Generalized estimating equations for longitudinal generalized Poisson count data with regression effects on the mean and dispersion level

Generalized estimating equations (GEE) fit parameters based on sums of weighted residuals, which may be applied for example to the Poisson distribution. We discuss Generalized Poisson (GP) response data. This distribution has a more flexible variance function than the Poisson distribution and has an additional dispersion parameter. To fit this parameter, second level estimating equations based on covariance residuals are necessary. This requires knowledge of variances of empirical covariances, which for most discrete distributions except the binary cannot be derived from first level GEE. We approximate them by a novel approach. We allow for regression on mean and overdispersion parameters. In an application we deal with the outsourcing of patent filing processes. Exploratory data analysis tools developed earlier by the authors are utilized to choose regression for the dispersion parameters. For the given data, our approach will outperform longitudinal Poisson regression and GP setups with constant dispersion.