Relative efficiencies of two‐stage sampling schemes for mean estimation in multilevel populations when cluster size is informative

Abstract

In multilevel populations, there are two types of population means of an outcome variable ie, the average of all individual outcomes ignoring cluster membership and the average of cluster‐specific means. To estimate the first mean, individuals can be sampled directly with simple random sampling or with two‐stage sampling (TSS), that is, sampling clusters first, and then individuals within the sampled clusters. When cluster size varies in the population, three TSS schemes can be considered, ie, sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster; sampling clusters with equal probability and then sampling the same percentage of individuals per cluster; and sampling clusters with equal probability and then sampling the same number of individuals per cluster. Unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative. Furthermore, the three TSS schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size. The relative efficiency of the sampling schemes is shown to vary across different cluster size distributions. However, sampling clusters with probability proportional to size is the most efficient TSS scheme for many cluster size distributions. Model‐based and design‐based inference are compared and are shown to give similar results. The results are applied to the distribution of high school size in Italy and the distribution of patient list size for general practices in England.

Optimal two-stage sampling for mean estimation in multilevel populations when cluster size is informative

“…There is no such equivalence under TSS2 due to sample size variation between clusters, and under TSS3 due to weighting cluster means by cluster size if informative cluster size is assumed in the design phase. Indeed, under noninformative cluster size, no weighting is needed under TSS3, 15 and then the optimal design equations for TSS1 apply to TSS3 as well. Note from Table 2 that the optimal number of clusters k à and the optimal number of individuals per cluster n à are inversely related, and that n à is an increasing function of the cluster-to-individual cost ratio c r ¼ c 2 c 1 > 1 and a decreasing function of q and w. These relations between the optimal design and c r , q, and w hold, under TSS1, for…”

“…Innocenti et al. 15 show that β0 in model (1) is the average of all cluster-specific means in the population, and differs from the average of all individual outcomes in the population μ, unless cluster size is non-informative ( α1=0) or constant across clusters, as can be seen from the following expression where θN, τN=σNθN, and σN2 are, respectively, the population mean, the coefficient of variation, and the variance of cluster size. The distinction between β0 and μ comes from considering the distribution of cluster effect uj over either the population of clusters (which yields β0) or the population of individuals (which yields μ).…”

“…The distinction between β0 and μ comes from considering the distribution of cluster effect uj over either the population of clusters (which yields β0) or the population of individuals (which yields μ). 15 This paper focuses on μ.…”

“…Note that this only holds if informative cluster size ( ψ≠0) is assumed at the design stage, such that in TSS3 cluster means are weighted by cluster size to estimate μ (Table 1). If non-informative cluster size ( ψ=0) is assumed already in the design stage, then no weighting is needed for TSS3, 15 and TSS3 then is as efficient as TSS1.…”

Efficient designs for mean estimation in multilevel populations and test norming

“…For instance, Seaman et al (2014) have discussed several methods to make clusterspecific inferences with Generalized Linear Mixed Models and population-average inferences with Generalized Estimating Equations when cluster size is informative. Innocenti et al (2019), instead, have investigated a different topic: the implications of informative cluster size for unbiased and efficient estimation of a population mean in surveys conducted with the three aforementioned TSS schemes. The present paper is also about mean estimation for these three TSS schemes when cluster size is informative, but focuses instead on sample size planning, and the consequences of informative cluster size for the required sample sizes and budget.…”