Crash Modification Factors in the Highway Safety Manual: Resources for Evaluation (2023) / Chapter Skim
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Appendix K Calibration of Prediction Models of Rural Segments for the 2nd Edition of the HSM
Appendix K Calibration of Prediction Models of Rural Segments for the 2nd Edition of the HSM
Pages 252-288
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From page 252... ...
Appendix K Calibration of Prediction Models of Rural Segments for the 2nd Edition of the HSM K-1
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K-8 Table 8. Base Condition SPFs, Four-Lane Undivided Segments ..............................................................
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Rural Two-Lane Undivided Shoulder Width KABCO Predictions ......................................... K-26 Figure 6.
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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)
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From page 256... ...
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.
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From page 257... ...
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.
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From page 258... ...
,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)
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Table 7. Base Condition SPFs, Two-Lane Undivided Segments Crash Type Severity b0 b1 c -7.463 0.927 1.999 KABCO (0.520)
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(0.379) -6.738 0.545 13.202 Single vehicle KABC (1.558)
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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.
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From page 262... ...
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
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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)
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From page 264... ...
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.
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From page 265... ...
= 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.
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From page 266... ...
(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.
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From page 267... ...
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.
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(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.
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From page 269... ...
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.
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(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.
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From page 271... ...
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.
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From page 272... ...
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.
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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.
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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.
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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-directionsideswipe and same-direction crashes Table 27. Rural Two-Lane Undivided Shoulder Width Formulae AADT (veh/day)
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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.
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Rural Two-Lane Undivided Shoulder Width SD KABCO Predictions K-26
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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.
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From page 279... ...
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.
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Rural Four-Lane Undivided Shoulder Width KABCO Predictions K-29
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Rural Four-Lane Undivided Shoulder Width OD KABCO Predictions K-30
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From page 282... ...
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.
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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)
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From page 284... ...
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.
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Rural Four-Lane Divided Shoulder Width SD KABCO Predictions K-34
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From page 286... ...
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.
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From page 287... ...
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.
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From page 288... ...
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.
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