Estimating the Variance 1 Running head: ESTIMATING THE VARIANCE Estimating the Variance in Before-After Studies

Problem: To simplify the computation of the variance in before-after studies, it is generally assumed that the observed crash data for each entity (or observation) are Poisson distributed. Given the characteristics of this distribution, the observed value ( i x ) for each entity is implicitly made equal to its variance. However, the variance should be estimated using the conditional properties of this observed value (defined as a random variable), i.e. ( ) | i i f x μ , since the mean of the observed value is in fact unknown. Method: parametric and non-parametric bootstrap methods were investigated to evaluate the conditional assumption using simulated and observed data. Results: The results of this study show that observed data should not be used as a substitute for the variance, even if the entities are assumed to be Poisson distributed. Consequently, the estimated variance for the parameters under study in traditional before-after studies is likely to be underestimated. Conclusions: the proposed methods offer more accurate approaches for estimating the variance in before-after studies.