The First Order Autoregressive (AR(1)) Mixed Effects Zero Inflated Poisson (ZIP) Model was developed to analyze longitudinal zero inflated Poisson data through the Bayesian Approach. The model describes the effect of covariates via regression and time varying correlations within subject. Subjects are classified into a "perfect" state with response equal to zero and a Poisson state with response following a Poisson regression model. The probability of belonging to the perfect state or Poisson state is governed by a logistic regression model. Both models include autocorrelated random effects, and there is correlation between random effects in the logistic and Poisson regressions.

Parameter estimation is investigated using simulation studies and analyses (both frequentist and Bayesian) of simpler mixed effect models. In the large sample setting we investigate the Fisher information of the model. The Fisher information matrix is then used to determine an adequate sample size for the AR(1) ZIP model. Simulation studies demonstrate the capability of Bayesian methods to estimate the parameters of the AR(1) ZIP model for longitudinal zero inflated Poisson data. However, a tremendous computation time and a huge sample size are required by the full AR(1) ZIP model.

Simpler models were fitted to simulated AR(1) ZIP data to investigate whether simplifying the assumed random structure could permit accurate estimates of fixed effect parameters. However, simulations showed that the bias of two nested models, ZIP model and mixed effects ZIP model, are too large to be acceptable. The AR(1) ZIP model was fitted to data on numbers of cigarettes smoked, collected in the National Longitudinal Study of Youth. It was found that decisions on whether to smoke and on the number of cigarettes to smoke were significantly related to age, sex, race and smoking behavior by peers. The random effect variances, autocorrelation coefficients and correlation between logistic and Poisson random effect were all significant.