  Many countries conduct a full census survey to report official population
statistics. As no census survey ever achieves 100 per cent response rate, a
post-enumeration survey (PES) is usually conducted and analysed to assess
census coverage and produce official population estimates by geographic area
and demographic attributes. Considering the usually small size of PES, direct
estimation at the desired level of disaggregation is not feasible. Design-based
estimation with sampling weight adjustment is a commonly used method but is
difficult to implement when survey non-response patterns cannot be fully
documented and population benchmarks are not available. We overcome these
limitations with a fully model-based Bayesian approach applied to the New
Zealand PES. Although theory for the Bayesian treatment of complex surveys has
been described, published applications of individual level Bayesian models for
complex survey data remain scarce. We provide such an application through a
case study of the 2018 census and PES surveys. We implement a multilevel model
that accounts for the complex design of PES. We then illustrate how mixed
posterior predictive checking and cross-validation can assist with model
building and model selection. Finally, we discuss potential methodological
improvements to the model and potential solutions to mitigate dependence
between the two surveys.
