A Weibull-count approach for handling under- and overdispersed longitudinal/clustered data structures

Abstract

A Weibull-model-based approach is examined to handle under- and overdispersed discrete data in a hierarchical framework. This methodology was first introduced by Nakagawa and Osaki (1975, IEEE Transactions on Reliability, 24, 300–301), and later examined for under- and overdispersion by Klakattawi et al. (2018, Entropy, 20, 142) in the univariate case. Extensions to hierarchical approaches with under- and overdispersion were left unnoted, even though they can be obtained in a simple manner. This is of particular interest when analysing clustered/longitudinal data structures, where the underlying correlation structure is often more complex compared to cross-sectional studies. In this article, a random-effects extension of the Weibull-count model is proposed and applied to two motivating case studies, originating from the clinical and sociological research fields. A goodness-of-fit evaluation of the model is provided through a comparison of some well-known count models, that is, the negative binomial, Conway–Maxwell–Poisson and double Poisson models. Empirical results show that the proposed extension flexibly fits the data, more specifically, for heavy-tailed, zero-inflated, overdispersed and correlated count data. Discrete left-skewed time-to-event data structures are also flexibly modelled using the approach, with the ability to derive direct interpretations on the median scale, provided the complementary log–log link is used. Finally, a large simulated set of data is created to examine other characteristics such as computational ease and orthogonality properties of the model, with the conclusion that the approach behaves best for highly overdispersed cases.

A robust Bayesian mixed effects approach for zero inflated and highly skewed longitudinal count data emanating from the zero inflated discrete Weibull distribution

“…An advantage of the Bayesian implementation of the mixed effects DW regression model is that it does not involve asymptotic approximations, unlike the frequentist method by Luyts et al (ie, the SAS procedure NLMIXED), and therefore may be more suitable for small sample problems. Moreover, the implementation of the ZIDW regression model (using JAGS) is user friendly and competitive with the ZINB regression model in terms of computational speed and model convergence.…”

“…This article proposed a Bayesian mixed effects ZIDW regression model for longitudinal count data as an alternative to mixed effects regression models that are based on the usual NB, ZINB, and conventional DW distribution. The mixed effects ZIDW regression model is an extension of the method proposed by Luyts et al 11 and uses the log-link function to specify the relationship between the linear predictors and the median counts, therefore offering a robust characteristic of central tendency, compared to the mean count, when the distribution of the data is skew. An advantage of the Bayesian implementation of the mixed effects DW regression model is that it does not involve asymptotic approximations, unlike the frequentist method by Luyts et al 11 (ie, the SAS procedure NLMIXED 12 ), and therefore may be more suitable for small sample problems.…”

“…These distributions, however, lack robustness when modeling regression functions for longitudinal count data that exhibit high skewness. As an alternative, Luyts et al recently introduced a flexible approach that uses the discrete Weibull (DW) distribution in a mixed effects regression model in order to accommodate underdispersed, overdispersed, and highly skewed data using the SAS procedure NLMIXED…”

“…Luyts et al argued that the DW distribution, according to the zero inflation index by Puig and Valero, is able to accommodate count data that are zero inflated and zero deflated, and therefore suggested that the DW distribution can be fitted to count data “without the need for introducing zero‐inflated or hurdle components.” However, as shown in the supplementary material of this article, the zero inflation index by Puig and Valero would also suggest that the conventional NB distribution is able to accommodate zero inflated counts while it is well‐known that this distribution cannot handle all forms of excess zeros (eg, structural zeros). These observations suggest that the conventional DW distribution is not necessarily fully robust to excess zeros.…”

“…10 These distributions, however, lack robustness when modeling regression functions for longitudinal count data that exhibit high skewness. As an alternative, Luyts et al 11 recently introduced a flexible approach that uses the discrete Weibull (DW) distribution in a mixed effects regression model in order to accommodate underdispersed, overdispersed, and highly skewed data using the SAS procedure NLMIXED. 12 The mixed effects DW regression model is flexible in the sense that it allows one to model the conditional median as an alternative to the conditional mean, which, in turn, offers a robust characteristic of central tendency when the distribution of the data is skew.…”