3. RECALIBRATION (Continuation)

3.3 AADT MODEL ESTIMATION RESULTS

This section discusses the development of AADT-only models. Two types of models are presented for Types I and II sites, and three types of models are given for Type III, IV, and V sites in this report. First, models were calibrated using all available data from the HSIS California database, the original sites from Minnesota and Michigan, and the Georgia validation data. This group of data is referred to as Group B. Second, Types I to IV models were developed for a subset of these sites that met specified conditions for possible use as base models in the IHSDM accident prediction algorithm. For Types III, IV, V sites, additional AADT models were calibrated from a dataset that met the base conditions of the significant variables in the full models. These AADT-only models used the data for the original sites from California and Michigan along with the Georgia validation data. This group of data is referred to as Group A. Table 124 shows the summary of the data used for AADT-only models.

Table 124. Summary of AADT Models Recalibrated and Data Used

¹ The California project data for Types III, IV, and V were not used

Models were developed for total accidents (TOTACC) and injury (fatal + nonfatal injury) accidents (INJACC) using accidents within 76.25 m (250 ft) of the intersection center.

3.3.1 Description of Base Conditions

The accident prediction algorithm outlined in the "Red Book" provides AMFs for four variables: intersection skew angle; left-turn lanes on major approach; right-turn lanes on major approach; and the number of intersection quadrants with inadequate sight distance.⁽³⁾ It was sought to develop AADT-only base models using sites which met the base condition criteria for these AMFs and for any other variables that logically and practically could provide additional AMFs.

Base conditions were defined by examining the distribution of variables and selecting the most common condition, keeping in mind that enough sites must remain to calibrate reliable models. Whether a variable exhibited an impact on safety was also considered in defining base conditions.

Group A and B Data

For type I and II intersections, the base models were developed using all of the Minnesota, Georgia, and California HSIS data, the combination of which is referred to as Group B. For Type III and IV intersections, base models were developed from two datasets. The first included the Michigan, original California, and Georgia sites (Group A). The second included the Michigan, Georgia, and California HSIS sites (Group B).

The Group B dataset has more sample sites but fewer variables than the Group A dataset for Type I to IV sites. Therefore, while the Group B dataset is more useful for the AADT model development, the Group A datasets benefit the full model development because of the large number of variables. Type III and IV considered both Group A and B sites for the full model and AADT model development. Type V base models were calibrated only for the Group A sites because only five sites could have been added from the Group B California HSIS data.

Group B Base Conditions

Intersection skew angle was not included as a base condition because the California HSIS data does not contain this variable and the skew angle at those sites where it is known is in fact highly variable. Selecting a skew angle of zero as a base condition would have left few sites for calibrating a base condition model.

For the Minnesota sites and the California HSIS sites, sight distance information is not available, and was therefore not included as a base condition. However, it is reasonable to believe that the majority of sites have adequate sight distance since roads are constructed to design standards and exceptions are made only where necessary. Thus, the base models developed could be applied assuming they represent sites with adequate sight distance, and that inadequate sight distance could be taken into account with the use of an AMF. In the event that many of the sites did, in fact, have deficient site distance, the base models calibrated with a contrary assumption would produce artificially high predictions if crashes actually increased with deficient sight distance.

The Group B nominal or base conditions and the number of sites meeting these conditions for model types I to IV are presented in table 125. For Types I and II, approximately 65 percent of all the sites met all of the specified base conditions, while for Types III and IV, the percentage of sites that met all of the base conditions was approximately 20 percent and 15 percent, respectively.

For Types I and II, no turning lanes on the major or minor road and no medians on the major road were selected as base conditions.

Table 125. Group B Base Conditions for Type I to IV AADT Models

¹ N/A: not available

Unlike Types I and II, terrain on major road was included as a base condition for Types III and IV because it showed a significant impact on safety. Also for Types III and IV, and unlike the cases for Types I and II, the base conditions included the presence of a left-turn lane on the major road and the presence of a median on the major road.

Group A Base Conditions

The Group A nominal or base conditions and the number of sites meeting these conditions for model Types III to V are presented in tables 126 to 131. Separate base conditions were defined for TOTACC and INJACC models. The percentage of sites meeting all of the base conditions ranged from approximately 26 percent to 54 percent.

Type III

For total accidents, vertical curves on the major road, commercial driveways on the major road, and intersection angle were selected as significant variables from the full models.

Table 126 shows the base conditions for these variables. For vertical curves, only a few sites met the "no vertical curve" condition and, therefore, a VEI less than 1 degree per 30.5 m (100 ft) (which is relatively flat) was used for the base condition. Similarly, an intersection angle between plus or minus 5 degrees was defined as the base condition representing "no skew." No commercial driveways within 76.25 m (250 ft) of the intersection center is the final base condition.

Table 126. Group A Base Conditions for Type III TOTACC AADT Models

For injury accidents, hazard rating on the major road, commercial driveways on the major road, and intersection angle were selected as significant variables from the full models. Table 127 shows base conditions for these variables. For hazard rating, values of 1 and 2 were taken as the base condition. An intersection angle between plus and minus 5 degrees and no commercial driveways within 76.25 m (250 ft) of the intersection center were the other base conditions.

Table 127. Group A Base Conditions for Type III INJACC AADT Models

Type IV

For total accidents, "right" sight distance from minor road and median type on major road were selected as significant variables from the full models. Table 128 shows the base conditions for these variables. Adequate sight distance and no median are the base conditions. The same rule was applied to judge adequate sight distance as the rule described on page 48 of the Harwood et al. report.⁽³⁾

Table 128. Group A Base Conditions for Type IV TOTACC AADT Models

For injury accidents, right sight distance from minor road, median type on major roads, and posted speed limit on minor road were selected as significant variables from the full models. Table 129 shows the base conditions for these variables. Adequate sight distance, no median type, and speed limit between 48 and 56 kilometers per hour (km/h) (30 and 35 miles per hour (mi/h)) are the base conditions. As for total accidents, the same rule was applied to estimate adequate sight distance as was described on page 48 of the Harwood et al. report.⁽³⁾ For posted speed limit, not enough sites with a single posted speed limit could represent a base condition. Therefore, a range with the highest frequency (48 to 56 km/h (30 to 35 mi/h)) was considered as the base condition.

Table 129. Group A Base Conditions for Type IV INJACC AADT Models

Type V

Total accidents HEICOM for major and minor roads, median type on major roads, and posted speed limit on major roads were estimated as significant in the full models and used for base conditions. Table 130 shows the base conditions for these variables. For HEICOM, radium larger than 458 m (1500 ft) was considered as the base condition. For posted speed limit, the speed range between 72-88 km/h (45-55 mi/h) was used as the base condition since this range had the highest frequency.

Table 130. Group A Base Conditions for Type V TOTACC AADT Models

For injury accidents, posted speed limit on major road and presence of lighting at intersection were selected as significant variables from the full models. Table 131 shows the base conditions for these variables. Presence of lighting and speed range between 72 and 89 km/h (45 and 55 mi/h) were considered as the base conditions.

Table 131. Group A Base Conditions for Type V INJACC AADT Models

3.3.2 Model Results

The AADT-only modeling results are discussed next. In the tables below, the data for and models calibrated using all available sites are referred to in the table headings as main models.

Type I Models

The datasets used to develop Type I AADT-only models included the Minnesota sites from the original study, with the additional years of accident data, the Georgia sites, and sites extracted from the California HSIS database. Sites with very low volume typically are not representative of sites in the general population. These low volume counts, sometimes as low as one vehicle per day, appear to be of suspect quality. With a large number of available sites, intersections with a major road AADT below 400 or a minor road AADT below 100 were removed from the data. The total number of sites and the number of Group B base condition sites are given in table 132. Summary statistics on these datasets are presented in table 133.

Table 132. Number of Sites Used for Type I Main and Group B Base AADT Models

Table 133. Summary Statistics for Type I Sites: Main and Group B Base AADT Models

Tables 134 and 135 report the parameter estimates for the Type I TOTACC and INJACC models calibrated using all sites as well as for those sites meeting the Group B base conditions. A unique constant term was estimated for each location to account for the differences in accident reporting and other characteristics between jurisdictions. Separate values of K, the overdispersion parameter, have been estimated for each location using a specially written maximum likelihood program.

Vogt and Bared did not report AADT-only models for Type I intersections, but AADT-only base models subsequently were derived by Harwood et al. by setting default values of two for HAZRAT1 and no right-turn lane on the major road in a model for predicting total intersection related accidents.^(1,3) A comparison between this model applied to the recalibration data and the recalibrated model for TOTACC is given. A similar model for INJACC from the original calibration is not available for comparison. The comparison indicates the recalibrated model for total accidents is improved as measured by the GOF using the K value.

The CURE method described earlier was used to suggest any alternate model forms that could provide an improved fit to the data. The results indicated that the recalibrated model using the original exponential model form adequately fit the data and that the various adjustments to this model form did not improve the AADT models. Specifically, the Pearson product-moment correlation coefficient, MPB per year, and MAD per year were negligibly different. Therefore it was decided to retain the original model form. Appendix D shows the CURE plot for the Type I AADT TOTACC model.

Tables 134 and 135 indicate that the coefficient of the log of major road AADT is about two times that for minor road AADT, which seems to be a reasonable expectation on the basis of other models reported in the literature. For the TOTACC model, the base conditions model was estimated with a lower overall K than the model using all sites. This would be expected, because the base condition sites should be more homogeneous in their design characteristics. For the INJACC model, the overall K was the same for the two AADT models.

Table 134. Parameter Estimates for Recalibrated TOTACC Type I: Main and Group B AADT Models

¹ The base model published in Harwood et al., 2000, (p. 21)

² N/A: not available

Table 135. Parameter Estimates for Recalibrated INJACC Type I: Main and Group B AADT Models¹

¹No previously calibrated AADT-only model for injury accidents exists for comparison for Type I sites

Type II Models

The datasets used to develop the Type II main and Group B base models included the Minnesota sites from the original study, with the additional years of accident data, the Georgia sites, and sites extracted from the California HSIS database. As for Type I, the large number of sites available allowed for the removal of those sites with a major road AADT less than 400 or a minor road AADT less than 100, the rationale again being that the omitted sites either have errors in AADTs or are unrepresentative of those in the general population. The total number of sites and the number of Group B base condition sites are given in table 136. Summary statistics on these datasets are presented in table 137.

Table 136. Number of Sites Used for Type II Main and Group B Base AADT Models

Table 137. Summary Statistics for Type II Sites: Main and Group B Base AADT Models

Tables 138 and 139 report the parameter estimates for the Type II TOTACC and INJACC models, which were calibrated using all sites and those meeting the base conditions. A unique constant term was estimated for each location to account for the differences in accident reporting and other characteristics between jurisdictions. Separate values of K, the overdispersion parameter, have been estimated for each location using the specially developed maximum likelihood program.

Vogt and Bared did not report AADT-only models for Type II intersections.⁽¹⁾ However, AADT-only base models were subsequently derived by setting default values of "no intersection skew angle" and "no driveways within 76.25 m (250 ft) of the intersection" on the major road for a model for predicting total intersection related accidents.⁽³⁾ A comparison between this model applied to the recalibration data and the recalibrated model for TOTACC is given. A similar model for INJACC for the original calibration is not available for comparison. The comparison indicates the recalibrated model for total accidents has an inferior GOF compared to the Vogt and Bared models using the K value for all States, both combined and individually. In particular, the model does a poorer job for the Georgia and California sites.

The CURE method was again used to suggest any alternate model forms that could provide an improved fit the data. The results indicated that the recalibrated model using the original exponential model form adequately fit the data, and the various adjustments to this model form did not improve the AADT models. Specifically, there was a negligible difference in Pearson product-moment correlation coefficient, MPB per year, and MAD per year. Therefore it was decided to retain the original model form. Appendix D shows the CURE plot for the Type II AADT TOTACC model.

Table 138 indicates that the coefficient of the log of major road AADT is about 35 percent higher than that for minor road AADT value. Table 139 indicates that the coefficient of the log of major road AADT is about 10 to 20 percent higher than that for minor road AADT, which is consistent with previous modeling efforts. For both the TOTACC and INJACC models, the base condition models were estimated with a lower overall K than the model using all sites. Again, this would be expected, because the base condition sites should be more homogeneous in their design characteristics.

Table 138. Parameter Estimates for Recalibrated TOTACC Type II: Main and Group B AADT Models

¹ The base model published in Harwood et al., 2000 (p. 22)

² N/A: not available

Table 139. Parameter Estimates for Recalibrated INJACC Type II: Main and Group B AADT Models ¹

¹ No previously calibrated AADT-only model for injury accidents exists for comparison for type II sites

Type III Models

The dataset used to develop the main Type III model included the Michigan sites from the original study, with the additional years of accident data, the Georgia sites, and those extracted from the California HSIS database. Unlike the case for Types I and II models, the researchers did not have the luxury of removing sites with a minor road AADT less than 100, because as many as 87 out of the 294 available sites had an AADT on the minor road of less than 100.

Base models were calibrated using both the Group A and Group B datasets. The total number of sites and the number of Group B base condition sites are given in table 140. Summary statistics on these datasets are presented in table 141.

Table 140. Number of Sites Used for Type III Main and Group B Base Condition AADT Models

Table 141. Summary Statistics for Type III Sites: Main and Group B Base AADT Models

Tables 142 and 143 report parameter estimates for the main Type III TOTACC and INJACC models, calibrated using all sites and as well as those meeting the Group B base conditions. Unlike models for Type I and II intersections, a unique constant term was not included for each State, because the State indicator variables were insignificant.

For reference, comparisons between the Vogt AADT models applied to the recalibration data and the recalibrated models for TOTACC and INJACC are also given in tables 142, 143, and 145. The comparisons indicate the recalibrated models for TOTACC and INJACC have a better GOF measures compared to the Vogt models.

The CURE plot method proposed by Hauer was also used to suggest any alternate model forms that could provide an improved fit for the data. The results indicated that the recalibrated models using the original exponential model form adequately fit the data and that the various adjustments to this model form did not improve the AADT models. Specifically, the differences in Pearson product-moment correlation coefficient, MPB per year, and MAD per year were negligible. Therefore the original model form was retained. Appendix D shows the CURE plot for the Type III AADT TOTACC model.

Table 142 indicates that the coefficient of the log of major road AADT is about two to three times that for minor road AADT, a reasonable expectation. Table 143 shows that coefficient of the log of major road AADT is about three to four times that for minor road AADT, which is also consistent with previous efforts. Pearson product-moment correlation coefficients for TOTACC and INJACC models were approximately 0.65.

Table 142. Parameter Estimates for Recalibrated TOTACC Type III: Main and Group B Base AADT Models

¹ The AADT model published in Vogt, 1999, (p. 111)

Table 143. Parameter Estimates for Recalibrated INJACC Type III: Main and Group B Base AADT Models

¹ The AADT model published in Vogt, 1999, (p.113)

Summary statistics on the Group A base condition datasets are presented in table 144.

Table 144. Summary Statistics for Type III Sites: Group A Base AADT Models

Table 145 reports the parameter estimates for the Group A Type III TOTACC and INJACC base models. The coefficient of the log of major road AADT is about three to four times that for minor road AADT, which is reasonable. The INJACC models differed from the TOTACC models in the use of LOG (AADT1 * AADT2) rather than the individual logs of the major and minor road AADTs. For INJACC, the coefficient of the log of AADT2 was quite insignificant because of large standard error. Pearson product-moment correlation coefficients of TOTACC and INJACC AADT models were 0.65 and 0.47, respectively.

Table 145. Parameter Estimates for Type III Group A TOTACC and INJACC Base Models

¹ The TOTACC AADT model published in Vogt, 1999, (p. 111)
² The INJACC AADT model published in Vogt, 1999, (p. 113)
³ N/A: not available

Type IV Models

The datasets used to develop the main Type IV model included the Michigan sites from the original study, with the additional years of accident data, the Georgia sites, and those extracted from the California HSIS database. Again, a large number of sites (52 out of 222) had a minor road AADT of less than 100, so, unlike the case for Types I and II, the researchers could not remove sites with a minor road AADT less than 100.

Base models were calibrated using both the Group A and Group B datasets. The total number of sites and the number of Group B base condition sites are given in table 146. Summary statistics on these datasets are presented in table 147.

Table 146. Number of Sites Used for Type IV Main and Group B Base Condition AADT Models

Table 147. Summary Statistics for Type IV Sites: Main and Group B Base AADT Models

Tables 148 and 149 report the parameter estimates for the main Type IV TOTACC and INJACC models, calibrated using all sites and those sites meeting the Group B base conditions. Unlike models for Type I and II intersections, a State indicator variable was insignificant. For reference, the estimated overdispersion parameter for the original full models is also given.

The CURE method was again used to suggest any alternate model forms that could provide an improved fit the data. The results indicated that the various adjustments to this model form did not improve the AADT models. Specifically, the Pearson product-moment correlation coefficient, MPB per year, and MAD per year were negligibly different. Therefore, the original model form was retrained. Appendix D shows the CURE plot for the Type IV AADT TOTACC model.

Table 148 indicates that, as expected, the coefficient of the log of major road AADT is significantly larger than that for minor road AADT. Table 149 shows that, again, the coefficient of the log of major road AADT for the INJACC model is also significantly larger than that for minor road AADT. Pearson product-moment correlation coefficients for TOTACC and INJACC models were 0.75 and 0.63 using all sites, and 0.90 and 0.89 for the base sites.

Comparisons between the Vogt AADT models applied to the recalibration data and the recalibrated models for TOTACC and INJACC are also given in tables 148, 149, and 151. The comparisons indicate the recalibrated models for TOTACC and INJACC have better GOF measures compared to the Vogt models.

Table 148. Parameter Estimates for Recalibrated TOTACC Type IV: Main and Group B Base AADT Models

¹ The AADT model published in Vogt, 1999, (p. 116)

Table 149. Parameter Estimates for Recalibrated INJACC Type IV: Main and Group B Base AADT Models

¹ The AADT model published in the Vogt's report, 1999, (p. 118)

Summary statistics on the Group A base condition datasets are presented in table 150.

Table 150. Summary Statistics for Type IV Sites: Group A Base AADT Models

Table 151 reports the parameter estimates for the Group A Type IV TOTACC and INJACC base models. This table indicates that the coefficient of the log of major road AADT is about one and one half times that for minor road AADT, conforming to expectations. As was the case for Type III, the INJACC model used LOG (AADT1 * AADT2) because log of AADT2 was quite insignificant. However, even with LOG (AADT1 * AADT2), the p-value was not statistically significant. It appears that the sample size of 38, with a relatively low number of injury accidents, was not sufficient to provide reasonable coefficients and p-values for the INJACC model.

Table 151. Parameter Estimates for Type IV Group A TOTACC and INJACC Base Models

¹ The TOTACC AADT model published in the Vogt's report, 1999, (p. 116)
² The INJACC AADT model published in the Vogt's report, 1999, (p. 118)
³ N/A: not available

Type V Models

The datasets used to develop the Type V main models included the Michigan sites from the original study, the additional years of accident data, the Georgia sites, and those extracted from the California HSIS database. Only five sites were available from the California HSIS database, so it was decided not to calibrate Group B base models that would have included these data. Only Group A base models, using the Georgia data and the original sites for Michigan and California, were calibrated.

The numbers of sites comprising the dataset used for the main model are given in table 152. Summary statistics are presented in table 153. As seen, for Michigan, the INJACC model used only the additional years of accident data, because injury data for the original years were not available.

Table 152. Number of Sites Used for Type V Main AADT Model

Table 153. Summary Statistics for Type V Sites: Main AADT Model

Tables 154 and 155 report the parameter estimates for the main Type V TOTACC and INJACC models calibrated using all sites. Unlike models for Type I and II intersections, a State indicator variable was insignificant. Both tables show that the coefficient of the log of major road AADT is, in accord with expectations, about two to three times that for minor road AADT. Pearson product-moment correlation coefficients for TOTACC and INJACC were 0.66 and 0.44, respectively.

Unlike the case for Types III and IV, the Vogt report does not provide AADT-only models for Type V. Therefore, a comparison between the Vogt models and the newly calibrated AADT models for TOTACC and INJACC could not be done.

The CURE plot method proposed by Hauer was used to explore alternate model forms that could provide an improved fit the data. The results indicated that the recalibrated models using the original exponential model form adequately fit the data and that the various adjustments to this model form did not improve the AADT models. Specifically, the differences in Pearson product-moment correlation coefficient, MPB per year, and MAD per year were negligible. Therefore, the original model form was retained. Appendix D shows the CURE plot for the Type V AADT TOTACC model.

Table 154. Parameter Estimates for Recalibrated TOTACC Type V: Main AADT Model

* There are no Type V AADT models for TOTACC in the Vogt report, 1999

Table 155. Parameter Estimates for Recalibrated INJACC Type V: Main AADT Model

* There are no Type V AADT models for INJACC in the Vogt report, 1999

Summary statistics on the Group A base condition datasets are presented in table 156.

Table 156. Summary Statistics for Type V Sites: Group A Base AADT Models

Table 157 reports the parameter estimates for the Group A Type V TOTACC and INJACC base models. This table indicates that the coefficient of the log of major road AADT is about two to three times greater than that for minor road AADT, which is reasonable. As was the case for Type III and IV models, the INJACC model used LOG (AADT1 * AADT2) because the individual logs for separate AADT terms were insignificant. Pearson product-moment correlation coefficients of TOTACC and INJACC AADT models were 0.70 and 0.57, respectively.

Table 157. Parameter Estimates for Type V Group A TOTACC and INJACC Base Models

* No AADT models previously calibrated

¹ N/A: not available

3.4 FULLY PARAMETERIZED STATISTICAL MODEL ESTIMATION RESULTS

This section discusses the development of the fully parameterized statistical models. Unlike the AADT models, these models include many variables with the intent to explain as much of variation in crash occurrence as possible, given the available set of potential explanatory variables. For Types I and II, models were developed using two different datasets. Types III, IV, and V used only one dataset. Table 158 summarizes what data was used for each model.

Table 158. Summary of Models Recalibrated and Data Used

^* The California project data for Types II, IV, and V were not used

As before, models for total accidents (TOTACC) and injury (fatal + nonfatal injury) accidents (INJACC) within 76.25 m (250 ft) of the intersection were developed.

3.4.1 Type I Models

Full models were developed using two groups of data. As indicated earlier, the first, Group A, was comprised of the sites from Minnesota and Georgia and consisted of many variables, including horizontal and vertical curvature. The California HSIS sites were not in this group because many of the variables were not available. The second, Group B, included the California HSIS sites, but fewer variables were available for modeling.

As was the case for the AADT-only models, sites with a major road AADT below 400 or a minor road AADT below 100 were not used. 270 sites from Minnesota and 116 sites from Georgia were used. The Group B data set contained an additional 1432 sites from the California HSIS database. Summary statistics on these datasets by State are available in appendix C.

Two model variants are reported. The first includes a State indicator term and the second does not. For reference the recalibrated models are compared to the models calibrated using data from Minnesota and recommended by Vogt and Bared.⁽¹⁾

Group A

Tables 159 and 160 report the parameter estimates for the Type I TOTACC and INJACC models for the Group A data.

Table 159. Parameter Estimates for Recalibrated TOTACC Full Models: Type I Group A

¹ Vogt and Bared, 1998, (p. 115)
² N/A: not available

Table 160. Parameter Estimates for Recalibrated INJACC Full Models: Type I Group A

¹ Vogt and Bared, 1998, (p. 116)
² N/A: not available

There were similarities and differences between the recalibrated models and the Vogt and Bared models. For total accidents, posted speed on major roads and the angle variable HAU were not included in the recalibrated model, while right-turn lane on minor roads and left-turn lanes on major roads were included. For injury accidents, right-turn on major roads, posted speed on major roads, number of driveways on major roads, and the angle variable HAU were not included, while left-turn lane on major roads were included.

When a State indicator variable was used in the models in addition to the logs of AADTs, only the variable HI1 proved to be significant. Without the State indicator, more geometric variables were statistically significant. The variables RT MAJ, RT MIN, LT MAJ, HAZRAT1, HI1, and VCI1 were all estimated to be significant at the 10 percent level or better in one or both of the TOTACC and INJACC models. However, the overdispersion parameter K is higher for the models without the State indicator variable and with more variables.

The data summary in appendix C shows that the Georgia sites experience, on average, more accidents than sites in Minnesota and also are higher in values for HI1, VCI1, and HAZRAT1, while being less likely to have a turning lane on the major road. The question arises then, whether the significantly higher accident risk in Georgia is due to differences in these geometric features or other fundamental reasons such as reporting levels, weather, and sociodemographics. Because the models, including the State location term, have lower overdispersion parameters, it can be concluded that this model should be used. The following analysis further supports this conclusion.

Figure 21 plots the cumulative residuals (Y-axis) versus major AADT (X-axis) for the TOTACC model without the State location variable. The cumulative residuals on occasion go outside the 95 percent confidence intervals for a random walk around 0, indicating that the model is performing poorly.

Figure 21. CURE Plot for Alternate TOTACC Type I Group A Model

Figure 22 plots the cumulative residuals (Y-axis) versus major AADT (X-axis) for the TOTACC model, including the State location variable. Although the cumulative residuals go outside the 95 percent plots for few sites at low volumes and AADTs above 15000, the results are a slight improvement over the model without the State location variable in that the oscillations tend to be closer to the x-axis.

Figure 22. CURE Plot for TOTACC Type I Group A Model

The GOF as measured by K was improved over the Vogt and Bared models, although the measures of MPB per year and MAD per year were better for the original model with the exception of the MPB per year for the TOTACC and INJACC Variant 1 models.

Group B

Tables 161 and 162 report the parameter estimates for the Type I TOTACC and INJACC models for the Group B data.

Table 161. Parameter Estimates for Recalibrated TOTACC Full Models: Type I Group B

¹ Vogt and Bared, 1998, (p. 115)
² These variables area not available in the Group B data
³ N/A: not available

Table 162. Parameter Estimates for Recalibrated INJACC Full Models: Type I Group B

¹ Vogt and Bared, 1998, (p. 116)
² These variables area not available in the Group B data
³ N/A: not available

The GOF as measured by K was improved over the Vogt and Bared models. In the recalibrated models, right-turn lanes on minor roads and left-turn lanes on major roads are significant in addition to right-turn lanes on major roads.

3.4.2 Type II Models

As for Type I, full models were developed using two groups of data. The first, Group A, consisted only of the sites from Minnesota and Georgia and included many variables, such as horizontal and vertical curvature. The California HSIS sites were not included in this group because many of the variables were not available. The second, Group B, included the California HSIS sites, but fewer variables were available for modeling.

As was the case for the AADT-only models, sites with a major road AADT below 400 or a minor road AADT below 100 were not included. 250 sites from Minnesota and 108 sites from Georgia were used. The Group B data contained an additional 748 sites from the California HSIS database. Summary statistics on these datasets are available in chapter 3.2, and listed by State in appendix C.

Two variants of models were calibrated. As for Type I, the first variant included a State indicator term and the second did not. For reference, the recalibrated models are compared to the models calibrated by Vogt and Bared.⁽¹⁾

Group A

Tables 163 and 164 report the parameter estimates for the Type II TOTACC and INJACC models for the Group A data.

Table 163. Parameter Estimates for Recalibrated TOTACC Full Models: Type II Group A

¹ Vogt and Bared, 1998, (p. 115)
² N/A: not available

Table 164. Parameter Estimates for Recalibrated INJACC Full Models: Type II Group A

¹ Vogt and Bared, 1998, (p. 117)
² N/A: not available

There were some similarities and differences between the recalibrated models and the Vogt and Bared models. For the recalibrated models, significant variables at approximately the 85 percent level or better for TOTACC included major and minor road AADTs, right-turn lanes on major roads, and the number of driveways for the variant that included a State indicator variable, and number of driveways and the vertical curvature variable VCI1 for the variant without the State indicator variable. The Vogt and Bared model also included the angle variable HAU, the major road posted speed, and the horizontal curvature variable HI1, although the last two were of low significance. For the INJACC models, number of driveways and horizontal curvature within 76.25 m (250 ft) of the intersection center were significant at the 10 percent level or better for both the State indicator and non-State indicator variants. The Vogt model included roadside hazard rating as a significant variable and others that were not significant. The GOF as measured by K was not as good as for the Vogt and Bared model.

Group B

Tables 165 and 166 report the parameter estimates for the Type II TOTACC and INJACC models using the Group B data.

Table 165. Parameter Estimates for Recalibrated TOTACC Full Models: Type II Group B

¹ Vogt and Bared, 1998, (p. 115)
² These variables area not available in the Group B data
³ N/A: not available

Table 166. Parameter Estimates for Recalibrated INJACC Full Models: Type II Group B

¹ Vogt and Bared, 1998, (p. 117)
² These variables area not available in the Group B data
³ N/A: not available

For TOTACC, significant variables in addition to major and minor road AADTs, include right-turn lane on major road for the variant with the State indicator variable and right-turn lane on major road and left-turn lane on major road for the variant without the State indicator variable. For INJACC, median and right-turn lane on major road were significant for both the State indicator and non-State indicator variants. The GOF as measured by K was not as good as that for the Vogt and Bared model.

3.4.3 Type III Models

The data used to calibrate full models consisted of the California and Michigan sites from the original study, additional years of accident data from the California and Michigan sites, and the Georgia sites. Some of the California sites experienced changes in their design features during the 1996-98 period. For such sites, accidents during the 1996-98 period were not included, and only 1993-95 accident data were used. Summary statistics on the data are provided in section 3.2.

Two models each are reported for TOTACC and INJACC. The main model was selected based on the highest Pearson product-moment correlation coefficient, lowest overdispersion, MPB per year, and MAD per year. The other model was the one judged to be next best on in terms of these measures.

For TOTACC, major and minor AADTs, vertical curve rate on major roads, intersection angle, commercial driveways on major roads, median width on major roads, and painted medians on major roads were found to be significant in the main models (15 percent level or better). State indicator variables were statistically insignificant and therefore not included in the models.

For INJACC, major and minor AADTs, roadside hazard rating on major road, intersection angle, commercial driveways on major road, peak turning percentage, and peak truck percentage were found to be significant in the main model at the 15 percent level or better. State indicator variables were again statistically insignificant.

For reference a comparison between Vogt's full models applied to the recalibration data and the newly recalibrated models for TOTACC and INJACC also is given. As expected, there were some differences between the recalibrated models and the Vogt models. For TOTACC, the recalibrated main model had the additional terms COMDRWY1, VEI1, HAU, and MEDTYPE1 compared to the Vogt model but did not include DRWY1, which was in the Vogt model. For the INJACC, the recalibrated main model had the additional terms COMDRWY1, HAZRAT1, PKTRUCK, and PKTURN compared to the Vogt model. The GOF as measured by Pearson product-moment correlation coefficients, MPB per year, and MAD per hear indicates that the recalibrated main models for TOTACC and INJACC provide better GOF than the Vogt models. However, the GOF as measured by K was not as good as those for the Vogt models.

Tables 167 and 168 report the model results for TOTACC and INJACC.

Table 167. Parameter Estimates for Recalibrated TOTACC Full Models: Type III

¹ The main model published in Vogt, 1999, (p. 111)
² Median Type 1 (painted) on major roads
³ N/A: not available

Table 168. Parameter Estimates for Recalibrated INJACC Full Models: Type III

¹ INJACC Variant 1 Model; there is no main INJACC model in Vogt, 1999
² N/A: not available

3.4.4 Type IV Models

The data used to calibrate full models include the California and Michigan sites from the original study, with the additional years of accident data, and the Georgia sites. Some of the California sites experienced changes in their design features during the 1996-98 period. For such sites, accidents during the 1996-98 period were ignored, and only 1993-95 accident data were used. Summary statistics on the data are provided in section 3.2.

Two models were developed for both TOTACC and INJACC. Again, the main model was selected based on the highest Pearson product-moment correlation coefficient, and lowest overdispersion, MPB per year, and MAD per year.

For TOTACC, major and minor road AADTs, peak left-turn percentages, peak through percentages on minor roads, peak truck percentages, and right-side sight distance on minor roads were found to be significant in the main model at the 10 percent level or better. State indicator variables were statistically insignificant in the main model and were therefore not included. However, Variant 1, the second best model, includes a Michigan indicator variable, which means there was more influence of the Michigan data on the model.

For INJACC, major and minor road AADTs, peak left-turn percentages on major roads, peak truck percentages, and speed limits on minor roads were selected as significant in the main model at the 10 percent level or better. State indicator variables were statistically insignificant.

For reference, a comparison between Vogt's full models applied to the recalibration data and the newly recalibrated models for TOTACC and INJACC is given. As expected, there were some differences between the recalibrated models and the Vogt's models. For TOTACC, the recalibrated main model had additional terms SDR2, PKTRUCK, PKTHRU2, and PKLEFT compared to the Vogt model, but did not include PKLEFT1 and LTLN1S, which were in the Vogt model. For the INJACC, the recalibrated main model included the additional term PKTRUCK compared to the Vogt model. The GOF as measured by Pearson product-moment correlation coefficients, MPB per year, and MAD per year indicates that the recalibrated models for TOTACC and the main model for INJACC provide better GOF than the Vogt models. However, the GOF as measured by K was not as good as those for the Vogt models.

Tables 169 and 170 report the model results for TOTACC and INJACC, respectively.

Table 169. Parameter Estimates for Recalibrated TOTACC Full Models: Type IV

¹ The main model published in Vogt, 1999, (p. 116)
² Median Type 1 (painted) on major roads
³ Indicator variable for Michigan
⁴ N/A: not available

Table 170. Parameter Estimates for Recalibrated INJACC Type IV: Full Models

¹ INJACC Variant 1 Model; there is no main INJACC model in Vogt, 1999
² Median Type 1 (painted) on major roads
³ N/A: not available

3.4.5 Type V Models

The data used to calibrate full models include the California and Michigan sites from the original study, additional years of accident data from the California and Michigan sites, and the Georgia sites. Some of California sites experienced changes in their design features during the 1996-98 period. For such sites, accidents during the 1996-98 period were ignored in the database, and only 1993-95 accident data were used. Summary statistics on the data are provided in section 3.2 of this report.

Two models were developed for both TOTACC and INJACC. Again, the main model was selected by examining the Pearson product-moment correlation coefficient, overdispersion, MPB per year, and MAD per year.

For TOTACC, major and minor AADTs, commercial driveways on major roads, speed limits on major roads, light, and horizontal curvature variables were found to be significant in the main model. Compared to the main model, Variant 1 in table 171 yields improvement in overdispersion, but not in other GOF measures. Variant 1 showed a lower Pearson product-moment correlation coefficient than the main model, but MPB per year and MAD per year were higher.

For INJACC, major and minor AADTs, presence of lighting, speed limits on major roads, and horizontal curve on minor roads within 244 m (800 ft) of intersection were significant variables in the main model. Variant 1 in table 172 was superior to the main model in terms of lower overdispersion and other GOF measures, with the exception of MPB per year. However, the model includes a Michigan indicator variable, which means more influence of the Michigan data on the model. Because the IHSDM requires the main model to be recalibrated to work in any State, the model with the State indicator was selected as a variant, not as the main model.

For reference, a comparison between the Vogt full models applied to the recalibration data and the newly recalibrated models for TOTACC and INJACC also is given. As expected, there were some differences between the recalibrated models and the Vogt models. For TOTACC, the recalibrated main model included additional terms COMDRWY1, SPD1, LIGHT, and HEICOM compared to the Vogt model but did not include PKTRUCK, PKLEFT2, PROT_LT, and VEICOM, all of which were in the Vogt model. For the INJACC model, the recalibrated main model separated the safety effect of major and minor road AADTs. The main model included LIGHT, SPD1, and HEI2 but excluded PKLEFT2, PKTRUCK, PROT_LT, and VEICOM when compared to the Vogt model. The GOF as measured by MPB per year and MAD per year indicates that the recalibrated models for TOTACC and INJACC provide better GOF than the Vogt models. Pearson product-moment correlation coefficients were also higher for INJACC but were a little lower for TOTACC. The GOF as measured by K was not as good as that for the Vogt models.

Tables 171 and 172 show the model results for TOTACC and INJACC, respectively.

Table 171. Parameter Estimates for Recalibrated TOTACC Full Models: Type V

¹ The main model published in Vogt, 1999, (p. 122)
² Median Type 1 (painted) on major roads
³ N/A: not available

Table 172. Parameter Estimates for Recalibrated INJACC Full Models: Type V

¹ The main model published in Vogt, 1999, (p. 124)
² Indicator variable for Michigan
³ N/A: not available

3.5 ESTIMATION OF ACCIDENT MODIFICATION FACTORS

This section discusses the derivation of AMFs for both total and injury (fatal and nonfatal injury) accidents.

FHWA and its contractors have developed a new approach that combines historical accident data, regression analysis, before-and-after studies, and expert judgment to make safety performance predictions that are better than those obtained by any of the individual approaches. A recent report documents an accident prediction algorithm for implementing the new approach for two-lane rural highway sections that include road segments and intersections.⁽³⁾ Ongoing efforts aim to produce similar documents for other types of facilities.

The accident prediction algorithm has been developed for incorporation in the IHSDM as the Crash Prediction Module, but is suitable for stand-alone applications. As indicated earlier, the structure of the accident prediction algorithm for the five types of rural at-grade intersections is as follows:

(15)

where

N_int = predicted number of total intersection-related accidents per year after application of AMFs;

N_b = predicted number of total intersection-related accidents per year for base conditions; and

AMF₁ AMF₂ ... AMF_n = AMFs for various intersection features

The accident algorithm report, referred to as the "Red Book," provides AMFs for two-lane rural roads and intersections.⁽³⁾ An additional report provided AMFs for turning lanes at intersections.⁽⁵⁾ As part of this project, the available data have been used to attempt to derive AMFs for these and any other variables at rural intersections to compare and refine the current AMFs.

The AMFs provided in the "Red Book" are shown below in table 173. These AMFs apply to total intersection accidents (TOTACC). Also shown are the AMFs relevant to intersection types I, II, and V from Harwood et al.⁽⁵⁾

Table 173. AMFs from Previous Studies

Derivation of AMFs in this project was attempted in two ways. First, AMFs were inferred from the parameter estimates of the full models presented in section 3.4. Second, a relatively new and untested regression procedure that relates the difference between the observed number of crashes at a site and is predicted by a base model to the nonbase condition factors was applied. This procedure is explained in detail later in this section.

3.5.1 AMFs Derived from Recalibration Full Models

One approach to deriving AMFs is to apply a model using the estimated parameter values from only statistically significant variables in accident prediction models. This approach suffers from correlation between geometric variables and traffic, and the difference in accident experience between sites is possibly due to the substantial unexplained variation resulting from omitted factors. Nevertheless, AMFs derived in this manner from the full models of section 3.4 are presented in tables 174 and 175.

Table 174. AMFs Derived from Type I and II Full Models

¹ As a continuous variable, the AMF for HI1 is a function and not a factor

Table 175. AMFs Derived from Type III, IV, and V Full Models

¹ Medtype1: Painted

3.5.2 AMFs Derived from Regression Models

Recalibration of AMFs was also undertaken using a relatively untested regression analysis procedure in which the following steps were taken:

1. Suitable base conditions for qualifying the current AMFs were selected, i.e., for a turning lane AMF, the base condition is no turning lane.
2. The AADT base models that meet all of the base conditions were estimated.
3. The difference between a site's accident count per year and the expected value per year was used as an estimate of the dependent variable, Y.
4. Any factor (for example, left-turn lane on major road) that differs from the base conditions was specified as an independent variable.
5. The coefficient of a factor was estimated from a simple linear regression model.
6. AMFs were estimated from the following equations:

If perfect knowledge for base models and AMFs is available, Y in the accident prediction algorithm should be the observed Y.

(16)

Equation (16) can be rewritten as:

(17)

From steps 3 and 5:

(18)

Equations (17) and (18) can be combined to estimate an AMF:

(19)

7. n the case that coefficients in equation (19) are insignificant (at the 90 percent level in this research), it was assumed that these factors have no effect on the safety of the intersection. This offered a larger sample size to re-estimate the insignificant AMFs.
8. The coefficients of the insignificant AMFs were re-estimated by repeating step 2 and subsequent steps.
9. If the coefficients became significant, the estimated coefficients were adopted as the relevant AMFs. If not, they were set to a value of 1.

Types I and II

For types I and II, the base conditions and base models described in section 3.3 and which were developed using the Group B dataset were used. AMFs derived from this procedure are shown in table 176. The base conditions include no turning lanes on major or minor road and no median on the major road. As mentioned, where AMFs were not estimated to be significantly different from 1, a value of 1 has been assigned.

Table 176. AMFs Estimated from Regression Procedure

Few AMFs could be derived with statistically significant results. For Type I, the AMF for right-turn lane on the minor road increases the expected number of crashes from the base model. For Type II, right- and left-turn lanes on major and median on major are expected to reduce the expected number of crashes from the base model.

Type III

Unlike the case for Type I and II, for which the Group B dataset was applied, the Group A dataset was used to calibrate AMFs for Type III intersections because the Group B dataset does not provide sight distance and angle data. As a result, new base models had to be calibrated for the AMFs.

The nominal or base condition for "intersection left-turn lane" is the presence of left-lane on the major-road approach because as many as 79.4 percent of the available sites have left-turn lanes. For "right-turn lane," the absence of right-turn lane is the base condition. The base condition for intersection sight distance is the availability of adequate intersection sight distance along the major road in all quadrants of the intersection. The same definition regarding sight distance in a quadrant as described on page 48 of the Harwood et al. report was used to define adequate sight distance.⁽³⁾ The base condition for intersection skew angle was plus or minus 5 degrees of skew to have more base sites and to accommodate any possible measurement errors.

Table 177 shows the base conditions for Type III AMFs. Table 178 displays the base model estimated using the sites meeting all of the base conditions. Because the coefficient of log of AADT1 was quite insignificant, log of (AADT1 * AADT2) was used as the traffic variable.

AMFs derived from this procedure are given in table 179. These AMFs apply to total accidents and total injury accidents.

Table 177. Base Conditions for Type III Sites

¹ Total means sites meeting all of the base conditions

Table 178. Base Model for Type III Sites

¹ Logs of AADT1 and AADT2 were combined because the coefficient of log of AADT1 was insignificant

Table 179. AMFs for Type III Sites

¹ SKEW = intersection skew angle (degrees), expressed as the absolute value of the difference between 90 degrees and the actual intersection angle
² 0.98 = mean of the observed TOTACC accidents per year of the sites meeting no angle, no right lane, and presence of left lane
³ 0.52 = mean of the observed INJACC accidents per year of the sites meeting no angle, no right lane, and presence of left lane

Type IV

As was the case for Type III, the Group A dataset was used to calibrate AMFs for Type IV intersections because the Group B dataset does not provide sight distance and angle data. New base models were calibrated for the AMFs.

The base condition for "intersection left-turn lane" is the presence of left-lane on the major road approaches because as many as 72.6 percent of the available sites have left-turn lane on both approaches. For "intersection right-turn lane," the absence of right-turn lane is the base condition. The base condition for intersection sight distance is the availability of adequate intersection sight distance along the major road in all quadrants of the intersection. As for Type III, the base condition for intersection skew angle was plus or minus 5 degrees of skew to have more base sites and to accommodate any possible measurement errors. The base conditions for Type IV AMFs are shown in table 180. Table 181 represents the base model estimated using the sites meeting all of the base conditions. Since the coefficient of log of AADT1 was quite insignificant, log of (AADT1 * AADT2) was used as the traffic volume variable.

The AMFs derived are shown in table 182. These AMFs apply to total accidents and total injury accidents.

Table 180. Base Conditions for Type IV Sites

¹ Total means sites meeting all of the base conditions

Table 181. Base Model for Type IV Sites

¹ Logs of AADT1 and AADT2 were combined because the coefficient of log of AADT1 was insignificant

Table 182. AMFs for Type IV Sites

¹ SKEW = intersection skew angle (degrees), expressed as the absolute value of the difference between 90 degrees and the actual intersection angle
² 0.43 = mean of the observed TOTACC accidents per year of the sites meeting no angle, no right lane, and presence of left lane
³ 0.72 = mean of the observed INJACC accidents per year of the sites meeting no angle, no right lane, and presence of left lane

Type V

Because the Group B dataset does not provide sight distance and angle, the Group A dataset was used to recalibrate the current AMFs for Type V intersections, and new base models were calibrated for applying the AMFs.

As many as 75 percent of the available sites have left-turn lane on both approaches. Therefore, the base condition for "intersection left-turn lane" is the presence of left-lane on the major road approaches. For "intersection right-turn lane," the absence of right-turn lane is the base condition. The base condition for intersection skew angle was plus or minus 5 degrees of skew for the same reasons as those for Type III and IV. Sight distance was ignored for the base conditions to develop a base model because the number of sites with adequate sight distance was almost equal to the ones with sight distance limited in one or two quadrants. As a result, if one of the two sight distance levels was considered as a base condition, there would be insufficient sites for a base model. In addition, sight distance is believed to have no effect on safety because conflicting traffic movements are controlled by signals. The base conditions for Type V AMFs are shown in table 183. Table 184 represents the base model estimated using the sites meeting all of the base conditions. Because the coefficient of log of AADT1 was insignificant, log of (AADT1 * AADT2) was used to represent the traffic volume variable.

No AMFs were statistically significant using this procedure for Type V sites. Therefore, a value of 1 was assigned, which indicates that the variable was not found to have a significant impact on the safety of the intersection.

Table 183. Base Conditions for Type V Sites

¹ Total means sites meeting all of the base conditions

Table 184. Base Model for Type V Sites

¹ Log of AADT1 was insignificant, so logs of AADT1 and AADT2 were combined

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