The effect of sample design on principal component analysis

Most sample surveys are multivariate and many lend themselves to multivariate methods of analysis. The most usual mode of such analysis is a standard statistical package, such as BMDP or SPSS, in which the multivariate analyses are based on the underlying assumption that the data are generated as independent observations from a common probability distribution. This assumption ignores the sample selection procedure involved in the survey, which leads to the following basic questions. What effects can the sample design have on methods of multivariate analysis? How should such effects be taken into account? This article considers the case of principal component analysis and, in particular, the point estimation of the eigenvalues and eigenvectors of a covariance matrix. It is assumed that the selection of the sample depends on the population values of auxiliary variables as, for example, in stratified sampling. The conventional estimators, based on the assumption of simple random sampling, are compared with alternative probability-weighted and maximum likelihood estimators. Under a multivariate normal model, simple expressions are presented for the approximate model bias of the different estimators. The validity of these results is assessed in a simulation study involving a disproportionate stratified design.