On the nature of over-dispersion in motor vehicle crash prediction models

Abstract

Statistical modeling of traffic crashes has been of interest to researchers for decades. Over the most recent decade many crash models have accounted for extra-variation in crash counts-variation over and above that accounted for by the Poisson density. The extra-variation -or dispersion -is theorized to capture unaccounted for variation in crashes across sites. The majority of studies have assumed fixed dispersion parameters in over-dispersed crash models-tantamount to assuming that unaccounted for variation is proportional to the expected crash count. Miaou and Lord [Miaou, S.P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes versus empirical Bayes methods. Transport. Res. Rec. 1840, 31-40] challenged the fixed dispersion parameter assumption, and examined various dispersion parameter relationships when modeling urban signalized intersection accidents in Toronto. They suggested that further work is needed to determine the appropriateness of the findings for rural as well as other intersection types, to corroborate their findings, and to explore alternative dispersion functions.This study builds upon the work of Miaou and Lord, with exploration of additional dispersion functions, the use of an independent data set, and presents an opportunity to corroborate their findings. Data from Georgia are used in this study. A Bayesian modeling approach with noninformative priors is adopted, using sampling-based estimation via Markov Chain Monte Carlo (MCMC) and the Gibbs sampler. A total of eight model specifications were developed; four of them employed traffic flows as explanatory factors in mean structure while the remainder of them included geometric factors in addition to major and minor road traffic flows. The models were compared and contrasted using the significance of coefficients, standard deviance, chi-square goodness-of-fit, and deviance information criteria (DIC) statistics. The findings indicate that the modeling of the dispersion parameter, which essentially explains the extra-variance structure, depends greatly on how the mean structure is modeled. In the presence of a well-defined mean function, the extra-variance structure generally becomes insignificant, i.e. the variance structure is a simple function of the mean. It appears that extra-variation is a function of covariates when the mean structure (expected crash count) is poorly specified and suffers from omitted variables. In contrast, when sufficient explanatory variables are used to model the mean (expected crash count), extra-Poisson variation is not significantly related to these variables. If these results are generalizable, they suggest that model specification may be improved by testing extra-variation functions for significance. They also suggest that known influences of expected crash counts are likely to be different than factors that might help to explain unaccounted for variation in crashes across sites.

The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives

“…While model estimates of overdispersion can be attributed to these factors, Lord et al (2005b) argued that there is a fundamental explanation for overdispersion that can be shown by viewing crash data as the product of Bernoulli trials with an unequal probability of events (this is also known as Poisson trials). Recently, some researchers have reported that modelestimated overdispersion can be greatly minimized by improving the model specification Mitra and Washington, 2007). data that is much more aggregated and thus important information is lost by using discrete time intervals -with larger intervals resulting in more information loss.…”

Modelling area-wide count outcomes with spatial correlation and heterogeneity: An analysis of London crash data

“…Overdispersion may result from unobserved heterogeneity that is inadequately captured by the explanatory variables (X) in the conditional mean function (Cameron and Trivedi, 1998). Using a Bayesian modelling approach, Mitra and Washington (2007) reported that extra-variation is a function of explanatory variables when the mean function (expected crash count) is poorly defined and suffers from omitted variables.…”

The Poisson inverse Gaussian (PIG) generalized linear regression model for analyzing motor vehicle crash data

“…Previous studies have shown that the dispersion parameter can also be modeled as a function of the explanatory variables to explain more variation. The link functional form for dispersion parameters was also investigated by authors (Hauer, 2001, Heydecker and Wu, 2001, Miaou and Lord, 2003, Mitra and Washington, 2007 . Considering that Equation (15) contains all the covariates (Mitra and Washington, 2007), we simply modeled the dispersion parameters of NB and PIG models as the function of segment length in the format recommended by Geedipally and Lord (2011) The nonlinear relationship provides more flexibility to capture the variance of the data.…”

“…The commonly used link function which associated crash with covariates such as daily traffic flow, lane width was chosen. Both models were developed using varying dispersion parameters for the flexibility of accounting for the variation and better statistical fit (Hauer, 2001, Heydecker and Wu, 2001, Miaou and Lord, 2003, Mitra and Washington, 2007, Geedipally and Lord, 2011. Models with fixed dispersion parameters were also listed for comparison purposes.…”