Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling

Bhuwaneshwar Kumar Gupt,F Lalthlamuanpuii,Md Irphan Ahamed

doi:10.13189/ms.2021.090509

Abstract

In survey planning, sometimes, there arises situation to use cluster sampling because of nature of spatial relationship between elements of population or physical feature of land over which elements are dispersed or unavailability of reliable list of elements. At the same time, there requires technique and strategy for ensuring precision of the sample in representing the parent population. Although several theoretical cum practical works have been done in cluster sampling, stratified sampling and stratified cluster sampling, so far, the problem of stratified cluster sampling for a study variable based on an auxiliary variable, which is required in practice, has never been approached. For the first time, this paper deals with the problem of optimum stratification of population of clusters in cluster sampling with clusters of equal size of a characteristic y under study based on highly correlated concomitant variable x for allocation proportional to stratum cluster totals under a super population model. Equations giving optimum strata boundaries (OSB) for dividing population, in which sampling unit of the population is a cluster, are obtained by minimising sampling variance of the estimator of population mean. As the equations are implicit in nature, a few methods of finding approximately optimum strata boundaries (AOSB) are deduced from the equations giving OSB. In deriving the equations, mathematical tools of calculus and algebra are used in addition to statistical methods of finding conditional expectation of variance. All the proposed methods of stratification are empirically examined by illustrating in live data, population of villages in Lunglei and Serchhip districts of Mizoram State, India, and found to perform efficiently in stratifying the population. The proposed methods may provide practically feasible solution in planning socio-economic survey.

Highlights

A heterogeneous population is divided into a number of groups called strata which are within strata homogeneous and sample is selected from strata using suitable sample selection method; the method is used for administrative convenience and enhancing the precision of representation of the sample for the parent population
When the clusters are considered as sampling units of population and stratified by methods of stratified sampling, the inter cluster homogeneity is increased within strata of clusters which in turn serves the purpose of scheming in cluster sampling for enhancing the precision of representation of sample for the parent population
Ever since Dalenius [1] introduced the problem of finding optimum strata boundaries (OBS) based on Tschuprow [2] and Neyman [3] optimum allocation (TNOA) for enhancing homogeneity within strata, the vastness of research in the area has been increasing as a number of researchers, inter alia, Dalenius and Gurney [4], Mahalanobis [5], Hansen et al [6], Dalenius and Hodges [7,8], Ekman [9], etc., embarked on the work who initially used study variable as stratification variable

Summary

Introduction

A heterogeneous population is divided into a number of groups called strata which are within strata homogeneous and sample is selected from strata using suitable sample selection method; the method is used for administrative convenience and enhancing the precision of representation of the sample for the parent population. We have introduced in this paper the problem of optimum stratification for a characteristic under study y based on a highly correlated auxiliary variable x in stratified cluster sampling with clusters of equal size under the following superpopulation model which is a modified form of the model used by Hanurav [24] and Rao [25]. We use information on the auxiliary variable x which is highly correlated with study variable y to stratify population whose units are clusters whereas clusters are formed by grouping the elements in the way discussed above elaborately; the allocation and sample selection procedure used in this work are allocation proportional to stratum total and SRSWR, which will hold true for SRSWOR too when finite population correction is neglected.

Derivation of Methods of Finding Optimum Strata Boundaries

Approximation Based on Series Expansion

Other Approximations Using Assumptions on Coefficient of Variation

Empirical Illustration

Findings

Conclusions