# About the covariance between the number of accidents and the number of victims Bijleveld, FD

Publication details: Leidschendam Stichting wetenschappelijk onderzoek verkeersveiligheid, SWOV, 2002; Report R-2002-24, Description: 42 sSubject(s): Online resources: Abstract: Traffic safety is not only indicated by the number of accidents, but also by numerous accident-related outcomes like the number of people killed, the number of people seriously injured, the amount of material damage, etc. When assessing the possible effect of, for example, a road safety measure on traffic safety, it is therefore important to be able to investigate the differentiated effect of such a measure on accidents and accident-related outcomes. In this study some statistical issues involved in the simultaneous analysis of accident-related outcomes (such as the number of victims, fatalities or accidents) of the traffic process were studied. The main focus of this study was the covariation of the outcomes: the interdependencies of accidentrelated outcomes were investigated by establishing their (theoretical) covariance structure. Estimates of the covariances of normal approximations of joint distributions were derived for the following cases: a. The total number of accidents, victims and fatalities in a certain class. Examples of classes are monthly data, car-only accidents. b. The logarithm of the total number of accidents, victims and fatalities in a certain class. c. The logarithm of the total number of accidents, the logarithm of the ratio of the number of victims to the number of accidents, the logarithm of the ratio of the number of fatalities to the number of victims. The quality of these estimates was evaluated using samples of real-life data from the Netherlands. Distributional aspects like effects on the estimates of small numbers, of small numbers of fatalities per accident, and of different types of accidents were also investigated in this study. It turns out that deviations are generally modest in most cases but may become serious when the counts are smaller. The following results were found: - It is possible to derive relatively simple expressions for the variances and covariances of (logarithms and ratios of) accidents and victim counts. As regards usability, some information needed to compute estimates of the covariance matrices may not be available over a longer period of time. However, in some cases this information can be estimated. - When performing a multivariate analysis using numbers of accidents, victims, and fatalities as outcome variables, or any of the other outcome variables mentioned above under b) and c), all three variables must be used. This follows from the finding that each of the three variables carries unique information that cannot be estimated from the other two. - The logarithm of the total number of accidents is (approximately) uncorrelated to the other two variables mentioned above under c). This means that the effect of explanatory variables on the logarithm of the total number of accidents can be (approximately) assessed independently, with no regard to the other two variables. - The approximation of the logarithm needed when log-counts of accidents victims or other accident counts are analysed, is usually sufficiently close. In practice but depending on circumstances, problems caused by the approximation are unlikely when counts are higher than 30.Traffic safety is not only indicated by the number of accidents, but also by numerous accident-related outcomes like the number of people killed, the number of people seriously injured, the amount of material damage, etc. When assessing the possible effect of, for example, a road safety measure on traffic safety, it is therefore important to be able to investigate the differentiated effect of such a measure on accidents and accident-related outcomes. In this study some statistical issues involved in the simultaneous analysis of accident-related outcomes (such as the number of victims, fatalities or accidents) of the traffic process were studied. The main focus of this study was the covariation of the outcomes: the interdependencies of accidentrelated outcomes were investigated by establishing their (theoretical) covariance structure. Estimates of the covariances of normal approximations of joint distributions were derived for the following cases: a. The total number of accidents, victims and fatalities in a certain class. Examples of classes are monthly data, car-only accidents. b. The logarithm of the total number of accidents, victims and fatalities in a certain class. c. The logarithm of the total number of accidents, the logarithm of the ratio of the number of victims to the number of accidents, the logarithm of the ratio of the number of fatalities to the number of victims. The quality of these estimates was evaluated using samples of real-life data from the Netherlands. Distributional aspects like effects on the estimates of small numbers, of small numbers of fatalities per accident, and of different types of accidents were also investigated in this study. It turns out that deviations are generally modest in most cases but may become serious when the counts are smaller. The following results were found: - It is possible to derive relatively simple expressions for the variances and covariances of (logarithms and ratios of) accidents and victim counts. As regards usability, some information needed to compute estimates of the covariance matrices may not be available over a longer period of time. However, in some cases this information can be estimated. - When performing a multivariate analysis using numbers of accidents, victims, and fatalities as outcome variables, or any of the other outcome variables mentioned above under b) and c), all three variables must be used. This follows from the finding that each of the three variables carries unique information that cannot be estimated from the other two. - The logarithm of the total number of accidents is (approximately) uncorrelated to the other two variables mentioned above under c). This means that the effect of explanatory variables on the logarithm of the total number of accidents can be (approximately) assessed independently, with no regard to the other two variables. - The approximation of the logarithm needed when log-counts of accidents victims or other accident counts are analysed, is usually sufficiently close. In practice but depending on circumstances, problems caused by the approximation are unlikely when counts are higher than 30.