Two-phase sampling for selection with probability proportional to size in sample surveys

Abstract

SUMMARY Sarndal & Swensson (1987) have given a general framework of estimation in two-phase sampling assuming arbitrary probability by sampling designs in both the phases where the selections are without replacement. This note discusses a specific with replacement version of the scheme where a ratio of two variables is suggested as a measure of size of the units and a mean of the ratios-type estimator is proposed. A comparison is made with the Raj (1965) scheme employing the difference method of estimation. Two-phase sampling requires collection of information on an auxiliary character x for the first phase sample and on the study character y for the second phase sample. A variety of schemes have been proposed in this context. Most of them suggest selection of the initial sample with equal probabilities. The value of x is ascertained for the units in this sample. This information may be used either for stratifying the initial sample or as a size measure for drawing a subsample or even to construct ratio or regression type estimators. Rao (1973) discusses the use of two-phase sampling in analytical surveys and Chaudhuri & Adhikary (1983) examine the optimality of such strategies with varying probabilities. Sdrndal (1982) develops a generalized regression approach for the situation. Sdrndal & Swensson (1987) give a general framework for estimation in two-phase sampling assuming arbitrary probability designs in the two phases and outline a general approach to estimating the variance of the two-phase estimator. Sometimes information on a second auxiliary character z may be readily available. For instance, while estimating the total yield of wheat in a village, the yield and area under the crop are likely to be unknown. But the total area of each farm may be known from village records or may be obtained at a low cost. Then y, x and z are respectively yield, area under wheat and area under cultivation. Raj (1965) suggests selection of the initial sample s' of size n' with replacement with probabilities pi proportional to zi (i = 1, . . ., N), where N is the population size. Information on the character x is gathered on s'. The second phase sample s of size n is a subsample of s', selected with equal probabilities without replacement on which information on y is collected. Call this scheme 1. Under this scheme, using a difference estimator, an unbiased estimator of the population total for y is provided by n n n'