Cluster sampling is common in survey practice, and the corresponding
inference has been predominantly design-based. We develop a Bayesian framework
for cluster sampling and account for the design effect in the outcome modeling.
We consider a two-stage cluster sampling design where the clusters are first
selected with probability proportional to cluster size, and then units are
randomly sampled inside selected clusters. Challenges arise when the sizes of
nonsampled cluster are unknown. We propose nonparametric and parametric
Bayesian approaches for predicting the unknown cluster sizes, with this
inference performed simultaneously with the model for survey outcome.
Simulation studies show that the integrated Bayesian approach outperforms
classical methods with efficiency gains. We use Stan for computing and apply
the proposal to the Fragile Families and Child Wellbeing study as an
illustration of complex survey inference in health surveys