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Validation of Accident Models for Intersections

FHWA Contact: John Doremi,
HRDI-10, (202) 493-3052, John.doremi@dot.gov

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3. RECALIBRATION

This chapter presents recalibration results for the five types of rural intersections that were the subject of the validation exercise undertaken in the first part of the project. The first section provides a discussion of the recalibration approach. In the second section, the data and related issues are discussed. Third, AADT model estimation results are presented, followed by fully parameterized model estimation results. Sensitivity analysis results for the AMFs derived in this research then are given. Finally, a discussion and conclusions as a result of model recalibration are provided.

3.1 RESEARCH APPROACH

This model recalibration effort complemented the comprehensive model validation previously conducted as part of a larger technical evaluation of crash prediction models. It should be acknowledged that several anticipated end-uses of the crash prediction models guided all decisions made throughout this careful evaluation, which resulted in some specific overriding considerations while conducting the model recalibration:

• The most likely end-use of the crash prediction models is embedded code within the IHSDM, with the sole intent to predict future crashes at intersections throughout the United States.
• The models need to be able to predict the change in safety as a result of changes in traffic and geometric features relative to nominal conditions, corrected for intersection type and State- or regional-specific effects.
• Environmental effects on safety, such as adverse weather and lighting conditions, while important factors, will be accounted for in State or regional correction factors.

Considering the likely end uses of the crash prediction models within the IHSDM, considerable time was spent identifying a strategy for recalibrating statistical models. A strategy was needed for several reasons. First, there were multiple levels and types of models in the source documents-requiring a prioritization of models to be calibrated. Second, there are numerous methodological approaches reflected in the source documents, which need to be prioritized. Finally, the treatment of explanatory variables is dependent upon the methodological approach taken. Before describing the research technical strategy, some guiding philosophical principles used to guide the model recalibration effort are presented.

It was felt that the majority of effort in the recalibration should be devoted to refinements to existing models. This includes changes to parameter estimates, and perhaps minor changes to model functional forms. This approach is based on the collective opinion that prior work, including the estimation of statistical models, was done carefully by experts in the field of transportation safety, and decisions such as variable selection, model functional form, and statistical model selection represent state-of-the-art knowledge with respect to intersection crash prediction models. Past documentation, critical evaluation, and discussion with other experts in the field confirm prior beliefs that the existing set of models represents a defensible and sound starting point. It is believed that moderate to serious departures from existing models should be accompanied by detailed and defensible descriptions of the how, why, and in what cases departures from previous methods and/or models were thought necessary and useful. Finally, capabilities with regard to model recalibration are limited, simply because of existing data limitations, availability of explanatory variables, and intersection representativeness across States. When these limitations are thought to be critical they are identified and discussed.

The technical strategy applied in this research effort is now described. Each of the strategies represents different possible end uses of the models, influenced by the stated guiding philosophical principles.

AADT Models: One set of models represents intersection crash models that forecast crashes in frequency-per-year based on minor and major road AADT-only. There are no other independent variables in these models. The intended use of these models is to provide a baseline crash forecast, which can then be modified with AMFs representing the effects of various geometric, roadside, and other relevant safety-related factors. The sample available for calibrating these models was much larger than the sample available for calibrating full models that, in a sense, partly compensates for the loss of statistical precision resulting from the omission of variables other than AADT.

Full Models: Another set of models represents statistical models with a full set of explanatory variables, including major and minor road AADT. These models are meant to provide a fuller understanding of the geometric, roadside, and operational features of intersections that influence on crashes. Another use might be to develop or infer additional crash modification factors for the various types of intersections examined in this research.

AMFs: A final set of "models" represents estimated effects of various geometric, roadside, and operational features. These provide a complement to the AADT models. The intended use of the AMFs is to provide percentage corrections to expected crash frequencies that result from the application of various crash countermeasures. AMFs represent a fairly intuitive approach to evaluating safety countermeasures, and are handled rather simply in the IHSDM.

When comparing and refining the three types models, several GOF measures were used in addition to inspection of model coefficients, collection of explanatory variables, and t-statistics and their associated p-values. Numerous measures are relied upon to avoid basing decisions on one single measure. Unfortunately, there is no one single criterion that dominates to the point of rendering the remaining measures as invalid or unimportant. It is through the assessment of many measures that a "best" model is chosen, and it is not always a clear winner.

3.1.1 Model Functional Forms

The negative binomial model form, which is identical to that used in previous efforts, was used to provide the best fit to the data.^(1,2) The following model form and error distribution were assumed to represent the underlying phenomenon:

AADT Only Models

(12)

where

= the mean number of accidents to be expected at site i in a given time period;

= the estimated intercept term; and

_{1 2}, estimated coefficients.

Fully Parameterized Models

The following model form and error distribution were assumed to represent the underlying phenomenon:

(13)

where

= the mean number of accidents to be expected at site i in a given time period;

= the estimated intercept term;

X_i1, X_i2....X_in, = the values of the non-traffic highway variables at site i during that time period; and

_{i1 i2}....._in, = estimated coefficients.

(14)

where

Var{m} = the estimated variance of the mean accident rate;

E{m} = the estimated mean accident rate from the model; and

K = the estimated overdispersion constant.

3.1.2 Goodness-of-Fit Evaluation

Four GOF measures were used in the model selection process (refer to chapter 2 for a description of the GOF measures.). A fifth approach to evaluating the GOF and in particular the suitability of alternate model forms was the Cumulative Residuals (CURE) method, proposed by Hauer and Hauer and Bamfo, in which the cumulative residuals (the difference between the actual and fitted values for each intersection) are plotted in increasing order for each covariate separately.^(8,9) The graph shows how well the model fits the data with respect to each individual covariate. Figure 19 illustrates the CURE plot for the covariate AADT1 for the total accidents for the selected AADT-only model for Type III intersections (presented in table 142). The indication is that the fit is very good for this covariate in that the cumulative residuals oscillate around the value of zero and lie between the two standard deviation boundaries. Figure 20 is a CURE plot for an alternate model. Clearly, the alternate model cannot be judged to be an improvement over the selected model. Appendix D contains CURE plots for the TOTACC AADT models for all intersection types.

Figure 19. CURE Plot for Type III TOTACC AADT Model

Figure 20. CURE Plot for Type III TOTACC AADT Model Using the CURE Method: Alternative Model

Now that the model's end uses, guiding research philosophies, and technical modeling strategy have been described, the details of the technical modeling efforts are presented and discussed. It is useful to first describe the data that were used in the model recalibration efforts, and to identify any difficulties, anomalies, and peculiar circumstances that needed to be remedied in the effort.

3.2 DESCRIPTION OF DATA

Different variables were used in developing statistical models for Types I and II compared to Types III, IV, and V. Although average daily traffic variables are common to all models, in general there were a larger number of variables available for estimation of model Types III, IV, and V. The abbreviation employed in the modeling efforts and their descriptions are provided in the following section.

3.2.1 Summary of Datasets

The data used for recalibration were obtained from three sources. The first two sets were identical to the data used for the validation exercise described in chapter 2. The first set was the original calibration data used by Vogt and Bared from Minnesota and Vogt from California and Michigan.^(1,2) Additional years of accident and traffic data were obtained for those sites which did not experience a change in major variables, such as traffic control or number of legs. There were primarily minor differences in the summary statistics between those calculated on the available data and those stated in the reports, particularly for the vertical curvature variables for Type V sites. However, existing differences are sufficiently minor that further clarification was not necessary. The accident data obtained for the original sites included data for both the original and additional years. Differences were found in the accident counts between the original data obtained and this new dataset for the original years. Again, although small differences exist, their causes are unknown and these discrepancies were small enough that the data could confidently be used for recalibration. The second source of data was for those sites selected in Georgia to provide and an independent set of validation data. The third source of data was the California HSIS database. This data set was acquired to increase the size of the recalibration datasets with the aim of providing improved models with smaller standard errors of parameter estimates. These data were collected with a minimum amount of effort with assistance from the HSIS staff. However, as site visits were not conducted, fewer variables were available for these sites. Table 108 summarizes the sources of data used for recalibrating Models I to V.

Table 108. Sources of Data

In this section, summary statistics are provided for the data available for recalibrating the full models (i.e. models with explanatory variables other than traffic volumes). For model Types III, IV, and V, the California HSIS sites were not included due to the limited availability of variables relevant to these models.

It is also appropriate and useful to examine which variables strongly correlate positively or negatively with crashes and which potential independent variables are correlated to one another. These statistics are also provided in this section of the report.

3.2.2 Type I

A summary of the full data for Type I intersections is shown in table 109. This dataset includes the original sites in Minnesota, with the additional years of accident and traffic data, the Georgia sites and the California HSIS sites. Some of the Minnesota sites experienced changes in some design feature or location information during the 1990-98 period and were not included in the analysis. Note that some variables are not available in the data for the Minnesota sites and California sites. The frequency column indicates the number of sites for which the information was available. Summary statistics by State are available in appendix C.

Table 109. Summary Statistics for Type I Sites

¹ N/A: not available

Table 110 shows correlation statistics and p-values that indicate the association between crash counts and the independent variables for type I intersections. Table 111 shows correlations between the independent variables. Only those correlations that are significant at the 90 percent level are shown.

As expected, major and minor road AADTs correlate positively with crashes. Turning lanes on the major and minor roads are also positively correlated with crashes, although this correlation is much less than that of vehicle volumes and the correlation for right-turn lane on major roads is not significant. Surprisingly, terrain and posted speed are negatively correlated with crashes, meaning that areas with rolling or mountainous terrain experience a lower crash risk than flatter terrains and that higher speeds are associated with fewer crashes. This counterintuitive result may arise because, as shown in Appendix C, Georgia sites have higher accident frequencies than California and Minnesota sites, as well as lower average posted speeds and a higher percentage of sites in rolling or mountainous terrain. With the presence of a median, VCI1 and HI1 were positively correlated with crashes, while HAU was negatively correlated with crashes although this correlation was not as strong. Shoulder width and number of driveways were not significantly correlated with crashes.

Table 110. Correlation Between Crashes and Independent Variables for Type I Sites

Table 111. Summary of Correlations for Independent Variables for Type I Sites

3.2.3 Type II

A summary of the full data for Type II intersections is shown in table 112. This dataset includes the original sites in Minnesota, with additional years of accident and traffic data, the Georgia sites and the California HSIS sites. Some of the Minnesota sites experienced changes in some design feature or location information during 1990-98 and were not included in the analysis. Note that some variables are not available in the data for the Minnesota sites and California sites. The frequency column indicates the number of sites for which the information was available. Summary statistics by State are available in appendix C.

Table 112. Summary Statistics for Type II Sites

¹ N/A: not available

Table 113 shows correlation statistics and p-values that indicate the association between crash counts and the independent variables for Type II intersections. Table 114 shows correlations between the independent variables. Only those correlations that are significant at the 90 pecent level are shown. Note that some variables are not included in the data for the Minnesota and California sites.

As expected, major and minor road AADTs correlate positively with crashes. Right-turn lanes on the major roads were negatively correlated with crashes, while right-turn lanes on the minor roads were positively correlated with crashes. Left-turn lanes on the major roads were positively correlated with crashes, however left-turn lanes on the minor roads were not significantly correlated with crashes. Again, terrain and posted speed are negatively correlated with crashes, meaning that areas with rolling or mountainous terrain experience a higher crash risk than flatter geographies and that higher speeds are associated with less crashes. Presence of a median, number of driveways, HI1, and roadside hazard rating on the major roads were all positively correlated with crashes. Intersection angle (HAU), shoulder width, and VCI1 were not significantly correlated with crashes.

Table 113. Correlation Between Crashes and Independent Variables for Type II Sites

Table 114. Summary of Correlations for Independent Variables for Type II Sites

3.2.4 Type III

A summary of the full data for Type III intersections is shown in table 115. In total, 42 variables were available for model development. The HSIS California data were excluded in developing Type III full models because this data set has only a few variables (turning lanes, median, terrain, etc) of relevance. This left the California and Michigan sites from the original study, with the additional years of accident data, for inclusion in the database. Some California sites experienced changes in some design features during 1996-98. For these, only 1993-95 data were used. As before the frequency column indicates the number of sites for which the information was available.

Table 115. Summary Statistics for Type III Sites

¹ N/A: not available

Table 115. Summary Statistics for Type III Sites (Continued)

¹ N/A: not available

Table 115. Summary Statistics for Type III Sites (Continued)

Table 116 shows correlation statistics and p-values that indicate the association between crash counts and the independent variables for Type III intersections. Table 114 shows correlations between the independent variables. Only those correlations that are significant at the 90 percent level are shown.

Major and minor road AADTs correlate positively with crashes as expected. Peak turning movement volumes also correlate with crashes, both positively and negatively. PKTURN, PKLEFT, and PKLEFT1 correlate positively with crashes, while PKTRUCK and PKLEFT2 correlate negatively with crashes. According to table 114, PKTRUCK correlates negatively with the AADT variables. This suggests that the negative correlation of crashes with PKTRUCK may, in part, be a consequence of the positive correlation of crashes with AADT variables. PKLEFT1 and PKLEFT2 are also negatively correlated with each other. There are several variables for which the correlation results are unexpected. Roadside hazard rating on major and minor roads, number of residential driveways on major and minor roads, posted speed limits on major and minor roads, terrain on major roads, shoulder width on major roads, "LIGHT," and the presence of left-and right-turn lane on minor roads, as well as other variables are correlated with crashes in the opposite direction to that expected, although many of these correlations are insignificant.

Table 116. Correlation Between Crashes and Independent Variables for Type III Sites

Table 116 . Correlation Between Crashes and Independent Variables for Type III Sites (Continued)

Table 117. Summary of Correlations for Independent Variables for Type III Sites

¹Only those correlations are shown for which p-values are less than 0.10

3.2.5 Type IV

A summary of the full data for type IV intersections is shown in table 118. In total, 53 variables were available for model development. The HSIS California data were again excluded because of a lack of sufficient variables (turning lanes, median, terrain, etc.) of relevance. Instead, the California and Michigan sites from the original study, with the additional years of accident data were included in the database. Some California sites experienced changes in some design features during 1996-98. For these, only 1993-95 data were used. As before, frequency indicates the number of sites for which the information was available.

Table 118. Summary Statistics for Type IV Sites

¹ N/A: not available

Table 118 . Summary Statistics for Type IV Sites (Continued)

¹ N/A: not available

Table 118 . Summary Statistics for Type IV Sites (Continued)

¹ N/A: not available

Table 119 shows correlation statistics and p-values that indicate the association between crash counts and the independent variables for Type IV intersections. Table 120 shows correlations between the independent variables. Only those correlations that are significant at the 90 percent level are shown.

Major and minor road AADTs correlate positively with crashes, as expected. Peak turning movements also correlate with crashes, both positively and negatively. There are several variables for which the correlation results are contrary to expectations. Shoulder width on the road, right-and left-turn lane on minor roads, acceleration lane on major roads, residential driveway and total driveway on minor roads, light, terrain on major and minor roads, vertical curves on major and minor roads, horizontal curves on major roads, absolute grades on major and minor roads, intersection angle, posted speed limit on major roads, and others are correlated with crashes in the opposite direction than expected, although many of these correlations are insignificant. For example, median width on major road is insignificant with a counterintuitive sign. However, as table 120 shows, there is a negative correlation between median width on major roads and median types, the result of which is that median type is skewing the effect of median width at Type IV intersections.

Table 119. Correlation Between Crashes and Independent Variables for Type IV Sites

Table 120. Summary of Correlations for Independent Variables for Type IV Sites

Table 120 . Summary of Correlations for Independent Variables for Type IV Sites (Continued)

¹ Only those correlations are shown for which p-values are less than 0.10.

3.2.6 Type V

A summary of the full data for Type V intersections is shown in table 121. In total, 53 variables were available for model development. The HSIS California data were again excluded because only five Type V sites were available. This left the California and Michigan sites from the original study, with the additional years of accident data for inclusion in the database. Some California sites experienced changes in some design features during 1996-98 period. For these, only 1993-95 data were used. As before, the frequency column indicates the number of sites for which the information was available.

Table 121. Summary Statistics for Type V Sites

Table 121 . Summary Statistics for Type V Sites (Continued)

¹ N/A: not available

Table 121 . Summary Statistics for Type V Sites (Continued)

¹ N/A: not available

Table 121 . Summary Statistics for Type V Sites (Continued)

¹ N/A: not available

Table 122 shows correlation statistics and p-values that indicate the association between crash counts and the independent variables for type V intersections. Table 123 shows correlations between the independent variables. Only those correlations that are significant at the 90 percent level are shown.

Again, as expected, major and minor road AADTs correlate positively with crashes. Peak turning movement volume also correlates with crashes, both positively and negatively. Shoulder width on major and minor roads, left-and right-lane on major and minor roads, acceleration lane on major and minor roads, protected left lane on major and minor roads, residential driveway on major and minor roads, terrains, sight distance, vertical curves, absolute grades, horizontal curves, intersection angle, and other variables are correlated with crashes in the opposite direction than expected, although many of these correlations are insignificant.

Table 122. Correlation Between Crashes and Independent Variables for Type V Sites

Table 122 . Correlation Between Crashes and Independent Variables for Type V Sites (Continued)

Table 123. Summary of Correlations for Independent Variables for Type V Sites

Table 123 . Summary of Correlations for Independent Variables for Type V Sites (Continued)

Table 123 . Summary of Correlations for Independent Variables for Type V Sites (Continued)

¹ Variables only significant with p-value of 0.1 were selected

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