A family of multi-level models with different types of random error components in the linear predictor is presented for analysing longitudinal count data in clinical trials. These models account for overdispersion, heterogeneity, serial correlation, and heteroscedasticity. The proposed models are applied to epileptic seizure count data and illustrated in a simulation study. The effects of omitted variables, link function, outliers, and initial conditions on overdispersion are investigated. It has been shown that proper introduction of the error component in the linear predictor overcomes the problem of overdispersion arising from the omitted variables. We use three model checking criteria deviance, variance inflation factor, and global goodness-of-fit tests based on Bayesian probability to identify the best structure of the error term in the linear predictor. Further, Markov Chain Monte Carlo method using Gibbs sampling is used as estimation approach.