Modeling the Relationship of Accidents to Miles Traveled PAUL P. JOVANIS and HSIN-LI CHANG ABSTRACT Consideration of highway safety studies in a time-space domain is used to in- troduce the concept that different study designs result in different underlying probability distributions describing accident occurrence. Poisson regression is proposed as a superior alternative to conventional linear regression for many safety studies because it requires smaller sample sizes and has other desirable statistical properties. Models are estimated using accident, travel mileage, and environmental data from the Indiana Toll Road. A pooled model including all accidents revealed that accident occurrence increases with automobile vehicle miles of travel (VMT), truck VMT, and hours of snowfall. Segmentation of the data into subsets that describe different types of collisions revealed that automobile accidents are much more sensitive to environmental conditions than are truck accidents. Use of the segmentation technique allowed a much clearer understanding of the effects of travel mileage on accident occurrence than could have been obtained from the pooled data alone. 42 Transportation Research Record 1 It is generally recognized that the occurrence of accidents results from the complex interaction of characteristics of the driver, vehicle, roadway, and environment. The number of accidents (accident fre- quency) is also clearly related to the amount of travel that occurs. Quantity of travel may be mea- sured in any of several ways including hourly volume, average daily traffic, or vehicle miles of travel (VMT), among others. These measures of quantity of travel can be used to describe traffic conditions that exist during exposure to accident risk. A more precise definition of exposure is, "...the amount or opportunity for accidents which the driver or traffic system experiences" (1). This broader interpretation of exposure has led some researchers to explore the effects on accident occurrence of environmental con- ditions during which the driving occurred (2). Previous studies relating accident occurrence to level of traffic have used a variety of measures of travel quantity. Belmont (3) found the accident rate (accident per million VMT) for two-lane sections almost linear with hourly traffic flow during day- light. For four-lane divided sections, Leutzbach (4) and Gwynn (5) found that a U-shaped relation exists between accident rate and hourly traffic flow, where the minimum values of the accident rates happened at approximately 600 to 1300 vehicles/hr per two lanes. In another study the accident rate increased rapidly when the traffic volume was below 550 vehicles/hr per two lanes, but showed little variation beyond this flow value (6). Smeed (7) considered the problem On a much broader scale, studying national yearly accident rates. He found that the total accident rate showed little variation with annual traffic volumes. When he separately considered single-vehicle and mutiple- vehicle crashes he found that the single-vehicle accident rate decreased with annual traffic while the multiple-vehicle rate increased. Ceder and Livneh (8) used both time-sequence analysis and cross-sectional analysis to study single- and multivehicle accidents for a series of eight roadway segments over an 8-year period. Ceder Transportation Center and Department of Civil Engi- neering, Northwestern University, Evanston, Ill. 60201. (9) expanded on this work by considering accident rates in Conjunction with free-flow and congested- flow conditions. He found that the total accident rate versus hourly flow curve followed a U-shaped configuration for the free flow, which is the result of a convex downward and a convex upward curve for single- and multivehicle accidents, respectively. For congested flow, the accident rate for multivehi- cle accidents increases rapidly with hourly traffic flow. A TIME-SPACE PERSPECTIVE ON ACCIDENT OCCURRENCE These studies can be considered as representing dif- ferent areas in a space-time plane (see Figure 1). , Smeed's research represents a very large area because he used national statistics on an annual basis 0• ' The use of horizontal lines indicates that the anal ysis was cross-sectional, comparing the accidents experiences of different countries (a spatial anairt sis). Gwynn considered only one route but conducted, his comparisons on an hourly basis in the tine domain, so there are vertical lines within the domaM defining his study (5). These areas are not drawn 00' , scale but are used to illustrate how these two studies would be represented in the space-time plane.. Different types of accident studies result in dit- ferent shapes in the space-time plane. Each of theile, shapes can be linked to particular probability die', tributions that describe the probability of accidenti occurrence. For example, with a long time period mWP large study section it may be reasonable to approxi' mate the occurrence of accidents by a normal distri bution. If the time and space domains are large there is a very small likelihood of zero acciden in a time interval. The normal distribution ei then have a large mean and a comparatively soal variance that will make zero or negative values 00 likely. This appears to be a reasonable distributioe to use in this context. The spatial and temporal aggregation required these large time-space areas makes it difficult 4 not impossible to isolate the influence of drill ° and environmental characteristics. The normal di tribution that is assumed in linear regression analysis of variance (ANOVA) may be appropriat e ARonn9q691