Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

The negative binomial (NB) (or Poisson–gamma) model has been used extensively by highway safety analysts because it can accommodate the overdispersion often exhibited in crash data. However, it has been reported in the literature that the maximum likelihood estimate of the dispersion parameter of NB models can be significantly affected when the data are characterized by small sample size and low sample mean. Given the important roles of the dispersion parameter in various types of highway safety analyses, there is a need to determine whether the bias could be potentially corrected or minimized. The objectives of this study are to explore whether a systematic relationship exists between the estimated and true dispersion parameters, determine the bias as a function of the sample size and sample mean, and develop a procedure for correcting the bias caused by these two conditions. For this purpose, simulated data were used to derive the relationship under the various combinations of sample mean, dispersion parameter, and sample size, which encompass all simulation conditions performed in previous research. The dispersion parameter was estimated by using the maximum likelihood method. The results confirmed previous studies and developed a reasonable relationship between the estimated and true dispersion parameters for reducing the bias. Details for the application of the correction procedure were also provided by using the crash data collected at 458 three-leg unsignalized intersections in California. Finally, the study provided several discussion points for further work.