Modelling of highly skewed longitudinal count data based on the discrete Weibull distribution

Abstract

Longitudinal data refer to multiple observations collected on the same subject (or unit) over time. Zero-inflated data (containing many zeros) frequently occur, resulting in overdispersion in count data. Regression models used to analyze count data are often based on the Poisson and negative binomial (NB) distribution. The Poisson distribution is restrictive when count data are overdispersed; the regression model can, therefore, give inappropriate fits when the variability in the data is larger or smaller than the theoretical variance. These two cases are, respectively, referred to as overdispersion and underdispersion. The NB distribution handles overdispersed data better compared to the Poisson distribution, but not underdispersed data. Another problem with the NB distribution is that it does not accommodate heavy-tailed or highly skewed data well. In this study, the discrete Weibull (DW) and the zero-inflated DW (ZIDW) distributions are explored in a mixed model context that models the median using a Bayesian approach. In contrast, the conventional NB and ZINB mixed-effects regression models model the mean counts over time. Results from the four mixed-effects regression models are compared. It is observed that the Bayesian DW and ZIDW mixed-effects regression models are computationally competitive with the Bayesian NB and ZINB mixed-effects regression models concerning flexibility, implementation, and convergence speed. The DW and ZIDW models are found to be excellent choices to model highly skewed longitudinal count data.