Mixed Models For Complex Survey Data

I want to fit a mixed model to a population distribution, but I have data from a complex (multistage) sample. The sampling is informative, that is, the model holding for the population is different from the model holding for the (biased) sample. Ignoring the sampling design and just fitting the mixed model to the sample distribution will lead to biased inference. Although both the model and sampling involve “clusters”, the model clusters and sample clusters need not be the same. I will use a pairwise composite likelihood method to estimate the parameters of the population model under this setting. In particular, consistency and asymptotic normality can be established. Variance estimation in this problem is challenging. I will talk about a variance estimator and how to show it is consistent.