Modelling overdispersion for complex survey data

The population characteristics observed by selecting a complex sample from a finite identified population are the result of at least two processes: the process which generates the values attached to the units in the finite population, and the process of selecting the sample of units from the population. In this paper we propose that the resulting observations be viewed as the joint realization of both processes. We overcome the inherent difficulty in modelling the joint processes of generation and selection by exploring second moment and other simplifying assumptions. We obtain general expressions for the mean and covariance function of the joint processes and show that several overdispersion models discussed in the literature for the analysis of complex surveys are a direct consequence of our formulation, under particular sampling schemes and population structures.