The asymptotic bias of minimum trace factor analysis, with applications tothe greatest lower bound to reliability

In theory, the greatest lower bound (g.l.b.) to reliability is the best pos sible lower bound to the reliability based on single test administration. Y et the practical use of the g.l.b. has been severely hindered by sampling b ias problems. It is well known that the g.l.b. based on small samples (even a sample of one thousand subjects is not generally enough) may severely ov erestimate the population value, and statistical treatment of the bias has been badly missing. The only results obtained so far are concerned with the asymptotic variance of the g.l.b. and of its numerator (the maximum possib le error variance of a test), based on first order derivatives and the asum ption of multivariate normality. The present paper extends these results by offering explicit expressions for the second order derivatives. This yield s a closed form expression For the asymptotic bias of both the g.l.b, and i ts numerator, under the assumptions that the rank of the reduced covariance matrix is at or above the Ledermann bound, and that the nonnegativity cons traints on the diagonal elements of the matrix of unique variances are inac tive. It is also shown that, when the reduced rank is at its highest possib le value (i.e., the number of variables minus one), the numerator of the g. l.b. is asymptotically unbiased, and the asymptotic bias of the g.l.b, is n egative. The latter results are contrary to common belief, bur apply only t o cases where the number of variables is small. The asymptotic results are illustrated by numerical examples.