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

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 BLUES 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.

Plane Answers to Complex Questions

“…In fact, Christensen (1987b) is probably easier to read than Section 2 because it includes more introductory material and fewer of the mathematical details. Closely related work is contained in Monlezun and Blouin (1988) and Mathew and Sinha (1992).…”

Theory and Methods: Hypothesis Testing in Two‐Stage Cluster Sampling

“…Additional work on this topic can be found in Christensen (1987), Thomas & Rao (1987), and Hwang & Perng (1990).…”

Environmetric analysis in complex surveys

“…Part of the growing literature on the subject is based on the traditional sampling framework such as, for example, the works of Konijn (1962) and Kish and Frankel (1974). Others such as Scott and Holt (1982), Deaton and Irish (1983), King and Evans (1986), Christensen (1984Christensen ( , 1986Christensen ( , 1987a, Moulton (1986Moulton ( , 1990), Dorfman (1993), Rao et al (1993), and more recently, Bhatti (1995Bhatti ( , 1996Bhatti ( , 1997Bhatti ( , 1999 have focused on the model-based approach.…”