The Determinants of the Bias in Minimum Rank Factor Analysis (MRFA)

Summary

Minimum Rank Factor Analysis (MRFA), see Ten Berge (1998), and Ten Berge and Kiers (1991), is a method of common factor analysis which yields, for any given covariance matrix Σ, a diagonal matrix Ψ of unique variances which are nonnegative and which entail a reduced covariance matrix Σ—Ψ which is positive semidefinite. Subject to the above constraints, MRFA minimizes the amount of common variance left unexplained when we take any fixed small number of common factors. Shapiro and Ten Berge (2002) have derived the asymptotic bias of the unexplained common variance, its variance, and also the asymptotic covariance matrix of the unique variances. The present research deals with the impact of sample size, population minimum rank, number of extracted factors and standardization of the sample covariance matrix on the bias of unexplained common variance and total common variance. Special attention was paid to situations where the asymptotic theory does not apply. The results indicate that the bias could present a practical problem only if the population minimum rank was unnaturally low or if the sample size was small.