Models for Zero-Inflated and Overdispersed Correlated Count Data: An Application to Cigarette Use

Count outcomes in tobacco research are often analyzed with the Poisson distribution. However, they often exhibit features such as overdispersion (variance larger than expected) and zero inflation (extra zeros) that violate model assumptions. Furthermore, longitudinal studies have repeated measures that generate correlated counts. Failure to account for overdispersion, zero inflation, and correlation can yield incorrect statistical inferences. Thus, it is important to familiarize researchers with proper models for such data.

Poisson and Negative Binomial models with correlated random effects with and without zero inflation are presented. The illustrative data comes from a study comparing a mindfulness training app (Craving to Quit [C2Q], n = 60) with a control app (experience sampling-only app, n = 66) on smoking frequency at 1, 3, and 6 months. Predictors include app, time, the app-by-time interaction, and baseline smoking. Each model is evaluated in terms of accounting for zero inflation, overdispersion, and correlation in the data. Emphasis is placed on evaluating model fit, subject-specific interpretation of effects, and choosing an appropriate model.

The hurdle Poisson model provided the best fit to the data. Smoking abstinence rates were 33%, 32%, and 28% at 1, 3, and 6 months, respectively, with variance larger than expected by a factor >7 at each follow-up. Individuals on C2Q were less likely to achieve abstinence across time but likely to smoke fewer cigarettes if smoking.

The models presented are specifically suited for analyzing correlated count outcomes and account for zero inflation and overdispersion. We provide guidance to researchers on the use of these models to better inform nicotine and tobacco research.

In tobacco research, count outcomes are often measured repeatedly on the same subject and thus correlated. Such outcomes often have many zeros and exhibit large variances relative to the mean. Analyzing such data require models specifically suited for correlated counts. The presented models and guidelines could improve the rigor of the analysis of correlated count data and thus increase the impact of studies in nicotine and tobacco research using such outcomes.