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Understanding and Communicating Reliability of Crash Prediction Models (2021)

Chapter:Chapter 3. 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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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
×
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Suggested Citation:"Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions." National Academies of Sciences, Engineering, and Medicine. 2021. Understanding and Communicating Reliability of Crash Prediction Models. Washington, DC: The National Academies Press. doi: 10.17226/26440.
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19 Chapter 3. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Mismatch Between CMFs and SPF Base Conditions Introduction The crash prediction models (CPMs) in Part C of the HSM include a safety performance function (SPF), one or more crash modification factors (CMFs), and a local calibration factor (C). Several model- and application-related factors have been described in Table 1 and Table 2 as having an influence on the reliability of the predicted crash frequency obtained from the CPM. This Chapter examines two of these factors. They are 1) add CMFs from other sources (consistent with base conditions), and 2) use of CMFs that are inconsistent with SPF base conditions. 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. Variance describes the extent of uncertainty in the estimate due to unexplained or random influences. Repeatability describes the extent to which multiple analysts using the same CPM with the same training, data sources, and site of interest obtain the same results (as measured by the number of significant figures showing agreement among results). A more reliable estimate has little bias, a smaller variance, and is likely to have results that show several significant figures in agreement (should there be repeated independent applications). As mentioned in Chapter 1, there are two categories of factors that influence the reliability of a CPM estimate. These categories include model-related factors and application-related factors. The factors in both categories combine to define the reliability of a CPM estimate when it is used to evaluate a given site. In this regard, the estimate’s reliability is likely to vary from site-to-site and analyst-to-analyst, depending on how the model is configured and applied by the analyst to a given site. Objectives The objectives of this chapter are to (1) describe procedures for quantifying the safety effect of specific two factors that influence CPM reliability, and (2) demonstrate how the procedures can be used to quantify and interpret CPM estimate reliability. The research undertaken to develop these procedures is documented in Appendix A. The two factors of interest relate to the use of CMFs from other sources (e.g., HSM Part D, FHWA CMF Clearinghouse) with the HSM Part C CPMs. These two factors are identified in Table 4. Table 4. Factors Related to CMFs that Influence the Reliability of an Estimated Value Using a CPM. Influence Category Factor Effect of Factor on Reliability Measures Bias Variance Repeatability Model-related factors influencing reliability 1. Add CMFs from other sources (consistent with base conditions) No effect; possibly less reliable Less reliable No effect if CMFs are well documented. Less reliable if CMFs are not well documented. Application-related factors influencing reliability 2. Use of CMFs that are inconsistent with SPF base conditions Less reliable Less reliable Less reliable if CMFs are not well documented. The first factor (i.e., add CMFs from other sources) represents the case were the analyst desires to include a CMF from HSM Part D (or other source) in the CPM, where the CMF is consistent with the

20 SPF base conditions. The motivation for type of application can vary (e.g., the analyst desires to evaluate the benefits of a new treatment that has yet to be deployed system-wide but for which a CMF has been developed from studies of its use at a few trial locations). The second factor (i.e., use of CMFs that are inconsistent with SPF base conditions) represents the case where the analyst desires to include a CMF from HSM Part D (or other source) in the CPM, where the CMF does not match any of the SPF base conditions. The motivation for this type of application often stems from the case where there is a CMF describing the safety effect of the treatment of interest but this CMF is not included in the HSM Part C CPM. Background Part C (Section C.7) of the HSM describes four methods for estimating the average crash frequency for a site. The methods are indicated to provide different levels of predictive reliability. However, the HSM does not quantify the reduction in reliability associated with each of the four methods, so analysts do not have the information they need to make an informed choice among the methods. These methods are identified in the following list in order of predictive reliability, with the most reliable method listed first:  Method 1 – Apply the Part C CPM to evaluate the existing and proposed conditions.  Method 2 – Apply the Part C CPM to evaluate the existing condition. Use a Part D CMF with the Part C CPM to evaluate the proposed condition.  Method 3 – Apply a jurisdiction-specific SPF to evaluate the existing condition. Use a Part D CMF with this SPF to evaluate the proposed condition.  Method 4 – Use observed crash frequency to evaluate the existing condition. Use a Part D CMF with the observed crash frequency to evaluate the proposed condition. 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). In this regard, a balanced CPM application occurs when the set of CMFs used collectively match all of the SPF’s base conditions. Method 2 presents the situation where there is a lack of balance between the CMFs used and the SPF. In this situation, the Part C SPF base conditions do not include those associated with the Part D CMF. This method corresponds to Factor 2 in Table 4. Method 3 can be implemented in one of two ways. In the first way, there is a lack of balance between the CMF obtained from Part D and the jurisdiction-specific SPF (similar to that described for Method 2). This variation of Method 3 also corresponds to Factor 2 in Table 4. In the second way, there is a balance between the CMF from Part D and the jurisdiction-specific SPF. This variation of Method 3 corresponds to Factor 1 in Table 4. There is a variation of Methods 1 or 2 that can sometimes occur in application. For this variation, one or more CMFs that are part of the Part C CPM are not used (i.e., omitted from calculations). This situation may occur when the analyst is interested in using the CPM to evaluate a site but does not have ready access to the data needed for one or more of the other CMFs in the CPM. Method 4 is not based on the use of an SPF. Rather, the observed crash frequency for the site of interest is used to estimate the expected crash frequency. Thus, the issue of CMF-SPF balance does not apply. For this reason, the reliability of Method 4 is not addressed in this report. CMF Applications That Affect CPM Reliability The discussion in the preceding section identifies three cases where the analyst’s selection of the CMFs used with a CPM results in a reduction in estimate reliability. These three cases are identified in Table 5. The procedures for quantifying the reliability associated with each application case are described in the next section.

21 Table 5. Summary of Applications Associated with Reliability Reduction in Predicted Value. Case Description Associated HSM Method CMF-SPF Balance* A CMF from Part D used with jurisdiction-specific SPF (CMF is consistent with SPF base conditions) 3 Yes B One or more CMFs used in the CPM do not have a corresponding base condition in the SPF 2, 3 No C One or more CMFs are not used in the CPM yet the corresponding base condition exists in the SPF 1, 2 No * A balanced CPM application occurs when the set of CMFs used collectively match all of the SPF’s base conditions Procedures to Assess Potential Reliability This section describes procedures for quantifying CPM estimate reliability for each of three CMF application cases. These cases are identified in Table 5. All of these procedures are based on theoretically- based equations that predict the bias in, and increased variance, of the CPM estimate of predicted crash frequency. The predicted bias and increased variance can be used in two ways: (1) they can be used to correct the CPM estimates by removing the bias and increased variance; and (2) alternatively, the CPM estimates be retained (i.e., bias not removed) and used only to quantify their overall reliability. The former use may be appropriate when evaluating a specific site. The latter use may be appropriate when making policy decisions about CPM development and application. Goodness-of-Fit Measure Definitions In this Chapter, each procedure applies two goodness-of-fit (GOF) measures. One measure is “increased root mean square error” and the second measure is “percent bias.” Both measures are described in the following two subsections. Increased Root Mean Square Error The increased root mean square error measure is an indication of overall reliability of the CPM estimate when it is used as described by one of the three application cases listed in Table 5. It is a measure of the uncertainty added to the estimate due to the use of a biased value of the overdispersion1 parameter, predicted crash frequency, or both. Equation 14 𝜎 , 𝜎 𝑒 . with Equation 15 𝜎 𝑘 𝑁 𝑘 , 𝑁 ,   Equation 16 𝑒 𝑁 𝑁 ,   where σe,I = increased root mean square error; e = error in predicted crash frequency; 𝜎 = absolute difference of the change in variance of the predicted value; 1 The terms overdispersion and dispersion are used interchangeably in this document.

22 kreported = reported overdispersion parameter for CPM; kp, true = predicted true overdispersion parameter; Np = predicted crash frequency from CPM, crashes/yr; and Np,true = predicted true crash frequency, crashes/yr; The reported overdispersion parameter is the value obtained from HSM Part C for the CPM that is used for the reliability evaluation. The predicted true overdispersion parameter is obtained from the equations provided in the procedure described in the next section (i.e., Procedural Steps). The predicted crash frequency from the CPM is the value obtained from the CPM when used in a manner described by one of the three application cases listed in Table 5. The predicted true crash frequency is the value obtained from the equations provided in the procedure described in the next section. The increased root mean square error can be normalized by dividing it by the predicted true crash frequency. This division produces a coefficient of variation that facilitates the relative comparison of alternative CPMs and alternative applications of a given CPM. Equation 17 𝐶𝑉 𝜎 ,𝑁 , where CVI is the coefficient of variation for the increased root mean square error and all other variables are as previously defined. A coefficient value of 0 indicates that there is no bias or additional uncertainty in the predicted crash frequency obtained from the CPM. As the coefficient of variation value increases, the prediction becomes less reliable because the bias has increased, the uncertainty has increased, or both. Values in excess of 0.20 are considered to be unreliable for most applications. Percent Bias The percent bias measure provides an indication of the relative error in the prediction. This measure is expressed as a percentage because the magnitude of the error is often correlated with the true (i.e., unbiased) value of the estimate. This characteristic facilitates the relative comparison of alternative CPMs and alternative applications of a given CPM. The percent bias measure is computed using the following equation. Equation 18 𝐵𝑖𝑎𝑠 100 𝑁 𝑁 ,𝑁 , where Bias = percent bias in reported value; Np = predicted crash frequency from CPM, crashes/yr; and Np,true = predicted true crash frequency, crashes/yr; The percent bias measure can be used to describe the predicted crash frequency obtained from the CPM, for a given application case. A percent bias value of 0 indicates there is no bias in the reported overdispersion parameter or the predicted crash frequency. As the bias increases, the overdispersion parameter or CPM prediction becomes less reliable. Bias percentages in excess of 10 percent are considered to be unreliable for most applications.

23 Procedural Steps This section describes the procedures for quantifying the bias and added uncertainty (i.e., variance) associated with each of the three application cases identified in Table 5. The procedure associated with each case is described in a separate subsection. Case A. CMF from Part D used with SPF (CMF is consistent with SPF base conditions) For this CPM application, a CMF from HSM Part D is used with a jurisdiction-specific SPF. The CMF of interest could also be obtained from another source (e.g., FHWA CMF Clearinghouse); however, this CMF is referred herein to as the “HSM Part D CMF” for consistency with HSM Method 3. For this application, there is a “balance” between the CMF and the base conditions associated with the SPF. In this Case A presentation, the CMF obtained from HSM Part D represents the “CMF of interest.” It describes the safety effect of a treatment of interest. The goodness-of-fit measures that are computed describe the reliability of the estimated average crash frequency when the treatment is applied to one or more sites of interest. There are seven steps in this procedure. Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 6. The data needed to apply the SPF are not listed, but they are also needed. If there are several sites of interest, the coefficient of variation CVI and bias percent Bias computed in Steps 6 and 7, respectively, were derived to have the characteristic that they are insensitive to the value of the predicted crash frequency (NP). As a result, the predicted coefficient of variation and bias percent reasonably describe the reliability of the predicted crash frequency for each of the sites of interest, regardless of their AADT or segment length. Therefore, for purposes of the reliability evaluation when there are several sites of interest, a typical site representing the collective set of sites can be used to compute the predicted crash frequency in Step 4. In this regard, typical values of AADT, segment length, etc. can be used to apply the SPF in Step 4. Table 6. Required data for Case A. Required Data and Relationships Potential Data Sources Input values needed for jurisdiction-specific SPF Agency files Overdispersion parameter for jurisdiction-specific SPF, kreported SPF development report HSM Part D CMF for geometric design element or traffic control feature of interest, CMFD HSM Part D; CMF Clearinghouse Average independent variable value at sites used to estimate the SPF, 𝑋 SPF development report, archived SPF database, agency files, or field meas. Standard deviation of the independent variable at sites used to estimate NSPF,j-s, σx,b SPF development report, archived SPF database, agency files, or field meas. Average independent variable value associated with CMF of interest at sites of interest, 𝑋 Site data from agency files, plans, or field measurements Standard deviation of the independent variable at sites of interest, σx,n Site data from agency files, plans, or field measurements The average of the independent variable at the sites used to estimate the SPF (𝑋𝑆𝑃𝐹) can be obtained from one of several sources. It can be obtained from the SPF development report if that report provides summary statistics of the variable X in the data used to estimate the SPF. Alternatively, the average of X can be obtained through access to the original database used to estimate the SPF. Finally, if this database is not available or it does not contain the variable of interest X, then it may be possible to identify the regions from which the data were obtained to develop the SPF and obtain values of the variable X at a representative set of sites. These values would be obtained from agency files or field measurements. The

24 value 𝑋𝑆𝑃𝐹 would then be computed using this representative set of sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋𝑆𝑃𝐹 is then computed using the 1 or 0 values at the collective set of study sites. The standard deviation of the independent variable at the sites used to estimate the SPF (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋𝑆𝑃𝐹. When One Site of Interest is Being Evaluated. The average of the independent variable value associated with the CMF of interest (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the site that is planned for treatment (or has been treated). Another source is field measurement at the site that is planned for treatment (or has been treated). Regardless of the source, 𝑋 equals the variable value for the site of interest. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The standard deviation of the independent variable at the sites interest (𝜎 , ) is equal to 0.0 if X is known with certainty. If X has been judged or measured, then the standard deviation should equal a small positive value reflecting the analyst’s uncertainty in the judgment or measurement. When Several Sites of Interest are Being Evaluated. The average of the independent variable value associated with the CMF of interest (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the sites that are planned for treatment (or have been treated). Another source is field measurements at the sites that are planned for treatment (or have been treated). Regardless of the source, if the treatment is applied uniformly to the sites of interest as part of a planned change, then X has the same value at each site and 𝑋 equals X. Alternatively, if the treatment is not applied uniformly to all of the sites of interest, then 𝑋 equals the value of X averaged for all sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋 is then computed using the 1 or 0 values for the collective set of sites. The standard deviation of the independent variable at the sites of interest (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋. Step 2. Compute Estimation Coefficient. First, compute the value of the CMF of interest for the sites used to estimate the SPF (𝐶𝑀𝐹 𝑋 ). If the CMF of interest corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. For example, if the CMF for “add beacon” is 0.95 and 30 percent of the sites used to estimate the SPF have the beacon, then the value of 𝐶𝑀𝐹 𝑋 is 0.985 (= 0.95 × 0.30 + 1.0 × [1− 0.30]). If the CMF of interest corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋 ). Second, compute the value of the CMF of interest for the sites of interest (𝐶𝑀𝐹 𝑋 ). If the CMF of interest corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. If all of the sites of interest are planned to have the beacon added, then the value of 𝐶𝑀𝐹 𝑋 is 0.95 (= 0.95 × 1.0 + 1.0 × [1− 1.0]). If the CMF of interest corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋). If 𝑋 is equal to 𝑋 then multiply 𝑋 by 1.01 and use the resulting value of 𝑋 to compute 𝐶𝑀𝐹 𝑋 . Use this computed CMF value and the resulting value of 𝑋 in the equation provided in the next calculation. Multiplication by the constant 1.01 is used to avoid division by “0” in this equation. Third, use the following equation to compute the estimation coefficient b.

25 Equation 19 𝑏 𝐿𝑛 𝐶𝑀𝐹 𝑋 𝐿𝑛 𝐶𝑀𝐹 𝑋𝑋 𝑋 where b = estimation coefficient; CMFD = HSM Part D CMF for geometric design element, or traffic control feature of interest; 𝑋 = average independent variable value associated with CMF of interest at sites of interest; 𝑋 = average independent variable value at sites used to estimate the SPF; CMFD(𝑋) = HSM Part D CMF value associated with 𝑋; and CMFD(𝑋 ) = HSM Part D CMF value associated with 𝑋 . Step 3. Compute Bias Adjustment Factor. Determine the correction term ct based on the sign of the estimation coefficient b. If b is greater than 0, then ct equals 1.120; otherwise ct equals 0.880. Then, use this term in the following equation to compute the bias adjustment factor fA. Equation 20 𝑓 1 0.5 𝑏 𝜎 , 𝜎 , 𝑐 with Equation 21 𝑐 1.120 𝑖𝑓 𝑏 0; 0.880 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where fA = bias adjustment factor for Case A; 𝜎 , = variance of the independent variable at sites of interest; 𝜎 , = variance of the independent variable at sites used to estimate the jurisdiction-specific SPF; b = estimation coefficient; and ct = correction term. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The following equation is used to compute the predicted crash frequency for the site of interest. Equation 22 𝑁 𝑁 , 𝐶𝑀𝐹 𝑋 where Np = predicted average crash frequency, crashes/yr; NSPF,j-s = predicted crash frequency for site with base conditions that are in balance with the CMFs (jurisdiction-specific); and CMFD(𝑋) = HSM Part D CMF value associated with 𝑋. Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The following equation is used to compute the unbiased predicted crash frequency for the site of interest. Equation 23 𝑁 , 𝑁 𝑓 where Np,true is the predicted true crash frequency, crashes/yr; and all other variables are as previously defined. Step 6. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, compute the error in the predicted crash frequency using the following equation.

26 Equation 24 𝑒 𝑁 𝑁 ,   where e is the error in predicted crash frequency; and all other variables are as previously defined. Second, compute the absolute difference in the change in variance of the predicted value using the following equation. Equation 25 𝜎 𝑘 𝑁 𝑘 𝑁 ,   where 𝜎 is the absolute difference of the change in variance of the predicted value; kreported is the reported overdispersion parameter for the jurisdiction-specific SPF; and all other variables are as previously defined. Third, compute the increased root mean square error using the following equation. Equation 26 𝜎 , 𝜎 𝑒 .   where σe,I is the increased root mean square error; and all other variables are as previously defined. Finally, compute the coefficient of variation using the following equation. Equation 27 𝐶𝑉 𝜎 ,𝑁 , where CVI is the coefficient of variation for the increased root mean square error and all other variables are as previously defined. Step 7. Compute the Amount of Bias The percent bias in the predicted crash frequency is computed using the following equation. Equation 28 𝐵𝑖𝑎𝑠 100 𝑁 𝑁 ,𝑁 , where Bias is the percent bias in the reported value; and all other variables are as previously defined. Case B. CMFs Do Not Have a Corresponding Base Condition in the SPF For this CPM application, one or more CMFs used with a CPM do not have a corresponding base condition in the SPF. These CMFs are called herein “external CMFs” because their variables were not considered when the SPF was developed and its base conditions were established. An example of this application is when a Part D CMF is used with a Part C CPM (and the Part D CMF’s variables are not included in the CPM’s base conditions). The likely source of the CPM is HSM Part C; however, this procedure is sufficiently general that it can be applied to CPMs from other sources (e.g., a CPM developed for a specific jurisdiction). There are eight steps in this procedure. Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 7. The data needed to apply the CPM and the HSM Part C CMFs are not listed, but they are also needed. If there are several sites of interest, the coefficient of variation CVI and bias percent Bias computed in Steps 7 and 8, respectively, were derived to have the characteristic that they are insensitive to the value of the predicted crash frequency (NP). As a result, the predicted coefficient of variation and bias percent reasonably describe the reliability of the predicted crash frequency for each of the sites of interest, regardless of their AADT, segment length, etc. Therefore, for purposes of the reliability evaluation when there are several sites of interest, a typical site representing the collective set of sites can be used to compute the predicted crash frequency in Step 4. In

27 this regard, typical values of AADT, segment length, and variables in the CMFs (other than the external CMF) can be used to apply the CPM in Step 4. Table 7. Required Data for Case B. Required Data and Relationships Potential Data Sources Input values needed for CPM Agency files Overdispersion parameter for CPM, kreported HSM Part C HSM Part C CMF for geometric design element or traffic control feature i, CMFi HSM Part C External CMF for geometric design element or traffic control feature of interest, CMFex HSM Part D; CMF Clearinghouse Average independent variable value at sites used to estimate the CPM, 𝑋 CPM development report, archived CPM database, agency files, or field meas. Standard deviation of the independent variable at sites used to estimate the CPM, σx,b CPM development report, archived CPM database, agency files, or field meas. Average independent variable value associated with external CMF at sites of interest, 𝑋 Site data from agency files, plans, or field measurements Standard deviation of the independent variable at sites of interest, σx,n Site data from agency files, plans, or field measurements The average of the independent variable at the sites used to estimate the CPM (𝑋𝐶𝑃𝑀) can be obtained from one of several sources. It can be obtained from the CPM development report if that report provides summary statistics of the variable X in the data used to estimate the CPM. Alternatively, the average of X can be obtained through access to the original database used to estimate the CPM. Finally, if this database is not available or it does not contain the variable of interest X, then it may be possible to identify the regions from which the data were obtained to develop the CPM and obtain values of the variable X at a representative set of sites. These values would be obtained from agency files or field measurements. The value 𝑋𝐶𝑃𝑀 would then be computed using this representative set of sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋𝐶𝑃𝑀 is then computed using the 1or 0 values at the collective set of study sites. The standard deviation of the independent variable at the sites used to estimate the CPM (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋𝐶𝑃𝑀. When One Site of Interest is Being Evaluated. The average of the independent variable value associated with the external CMF (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the site that is planned for treatment (or has been treated). Another source is field measurements at the site that is planned for treatment (or has been treated). Regardless of the source, 𝑋 equals the variable value for the site of interest. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The standard deviation of the independent variable at the sites interest (𝜎 , ) is equal to 0.0 if X is known with certainty. If X has been judged or measured, then the standard deviation should equal a small positive value reflecting the analyst’s uncertainty in the judgment or measurement. When Several Sites of Interest are Being Evaluated. The average of the independent variable value associated with the external CMF (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the sites that are planned for treatment (or have been treated). Another source is field measurements at the sites that are planned for treatment (or have been treated). Regardless of the source, if the treatment is applied uniformly to the sites of interest as part of a planned change, then X has the same value at each site and 𝑋 equals X. Alternatively, if the treatment is not applied uniformly

28 to all of the sites of interest, then 𝑋 equals the value of X averaged for all sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋 is then computed using the 1, 0 values for the collective set of sites. The standard deviation of the independent variable at the sites interest (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋. Step 2. Compute Estimation Coefficient. First, compute the value of the external CMF for the sites used to estimate the CPM (𝐶𝑀𝐹 𝑋 ). If the external CMF corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. For example, if the CMF for “add beacon” is 0.95 and 30 percent of the sites used to estimate the CPM have the beacon, then the value of 𝐶𝑀𝐹 𝑋 is 0.985 (= 0.95 × 0.30 + 1.0 × [1− 0.30]). If the external CMF corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋 ). Second, compute the value of the external CMF for the sites of interest (𝐶𝑀𝐹 𝑋 ). If the external CMF corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. If all of the sites of interest are planned to have the beacon added, then the value of 𝐶𝑀𝐹 𝑋 is 0.95 (= 0.95 × 1.0 + 1.0 × [1− 1.0]). If the external CMF corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋). If 𝑋 is equal to 𝑋 then multiply 𝑋 by 1.01 and use the resulting value of 𝑋 to compute 𝐶𝑀𝐹 𝑋 . Use this computed CMF value and the resulting value of 𝑋 in the equation provided in the next calculation. Third, use the following equation to compute the estimation coefficient b. Equation 29 𝑏 𝐿𝑛 𝐶𝑀𝐹 𝑋 𝐿𝑛 𝐶𝑀𝐹 𝑋𝑋 𝑋 where b = estimation coefficient; CMFex = external CMF (i.e., not associated with the SPF’s base conditions); 𝑋 = average independent variable associated with CMF of interest at sites of interest; 𝑋 = average independent variable value at sites used to estimate the CPM; CMFex(𝑋) = external CMF value associated with 𝑋; and CMFex(𝑋 ) = external CMF value associated with 𝑋 . Step 3. Compute Bias Adjustment Factor. Determine the correction term ct based on the sign of the estimation coefficient b. If b is greater than 0, then ct equals 1.120; otherwise ct equals 0.880. Then, use this term in the following equation to compute the bias adjustment factor fB. Equation 30 𝑓 1 0.5 𝑏 𝜎 , 𝑐 with Equation 31 𝑐 1.120 𝑖𝑓 𝑏 0; 0.880 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where fB = bias adjustment factor for Case B;

29 𝜎 , = variance of the independent variable at sites of interest; b = estimation coefficient; and ct = correction term. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The following equation is used to compute the predicted crash frequency for the site of interest. Equation 32 𝑁 𝐶 𝑁 𝐶𝑀𝐹 … 𝐶𝑀𝐹 𝐶𝑀𝐹 𝑋 where Np = predicted average crash frequency, crashes/yr; C = local calibration factor; NSPF = predicted crash frequency for site with base conditions, crashes/yr; CMFi = HSM Part C CMF for geometric design element, or traffic control feature i (i = 1 to n); n = total number of HSM Part C CMFs; and 𝐶𝑀𝐹 𝑋 = value associated with 𝑋 (i.e., not associated with the SPF’s base conditions). Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The following equation is used to compute the unbiased predicted crash frequency for the site of interest. Equation 33 𝑁 , 𝑁 𝐶𝑀𝐹 𝑋𝑓 𝐶𝑀𝐹 𝑋 where Np,true is the predicted true crash frequency, crashes/yr; and all other variables are as previously defined. Step 6. Compute the Unbiased Overdispersion Parameter for the CPM with the External CMF. First, compute the adjustment factor Δ0 using the second equation shown below (i.e., Equation 35). Then, compute the overdispersion parameter for the CPM with the external CMF using the following equation. Equation 34 𝑘 , 𝑘 1.13 𝑏 𝜎 , ∆ with Equation 35 ∆ 1 0.10 2 min 5,𝑝 1 where kp,true = predicted true overdispersion parameter for CPM with external CMF; kreported = reported overdispersion parameter for the CPM; 𝜎 , = variance of the independent variable at sites used to estimate the CPM; p = number of empirically derived constants in (a) the CMFs associated with the CPM and (b) the external CMF (exclude those in the SPF for intercept, AADT, and segment length) (refer to the text following this variable list); Δ0 = adjustment factor for the incremental effect of additional empirical coefficients; and all other variables are as previously defined. The number of empirically derived constants p is determined by inspecting the CMFs in the CPM. Regression constants in the SPF (e.g., intercept, AADT coefficient, segment length coefficient) are not considered when determining the value of p. If the CMF is associated with a discrete treatment (e.g., add beacon), then the CMF is often represented by a single empiric constant. If the CMF is associated with a continuous variable (e.g., lane width), then it is often a function that includes one or more empirical

30 constants. In general, there is at least one empirically derived constant for each CMF used to develop the CPM plus 1 for the external CMF. For practical applications of this procedure, the variable p can be estimated as equal to the number of CMFs used in Equation 32 plus 1 for the external CMF (i.e., p = n + 1). Step 7. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, compute the error in the predicted crash frequency using the following equation. Equation 36 𝑒 𝑁 𝑁 ,   where e is the error in predicted crash frequency; and all other variables are as previously defined. Second, compute the absolute difference in the change in variance of the predicted value using the following equation. Equation 37 𝜎 𝑘 𝑁 𝑘 , 𝑁 ,   where 𝜎 is the absolute difference of the change in variance of the predicted value; and all other variables are as previously defined. Third, compute the increased root mean square error using the following equation. Equation 38 𝜎 , 𝜎 𝑒 . where σe,I is the increased root mean square error; and all other variables are as previously defined. Finally, compute the coefficient of variation using the following equation. Equation 39 𝐶𝑉 𝜎 ,𝑁 , where CVI is the coefficient of variation for the increased root mean square error and all other variables are as previously defined. Step 8. Compute the Amount of Bias The percent bias in the predicted crash frequency is computed using the following equation. Equation 40 𝐵𝑖𝑎𝑠 100 𝑁 𝑁 ,𝑁 , where Bias is the percent bias in the reported value; and all other variables are as previously defined. Case C. CMF Not Used in CPM But Base Condition Accommodated in the SPF For this CPM application, the analyst chooses not to use one or more of the CMFs that were included in the CPM when it was developed. All these CMFs (whether used or not used) have a corresponding base condition in the SPF. The unused CMFs are called herein “omitted CMFs.” An example of this application is when the analyst does not have ready access to the data needed for a CMF and, as a result, chooses not to use the CMF when evaluating one or more sites. The likely source of the CPM is HSM Part C; however, this procedure is sufficiently general that it can be applied to CPMs from other sources (e.g., a CPM developed for a specific jurisdiction). There are eight steps in this procedure. Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 8. The data needed to apply the CPM and the HSM Part C CMFs are not listed, but they are also needed. If there are several sites of interest, the

31 coefficient of variation CVI and bias percent Bias computed in Steps 7 and 8, respectively, were derived to have the characteristic that they are insensitive to the value of the predicted crash frequency (NP). As a result, the predicted coefficient of variation and bias percent reasonably describe the reliability of the predicted crash frequency for each of the sites of interest, regardless of their AADT, segment length, etc. Therefore, for purposes of the reliability evaluation when there are several sites of interest, a typical site representing the collective set of sites can be used to compute the predicted crash frequency in Step 4. In this regard, typical values of AADT, segment length, and variables in the CMFs (other than the omitted CMF) can be used to apply the CPM in Step 4. Table 8. Required Data for Case C. Required Data and Relationships Potential Data Sources Input values needed for CPM Agency files Overdispersion parameter for CPM, kreported HSM Part C HSM Part C CMF for geometric design element or traffic control feature i, CMFi HSM Part C Omitted CMF for geometric design element or traffic control feature of interest, CMFom HSM Part C Average independent variable value at sites used to estimate the CPM, 𝑋 CPM development report, archived CPM database, agency files, or field meas. Standard deviation of the independent variable at sites used to estimate the CPM, σx,b CPM development report, archived CPM database, agency files, or field meas. Average independent variable value associated with omitted CMF at sites of interest, 𝑋 Site data from agency files, plans, or field measurements Standard deviation of the independent variable at sites of interest, σx,n Site data from agency files, plans, or field measurements The average of the independent variable at the sites used to estimate the CPM (𝑋𝐶𝑃𝑀) can be obtained from one of several sources. It can be obtained from the CPM development report if that report provides summary statistics of the variable X in the data used to estimate the CPM. Alternatively, the average of X can be obtained through access to the original database used to estimate the CPM. Finally, if this database is not available or it does not contain the variable of interest X, then it may be possible to identify the regions from which the data were obtained to develop the CPM and obtain values of the variable X at a representative set of sites. These values would be obtained from agency files or field measurements. The value 𝑋𝐶𝑃𝑀 would then be computed using this representative set of sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋𝐶𝑃𝑀 is then computed using the 1, 0 values at the collective set of study sites. The standard deviation of the independent variable at the sites used to estimate the CPM (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋𝐶𝑃𝑀. When One Site of Interest is Being Evaluated. The average of the independent variable value associated with the omitted CMF (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the site that is planned for treatment (or has been treated). Another source is field measurements at the site that is planned for treatment (or has been treated). Regardless of the source, 𝑋 equals the variable value at the site of interest. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The standard deviation of the independent variable at the sites interest (𝜎 , ) is equal to 0.0 if X is known with certainty. If X has been judged or measured, then the standard deviation should equal a small positive value reflecting the analyst’s uncertainty in the judgment or measurement.

32 When Several Sites of Interest are Being Evaluated. The average of the independent variable value associated with the omitted CMF (𝑋) can be obtained in several ways. One source of this information is agency files (e.g., design plans) for the sites that are planned for treatment (or have been treated). Another source is field measurements at the sites that are planned for treatment (or have been treated). Regardless of the source, if the treatment is applied uniformly to the sites of interest as part of a planned change, then X has the same value at each site and 𝑋 equals X. Alternatively, if the treatment is not applied uniformly to all of the sites of interest, then 𝑋 equals the value of X averaged for all sites. If the treatment is discrete (e.g., “add beacon”), an indicator variable is used to indicate treatment presence (where 1 corresponds to treatment present, and 0 corresponds to treatment not present). The average 𝑋 is then computed using the 1 or 0 values for the collective set of sites. The standard deviation of the independent variable at the sites interest (𝜎 , ) is obtained using the same data source that was used to obtain 𝑋. Step 2. Compute Estimation Coefficient. First, compute the value of the omitted CMF for the sites used to estimate the CPM (𝐶𝑀𝐹 𝑋 ). If the omitted CMF corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. For example, if the CMF for “add beacon” is 0.95 and 30 percent of the sites used to estimate the CPM have the beacon, then the value of 𝐶𝑀𝐹 𝑋 is 0.985 (= 0.95 × 0.30 + 1.0 × [1− 0.30]). If the omitted CMF corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋 ). Second, compute the value of the omitted CMF for the sites of interest (𝐶𝑀𝐹 𝑋 ). If the omitted CMF corresponds to a discrete treatment (e.g., “add beacon”), then the CMF value is computed using the corresponding site data, where each site with the treatment is assigned the CMF value and each site without the treatment is assigned a value of 1.0. If all of the sites of interest are planned to have the beacon added, then the value of 𝐶𝑀𝐹 𝑋 is 0.95 (= 0.95 × 1.0 + 1.0 × [1− 1.0]). If the omitted CMF corresponds to a continuous treatment (e.g., “lane width), then the CMF value used is computed using the CMF function with the average value of X (i.e., 𝑋). If 𝑋 is equal to 𝑋 then multiply 𝑋 by 1.01 and use the resulting value of 𝑋 to compute 𝐶𝑀𝐹 𝑋 . Use this computed CMF value and the resulting value of 𝑋 in the equation provided in the next calculation. Multiplication by the constant 1.01 is used to avoid division by “0” in this equation. Third, use the following equation to compute the estimation coefficient b. Equation 41 𝑏 𝐿𝑛 𝐶𝑀𝐹 𝑋 𝐿𝑛 𝐶𝑀𝐹 𝑋𝑋 𝑋 where b = estimation coefficient; CMFom = omitted CMF (i.e., associated with the SPF’s base conditions but excluded from CPM); 𝑋 = average independent variable associated with CMF of interest at sites of interest; 𝑋 = average independent variable value at sites used to estimate the CPM; CMFom(𝑋) = omitted CMF value associated with 𝑋; and CMFom(𝑋 ) = omitted CMF value associated with 𝑋 . Step 3. Compute Bias Adjustment Factor. Determine the correction term ct based on the sign of the estimation coefficient b. If b is greater than 0, then ct equals 1.120; otherwise ct equals 0.880. Then, use this term in the following equation to compute the bias adjustment factor fC. Equation 42 𝑓 1 0.5 𝑏 𝜎 , 𝑐

33 with Equation 43 𝑐 1.120 𝑖𝑓 𝑏 0; 0.880 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where fC = bias adjustment factor for Case C; 𝜎 , = variance of the independent variable at sites of interest; b = estimation coefficient; and ct = correction term. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The following equation is used to compute the predicted crash frequency for the site of interest. Equation 44 𝑁 𝐶 𝑁 𝐶𝑀𝐹 … 𝐶𝑀𝐹 ; 𝐶𝑀𝐹 𝐶𝑀𝐹 where Np = predicted average crash frequency, crashes/yr; C = local calibration factor; NSPF = predicted crash frequency for site with base conditions, crashes/yr; CMFi = HSM Part C CMF for geometric design element, or traffic control feature i (i = 1 to n); n = total number of HSM Part C CMFs; and CMFom = omitted CMF (i.e., associated with the SPF’s base conditions but excluded from CPM). Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The following equation is used to compute the unbiased predicted crash frequency for the site of interest. Equation 45 𝑁 , 𝑁 𝑓 𝐶𝑀𝐹 𝑋 𝐶𝑀𝐹 𝑋 where Np,true is the predicted true crash frequency, crashes/yr; and all other variables are as previously defined. Step 6. Compute the Unbiased Overdispersion Parameter for the CPM with the Omitted CMF. First, compute the adjustment factor Δ0 using the second equation shown below (i.e., Equation 47). Then, compute the overdispersion parameter for the CPM with the omitted CMF using the following equation. Equation 46 𝑘 , 𝑘 1.16 𝑏 𝜎 , ∆ with Equation 47 ∆ 1 0.10 2 min 5,𝑝 1 where kp,true = predicted true overdispersion parameter for CPM with omitted CMF; kreported = reported overdispersion parameter for the CPM; 𝜎 , = variance of the independent variable at sites used to estimate the CPM; p = number of empirically derived constants in the CMFs associated with the CPM which would include those in the omitted CMF (exclude those in the SPF for intercept, AADT, and segment length) (refer to the text following this variable list); Δ0 = adjustment factor for the incremental effect of additional empirical coefficients; and all other variables are as previously defined.

34 The number of empirically derived constants p is determined by inspecting the CMFs in the CPM. Regression constants in the SPF (e.g., intercept, AADT coefficient, segment length coefficient) are not considered when determining the value of p. If the CMF is associated with a discrete treatment (e.g., add beacon), then the CMF is often represented by a single empiric constant. If the CMF is associated with a continuous variable (e.g., lane width), then it is often a function that includes one or more empirical constants. In general, there is at least one empirically derived constant for each CMF used to develop the CPM. For practical applications of this procedure, the variable p can be estimated as equal to the number of CMFs used in Equation 44 (i.e., p = n). Step 7. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, compute the error in the predicted crash frequency using the following equation. Equation 48 𝑒 𝑁 𝑁 ,   where e is the error in predicted crash frequency; and all other variables are as previously defined. Second, compute the absolute difference in the change in variance of the predicted value using the following equation. Equation 49 𝜎 𝑘 𝑁 𝑘 , 𝑁 ,   where 𝜎 is the absolute difference of the change in variance of the predicted value; and all other variables are as previously defined. Third, compute the increased root mean square error using the following equation. Equation 50 𝜎 , 𝜎 𝑒 .   where σe,I is the increased root mean square error; and all other variables are as previously defined. Finally, compute the coefficient of variation using the following equation. Equation 51 𝐶𝑉 𝜎 ,𝑁 , where CVI is the coefficient of variation for the increased root mean square error and all other variables are as previously defined. Step 8. Compute the Amount of Bias The percent bias in the predicted crash frequency is computed using the following equation. Equation 52 𝐵𝑖𝑎𝑠 100 𝑁 𝑁 ,𝑁 , where Bias is the percent bias in the reported value; and all other variables are as previously defined. Example Applications The section provides an example application for each of the procedures described in the previous section. Specifically, one example is provided for each of the Case A, Case B, and Case C procedures. The example application associated with each procedure is described in a separate subsection.

35 Case A. CMF from Part D used with SPF (CMF is consistent with SPF base conditions) For this CPM application, a CMF from HSM Part D is used with a jurisdiction-specific SPF. The CMF of interest could also be obtained from another source (e.g., FHWA CMF Clearinghouse); however, this CMF is referred to herein as the “HSM Part D CMF” for consistency with HSM Method 3. For this application, there is a “balance” between the CMF and the base conditions associated with the SPF. The Question An agency desires to conduct a safety evaluation of a rural two-lane two-way road segment. The segment is of interest because it has a relatively narrow lane width. The results of the evaluation will be used to determine the site’s potential for safety improvement. From a prior project, the agency has developed a jurisdiction-specific SPF for rural two-lane two-way road segments. The SPF predicts the frequency of crashes of all types and severities. The typical lane width for rural two-lane two-way road segments in the agency’s jurisdiction is 12 ft. The agency has obtained a CMF from HSM Part D, Chapter 13 for lane width. This CMF is applicable to rural two-lane two-way road segments. The CMF is a function of lane width and segment AADT. The analyst desires to assess the reliability of the predicted crash frequency given that the Lane Width CMF used is obtained from HSM Part D. Outline of Solution Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 9. The data needed to apply the SPF and the HSM Part D CMF are also needed. Table 9. Required Data for Case A Example Application. Required Data and Relationships Potential Data Sources Input values needed for jurisdiction-specific SPF AADT = 10,000 veh/day Segment length = 0.5 mi Nspf, j-s = 1.34 crashes/yr Overdispersion parameter for jurisdiction-specific SPF, kreported kreported = 0.472 HSM Part D CMF for geometric design element or traffic control feature of interest, CMFD HSM Part D, Table 13-2 Average independent variable value at sites used to estimate the SPF, 𝑋 Average lane width, 𝑋 = 12 ft Standard deviation of the independent variable at sites used to estimate NSPF,j-s, σx,b Standard deviation, σx,b = 2.0 ft Average independent variable value associated with CMF of interest at sites of interest, 𝑋 Average lane width, 𝑋= 10 ft Standard deviation of the independent variable at sites of interest, σx,n Standard deviation, σx,n = 0.1 ft A review of agency files indicates that the segment of interest has an AADT of 10,000 veh/day and a segment length of 0.50 miles. The SPF was used (with an AADT of 10,000 veh/day and segment length of 0.5 mi) to compute the predicted crash frequency for a segment with 12-ft lane width. The computed value of NSPF,j-s is 1.34 crashes/yr. As indicated in the previous section, the SPF was developed using sites having a typical lane width of 12 ft. In fact, a review of the SPF development report prepared by the agency indicated that the sites used

36 to estimate the SPF had an average lane width of 12 ft and a standard deviation of 2.0 ft. Thus, the average independent variable value 𝑋 equals 12 ft and the standard deviation of lane width at these sites 𝜎 , equals 2.0 ft. Also, based on this SPF development approach, the base condition lane width is defined as 12 ft. The segment of interest was investigated using measurements from aerial photographs. From this investigation, it was learned that the average lane width at the site 𝑋 is 10 ft. However, measurements along the segment length suggested that the standard deviation 𝜎 , of the segment’s lane width is 0.10 ft. Step 2. Compute Estimation Coefficient. The value of the CMF of interest at the segments used to estimate the SPF (𝐶𝑀𝐹 𝑋 ) is obtained from HSM Part D. For an AADT of 10,000 veh/day, the CMF for an average lane width of 12 ft is 1.00 (i.e., 𝐶𝑀𝐹 12 = 1.00). The value of the CMF of interest for the segment of interest (𝐶𝑀𝐹 𝑋 ) is also obtained from HSM Part D. For the stated AADT, the CMF for an average lane width of 10 ft is 1.30 (i.e., 𝐶𝑀𝐹 10 = 1.30). These two CMF values are used in Equation 19 to compute the estimation coefficient b of -0.131. Step 3. Compute Bias Adjustment Factor. Initially, Equation 21 is used to determine that a correction factor ct value of 0.88 is needed. Then Equation 20 is used to compute the bias adjustment factor fA. The computed value for fA is 0.970. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The SPF value NSPF,j-s and the CMF value CMFD(𝑋) are used in Equation 22 to compute the predicted average crash frequency Np. The computed value of Np is 1.742 crashes/yr. Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The predicted crash frequency Np and bias adjustment factor fA are used in Equation 23 to compute the predicted true crash frequency Np,true. The computed value of Np,true is 1.796 crashes/yr. Step 6. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, the predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 24 to compute the error in the predicted crash frequency e. The computed value of e is -0.054 crashes/yr. Second, the reported overdispersion parameter kreported, predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 25 to compute the absolute difference in the change in variance of the predicted value 𝜎 . The predicted value of 𝜎 is 0.0906. Third, the predicted value of 𝜎 and the predicted error e are used in Equation 26 to estimate the increased root mean square error σe,I. The predicted value of σe,I is 0.306 crashes/yr. If the variance of the predicted value were computed, it should be increased by the square of σe,I. to include the error in the predicted value Np. Finally, the predicted value of σe,I and the predicted true crash frequency Np,true are used in Equation 27 to compute the coefficient of variation for the increased root mean square error CVI. The predicted value of CVI is 0.17. It suggests that there is a relatively large amount of error-related variability in the predicted crash frequency. Step 7. Compute the Amount of Bias The predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 28 to compute the percent bias in the predicted crash frequency (Bias). The predicted value of Bias is -3.0 percent. Case B. CMFs Do Not Have a Corresponding Base Condition in the SPF For this CPM application, one or more CMFs used with a CPM do not have a corresponding base condition in the SPF. These CMFs are called herein “external CMFs” because their variables were not

37 considered when the SPF was developed and its base conditions were established. An example of this application is when a Part D CMF is used with a Part C CPM (and the Part D CMF’s variables are not included in the CPM’s base conditions). The Question An agency desires to conduct a safety evaluation of a four-leg, two-way stop control (TWSC) intersection on a rural two-lane two-way road. The intersection is of interest because it has a relatively high crash frequency which suggests it may have potential for safety improvement. The results of the evaluation will be used to determine the safety benefit of adding flashing beacons at the intersection. The analyst has elected to use the CPM for TWSC intersections in Chapter 10 of the HSM. This CPM predicts the frequency of crashes of all types and severities. The CPM does not include a CMF for “add flashing beacon” at TWSC intersection. Hence, the CPM does not include a base condition that corresponds to this CMF. The agency has obtained a CMF from HSM Part D, Chapter 14 for “Provide Flashing Beacon at Stop Controlled Intersection.” The CMF for “all crash types and severities” is 0.95. The analyst desires to assess the reliability of the predicted crash frequency given that the CMF from HSM Part D does not have a corresponding base condition in the SPF. Outline of Solution Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 10. The data needed to apply the SPF and the HSM Part C CMFs are not listed, but they are also needed. Table 10. Required Data for Case B Example Application. Required Data and Relationships Potential Data Sources Input values needed for CPM AADTmajor = 10,000 veh/day AADTminor = 2,000 veh/day Nspf = 4.97 crashes/yr Overdispersion parameter for CPM, kreported kreported = 0.24 HSM Part C CMF for geometric design element or traffic control feature i, CMFi Product of CMF1 to CMFn = 1.00 External CMF for geometric design element or traffic control feature of interest, CMFex HSM Part D, Chapter 14, CMFex, add beacon = 0.95 Average independent variable value at sites used to estimate the CPM, 𝑋 0.10 Standard deviation of the independent variable at sites used to estimate the CPM, σx,b 0.30 Average independent variable value associated with external CMF at sites of interest, 𝑋 1.00 Standard deviation of the independent variable at sites of interest, σx,n 0.00 A review of agency files indicates that the major road AADT is 10,000 veh/day and the minor-road AADT is 2,000 veh/day. The SPF was used with these AADT values to compute the predicted crash frequency NSPF of 4.97 crashes/yr. The CMFs identified in Chapter 10 were individually considered. Their product is equal to 1.00. A review of the CPM development report prepared by the researchers that developed the CPM did not indicate whether the intersections used to estimate the CPM included flashing beacons. The database used

38 by these researchers was obtained and the location of the intersections identified. A review of Google Earth historical aerial photos for each intersection indicated that a few of the intersections had a flashing beacon during the period for which data were collected to develop the CPM. A “1” was added to the database for each site with a beacon and “0” was recorded for those sites without a beacon. The average of these values 𝑋𝐶𝑃𝑀 was computed as 0.10 and the standard deviation 𝜎 , was computed as 0.30. There is only one intersection of interest for this application. The safety effect of adding a flashing beacon at this intersection is being evaluated. This treatment is discrete so a “1” is used to indicate presence, which makes 𝑋 equal to 1.0. Because only one intersection is of interest and its treatment is specified with certainty, there is no uncertainty with 𝑋 so 𝜎 , is 0.0. Step 2. Compute Estimation Coefficient. The value of the CMF of interest at the intersections used to estimate the CPM (𝐶𝑀𝐹 𝑋 ) is computed as a weighted average for the collective set of sites. It is known that 10 percent of the sites have a beacon. At each of these sites the CMF value is 0.95. The remaining sites have no beacon, so the CMF value is effectively 1.0. The weighted average CMF is computed as 𝐶𝑀𝐹 0.10 is 0.995 (= 0.95 × 0.10 + 1.0 × [1− 0.10]). The value of the CMF of interest for the intersection of interest (𝐶𝑀𝐹 𝑋 ) is obtained from HSM Part D. The CMF for adding the beacon is 0.95 (i.e., 𝐶𝑀𝐹 1 = 0.95). These two CMF values are used in Equation 29 to compute the estimation coefficient b of -0.0514. Step 3. Compute Bias Adjustment Factor. Initially, Equation 31 is used to determine that a correction factor ct value of 0.88 is needed. Then Equation 30 is used to compute the bias adjustment factor fB. The computed value for fB is 1.00. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The SPF value NSPF of 4.97, the CMF product of 1.0, and the external CMF CMFex(𝑋) value of 0.95 are used in Equation 32 to compute the predicted average crash frequency Np. The computed value of Np is 4.72 crashes/yr. Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The predicted crash frequency Np, CMFex(𝑋 ), CMFex(𝑋), and bias adjustment factor fB are used in Equation 33 to compute the predicted true crash frequency Np,true. The computed value of Np,true is 4.94 crashes/yr. Step 6. Compute the Unbiased Overdispersion Parameter for the CPM with the External CMF. The CPM in HSM Chapter 10 for four-leg TWSC intersections has four CMFs, so the variable p equals 5 (= n + 1). The adjustment factor Δ0 is computed using Equation 35. The computed value of Δ0 is 0.10. This factor is then used with kreported, b, and σx,b in Equation 34 to compute kp,true. The computed value of kp,true is 0.240. Step 7. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, the predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 36 to compute the error in the predicted crash frequency e. The computed value of e is -0.224 crashes/yr. Second, the reported overdispersion parameter kreported, predicted true overdispersion parameter kp,true, predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 37 to compute the absolute difference in the change in variance of the predicted value 𝜎 . The predicted value of 𝜎 is 0.518. Third, the predicted value of 𝜎 and the predicted error e are used in Equation 38 to estimate the increased root mean square error σe,I. The predicted value of σe,I is 0.753 crashes/yr. If the variance of the predicted value were computed, it should be increased by the square of σe,I. to include the error in the predicted value Np.

39 Finally, the predicted value of σe,I and the predicted true crash frequency Np,true are used in Equation 39 to compute the coefficient of variation for the increased root mean square error CVI. The predicted value of CVI is 0.15. It suggests that there is a relatively large amount of error-related variability in the predicted crash frequency. Step 8. Compute the Amount of Bias The predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 40 to compute the percent bias in the predicted crash frequency (Bias). The predicted value of Bias is -4.5 percent. Case C. CMF Not Used in CPM But Base Condition Accommodated in the SPF For this CPM application, the analyst chooses not to use one or more of the CMFs that were included in the CPM when it was developed. All of these CMFs (whether used or not used) have a corresponding base condition in the SPF. The unused CMFs are called herein “omitted CMFs.” An example of this application is when the analyst does not have ready access to the data needed for a CMF and, as a result, chooses not to use the CMF when evaluating one or more sites. The Question An agency desires to conduct a safety evaluation of several four-leg, two-way stop control (TWSC) intersections on rural two-lane two-way roads in one region of its jurisdiction. The intersections of interest have a relatively high crash frequency which suggests they may have the potential for safety improvement. The analyst has elected to use the CPM for TWSC intersections in Chapter 10 of the HSM. This CPM predicts the frequency of crashes of all types and severities. The CPM includes a CMF for skew angle at four-leg TWSC intersections. However, data describing the skew angle at the intersections of interest is not readily available, so the analyst is inclined to assume that the skew angle is 0.0 degrees such that the Skew Angle CMF value is 1.0. Before accepting this inclination (which is equivalent to omitting the Skew Angle CMF from the analysis), the analyst assesses its impact on the reliability of the predicted crash frequency. Outline of Solution Step 1. Assemble the data needed to apply the procedure. The data needed to apply the procedure are listed in Table 11. The data needed to apply the SPF and the HSM Part C CMFs are not listed, but they are also needed.    

40 Table 11. Required Data for Case C Example Application. Required Data and Relationships Potential Data Sources Input values needed for CPM AADTmajor = 10,000 veh/day AADTminor = 2,000 veh/day Nspf = 4.97 crashes/yr Overdispersion parameter for CPM, kreported kreported = 0.24 HSM Part C CMF for geometric design element or traffic control feature i, CMFi Product of CMF1 to CMFn = 1.00 Omitted CMF for geometric design element or traffic control feature of interest, CMFex HSM Part C Average independent variable value at sites used to estimate the CPM, 𝑋 10 degrees Standard deviation of the independent variable at sites used to estimate the CPM, σx,b 15 degrees Average independent variable value associated with omitted CMF at sites of interest, 𝑋 3.0 degrees Standard deviation of the independent variable at sites of interest, σx,n 10 degrees A review of agency files indicates that the AADT varies among the intersections of interest. However, the coefficient of variation and bias percent computed in Steps 7 and 8, respectively, are derived to have the characteristic that they are insensitive to the predicted crash frequency (NP). As a result, the predicted coefficient of variation and bias percent reasonably describe the reliability of the predicted crash frequency for each of the intersections of interest, regardless of the intersection AADT. For purposes of the reliability evaluation, typical AADT values can be used for the calculations. In this regard, the typical intersection major road AADT is 10,000 veh/day and the typical minor-road AADT is 2,000 veh/day. The SPF was used with these AADTs to compute the predicted crash frequency NSPF of 4.97 crashes/yr. The CMFs identified in Chapter 10 were individually considered. The Skew Angle CMF was not included in this consideration. As noted in the previous paragraph, the key reliability measures are insensitive to the predicted crash frequency. Thus, as a simplifying assumption, the product of the considered CMFs was set to equal to 1.00 for the intersections of interest. A review of the CPM development report prepared by the researchers that developed the CPM did not include the skew angle at the intersections used to estimate the CPM. The database used by these researchers was obtained and skew angle for a sample of the intersections in the database was measured using Google Earth aerial imagery. The average of the measured skew angle values 𝑋𝐶𝑃𝑀 was computed as 10 degrees and the standard deviation 𝜎 , was computed as 15 degrees. A similar sampling process was used to measure the skew angle at some of the intersections of interest. The average of the measured skew angle values 𝑋 was computed as 3.0 degrees and the standard deviation 𝜎 , was computed as 10 degrees. Step 2. Compute Estimation Coefficient. The value of the CMF of interest at the intersections used to estimate the CPM (𝐶𝑀𝐹 𝑋 ) is computed using the Skew Angle CMF function provided in HSM Chapter 10. For an average angle of 10 degrees, the CMF is computed as 1.055 (i.e., 𝐶𝑀𝐹 10 = 1.055). The value of the CMF of interest for the intersection of interest (𝐶𝑀𝐹 𝑋 ) is computed using the same CMF function. For an average angle of 3 degrees, the CMF is computed as 1.016 (i.e., 𝐶𝑀𝐹 3 = 1.016). These two CMF values are used in Equation 41 to compute the estimation coefficient b of 0.0054.

41 Step 3. Compute Bias Adjustment Factor. Initially, Equation 43 is used to determine that a correction factor ct value of 1.12 is needed. Then Equation 42 is used to compute the bias adjustment factor fC. The computed value for fC is 1.002. Step 4. Compute the Predicted Crash Frequency for Site of Interest. The SPF value NSPF of 4.97 and the CMF product of 1.0 are used in Equation 44 to compute the predicted average crash frequency Np. The computed value of Np is 4.97 crashes/yr. Step 5. Compute the Unbiased Predicted Crash Frequency for Site of Interest. The predicted crash frequency Np, CMFom(𝑋 ), CMFom(𝑋), and bias adjustment factor fC are used in Equation 45 to compute the predicted true crash frequency Np,true. The computed value of Np,true is 4.79 crashes/yr. Step 6. Compute the Unbiased Overdispersion Parameter for the CPM with the External CMF. The CPM in HSM Chapter 10 for four-leg TWSC intersections has four CMFs, so the variable p equals 4 (= n). The adjustment factor Δ0 is computed using Equation 47. The computed value of Δ0 is 0.30. This factor is then used with kreported, b, and σx,b in Equation 46 to compute kp,true. The computed value of kp,true is 0.242. Step 7. Compute the Increased Root Mean Square Error and Coefficient of Variation. First, the predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 48 to compute the error in the predicted crash frequency e. The computed value of e is 0.176 crashes/yr. Second, the reported overdispersion parameter kreported, predicted true overdispersion parameter kp,true, predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 49 to compute the absolute difference in the change in variance of the predicted value 𝜎 . The predicted value of 𝜎 is 0.361. Third, the predicted value of 𝜎 and the predicted error e are used in Equation 50 to estimate the increased root mean square error σe,I. The predicted value of σe,I is 0.626 crashes/yr. If the variance of the predicted value were computed, it should be increased by the square of σe,I. to include the error in the predicted value Np. Finally, the predicted value of σe,I and the predicted true crash frequency Np,true are used in Equation 51 to compute the coefficient of variation for the increased root mean square error CVI. The predicted value of CVI is 0.13. It suggests that there is a relatively large amount of error-related variability in the predicted crash frequency. Step 8. Compute the Amount of Bias The predicted crash frequency Np and the predicted true crash frequency Np,true are used in Equation 52 to compute the percent bias in the predicted crash frequency (Bias). The predicted value of Bias is 3.7 percent.

Next: Chapter 4. Procedures for Quantifying the Reliability of Crash Prediction Model Estimates with a Focus on Error in Estimated Input Values »
Understanding and Communicating Reliability of Crash Prediction Models Get This Book
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Understanding and communicating consistently reliable crash prediction results are critical to credible analysis and to overcome barriers for some transportation agencies or professionals utilizing these models.

The TRB National Cooperative Highway Research Program's NCHRP Web-Only Document 303: Understanding and Communicating Reliability of Crash Prediction Models provides guidance on being able to assess and understand the reliability of Crash Prediction Models.

This document is supplemental to NCHRP Research Report 983: Reliability of Crash Prediction Models: A Guide for Quantifying and Improving the Reliability of Model Results.

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