SIMULATIONS AND DERIVED APPROXIMATIONS FOR THE MEANS AND STANDARD DEVIATIONS OF THE CHARACTERISTIC ROOTS OF A WISHART MATRIX

Abstract

Monte Carlo simulations were done to estimate the means and standard deviations of the characteristic roots of a Wishart matrix which can be used in computing tests of hypotheses concerning multiplicative terms in balanced linear-bilinear (multiplicative) models for an m × n table of data. In this report we extend the previous results (Mandel, 1971; Cornelius, 1980) to r ≤ 199, c ≤ 149 or r ≤ 149, c ≤ 199, where r and c are row and column degrees of freedom, respectively, of the two-way array of residuals (with total degrees of freedom rc) after fitting the linear effects. For 187 combinations of r and c at intervals over this domain, we used 5000 simulations to estimate expectations and standard deviations of the Wishart roots. Using weighted linear regression variable selection techniques, symmetric functions of r and c were obtained for approximating the simulated means and standard deviations. Use of these approximating functions will avoid the need for reference to tables for input to computer programs which require these values for tests of significance of sequentially fitted terms in the analyses of balanced linear-bilinear models.

Keywords:

• Biadditive model
• Degrees of freedom
• Interaction
• Linear-bilinear model
• Multiplicative model
• Nonadditive data

ACKNOWLEDGMENTS

Research reported in this paper is part of a project of the Kentucky Agricultural Experiment Station and is published as Journal Article No. 99-06-84 with the approval of the Director.