On Variances of Ratios and their Differences in Multi-Stage Samples

Abstract

We discuss the computation of variances for the estimators r = y/x and (r — r') where the random variables (variates) y and x are sample totals for two variates obtained from some multi-stage design. The variate x often represents the sample size; then the ratio r = y/x is the simple and usual sample mean or proportion and a common statistic for presenting the results of sample surveys. The difference (r — r') occurs frequently and importantly in multistage samples either as the change in the estimates of the same characteristic obtained from two different surveys, or as the comparison of some characteristic estimated for two subclasses from the same survey. Several useful computational forms are presented for var(r — r') = var(r) + var(r') — 2 cov(r, r'). The aims of the presentation are: (a) to be general enough to cover the complexities which arise frequently in practical sample designs; (b) to provide easy computing formulas for good approximations; and (c) to make the procedures comprehensible to nontechnicians.A revised version of a paper contributed to the 115tn Annual Meeting of the American Statistical Association in New York on December 29, 1955. It benefited from a special “off-campus” assignment for research granted in 1958 by the Survey Research Center, University of Michigan to the senior author. As a Visitor in the Department of Statistics at Harvard University he received helpful suggestions from William G. Cochran. © Taylor & Francis Group, LLC.