Optimal Sampling Design with Random Size Clusters for a Mixed Model with Measurement Errors

Abstract

Our investigation concerns sampling in epidemiological studies, in the presence of both strata and clusters; the problem is to choose the number of clusters to sample in each stratum given that the size of the clusters in general is a random variable. The issue of unplanned randomness in the design seems to have been scarcely addressed in the survey sampling literature. We were motivated by a sample survey — carried out in 1990–1995 by the Italian National Institute of Nutrition (INN-CA) — on the food habits of the Italian population, divided into four geographical areas: the household came in both as random factor which influenced the individual response and — due to the varying number of its members — as a random component of the design which affected the sample size. In this paper we assume various mixed models under different hypothesis on measurement errors (typically correlated) in the response and for each of them find the optimal designs under several optimality criteria, namely the determinant, the trace, the maximum eigenvalue of the unconditional Fisher information of the fixed effect parameters. In all the models we deal with in the present paper, the optimal design depends on just one unknown parameter τ, a given function of the variance components and correlation coefficients. The dependence of the design on τ is investigated through some simulations. The solutions given for the special cases motivated by the INN-CA study should be applicable to a wider variety of situations.