An informative sampling design leads to the selection of units whose inclusion probabilities are correlated with the response variable of interest. Inference under the population model performed on the resulting observed sample, without adjustment, 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.
"Bayesian pairwise estimation under dependent informative sampling." Electron. J. Statist. 12 (1) 1631 - 1661, 2018. https://doi.org/10.1214/18-EJS1435