  An informative sampling design leads to the selection of units whose
inclusion probabilities are correlated with the response variable of interest.
Model inference performed on the resulting observed sample will be biased for
the population generative model. One approach that produces asymptotically
unbiased inference employs marginal inclusion probabilities to form sampling
weights used to exponentiate each likelihood contribution of a pseudo
likelihood used to form a pseudo posterior distribution. Conditions for
posterior consistency restrict applicable sampling designs to those under which
pairwise inclusion dependencies asymptotically limit to 0. There are many
sampling designs excluded by this restriction; for example, a multi-stage
design that samples individuals within households. Viewing each household as a
population, the dependence among individuals does not attenuate. We propose a
more targeted approach in this paper for inference focused on pairs of
individuals or sampled units; for example, the substance use of one spouse in a
shared household, conditioned on the substance use of the other spouse. We
formulate the pseudo likelihood with weights based on pairwise or second order
probabilities and demonstrate consistency, removing the requirement for
asymptotic independence and replacing it with restrictions on higher order
selection probabilities. Our approach provides a nearly automated estimation
procedure applicable to any model specified by the data analyst. We demonstrate
our method on the National Survey on Drug Use and Health.

Methodology