Testes para a igualdade de matrizes de covariâncias de duas populações normais multivariadas dependentes

Abstract

In some experimental situations of biological, physical and human sciences is common the researcher be interested in making comparisons of variance and covariance matrices of two populations. If two populations samples are independent it is expected no covariance between them. However, in situations where a group of variable is measured before and after the performance of a particular treatment the sample data can be paired. For the case where only one variable is measured in each situation, pre (X) and post (Y) treatment, Morgan (1939) and Pitman (1939) proposed an exact t test based on the correlation between the normal variables X and Y and the correlation of the two new variables that are linear combinations of X and Y. However, the test proposed by Morgan (1939) and Pitman (1939) considers only situations where we have q = 2 populations and p = 1 variable. In the literature, some solutions are presented for the case of q 2 and p 1 variable. However, all are asymptotic tests. Therefore, the propose of this work is to generalize the Morgan (1939) and Pitman (1939) test for the multivariate case, considering the situation of q = 2 populations. We proposed covariance comparisons tests in the presence of correlation by the nonparametric bootstrap method (tb0 ). The UV was maximized to optimize the a parametric (ta) and for setting the a in an unitary vector (tc). Then we evaluate the performance of these tests and compare them with the others. We separated the conclusions regarding the performance of the tests in two cases. In the first case, which p = 2 was considered, it was found that, among the tests that controlled the type I error, the tests LRT3 and W2 were better than their competitors in all situations studied. The tb0 test was considered intermediate and the ta and tc tests were lower than the others. In the second case, which p = 4 and p = 10 were considered, it was found that the ta, tc and tb0 tests stood out for having a great performance, achieving 100% marks almost always when n 20. Therefore, in real situations, we recommend the application of the tc and tb0 tests proposed in this work.