Design effects and ratios of homogeneity in complex sampling designs

Abstract

In the literature, a simple formula has been proposed which relates the design effect, deff, to a ratio of homogeneity among population elements within primary sampling units, roh. The formula is deff=1+(m- 1) roh, where m is the average number of elements sampled per selected primary unit. In this paper, we analyse the validity of the above formula in a general framework of a complex multistage sampling design and from a randomization theory point of view. The analy-sis is carried out for expansion and ratio estimators of population and domain means. The chief analytical results consist in expressing the ratio of homogeneity as a sum of a population parameter and other components related to different design features. It is then shown that departures from self-weighting and from simple random sampling within primary units are the main design components that may not comply with the desidered independence of roh from m.