The Analysis of Two-Stage Sampling Data by Ordinary Least Squares

Abstract

Abstract A reasonably broad class of models for analyzing two-stage samples is considered. For models in this class, ordinary least squares (OLS) estimates are best linear unbiased estimates (BLUE's) and a large variety of exact tests and confidence intervals are available. The necessary computations can be made simply by fitting two models in the ordinary fashion using least squares. The most stringent assumption made is that there are equal numbers of observations taken in each cluster. For such a sampling scheme, OLS estimates are BLUE's iff for each variable in the model, the variable obtained by replacing each component with the corresponding cluster average is also, either implicitly or explicitly, contained in the model. A quite general condition on the parameterization (actually the design matrix) of such models is given that, if satisfied, assures the existence of exact inferential procedures for a wide variety of parametric functions. For any model in which OLS estimates are BLUE's there exist parameterizations that satisfy the condition in question. Results are also given for the problem of testing a given model against a reduced model.