Complete Sample Likelihood Estimation and Inference for Single and Multi-Stage Sample Surveys

In the analysis of sample surveys, accounting for informative sampling design is critical to ensure estimates of model parameters such as regression coefficients are not systematically biased. One popular solution is maximum pseudo-likelihood estimation, which incorporates weights given by functions of the selection probabilities. In this article, we propose an alternative method for analytical estimation and inference in sample surveys known as complete sample likelihood or CSL. This approach jointly postulates a population regression model for the responses, and a model for the selection probabilities conditional on the response and measured covariates. CSL can offer potentially substantial reductions in bias and standard errors compared to maximum pseudo-likelihood estimation, while also avoiding a major drawback of existing “plug-in" sample likelihood techniques which require burdensome resampling techniques to perform inference. We develop general forms for CSL and theoretically justify their usage in both single- and multi-stage sample surveys, before deriving several tractable cases involving generalized linear (mixed) models commonly seen in practice with sample surveys. Simulation studies and an application to a two-stage sample survey of smoking status in New Zealand demonstrate the superior finite sample performance of CSL, and its ability to facilitate deployment of likelihood-based techniques and diagnostic tools.