Estimating the variance in before-after studies
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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 (x(i)) 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), that is, f(x(i)/mu(i)), 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.
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