Effects of Sample Size on Goodness-of-Fit Statistic and Confidence Intervals of Crash Prediction Models Subjected to Low Sample Mean Values

Abstract

The statistical relationship between motor vehicle crashes and covariates can generally be modeled via generalized linear models (GLMs) with logarithmic links with errors distributed in a Poisson or Poisson-gamma manner. The scaled deviance and Pearson's χ2 have been proposed to test the statistical fit of GLMs. Recent studies have shown that these two estimators are not adequate for testing the goodness of fit (GOF) of GLMs when they are developed from data characterized by low sample mean values. To circumvent this problem, a testing method has been proposed to evaluate the GOF of such GLMs. Because this method can be time-consuming to implement, there is a need to determine whether it is sensitive to different sample sizes. The primary objective of this paper is to investigate the effects of decreasing sample sizes on the GOF testing technique. A secondary objective is to estimate how the reducing of sample size influences the confidence intervals of GLMs. To accomplish the objectives, GLMs were fit with the use of two data sets subjected to average and low sample means collected in Toronto, Ontario, Canada. Several models were estimated for different sample sizes. The results of the study show that the testing technique is more effective for smaller than for larger samples when data are subjected to low sample mean values. The results also show that the width of the confidence intervals increases, as expected, as the sample size decreases and can be extremely large for small sample sizes. Hence, statistical models characterized by low sample mean values should be developed on the basis of a large number of observations. Data sets containing at least 100 observations (e.g., intersections, segments) are recommended in the development of models. The paper concludes with recommendations for future studies involving such data sets.