The objective of this paper is to study the feasibility and consistency of complex survey estimation methods for population models incorporating shared cluster-level random-effect parameters, in contexts where sampling may be informative and where joint inclusion-probabilities are unavailable. There are several well-known papers in this area (Binder 1983, Pfeffermann et al. 1998, Korn and Graubard 2003, Rabe-Hesketh and Skrondal 2006) and several recent papers (Rao et al. 2013, Yi et al. 2016, Kim et al. 2017, Savitsky and Williams 2018, and Williams and Savitsky 2019), but the problem of model- and design-consistent estimation of variance-component parameters in surveys has not yet been solved in a practically effective way.

One contribution of this paper is to propose an EM algorithm applied to the pseudo-loglikelihood estimating an augmented census-loglikelihood incorporating cluster random effects. This algorithm consistently estimates superpopulation variance components under assumed mixed-effect models for survey data under probability sampling designs in which sampling of clusters may be informative but within-cluster sampling is not. A second contribution is to assess the performance of all of the competing proposed methods that use only single-inclusion weights, in the presence of informative sampling under the two-level one-way random-effects Analysis of Variance model. This comparison supports the conclusion that none of these methods is consistent under general informative sampling.