  We devise survey-weighted pseudo posterior distribution estimators under
two-stage informative sampling of both primary clusters and secondary nested
units for a one-way analysis of variance (ANOVA) population generating model as
a simple canonical case where population model random effects are defined to be
coincident with the primary clusters, for example student performance based on
a survey of schools and students such as the 2000 OECD Programme for
International Student Assessment (PISA). We consider estimation on an observed
informative sample under both an augmented pseudo likelihood that co-samples
the random effects, as well as an integrated likelihood that marginalizes out
the random effects from the survey-weighted augmented pseudo likelihood. This
paper includes a theoretical exposition that enumerates easily verified
conditions for which estimation under the augmented pseudo posterior is
guaranteed to be consistent at the true generating parameters. We reveal in
simulation that both approaches produce asymptotically unbiased estimation of
the generating hyperparameters for the random effects when a key condition on
the sum of within cluster weighted residuals is met. We present a comparison
with two frequentist alternatives, an expectation-maximization approach and a
composite likelihood method that requires pairwise sampling weights.
