Factor analysis of correlation matrices when the number of random variables exceeds the sample size

Factor analysis which studies correlation matrices is an effective means of data reduction whose inference on the correlation matrix typically requires the number of random variables, p, to be relatively small and the sample size, n, to be approaching infinity. In contemporary data collection for biomedical studies, disease surveillance and genetics, p > n limits the use of existing factor analysis methods to study the correlation matrix. The motivation for the research here comes from studying the correlation matrix of log annual cancer mortality rate change for p = 59 cancer types from 1969 to 2008 (n = 39) in the U.S.A. We formalise a test statistic to perform inference on the structure of the correlation matrix when p > n. We develop an approach based on group sequential theory to estimate the number of relevant factors to be extracted. To facilitate interpretation of the extracted factors, we propose a BIC (Bayesian Information Criterion)-type criterion to produce a sparse factor loading representation. The proposed methodology outperforms competing ad hoc methodologies in simulation analyses, and identifies three significant underlying factors responsible for the observed correlation between cancer mortality rate changes.