Prof. Carlos Manuel Agra Coelho
Prof. Carlos Manuel Agra Coelho
Universidade Nova de Lisboa, Portugal
Title: A likelihood ratio test for the equality of canonical correlations
Canonical correlations are a much interesting and useful measure of association between two sets of variables and their study and analysis lies as the basis of many of the linear models commonly used, from (Multivariate) Linear Regression to Discriminant Analysis and from MANOVA (Multivariate Analysis of Variance) to MANCOVA (Multivariate Analysis of Covariance). In a true multivariate setting, where we have several response variables, the test of equality of canonical correlations appears as a much useful and desired tool. However, devising such a test has not been easy given the rather complicated structure of the joint distribution of the sample canonical correlations, moreover when, under the null hypothesis, one has to consider the case of equality of some of the population canonical correlations. But, by first obtaining the likelihood ratio test (l.r.t.) for a particular diagonal-block covariance structure where the diagonal blocks are compound symmetric (that is, with equal variances and equal covariances) it is then possible to convert this test into a l.r.t. for equality of canonical correlations. In this talk will be presented the derivation of the test and it will be shown how by adequately splitting the null hypothesis into a set of conditionally independent hypotheses we can easily obtain the corresponding l.r.t. statistic. Also, from this split of the null hypothesis, near-exact distributions will be derived, which, given the non-manageable structure of the exact distribution in the general case, are a much useful tool from which near-exact quantiles and p-values may be obtained. A measure of proximity is used to evaluate the proximity of the near-exact distributions obtained to the exact distribution. The results show the extreme proximity of the exact and the near-exact distributions obtained. Also, for the particular case of the test of equality of only two canonical correlations the exact distribution of the l.r.t. statistic is obtained in a closed finite form from which we can easily compute exact quantiles and p-values.
Carlos A. Coelho is Professor of Statistics at the Mathematics Department of Faculdade de Ciências e Tecnologia of Universidade Nova de Lisboa. He holds a Ph.D. in Biostatistics by The University of Michigan, Ann Arbor, MI, U.S.A., and his main areas of research are Mathematical Statistics and Distribution Theory, namely the study and development of exact and near-exact distributions for likelihood ratio test statistics used in Multivariate Analysis. Other areas of interest are Estimation, Univariate and Multivariate Linear, Generalized Linear and Mixed Models, as well as Computational Statistics. In a 2004 paper published in the Journal of Multivariate Analysis, he laid the foundations for what he called ‘near-exact distributions’. Since then these have been successfully applied to a large number of statistics, with more than 40 papers published on this topic. The technique combines an adequately developed decomposition of the characteristic function of the statistic or random variable being studied or of its logarithm, decomposition which often is an adequate factorization, with the procedure of keeping the most of this characteristic function unchanged and replacing the remaining part by an adequate asymptotic approximation. All this is done in order to obtain a manageable and very well-fitting approximation, which may be used to compute very sharp p-values and quantiles. Carlos A. Coelho is an Elected Member of the International Statistical Institute and serves as Associate Editor in the Editorial Boards of Journal of Applied Statistics, REVSTAT, Journal of Statistical Theory and Practice, Journal of Interdisciplinary Mathematics, American Journal of Mathematical and Management Sciences and Discussiones Mathematicae-Probability and Statistics.