Fitting complex survey data to regression equations is explored under a design-sensitive model-based framework. A robust version of the standard model assumes that the expected value of the difference between the dependent variable and its model-based prediction is zero no matter what the values of the explanatory variables. The extended model assumes only that the difference is uncorrelated with the covariates. Little is assumed about the error structure of this difference under either model other than independence across primary sampling units. The standard model often fails in practice, but the extended model very rarely does. Under this framework some of the methods developed in the conventional design-based, pseudo-maximum-likelihood framework, such as fitting weighted estimating equations and sandwich mean-squared-error estimation, are retained but their interpretations change. Few of the ideas here are new to the refereed literature. The goal instead is to collect those ideas and put them into a unified conceptual framework.
