Empirical likelihood inference for regression parameters when modelling hierarchical complex survey data

The data used in social, behavioural, health or biological sciences may have a hierarchical structure due to the natural structure in the population of interest or due to the sampling design. Multilevel or marginal models are often used to analyse such hierarchical data. The data may include sample units selected with unequal probabilities from a clustered and stratified population. Inferences for the regression coefficients may be invalid when the sampling design is informative. We apply a profile empirical likelihood approach to the regression parameters, which are defined as the solutions of a generalised estimating equation. The effect of the sampling design is taken into account. This approach can be used for point estimation, hypothesis testing and confidence intervals for the subvector of parameters. It asymptotically provides valid inference for the finite population parameters under a set of regularity conditions. We consider a two–stage sampling design, where the first stage units may be selected with unequal probabilities. We assume that the model and sampling hierarchies are the same. We treat the first stage sampling units as the unit of interest, by using an ultimate cluster approach. The estimating functions are defined at the ultimate cluster level of the hierarchy.