Fitting H. F. Smith's Empirical Law to Cluster Variances for Use in Designing Multi-Stage Sample Surveys

Abstract

A number of methods are examined for using data from a multistage sample survey to estimate H. Fairfield Smith's heterogeneity coefficient, b . Smith's empirical law holds that cluster variances are proportional to the — b power of cluster size. The most reasonable method for estimating b seems to be generalized least squares applied to the linear model obtained by taking logs and then adding a lack-of-fit variance. The methods allow for simple measurement errors and systematic fixed effects, all in a finite population context. There is, however, a supposition of nearly equal or balanced sizes of the nested units. The b value so estimated can be used to derive optimum elementary cluster size and optimum subsampling rates for the multi-stage sample design. These applications are illustrated by numerical examples.