Rural Public Transportation Technologies: User Needs and Applications FR1798
OVERVIEW OF LITERATURE AND STATISTICAL MODELING APPROACHES
This section of the report presents a brief overview of previous studies of interchange ramp and speedchange lane accidents. The discussion also reviews nontraditional statistical approaches to accident modeling used in recent research and describes how those nontraditional approaches can be applied to models for ramp and speedchange lane accidents.
Previous Evaluations of Ramp and SpeedChange Lane Accidents
There has been substantial earlier research on the safety effects of various interchange elements, but none of this research has resulted in relationships that appear directly useful in estimating the effectiveness of various interchange improvements. The most recent summary of research on interchange safety was prepared in the early 1990s by Twomey, Heckman, and Hayward.^{(1)}Earlier sources include an annotated bibliography prepared in the early 1980s by Harwood et al. and a summary of research findings prepared by Oppenlander
and Dawson in 1970.^{(2,3,4)}
Statistical Modeling of Interchange Accidents
Several previous studies have undertaken statistical modeling of accidents in interchange areas. One of the best known efforts of this type was the Interstate System Accident Research (ISAR) study undertaken by FHWA in the late 1960s. A key summary of this work is presented by Cirillo, Deitz, and Beatty and includes 6 multiple regression models to predict accident frequencies for entire interchange areas and 13 models of specific interchange components, including ramps and speedchange lanes.^{(5)} Accident severity distributions were also examined. Traffic volumes were found to be a key variable in predicting
interchange accident experience. Geometric features of ramps considered in accident modeling included ramp type (onramp vs. offramp), ramp length, speedchange lane length, presence of curvature, maximum curvature on ramp, central angle of first curve on ramp, ramp grade, right and left shoulder widths, minimum stopping sight distance, and difference between ramp and speedchange lane design speeds. In their accident modeling work, Cirillo, Deitz, and Beatty developed separate models for ramps and their associated speedchange lanes, while most
other studies combined these features.
Multiple regression modeling of ramp accidents was also conducted by Morganstein and Edmonds, using the ISAR data base, and by Kim, using a data base developed in Michigan.^{(6,7)} Other studies of interchange safety focused on determining average accident rates for interchange features rather than developing statistical models.
Effect of Horizontal Alignment of Ramps
Research by Yates with the ISAR data base estimated the average accident rates as a function of curvature and traffic volume for loop and outer connection ramps in cloverleaf interchanges.^{(8)} As shown in table 1, ramps without curvature were found to have smaller accident rates than those with curvature in both urban and rural areas for all traffic volume levels except for 0 to 499 veh/day in urban areas. Rural loop ramps with low curvature were found to have higher accident rates than those with high curvature, as shown in table 2, while the opposite was true in urban areas.
A California study by Lundy completed in 1965 determined accident rates for ramps grouped by ramp type and curvature.^{(9)} Offramps were found to have consistently higher average accident rates than onramps, while the average accident rates of curved ramps were only slightly higher than straight ramps. No statistical analyses of these data were conducted.
A 1961 review of interchange accident experience in New Jersey by Fisher concluded that very few accidents could be attributed to loop ramps with radii over 31 m (100 ft).
Effect of Vertical Alignment of Ramps
The general ramp grade can be determined by whether the crossroad at an interchange goes over or under the mainline freeway. Lundy found that upgrade offramps had lower accident rates than downgrade offramps.^{(9)} However, the accident rates of onramps did not appear to depend on whether the onramp was on an upgrade or downgrade.
Effect of Ramp Configuration
Lundy determined the accident rates for ramps of different ramp types (onramp vs. offramp) and ramp configurations.^{(9)} These findings are summarized in table 3. They generally indicate that diamond ramps have the lowest accident rates. Loop ramps,
which involve higher curvature, were found to have higher accident rates. The highest accident rates were found for scissors connections, where ramps cross one another with stopsign control,
and for ramps that enter or leave the left side of the mainline freeway lanes. Although the Lundy data shown in the table were first developed in 1965, the California Department of Transportation (Caltrans) has continually updated these accident rate estimates by ramp type and configuration over the years as the basis for their accident surveillance program for freeway interchanges.
Table 1. Accident Rates on Outer Connection Ramps as a Function
of Curvature and Average Daily Traffic Volume^{(8)}
Average daily traffic volume (veh/day) 
Accident rate (per 100 million vehicles) 
Urban ramps 
Rural ramps 
Straight^{a} 
Curved^{b} 
Straight^{a} 
Curved^{b} 

0499 
0.74 
0.64 
0.00 
0.67 
5001,000 
0.34 
0.72 
0.13 
0.49 
1,0011,500 
0.64 
0.84 
0.00 
0.61 
1,5012,000 
0.15 
0.93 
0.00^{c} 
0.20 
>2,000 
0.49 
0.82 
0.00^{c} 
0.72 
COMBINED 
0.44 
0.81 
0.05 
0.56 
a Less than 1 degree of curvature.
b Greater than 1 degree of curvature.
c Based on less than 10 ramps.
Table 2. Accident Rates for Loop Ramps as a Function of Curvature
and Average Daily Traffic Volume^{(8)}
Average daily traffic volume (veh/day) 
Accident rate (per 100 million vehicles) 
Urban ramps 
Rural ramps 
Low curvature^{a} 
High curvature^{b} 
Low curvature^{a} 
High curvature^{b} 

0499 
0.000 
0.841 
1.000 
0.260 
5001,000 
0.000 
0.960 
0.810 
0.370 
1,0011,500 
1.320 
0.690 
0.000 
0.000 
1,5012,000 
0.000 
0.720 
0.000^{(9)} 
0.000 
>2,000 
0.141 
1.000 
0.000^{c} 
0.000 
COMBINED 
0.200 
0.940 
0.631 
0.250 
a Less than 12 degrees of curvature.
b Greater than 36 degrees of curvature.
c Based on less than 10 ramps.
Table 3. Accident Rates by Ramp Type and Configuration
^{(9)}
Ramp configuration 
Accident rate (per million vehicles) 
Onramp 
Offramp 
Combined 

Diamond 
0.40 
0.67 
0.53 
Cloverleaf outer connection with C/D roads^{a} 
0.45 
0.62 
0.61 
Direct connection 
0.50 
0.91 
0.67 
Cloverleaf loops with C/D roads^{a} 
0.38 
0.40 
0.69 
Buttonhook 
0.64 
0.96 
0.80 
Other loop with C/D roads 
0.78 
0.88 
0.83 
Cloverleaf outer connection without C/D roads 
0.72 
0.95 
0.84 
Trumpet ramps 
0.84 
0.85 
0.85 
Scissors ramps 
0.88 
1.48 
1.28 
Leftside ramps 
0.93 
2.19 
1.91 
AVERAGE 
0.59 
0.95 
0.79 
a) Only the combined on and offramp accident rates include accidents
on collector/distributor (C/D) roads.
Accident Locations Along Ramps
The Fisher study in New Jersey found that most accidents were associated with speedchange lanes and ramp terminals and very few accidents were associated with the main portion of the ramp.
Effect of Ramp Traffic Volumes
Virtually every study of ramp accidents has concluded that traffic volumes are the single strongest predictor of accident frequencies and accident rates. By contrast, geometric design features of ramps were found to have much less ability to predict ramp accidents.
SpeedChange Lanes and Weaving Areas
The safety performance of speedchange lanes and weaving areas was documented with the ISAR data by Cirillo.^{(11,12,13)} Table 4 summarizes the average accident rates of offramps, onramps, speedchange lanes (including both deceleration and acceleration lanes), and weaving areas from the ISAR data for both rural and urban areas. Statistical modeling by Cirillo concluded that accident rate decreases with increasing length of weaving areas and speedchange lanes. Separate multipleregression relationships of the relationship between length of weaving and speedchange lanes were developed for various traffic volume levels. Statistical relationships for weaving areas were based on the weaving volumes, and for speedchange lanes, they were based on the percentage of merging or diverging traffic. Accident rates in weaving areas and speedchange lanes generally increased with increasing traffic volumes. The effect of acceleration lane length on accident rate was found to be substantial when the percentage of merging traffic exceeded 6 percent. The effect of speedchange lane length
was not as great for deceleration lanes as for acceleration lanes.
Table 4. Accident Rates by Area Type and Interchange Unit
Rural

Interchange unit 
Vehiclemile of travel (100 million) 
Numbers of accidents 
Accident rate (per 100 million vehiclemiles) 
Deceleration lane 
2.51 
348 
137 
Offramp 
0.57 
199 
346 
Onramp 
0.59 
95 
161 
Acceleration lane 
3.86 
280 
76 
Mainline weaving area 
0.49 
87 
116 

AVERAGE 
 
 
109 

Urban 
Interchange unit 
Vehiclemiles of travel (100 million) 
Number of accidents 
Accident rate (per 100 million vehiclemiles) 
Deceleration lane 
5.83 
1,089 
186 
Offramp 
1.48 
546 
370 
Onramp 
1.61 
1,159 
719 
Acceleration lane 
8.40 
1,461 
174 
Mainline weaving area 
2.45 
555 
227 

AVERAGE 
 
 
214 

1 mi = 1.61 km
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Innovative Approaches to Statistical Modeling
In past research, including several of the studies discussed above, accident predictive models have often been developed with accident rates (i.e., accident frequencies per unit of exposure) as
the dependent variable using simple multiplelinear regression. In this traditional approach, the dependent variable (accident rate) was modeled as a linear combination of highwayrelated parameters, with or without interactions, under the assumption that the dependent variable follows
a normal distribution. The results obtained from this approach have generally been disappointing,
both in terms of the proportion of the variation in accident rates explained by the models and the
generally weak role of geometric design variables as accident predictors. Part of the reason for the disappointing results of past research may be that multiple regression is an inappropriate approach for developing such relationships.
There are several reasons for this concern. First, accident rates often do not follow a normal distribution. Traffic accidents are random, discrete events that are sporadic in nature. Normalizing accident frequencies with exposure estimates, such as million vehiclemiles of travel or million vehicles traveling on interchange ramps and speedchange lanes, to make accident rates appear to be a continuous random variable does not change the fundamentally discrete nature of accident data.
Second, accident frequencies for particular ramps and speedchange lanes or relatively small roadway sections are typically very small integers, even if several years of accident data are obtained for those interchange elements or roadway sections. In fact, it is not uncommon for a substantial proportion of the sites in an accident study to have experienced no accidents at all during the study period. Small integer counts, often zero or close to zero, do not typically follow a normal distribution. In fact, the Poisson and negative binomial distributions are often more appropriate for discrete counts of events that are likely to be zero or a small integer during a given time period.
Finally, accident frequencies and accident rates are necessarily nonnegative. However, there is nothing to constrain traditional multipleregression models from predicting negative accident frequencies or accident rates, which confronts the accident analyst trying to use the predictive model with a meaningless result.
Research to develop accident predictive models published in recent literature has moved away from approaches based on multiple regression and has begun to use underlying distributional assumptions other than the normal distributional assumptions. As stated above, the Poisson distribution is appropriate for rare events such as traffic accident counts where the number of events in a given time period is likely to be zero or a small integer.
Several recent studies have implemented these nontraditional statistical approaches. For example, Miaou and Lum investigated four types of regression models to evaluate the relationship between truck accidents and highway geometric design elements.^{(14)} The four models considered by the authors were two conventional linear regression models (one was normal or additive; the other was lognormal or multiplicative) and two multiplicative Poisson regression models (one using an exponential rate function; the other, a nonexponential rate function). Miaou and Lum concluded that of the four models tested, the Poisson model with the exponential rate
function provided the best form of the relationship between truck accidents and highway geometric design elements in their study. The authors also identified the inherent limitations in using a Poisson model, which are discussed below.
One of the basic assumptions when choosing a Poisson model is that the mean and the variance of the error distribution are equal. However, in many applications, including the work that will be presented in this report, the data exhibit extra variation (i.e., the variance is greater than the mean of the estimated Poisson model). This situation is referred to as overdispersion. An alternative statistical model for addressing error structures with overdispersion like that often found in accident data is the negative binomial distribution. This approach has been used recently
by several researchers, including Hauer et al., Knuiman et al., Miaou et al., Shankar et al., Hadi
et al., and Bauer and Harwood.
The performance of Poisson and negative binomial regression models was recently compared by Miaou.^{(21)} The author applied these models to define a relationship between truck accidents and geometric design of road sections. The author concluded that with moderate to high overdispersion in the data, the negative binomial model provides a sensible approach to modeling accidents in that particular application. However, with certain modeling estimation procedures, the regression coefficients are quite consistent between the Poisson and the negative binomial approach. In any case, Miaou suggests the use of Poisson regression as an initial step in the modeling effort, with the negative binomial model then being applied where appropriate. A 1987 paper by Lawless also examined the efficiency and robustness properties of the negative binomial and mixed Poisson regression models when applied to count data that exhibit extra variation.
None of these past studies have addressed ramp and speedchange lane accidents, but their results suggest that Poisson and negative binomial regression are likely to be appropriate approaches to statistical modeling. The results obtained from implementing these approaches are presented in the remainder of this report.
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