Understanding and Communicating Reliability of Crash Prediction Models (2021) / Chapter Skim
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From page 103... ...
101 Appendix A: The Development of Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions
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102 CONTENTS BACKGROUND .................................................................................................................................... 103Β ObjectiveΒ andΒ ScopeΒ ........................................................................................................................................Β 104Β LITERATURE REVIEW .....................................................................................................................
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103 Background The CPMs in Part C of the Highway Safety Manual (HSM) are used to estimate the predicted average crash frequency of a site with specific geometric design elements and traffic control features.
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104 Method 1 relates to the use of CPMs wherein there is a "balance" between the crash modification factors (CMFs) and the base conditions associated with the safety performance function (SPF)
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105 Literature Review The reliability of the prediction from a CPM can be described in terms of bias, variance, and repeatability. Bias represents the difference between the CPM estimate and the true value.
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106 Table 2. Variance estimates for various assumed conditions.
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107 However, there is also a practical reality that the correlation among independent variables will increase as more variables are added to the CMF. As a result, there may be an "optimum" number of variables that yields the smallest uncertainty in the predicted mean crash frequency.
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108 Table 3. Overdispersion parameter for several crash prediction models.
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109 a. Model fit to data.
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110 highway segments with 12-ft lanes. It then uses the CMF from Part D for "modify lane width" with this SPF to estimate the predicted crash frequency at sites with 11-ft lanes.
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111 π , = variance of the independent variable at sites used to establish the base condition (i.e., those sites used to estimate the jurisdiction-specific SPF)
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112 Table 4. Predicted bias for Case A
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113 Equation 13 π 1 0.5 π π , Equation 14 π πΏπ πΆππΉπ π where Np = predicted average crash frequency, crashes/yr; fB = bias adjustment factor for Case B; π , = variance of the independent variable at sites of interest; b = estimation coefficient; X = independent variable associated with CMF; and Xbase = independent variable associated with the base condition for the CMF of interest. Additional discussion on the use and meaning of Equation 14 is provided in the previous section associated with Case A
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114 wi = weight associated with site i; and wj = weight associated with site j. The average percent bias for a group of sites of interest is obtained by applying the CMF to each site and computing a weighted average of the values, where the weight used is the predicted average crash frequency based on the SPF.
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115 None of the sites in the roadway network of interest has lighting. Examination of the original database used to develop the CMF indicates that one-half of the sites therein have lighting and the other one-half have no lighting.
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116 Table 5. Predicted bias for Case B
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117 The CPM is used by the analyst with the external CMF. The analyst computes the variance of the predicted mean using the Condition 1 equations in Table 2 (i.e., π , π π )
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118 The database used to estimate π , is that database used to develop the CPM. This value may be difficult to quantify if the CPM was developed in a previous time period such that the database is not available or it does not contain the variable of interest X
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119 Table 6. Overdispersion parameter bias for Case B
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120 two CV values is also relatively constant for a range of variable values. The examination indicated that this ratio can be approximated by the following equation.
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121 independent variable (in the external CMF) at the sites used to estimate the CPM base conditions (π , )
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122 Equation 31 π π π with Equation 32 π 1 0.5 π π , Equation 33 π πΏπ πΆππΉπ π where Np = predicted average crash frequency, crashes/yr; fC = bias adjustment factor for Case C; π , = variance of the independent variable at sites of interest; b = estimation coefficient; X = independent variable associated with CMF; and Xbase = independent variable associated with the base condition for the CMF of interest. Additional discussion on the use and meaning of Equation 33 is provided in a previous section associated with Case A
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123 Equation 36 π΅πππ 100 πΆππΉ ππΆππΉ π 1 The use of Equation 36 can be demonstrated by an example. Consider the situation where the analyst is investigating the effect of reducing lane width on a roadway network.
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124 to estimate b as -0.105 (= Ln[0.90]
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125 In practice, when there is a Case C application, the overdispersion parameter reported for the CPM is not adjusted by the analyst to reflect the omission of the CMF. As a result, the overdispersion parameter in a Case C application is biased to a smaller value than the true value.
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126 Equation 46 π΅πππ , 100 π π ,π , This bias can be removed if Equation 43 is used for Case C applications to predict kp,r . This predicted value can then be used to estimate the variance of the predicted crash frequency.
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127 For a Case C application, the CPM is considered the "full" model and the CPM-with-omitted-CMF is the "reduced" model. If the database used to develop the CPM could be acquired, the independent variable associated with the omitted CMF removed, and the only CPM overdispersion parameter re-estimated using regression analysis, the resulting model would be considered the "reduced" model for this discussion.
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128 Table 10. Coefficient of variation ratio for Case C
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129 This section consists of four subsections. The experimental design is described in the next three subsections.
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130 Local-Jurisdiction Database To replicate the conditions associated with Application Cases A, B, and C, data for a second set of sites were also generated. These sites represent locations in the analyst's jurisdiction and to which the estimated CPM was applied.
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131 5. Compute the Desired Results The following values are computed for the sites in each site-specific database.
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132 ο§ Overall average of the predicted crash frequency in the CPM development database, Np ο§ Variance of the independent variable X at sites in the CPM development database, π , ο§ Estimation coefficient associated with variable X in full model, b ο§ Overall average of the independent variable X at sites in the CPM development database, π ο§ Overdispersion parameter associated with full model, kf ο§ Overall average of the predicted crash frequency in the local-jurisdiction database, NB ο§ Variance of the independent variable X at sites in the local-jurisdiction database, π , ο§ Overall average of the independent variable X at sites in the local-jurisdiction database, π ο§ Overdispersion parameter associated with reduced model, kr ο§ Overall average bias adjustment factor, fB ο§ Overall average CV ratio, CVR Case C CMF Not Used in CPM But Base Condition Accommodated in the SPF The process used to create the data for Case C is described as follows.
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133 7. Compute the Desired Results The following values are computed for the sites in each database.
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134 Table 11. Variables used to create the evaluation database.
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135 Equation 51 π 1.120 ππ π 0; 0.880 ππ‘βπππ€ππ π where fA = bias adjustment factor for Case A; ct = correction term; and all other terms are as previously defined. The correction factor values were determined using a search routine that sought to minimize the sum of the squared error in the individual site observations.
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136 where all variables are as previously defined and the correction term ct is obtained from Equation 51. A comparison of the estimates from Equation 52 with the ground-truth values is shown in Figure 3a.
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137 where all variables are as previously defined. Figure 3c shows the comparison of the predicted and true values of the overdispersion parameter.
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138 Figure 4d shows the comparison of the predicted and true ratios of the coefficient of variation. The predicted ratio was calculated with Equation 47.
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139 Miaou, S.P.
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