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Many countries conduct a full census survey to report official populationstatistics. As no census survey ever achieves 100 per cent response rate, apost-enumeration survey (PES) is usually conducted and analysed to assesscensus coverage and produce official population estimates by geographic areaand demographic attributes. Considering the usually small size of PES, directestimation at the desired level of disaggregation is not feasible. Design-basedestimation with sampling weight adjustment is a commonly used method but isdifficult to implement when survey non-response patterns cannot be fullydocumented and population benchmarks are not available. We overcome theselimitations with a fully model-based Bayesian approach applied to the NewZealand PES. Although theory for the Bayesian treatment of complex surveys hasbeen described, published applications of individual level Bayesian models forcomplex survey data remain scarce. We provide such an application through acase study of the 2018 census and PES surveys. We implement a multilevel modelthat accounts for the complex design of PES. We then illustrate how mixedposterior predictive checking and cross-validation can assist with modelbuilding and model selection. Finally, we discuss potential methodologicalimprovements to the model and potential solutions to mitigate dependencebetween the two surveys.

Small domain estimation of census coverage: A case study in Bayesian analysis of complex survey data