Out-of-sample prediction and interpretation for random parameter generalized linear models

Incorporating random parameters (RPs) into generalized linearized models (GLMs) - such as the negative binomial (NB) regression model used to predict crash frequencies - has been shown to improve model fit and better address issues such as unobserved heterogeneity. However, applying models with RPs to make predictions for observations outside the sample used to estimate the model is not straightforward. Recent studies have proposed various methods to incorporate RPs in out-of-sample predictions, but these tend to provide biased estimates or are computationally intensive to apply. This paper applies fundamental statistical theory to leverage properties of the underlying RP distributions incorporated into GLMs to provide more direct and accurate predictions, as well as directly estimate prediction variance for out-of-sample observations. Methods are provided for several common RP distributions - including the normal/Gaussian, lognormal, triangular, uniform, and gamma distributions - combined within log-link GLM framework. Additionally, closed-form equations for elasticities and marginal effects for the random parameters are provided. The proposed methods are tested using crash frequency prediction models developed using data from the Highway Safety Information System (HSIS). The results suggest that the proposed exact method provides more accurate predictions than the computational-intensive simulation-based approximation approaches while also being simple to apply. The method is suitable for the widespread use of RPs in research and in practical applications of GLMs.