Balanced repeated replications based on mixed orthogonal arrays

Abstract

SUMMARY For stratified samples with arbitrary numbers of primary sampling units per stratum, one can divide the units in the stratum into two groups and then apply the balanced half-samples method to the two groups thus formed. The efficiency loss of the grouped balanced half-samples variance estimator can be substantial. Use of a mixed orthogonal array, whose numbers of symbols correspond to the numbers of units per stratum, gives a set of balanced subsamples. Unlike that of Gupta & Nigam (1987), the proposed method of analysis gives a consistent variance estimator and bias reduction for nonlinear statistics. To further reduce the number of replicates, use of nearly orthogonal arrays is considered. The balanced half-samples method (McCarthy, 1966, 1969) is commonly used for estimating the variance of nonlinear statistics from stratified samples with two primary sampling units per stratum. These samples arise naturally if deep stratification is desired as in some large-scale surveys. The method is extended (Gurney & Jewett, 1975) to the case of p primary sampling units per stratum for prime p. This extension for p > 2 is not adopted in common practice since it requires a much larger number of replicates. Also, in many survey designs the numbers of primary sampling units are unequal. A common practice is to divide the units in the stratum into two groups and then to apply the balanced half-samples method to the two groups thus formed. This has gained popularity because the software for obtaining balanced half-samples is readily available and the number of replicates required is more economical. As shown in ? 2, the resulting variance estimators can be quite inefficient and their bias based on a conditional analysis can be considerable. The conditional analysis also suggests a method of grouping to reduce the bias. The degree of grouping depends on the tradeoff between computational economy and statistical efficiency. If the efficiency loss from grouping is not justifiable, one may draw replicates by using mixed orthogonal arrays with variable numbers of symbols to accommodate the different numbers of units or groups of units in the strata. As long as an orthogonal array is used for drawing replicates, we call the method balanced repeated replications. In ? 3 we show that the analysis method of Gupta & Nigam (1987) based on mixed orthogonal arrays leads to an inconsistent variance estimator for nonlinear statistics, and propose a method that is valid for variance estimation and bias reduction