with the exception of lighting to validate the adjustment factors. Even if all sites were used there are only 21 segments with 94 total crashes that have lighting. K-27 

Shoulder Width The recommended adjustment factors are determined using the equation below with the parameters in Table 29 and following the recommendations in Table 30. AF = exp(a(shldwidth-BC)) where shldwidth = paved right shoulder width in feet BC = Desired base condition of shoulder width in feet a = parameter estimate for shoulder width b = the standard error of the parameter estimate for shoulder width, assumed to be largest value giving statistical significance at the 95% confidence level Table 29. Rural Four-Lane Undivided Shoulder Width Parameters Divided Undivided KABCO SV KABCO MV KABCO KABCO MV KABCO a -0.118 -0.053 -0.137 -0.067 -0.111 b 0.060 0.027 0.070 0.034 0.057 Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor. Figures 9 to 12 plot the cumulative residuals (observed-predicted) versus shoulder width on the x-axis without and with applying the adjustment factors. In the figures the plot denoted CMF includes the adjustment factor application. For total KABCO and same direction KABCO crashes the predictions are slightly improved but still show fairly large residuals as a result of underpredicting crashes. For opposite direction KABCO crashes the cumulative residuals are almost identical with and without the adjustment factor and show some bias underpredicting crashes. For single-vehicle KABCO crashes use of the adjustment factor results on average in fewer crashes being predicted so the plot of cumulative residuals is shifted down but is comparable overall the plot without using the adjustment factors. K-28 

Table 30. Rural Two-Lane Undivided Lane Width Formulae Crash Crash Type AF Severity All KABCO KABC Use KABCO formula KAB KA Single vehicle KABCO Divided – Use SV KABCO formula KABC KAB Undivided – Use KABCO and MV KABCO formulae to estimate SV as AFSV = (AFKABCO-AFMV*PMV)/PSV KA *If AFSV is predicted to be less than 0 used a value of 0 Same direction KABCO KABC Use MV KABCO formula KAB KA KABCO Opposing KABC Use MV KABCO formula direction KAB KA KABCO 1.00 Intersecting KABC 1.00 direction KAB 1.00 KA 1.00 200.0000 180.0000 160.0000 140.0000 120.0000 100.0000 cumres KABCO 80.0000 cumres CMF KABCO 60.0000 40.0000 20.0000 0.0000 -2 -20.0000 0 2 4 6 8 10 Figure 9. Rural Four-Lane Undivided Shoulder Width KABCO Predictions K-29 

90.0000 80.0000 70.0000 60.0000 50.0000 cumres SDKABCO 40.0000 cumres CMF 30.0000 SDKABCO 20.0000 10.0000 0.0000 -2 0 2 4 6 8 10 -10.0000 Figure 10. Rural Four-Lane Undivided Shoulder Width SD KABCO Predictions 35.0000 30.0000 25.0000 20.0000 cumres ODKABCO 15.0000 cumres CMF 10.0000 ODKABCO 5.0000 0.0000 -2 0 2 4 6 8 10 -5.0000 Figure 11. Rural Four-Lane Undivided Shoulder Width OD KABCO Predictions K-30 

15.0000 10.0000 5.0000 cumres SVKABCO 0.0000 -2 0 2 4 6 8 10 cumres CMF SVKABCO -5.0000 -10.0000 -15.0000 Figure 12. Rural Four-Lane Undivided Shoulder Width SV KABCO Predictions For total KABCO and same direction KABCO crashes the predictions are slightly improved but still show fairly large residuals as a result of underpredicting crashes. For opposite direction KABCO crashes the cumulative residuals are almost identical with and without the adjustment factor and show some bias underpredicting crashes. For single-vehicle KABCO crashes use of the adjustment factor results on average in fewer crashes being predicted so the plot of cumulative residuals is shifted down but is comparable overall the plot without using the adjustment factors. Based on the results the use of the adjustment factors does not significantly improve the predictions which still show bias, generally underpredicting crashes at the non-base condition values of shoulder width, so it can be considered that the AFs do not validate well. 5.3 Rural Four-Lane Divided Roads Lighting Presence of lighting was not in fact one of the recommended adjustment factors but was nevertheless explored. The base condition for lighting is the absence of roadway segment lighting. The HSM 1st edition adjustment factor applies to total crashes and is determined using the formula: AFlighting = 1.0 – [(1.0-0.72*pinr-0.83*pnr) x pnr] pinr = proportion of total nighttime crashes for unlit segments that are KABC severity (0.382 is default) ppnr = proportion of total nighttime crashes for unlit segments that are PDO (0.618 is default) pnr = proportion of total crashes for unlit segments that occur at night (0.370 is default) With the default values used AFlighting is equal to 0.92. Approach 1 was applied. Table 31 shows the number of sites, observed crashes and results of the approach 1 analysis. The only crash category with close to 100 crashes was total crashes and the implied adjustment factor of presence of lighting is 2.18. This does not agree with the HSM lighting adjustment K-31 

factor of 0.92. It should be noted that these is the potential here for endogeneity bias in that the sites with lighting may have inherent safety issues that warranted the installation of lighting. Table 31. Rural Four-Lane Divided Lighting Approach 1 No. Observed Crashes Lighting Sites KA KAB KABC KABCO SD KABCO OD KABCO SV KABCO Not Present 159 33 122 166 397 151 34 131 Present 21 7 20 29 94 32 4 31 Calibration Factors No. Lighting C C C C C SD C OD C SV Sites KA KAB KABC KABCO KABCO KABCO KABCO Not Present 159 1.80 Present 21 3.92 Implied CMF 2.18 Based on the results from Approach 1 the adjustment factor is not validated for KABCO crashes. Shoulder Width The recommended adjustment factors are determined using the equation below with the parameters in Table 32 and following the recommendations in Table 33. AF = exp(a(shldwidth-BC)) where shldwidth = paved right shoulder width in feet BC = Desired base condition of shoulder width in feet a = parameter estimate for shoulder width b = the standard error of the parameter estimate for shoulder width, assumed to be largest value giving statistical significance at the 95% confidence level Table 32. Rural Four-Lane Divided Shoulder Width Parameters Divided Undivided KABCO SV KABCO MV KABCO KABCO MV KABCO a -0.118 -0.053 -0.137 -0.067 -0.111 b 0.060 0.027 0.070 0.034 0.057 K-32 

Table 33. Rural Two-Lane Undivided Lane Width Formulae Crash Crash Type AAF (std. err.) Severity All KABCO KABC Use KABCO formula KAB KA Single vehicle KABCO Divided – Use SV KABCO formula KABC KAB Undivided – Use KABCO and MV KABCO formulae to estimate SV as AAFSV = (AAFKABCO-AAFMV*PMV)/PSV KA *If AAFSV is predicted to be less than 0 used a value of 0 Same direction KABCO KABC Use MV KABCO formula KAB KA KABCO Opposing KABC Use MV KABCO formula direction KAB KA KABCO 1.00 Intersecting KABC 1.00 direction KAB 1.00 KA 1.00 Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor. Figures 13 to 16 plot the cumulative residuals (observed-predicted) versus shoulder width on the x-axis without and with applying the adjustment factors. In the figures the plot denoted CMF includes the adjustment factor application. For total and same direction KABCO crashes, between shoulder widths of 1 to 4 feet, the base model on its own under predicts crashes but, with the use of the adjustment factor, overpredicts crashes. In this range the residuals are smaller for the base model on its own. For opposite direction KABCO crashes the base model on its own predicts more accurately than when using the adjustment factor up to 4 feet and thereafter perform similarly. For single-vehicle KABCO crashes the use of the adjustment factor improves the predictions somewhat between 2 and 5 feet. K-33 

150.0000 100.0000 50.0000 cumres KABCO cumres CMF KABCO 0.0000 -2 0 2 4 6 8 10 -50.0000 -100.0000 Figure 13. Rural Four-Lane Divided Shoulder Width KABCO Predictions 50.0000 40.0000 30.0000 20.0000 cumres SDKABCO 10.0000 0.0000 cumres CMF -2 0 2 4 6 8 10 SDKABCO -10.0000 -20.0000 -30.0000 -40.0000 Figure 14. Rural Four-Lane Divided Shoulder Width SD KABCO Predictions K-34 

10.0000 8.0000 6.0000 cumres ODKABCO 4.0000 cumres CMF 2.0000 ODKABCO 0.0000 -2 0 2 4 6 8 10 -2.0000 -4.0000 Figure 15. Rural Four-Lane Divided Shoulder Width OD KABCO Predictions 70.0000 60.0000 50.0000 40.0000 30.0000 cumres SVKABCO 20.0000 10.0000 cumres CMF SVKABCO 0.0000 -2 -10.0000 0 2 4 6 8 10 -20.0000 -30.0000 -40.0000 Figure 16. Rural Four-Lane Divided Shoulder Width SV KABCO Predictions Based on the results the use of the adjustment factors does not significantly improve the predictions. Median Width The recommended adjustment factors are determined using the equation below with the parameters in Table 34 and following the recommendations in Table 34. AF = exp(a(MedianWidth-BC)) K-35 

where MedianWidth = median width width in feet BC = Desired base condition of median width in feet a = parameter estimate for median width b = the standard error of the parameter estimate for shoulder width, assumed to be largest value giving statistical significance at the 95% confidence level Only total KABCO crashes provided a large of enough sample size to investigate. Table 34. Rural Four-Lane Divided Median Width Parameters Total Crashes Cross-Median Crashes Site Type a b a b R4D -0.00461 0.00080 -0.01695 0.00200 U4D -0.00533 0.00090 -0.01340 0.00205 Table 35. Rural Two-Lane Undivided Median Width Formulae Crash Type Crash AF Severity All KABCO Use formula KABC KAB KA Single vehicle KABCO 1.00 KABC 1.00 KAB 1.00 KA 1.00 Same KABCO 1.00 direction KABC 1.00 KAB 1.00 KA 1.00 Opposing KABCO Use formula direction KABC KAB KA Because the adjustment factor is not a simple factor for one or more levels, Approach 1 and Approach 2 are not capable of making a straightforward validation. Approach 3 is applied which compares the predictive performance of the base model with and without using the adjustment factor. Figure 17 plots the cumulative residuals (observed-predicted) versus median width on the x-axis without and with applying the adjustment factors. In the figures the plot denoted CMF includes the adjustment factor application. K-36 

For total KABCO crashes the use of the adjustment factor does improve the plot of cumulative residuals to bring the plot closer to overall indicating that the adjustment factor improves the predictions so correctly adjusts for the effect of median width. 80.0000 70.0000 60.0000 50.0000 40.0000 30.0000 cumres KABCO 20.0000 cumres CMF KABCO 10.0000 0.0000 -10.0000 0 50 100 150 -20.0000 -30.0000 Figure 17. Rural Four-Lane Divided Median Width KABCO Predictions 5.4 Summary on Validation of Adjustment Factors In general, sample sizes were too small for a definitive validation exercise where it can be concluded with high confidence that the implied AFs are statistically similar to the ones being validated. Instead, the best that can be said in general is that one cannot reject a hypothesis that the implied AFs from the validation are statistically different from the HSM recommended AFs. That said, there can be more confidence in some AFs because a) there is relative consistency in the results from different approaches for validating the same AFs and b) there is relative consistency in the AF trends where the AFs are in the form of a function, as was the case for RHR. K-37