  An informative sampling design leads to unit inclusion probabilities that are
correlated with the response variable of interest. However, multistage sampling
designs may also induce higher order dependencies, which are typically ignored
in the literature when establishing consistency of estimators for survey data
under a condition requiring asymptotic independence among the unit inclusion
probabilities. We refine and relax this condition of asymptotic independence or
asymptotic factorization and demonstrate that consistency is still achieved in
the presence of residual sampling dependence. A popular approach for conducting
inference on a population based on a survey sample is the use of a
pseudo-posterior, which uses sampling weights based on first order inclusion
probabilities to exponentiate the likelihood. We show that the pseudo-posterior
is consistent not only for survey designs which have asymptotic factorization,
but also for designs with residual or unattenuated dependence. Using the
complex sampling design of the National Survey on Drug Use and Health, we
explore the impact of multistage designs and order based sampling. The use of
the survey-weighted pseudo-posterior together with our relaxed requirements for
the survey design establish a broad class of analysis models that can be
applied to a wide variety of survey data sets.
