Guidelines for the Development and Application of Crash Modification Factors (2022) / Chapter Skim
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From page 225... ...
C-1 C-2 Introduction C-3 Chapter 1 Guidelines for Developing CMFunctions from Cross-Sectional Regression Models C-3 1.1 Bias Due to Aggregation, Averaging, or Incompleteness in Data C-5 1.2 Functional Form for Effects of Independent Variables C-9 1.3 Model Structure -- Application of Hierarchical Modeling C-10 1.4 Tools for Assessing Model Fit and Choosing Among or Amalgamating Information from Competing Models C-14 1.5 Including Estimates from Previous Studies in the Estimation Methodology through Full Bayes Methods C-14 1.6 Addressing Multicollinearity Among Explanatory Variables C-15 1.7 Addressing Endogeneity C-16 1.8 Modeling Interactions, Especially for Estimating Effects of Combination Treatments C-18 1.9 Estimating Precision of CMFs from CMFunctions Derived C-18 1.10 Corroboration of Results C-21 1.11 Database Requirements C-23 Chapter 2 Guidelines for Developing CMFunctions from Models That Relate CMF Point Estimates to Application Circumstance C-23 2.1 Guidelines for Conducting Systematic Reviews C-26 2.2 Guidelines on Which Application Circumstances and Key Influential Factors to Collect Information on, Grouped by Treatment and Location Types C-26 2.3 Guidelines for Conducting Meta-Regression C-31 2.4 Fixed Versus Random Effects Models C-33 2.5 Selection of Estimation Method C-35 2.6 Guidelines for Creating Subgroups C-36 2.7 Estimating Precision of CMFs from CMFunctions Derived C-36 2.8 Improving Site and Study Level Estimates of CMFs C-37 Chapter 3 Example Applications C-37 Case Study 1 Simultaneous Application of Shoulder Rumble Strips and Centerline Rumble Strips on Rural Two-Lane Roads C-52 Case Study 2 Conversion of Conventional Intersections to Roundabouts C-63 Case Study 3 Safety Effects of Flattening a Horizontal Curve C-82 Case Study 4 Safety Effects of Left- and Right-Turn Lanes on Major Roads at Three-Legged Stop-Controlled Intersections C-94 References A P P E N D I X C Guidelines for Developing Crash Modification Functions
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C-2 The primary objective of this appendix is to provide guidelines for researchers to estimate future crash modification factors (CMFs) and crash modification functions (CMFunctions)
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C-3 CMFs derived from cross-sectional panel data are based on a single period under the assumption that the ratio of average crash frequencies for sites with and without a feature is an estimate of the CMF for implementing that feature. Cross-sectional designs are particularly useful for estimating CMFs where there are insufficient instances for a preferred before-after design where a feature is implemented.
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C-4 Guidelines for the Development and Application of Crash Modification Factors change significantly over time. Lord and Mannering (2010)
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Guidelines for Developing Crash Modification Functions C-5 crash type, severity etc. The factors affecting crash risk for different crash types may differ and the combining of different crash types may mask this relationship.
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C-6 Guidelines for the Development and Application of Crash Modification Factors In this model form, the effect on safety of non-AADT explanatory variables is treated as an exponential function. While the shape of the exponential curve is somewhat flexible it does not permit relationships that have turning points (peaks or valleys)
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Guidelines for Developing Crash Modification Functions C-7 Integrate-Differentiate Method Overview In the ID method, the integrate function is a cumulative function, F(x)
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C-8 Guidelines for the Development and Application of Crash Modification Factors order. Hauer (2015, p.
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Guidelines for Developing Crash Modification Functions C-9 variables, a table can be produced showing the prediction bias for each level of the variable as in the example in Table C1. In this example there are three categories of speed limit.
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C-10 Guidelines for the Development and Application of Crash Modification Factors Equation C6B = exp local factor0 1( ) β × β × Equation C7C = exp local factor0 1( )
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Guidelines for Developing Crash Modification Functions C-11 Mean Absolute Deviation (MAD) The MAD is the sum of the absolute value of predicted validation observations minus observed validation observations, divided by the number of validation observations.
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C-12 Guidelines for the Development and Application of Crash Modification Factors where Var{m} = the estimated variance of the mean accident rate E{m} = the estimated mean accident rate K = the estimated overdispersion parameter Variance overdispersion in a Poisson process can lead to a negative binomial dispersion of errors, particularly when the Poisson means are themselves approximately gamma distributed or possess gamma heterogeneity. The negative binomial distribution has been shown to adequately describe errors in motor vehicle crash models in many instances.
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Guidelines for Developing Crash Modification Functions C-13 CURE Plots CURE plots may also be used as a GOF measure to see if the predictions are biased for ranges of the independent variables; e.g., overpredicting crashes at high AADTs. CURE plots are discussed in Section 1.2, including guidelines from Hauer (2015)
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C-14 Guidelines for the Development and Application of Crash Modification Factors remain. Consideration should also be given to the implied relationship with crashes and the magnitude of this relationship to avoid illogical results based on engineering knowledge when deciding which variables to include in a model.
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Guidelines for Developing Crash Modification Functions C-15 • Removal of one of the variables if its effect is associative rather than causative, or if only one variable is typically available • Do nothing, with limitations documented • During study design ensure that all levels of correlated variables are collected and samples are stratified 1.7 Addressing Endogeneity Elvik (2011b) provides a succinct description of the concern of endogeneity in multivariable regression models.
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C-16 Guidelines for the Development and Application of Crash Modification Factors In yet another example, Austin and Carson (2002) investigated the safety impacts of warning treatments at highway-rail grade crossings.
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Guidelines for Developing Crash Modification Functions C-17 Equation C21 simplifies to: CMF c LW c LW c LW LW Equation C22 exp exp exp2 1 2 1[ ]
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C-18 Guidelines for the Development and Application of Crash Modification Factors G = absolute value of percent grade; 0% for level tangents, ≥ 1% otherwise R = curve radius (ft) ; missing for tangents IHC = horizontal curve indicator: 1 for horizontal curves; 0 otherwise LC = horizontal curve length (mi)
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Guidelines for Developing Crash Modification Functions C-19 2. The definition of target crashes should also be same, or as close as possible.
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C-20 Guidelines for the Development and Application of Crash Modification Factors 8. There should be a dose–response pattern in the relationship between treatment and effect (provided the treatment comes in different doses)
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Guidelines for Developing Crash Modification Functions C-21 frequencies of technical inspections, etc. The other case is when the size of the effect of a safety treatment on its target risk factor (or factors)
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C-22 Guidelines for the Development and Application of Crash Modification Factors variables that impact the CMF which still need to be included, from those that may still affect safety but do not impact the CMF. The approach needs to be tested, developed, and refined, and if found promising, widely used to exploit available cross-sectional data.
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C-23 2.1 Guidelines for Conducting Systematic Reviews The guidelines provided for conducting systematic reviews are relevant to the case where CMFunctions are being developed using CMF estimates from multiple studies. Elvik (2005)
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C-24 Guidelines for the Development and Application of Crash Modification Factors Elvik (2005) also recommends a sensitivity analysis be done to see if the outcome is dependent on the choices made in conducting the systematic review.
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Guidelines for Developing Crash Modification Functions C-25 Table C2 shows Elvik's (unpublished) proposed classification of study designs that identifies three levels of study quality for each design.
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C-26 Guidelines for the Development and Application of Crash Modification Factors As far as developing accident modification functions is concerned, country of origin and year of publication are potentially confounding variables. If estimates of effect are associated with these variables, the accident modification function may apply only to certain countries or a certain period.
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Guidelines for Developing Crash Modification Functions C-27 to extract this information. The second type is to use the site-level data from one or more studies with estimates of effect determined for each site.
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C-28 Guidelines for the Development and Application of Crash Modification Factors same weight to all estimates of effect, and include all independent variables in the regression model. The reason for not using the statistical weights assigned to each estimate of effect in metaanalysis is that the weights may confound the analysis.
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Guidelines for Developing Crash Modification Functions C-29 versus potential influential factors can be useful for revealing appropriate models forms. This approach is undertaken in Chapter 3, Case Studies 1 and 2.
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C-30 Guidelines for the Development and Application of Crash Modification Factors Elvik (unpublished) lists several model criteria.
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Guidelines for Developing Crash Modification Functions C-31 effect. If the confidence intervals for the individual estimates of effect overlap the confidence interval for the summary estimate of effect, the estimates are consistent, differing only in terms of statistical precision.
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C-32 Guidelines for the Development and Application of Crash Modification Factors The test statistic, Q, is chi-square distributed with g-1 degrees of freedom. If the test is statistically significant a random effects model should be used.
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Guidelines for Developing Crash Modification Functions C-33 Unfortunately, the average AADT was not provided. The plot indicates there may be a linear relationship between the CMF and volume variable with a lower CMF at higher values of AADT.
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0.81 0.20 Regression cross-section PA Divided by median NULL Rural 8267 18753 0.74 0.24 Regression cross-section PA Divided by median NULL Rural 8267 18753 0.76 0.09 Before/after using empirical Bayes or full Bayes PA Undivided Rural 948 9067 0.76 0.15 Regression cross-section PA Undivided Rural 910 10177 0.76 0.15 Regression cross-section PA Undivided Rural 910 10177 0.99 0.06 Before/after using empirical Bayes or full Bayes PA Divided by median NULL Urban 11254 59391 0.96 0.05 Regression cross-section PA Divided by median NULL Urban 11254 92757 0.95 0.06 Regression cross-section PA Divided by median NULL Urban 11254 92757 0.93 0.04 Regression cross-section PA All All CMF Standard Error Study Method State roadDivType numLanes areaType minTrafficVol maxTrafficVol 0.84 0.06 Simple before/after MN Divided Multilane Rural 2000 50000 1.10 0.15 Before/after using empirical Bayes or full Bayes MN Divided by median NULL Rural 4959 7459 1.16 0.09 Regression cross-section MN Divided by median NULL Rural 4959 31692 1.17 0.14 Regression cross-section MN Divided by median NULL Rural 4959 31692 1.14 0.08 Before/after using empirical Bayes or full Bayes MN Undivided Rural 782 10386 0.96 0.07 Regression cross-section MN Undivided Rural 180 10386 1.18 0.10 Regression cross-section MN Undivided Rural 180 10386 1.18 0.08 Before/after using empirical Bayes or full Bayes MN,MO,PA Divided by median NULL Rural 4959 20763 1.20 0.10 Regression cross-section MN,MO,PA Divided by median NULL Rural 4956 31692 1.28 0.11 Regression cross-section MN,MO,PA Divided by median NULL Rural 4956 31692 1.06 0.06 Before/after using empirical Bayes or full Bayes MN,MO,PA Undivided Rural 782 10386 0.86 0.10 Regression cross-section MN,MO,PA Undivided Rural 180 12776 0.94 0.13 Regression cross-section MN,MO,PA Undivided Rural 180 12776 1.08 0.04 Before/after using empirical Bayes or full Bayes MO Divided by median NULL Rural 11539 37112 1.22 0.09 Before/after using empirical Bayes or full Bayes MO Divided by median NULL Rural 5326 20763 1.08 0.07 Regression cross-section MO Divided by median NULL Rural 11539 37112 1.11 0.08 Regression cross-section MO Divided by median NULL Rural 11539 37112 1.28 0.14 Regression cross-section MO Divided by median NULL Rural 4956 20763 1.28 0.14 Regression cross-section MO Divided by median NULL Rural 4956 20763 1.40 0.18 Before/after using empirical Bayes or full Bayes MO Undivided Rural 861 6205 0.83 1.16 Regression cross-section MO Undivided Rural 861 12776 0.85 1.29 Regression cross-section MO Undivided Rural 861 12776 1.07 0.04 Before/after using empirical Bayes or full Bayes MO,PA Divided by median NULL Rural 6777 37112 1.01 0.07 Regression cross-section MO,PA Divided by median NULL Rural 6777 37112 1.07 0.08 Regression cross-section MO,PA Divided by median NULL Rural 6777 37112 0.85 0.20 Simple before/after ND Undivided NULL Rural 1.00 0.12 Before/after using empirical Bayes or full Bayes PA Divided by median NULL Rural 6777 24752 0.87 0.36 Before/after using empirical Bayes or full Bayes PA Divided by median NULL Rural 9653 18753 1.08 0.12 Regression cross-section PA Divided by median NULL Rural 6777 34406 1.06 0.16 Regression cross-section PA Divided by median NULL Rural 6777 24752 Table C4. CMFs for addition of shoulder rumble strips.
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Guidelines for Developing Crash Modification Functions C-35 comprised drivers aged 25–54. The pre-period commenced 2 years prior to GDL and lasted for one year.
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C-36 Guidelines for the Development and Application of Crash Modification Factors on a majority vote, where each doctor gets an equal vote. Now replace each expert by a classifier, and you have the basic idea behind bagging.
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C-37 The following example applications consist of four case studies that use real observational data to illustrate the issues in CMFunction development. Two case studies demonstrate methods for developing CMFunctions from application circumstance data while the other two case studies demonstrate methods for developing CMFunctions from cross-sectional data.
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C-38 Guidelines for the Development and Application of Crash Modification Factors – Tools for exploring the appropriate functional form of the model (Section 1.2) – Combining results from multiple sites (or groups)
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Guidelines for Developing Crash Modification Functions C-39 In the original analysis, to account for potential selection bias and regression-to-the-mean, an empirical Bayes (EB) before-after analysis was conducted, utilizing reference groups of untreated two-lane rural roads with similar characteristics to the treated sites.
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C-40 Guidelines for the Development and Application of Crash Modification Factors sites show an extremely large variation with many CMF estimates near 0 and a large value of the CMF for any site that happened to have a crash in the after period. For sites with no after period crashes, the CMF would be equal to 0 and it is not possible to estimate the standard error of the CMF because one of the terms in the variance equation would involve a division by 0.
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Guidelines for Developing Crash Modification Functions C-41 of the variance of the CMF. This weighting will give CMF estimates that are based on more data and thus are more precise more influence on the model.
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C-42 Guidelines for the Development and Application of Crash Modi cation Factors In the next step the data were grouped by both state and AADT, resulting in 12 groups. Figure C8 plots the CMFs versus AADT for these groupings.
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Guidelines for Developing Crash Modification Functions C-43 Next, a state indicator variable is included in the model for the data grouped by state and AADT. Thus, an additional term is estimated for each state that is added to the intercept.
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C-44 Guidelines for the Development and Application of Crash Modification Factors The results in Table C12 confirm that the CMF is lower at sites with higher crashes per mileyear before treatment. The parameter estimate for crash rate is statistically significant with a p-value of 0.0324.
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Guidelines for Developing Crash Modification Functions C-45 The next model, shown in Table C14, includes an indicator variable for State and is consistent with the previous model without the State indicator variable. The intercept terms for Pennsylvania and Kentucky are statistically different from each other, while the intercept for Missouri is somewhere in the middle of the term for the other two.
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C-46 Guidelines for the Development and Application of Crash Modification Factors appears to be associated with a lower CMF. It may also be that at narrow shoulder widths there is not enough room for drivers to recover even if alerted that they are leaving the travel lane.
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Guidelines for Developing Crash Modification Functions C-47 CMF ROR CMF vs. SHLDWID 0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 0.00 2.00 4.00 6.00 8.00 10.00 12.00 Figure C12.
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C-48 Guidelines for the Development and Application of Crash Modification Factors number of sites and crashes within groups is low. The correlation between the three variables may also contribute to this difficulty.
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Guidelines for Developing Crash Modification Functions C-49 Parameter Estimate Standard Error p-value Intercept 0.9062 0.1701 <0.0001 AADT -0.1053 0.1400 0.4520 SHLDWID -0.0252 0.0199 0.2042 Table C23. ROR CMFunction grouped by AADT and shoulder width.
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C-50 Guidelines for the Development and Application of Crash Modification Factors application circumstances between the states that is not represented in the model. Examining the results in Table C26 shows that there is a minimal difference in the parameter estimates between the two models.
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Volcat Rateb SHLDWID No. sites Observed Crashes Expected Crashes CMF Std Error 1 1 1 16 0 0.91 0.00 1 1 2 23 3 3.10 0.90 0.57 1 1 3 29 0 1.94 0.00 1 1 4 27 12 13.50 0.79 0.36 1 2 1 95 11 10.70 1.01 0.33 1 2 2 140 18 17.85 1.00 0.25 1 2 3 61 1 4.94 0.20 0.20 1 2 4 113 9 12.32 0.72 0.25 1 3 1 68 2 6.51 0.30 0.22 1 3 2 108 12 12.10 0.98 0.30 1 3 3 59 1 3.03 0.32 0.33 1 3 4 66 0 4.33 0.00 1 4 1 39 6 5.81 1.01 0.44 1 4 2 87 12 15.77 0.75 0.24 1 4 3 15 2 3.27 0.59 0.43 1 4 4 10 5 16.45 0.28 0.15 1 5 1 43 53 47.18 1.09 0.24 1 5 2 123 76 76.57 0.99 0.13 1 5 3 11 2 6.77 0.29 0.21 1 5 4 10 24 78.33 0.30 0.08 2 1 1 1 4 0.39 5.24 4.15 2 1 3 3 1 0.38 2.00 2.19 2 1 4 7 3 1.37 1.90 1.31 2 2 1 11 6 2.78 1.93 1.02 2 2 2 28 4 5.83 0.66 0.35 2 2 3 15 0 2.47 0.00 2 2 4 47 18 16.56 1.07 0.28 2 3 1 10 1 1.92 0.45 0.48 2 3 2 31 5 6.24 0.77 0.38 2 3 3 30 6 4.44 1.29 0.60 2 3 4 118 48 44.28 1.05 0.23 2 4 1 17 6 8.25 0.70 0.32 2 4 2 24 5 6.93 0.70 0.34 2 4 3 19 4 0.91 4.06 2.32 2 4 4 51 9 14.84 0.60 0.21 2 5 1 40 6 16.57 0.36 0.15 2 5 2 83 54 75.15 0.72 0.11 2 5 3 18 3 10.04 0.29 0.18 2 5 4 29 22 33.26 0.63 0.19 3 1 4 1 0 0.14 0.00 3 2 1 2 5 1.54 2.43 1.68 3 2 2 3 0 0.57 0.00 3 2 3 2 0 0.17 0.00 3 2 4 10 2 3.73 0.51 0.38 3 3 1 3 1 1.56 0.49 0.54 3 3 2 12 4 2.27 1.62 0.94 3 3 3 4 0 0.29 0.00 3 3 4 50 6 6.72 0.86 0.39 3 4 1 2 2 0.98 1.40 1.24 3 4 2 10 0 1.59 0.00 3 4 3 1 0 0.33 0.00 3 4 4 33 14 24.04 0.54 0.21 3 5 1 6 1 4.22 0.22 0.23 3 5 2 51 1 6.14 0.16 0.16 3 5 3 3 0 1.77 0.00 3 5 4 19 14 14.00 0.98 0.29 4 2 1 1 0 0.37 0.00 4 2 3 3 1 0.64 1.18 1.30 4 2 4 6 4 2.47 1.08 0.84 4 3 2 3 0 0.39 0.00 4 3 3 7 1 0.89 0.97 1.03 4 3 4 21 6 4.34 1.32 0.61 4 4 1 2 0 0.45 0.00 4 4 2 4 0 0.77 0.00 4 4 3 9 0 2.14 0.00 4 4 4 30 2 1.96 0.97 0.72 4 5 1 12 1 2.33 0.39 0.41 4 5 2 13 0 3.62 0.00 4 5 3 16 5 13.59 0.34 0.18 4 5 4 23 5 14.08 0.35 0.16 Table C27.
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C-52 Guidelines for the Development and Application of Crash Modification Factors Case Study 2 Conversion of Conventional Intersections to Roundabouts Preamble This is one of a series of four case studies to demonstrate the proposed guidelines for developing CMFunctions from either cross-sectional data or before-after data from actual safety treatment applications. The purposes of this specific case study are to: • Illustrate a heuristic methodology a future researcher may follow to derive a CMFunction from before-after application circumstance data on individual treatments sites using regression analysis.
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Guidelines for Developing Crash Modification Functions C-53 • Area type (urban versus rural) • Number of circulating lanes • Number of entering legs A key difficulty that would be typically encountered in developing CMFunctions in beforeafter evaluations is that the variables available for consideration are likely correlated.
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C-54 Guidelines for the Development and Application of Crash Modification Factors may show a large variation with many CMF estimates near 0 and a large value of the CMF for any site that happened to have a crash in the after period. For sites with no after-period crashes, the CMF would be equal to 0, and it is not possible to estimate the standard error of the CMF because one of the terms in the variance equation would involve a division by 0.
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Guidelines for Developing Crash Modification Functions C-55 4 Meta-Regression CMFunction Exploration As noted earlier, this is a heuristic approach illustration. In this, the first step taken was to look at the impact of entering AADT on the CMF estimate, followed by exploration of the impacts of expected number of crashes and other variables.
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C-56 Guidelines for the Development and Application of Crash Modification Factors The model estimates are shown in Table C29. The parameter estimate for the AADT variable is statistically significant.
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Guidelines for Developing Crash Modification Functions C-57 In this model the CMF increases as EBrate increases although the parameter estimate is insignificant. The indication of an increasing CMF with increasing EBrate is perhaps not surprising given that the expected crash frequency is correlated positively with traffic volumes.
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C-58 Guidelines for the Development and Application of Crash Modification Factors where wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i exp AADT EBrateintercept= β β Equation C38CMF 10,000 1 2 The parameter estimates in Table C33 show the same relationships with the expected CMF as for Model 3 and the precision of the estimated parameters is much improved. To compare Models 3 and 4, scatterplots were prepared showing the actual CMF value and the CMF predictions versus the two predictive variables.
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Guidelines for Developing Crash Modi cation Functions C-59 CMF Model 3 Model 4 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 0.00 10.00 20.00 30.00 40.00 50.00 Figure C17. Scatterplot for signal conversions -- CMF and predicted CMFs vs.
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C-60 Guidelines for the Development and Application of Crash Modification Factors The model indicates that the CMF is smaller for rural locations and even smaller at suburban locations compared to urban locations but only the difference for suburban was statistically significant at a level below 10%. It also indicates that the CMF is smaller for roundabouts with three approaches and for roundabouts with only one circulating lane.
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Guidelines for Developing Crash Modification Functions C-61 where AADT used in the model estimation in this case is the mean AADT for each group The parameter estimates for this model are shown in Table C36. The results for the AADT variable are consistent with site-level Model 5.
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C-62 Guidelines for the Development and Application of Crash Modification Factors 5 Discussion of Findings This case study sought to explain the variation in CMF findings for the conversion of conventional signalized intersections to roundabouts. The data consisted of site-level results from several empirical Bayes before-after analyses using data from various States.
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Guidelines for Developing Crash Modification Functions C-63 Case Study 3 Safety Effects of Flattening a Horizontal Curve Preamble This is one of a series of four case studies to demonstrate the proposed guidelines for developing CMFunctions from either cross-sectional data or before-after data from actual safety treatment applications. The purposes of this specific case study are to: 1.
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C-64 Guidelines for the Development and Application of Crash Modification Factors Figure C21 illustrates the geometric characteristics of horizontal curves, where PI = Point of tangent intersection PC = Point of curve (beginning of curve) PT = Point of tangent (ending of curve)
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Guidelines for Developing Crash Modification Functions C-65 The data used are from Washington State and were acquired from the Highway Safety Information System (HSIS)
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C-66 Guidelines for the Development and Application of Crash Modification Factors the T for the longest radii curve in the group was used to add on segments of roadway before and after the curve for those curves with smaller radii. In this way, the study area for all curves within an angle group is the same because the differences between deflection angles within an angle group are relatively small.
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Guidelines for Developing Crash Modification Functions C-67 0 1 2 3 4 5 6 7 8 0 5000 10000 15000 Crashes vs. Radius crashes Figure C22.
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C-68 Guidelines for the Development and Application of Crash Modification Factors Figure C25. Integrate-Differentiate plot for Angle Group 3.
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Guidelines for Developing Crash Modification Functions C-69 Figure C26. Integrate-Differentiate plot for Angle Group 4.
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C-70 Guidelines for the Development and Application of Crash Modification Factors The ID plots are not particularly informative, although they do seem to suggest that crash frequency does decline as the radius increases. This is most visible in the plots for Groups 2, 3, and 4.
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Guidelines for Developing Crash Modification Functions C-71 As Equation C48 shows, if the parameter c is negative (i.e., fewer crashes occur with a larger radius) , as the difference between the smaller and larger curve radii increases, the CMF gets smaller.
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C-72 Guidelines for the Development and Application of Crash Modification Factors The models in Table C40 show mixed success. Only two groups (1 and 4)
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Guidelines for Developing Crash Modification Functions C-73 4.3.1 Exponential Models (Equation C3) Model 1.
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C-74 Guidelines for the Development and Application of Crash Modification Factors tend to have larger study areas by this definition. The result being that without accounting for this fact it is not surprising to see that more crashes are associated with the larger deflection angle sites for a given radius.
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Guidelines for Developing Crash Modification Functions C-75 Model 3. As a check on the relationship between radius and crashes for different levels of deflection angle, for this model we also allow the parameter for radius to be dependent on the deflection angle as was done in Model 1.
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C-76 Guidelines for the Development and Application of Crash Modification Factors separate intercept term for each angle group to account for the fact that the study areas vary in size. The model is: Crashes per year exp AADT c d Rbanggrp Equation C551000( )
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Guidelines for Developing Crash Modification Functions C-77 a statistically significant result for the radius term. For groups 3 and 6, the results indicate an increasing crash frequency with increasing radius, contrary to what is expected, although the results are not statistically significant.
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C-78 Guidelines for the Development and Application of Crash Modification Factors The parameter estimates are shown in the reproduced Table C51. The loglikelihood for this model is 443.
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Guidelines for Developing Crash Modification Functions C-79 where K = the number of parameters estimated in the model loglikelihood = the loglikelihood calculated for the model When determining the value of K, the number of parameters estimated in the model, if a negative binomial model is applied the overdispersion parameter is included in the count of estimated parameters. The loglikelihood is calculated by taking the natural logarithm of the likelihood function.
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C-80 Guidelines for the Development and Application of Crash Modification Factors Figure C30. CURE plot for Exponential Model and Radius/1000.
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Guidelines for Developing Crash Modification Functions C-81 Recommended Model Form Although the two models perform similarly for each of the goodness-of-fit measures, the exponential model in Equation C60 is consistently preferred even though the differences are not very large. Crashes per year exp AADT expb c Ranggrp Equation C601000= α × where αanggrp = intercept term specific to each angle group AADT = average annual daily traffic avgangle = average deflection angle for the angle group R = radius in feet The authors tried to enhance Equation C60 by allowing the c parameter to vary by other explanatory variables in a multilevel fashion, including lane width and shoulder width under the hypothesis that the impact of increasing the curve radius may be greater on roadways with narrow lane and/or shoulder widths.
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C-82 Guidelines for the Development and Application of Crash Modification Factors Two issues were identified as important considerations: Issue 1 Minimizing Confounding Factors Issue 2 Determining the Appropriate Model Form To handle the confounding factor of deflection angle sites were group by similar values of this variable. Based on the data and previous research, two alternate assumptions of model form were pursued, one with an exponential relationship between crashes and radius and the second with a linear relationship.
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Guidelines for Developing Crash Modification Functions C-83 1. Illustrate a heuristic methodology a researcher may follow to derive a CMFunction from cross-sectional regression analysis.
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C-84 Guidelines for the Development and Application of Crash Modification Factors Issue 1 Minimizing Confounding Factors One of the most difficult aspects of developing CMFunctions through cross-sectional data is to minimize the differences between sites in variables that affect crash risk other than the variable(s) of interest.
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Guidelines for Developing Crash Modification Functions C-85 4 CMFunction Development by Regression Modeling The first step in the analysis was to estimate constant CMFunctions and explore whether the sites used should be constricted in some way to account for confounding factors prior to estimating the model. This is the focus of Chapter 3.1.
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From page 310... ...
C-86 Guidelines for the Development and Application of Crash Modification Factors It was also attempted to separately estimate CMFs for LTMAJ, RTMAJ, and the combined LTMAJ and RTMAJ treatments by eliminating sites with any other turn lanes on the major or minor road. For example, for estimating the CMF for LTMAJ, the data set included sites with no turning lanes at all or sites with a left-turn lane on the major road but no other turn lanes on the major or minor road.
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From page 311... ...
Guidelines for Developing Crash Modification Functions C-87 propensity score indicated a very strong overlap of probability of treatment between the treatment and comparison sites indicating that there is no site selection bias to be considered. Based on the results of the propensity score and CMFs estimated after further restricting the data as to account for potential confounding factors, it was concluded that the results exhibited no significant change and that all sites could be combined for further model estimation.
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From page 312... ...
C-88 Guidelines for the Development and Application of Crash Modification Factors where d = β1 + β3 × MINAADT e = β4 f = β7 + β8 × MAJAADT + β9 × MINAADT The parameter estimates are shown in Table C58. The relationships hold in that the CMF for LTMAJ increases with increasing MINAADT, the CMF for RTMAJ is constant and the CMF for LTMAJ+RTMAJ decreases with increasing MAJAADT and increases for increasing MINAADT.
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From page 313... ...
Guidelines for Developing Crash Modification Functions C-89 where: d = β1 + β2 × MINAADT + β3 × RTMIN e = β4 + β5 × RTMIN + β6 × Median f = β7 + β8 × MAJAADT + β9 × MINAADT The parameter estimates are shown in Table C59. The results in Table C59 are surprising as they indicate that the safety benefits of LTMAJ and RTMAJ are negated (i.e., CMF is greater than 1)
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From page 314... ...
C-90 Guidelines for the Development and Application of Crash Modification Factors The modeling was performed using the Winbugs software to apply full Bayes MCMC estimation techniques. Each parameter in the model is assigned an assumed distribution, mean and variance.
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From page 315... ...
Guidelines for Developing Crash Modification Functions C-91 In the next model more informative priors (i.e., priors with less variance) are applied.
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From page 316... ...
C-92 Guidelines for the Development and Application of Crash Modification Factors notable that the parameter estimates for a, b and c are similar between the subset data and full data which may be expected because MAJAADT and MINAADT are the largest predictor of crash risk and we may expect the constant term to be stable. Now the model using the same subset is estimated but including the parameter estimates that were estimated using the full dataset.
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Guidelines for Developing Crash Modification Functions C-93 The development of CMFunctions showed some success. The model estimated for Equation C64 appears to be most intuitive.
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From page 318... ...
C-94 Aul, N
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From page 319... ...
Guidelines for Developing Crash Modification Functions C-95 Hauer, E
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From page 320... ...
C-96 Guidelines for the Development and Application of Crash Modification Factors Washington, S., M Karlaftis, and F
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Key Terms
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