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Estimating the Variance in Before-After Studies

Publication Information


The before-after study is usually the method of choice for measuring the safety effects of a single treatment or a combination of treatments in highway safety. This type of study has become very popular over the last few years. It is usually carried out using a four-stage estimation process. One important assumption with this process is related to the computation of the variance when a series of composite entities are used to assess the effectiveness of the treatment. To simplify the computation of the variance, it is generally assumed that the observed crash data for each entity are Poisson distributed. This assumption allows the transportation safety analyst to utilize observed crashes as a substitute for the variance, given the characteristics associated with the Poisson distribution.

In the light of the recent work conducted on the nature of the crash process, there is a need to determine whether this assumption holds and how it may affect the computation of the variance in traditional before-after studies. The objectives of this paper are to evaluate whether or not the assumption that crash data should be assumed to be purely Poisson distributed is valid, and if not, to examine how this may affect the estimation of the variance for estimating the inferences associated with the parameters commonly estimated for this kind of study: the difference between the estimated and predicted crashes, ´, and the index of effectiveness..

To accomplish the objectives of this study, a non-parametric bootstrap method is investigated to evaluate this assumption. The bootstrap method is first applied to data simulated using a Poisson distribution, a Negative Binomial distribution, a mixture of these two distributions, and a NB distribution with a varying mean. Then, the method is applied to two datasets containing observed before and after crash data taken from the literature. The results of this study show that one should not automatically assume that crash data are Poisson distributed. Even if the data follow a Poison distribution, the crash counts observed at a given site may not actually be equal to its true mean. Consequently, the estimated variances for the difference between the estimated and predicted crashes, and the index of effectiveness are likely to be underestimated when the traditional method is used in before-after studies. The bias could potentially lead to erroneously believe that the treatment is effective, when in fact it is not.

Publishing Date:
January 2007
Lord, Dominique
FHWA Program(s):
AMRP Program(s):
Safety Data and Analysis
FHWA Activities:
Highway Safety Information System
Subject Area:
Safety and Human Factors