A Non-Parametric Method to Test Equality of Intermediate Latent Roots of Two Populations in a Principal Component Analysis

Abstract

We propose a nonparametric criterion to test the hypothesis that the j-th largest roots of two populations are equal, in Section 2. This testing procedure is distribution free, and in Section 3 and 4 we show that it is reliable when the sample sizes increase and the population latent roots are separate, under the multivariate normal distribution. We suggest which sample sizes are necessary to rely on the above test.

A Statistic for Testing the Equality of Eigenvalue of Covariance Matrix on Multipopulation

“…One of the assumptions for the Ansari-Bradley test is that the samples are independent. Sugiyama and Ushizawa (1998) prove that the degree of dependence between each principal component score is weak when the sample size is sufficiently large under normality. They show the Ansari-Bradley test could be applicable to test the equality of the eigenvalues using principal component scores.…”

“…Therefore we are interested in testing the equality of the j-th largest eigenvalues of the covariance matrix, and we treat the contaminated normal and skew normal populations as examples of non-normal populations in this paper. For bipopulations, Sugiyama and Ushizawa (1998) propose a testing procedure for the equality of the j-th largest eigenvalues by applying the Ansari-Bradley test (1960). In addition, Takeda (2001) proposed a test statistic and derived the exact distribution of the statistic under normality.…”

Nonparametric Test for Eigenvalues of Covariance Matrix in Multipopulation

“…However, there exist weak correlations between each principal component scores. Sugiyama and Ushizawa (1998) proved that the degree of dependence between each principal component score was weak when the sample size was sufficiently large under the multivariate normal distribution. Then they showed that the Ansari-Bradley test could be applicable to test the equality for variance of Y 1 and Y 2 (cf.…”

“…Since it is difficult to obtain the exact distribution of eigenvalues of a covariance matrix under the nonnormal population, we have not seen testing of the hypothesis that the j-th largest eigenvalues are equal under multipopulation. For two populations, Sugiyama and Ushizawa (1998) proposed the nonparametric procedure which is the Ansari-Bradley test by using the principal component scores. In this paper, we extend the testing procedure under the multipopulation.…”

Nonparametric Test for Equality of Intermediate Latent Roots in Non-Normal Distribution

“…Let .13 ~ be the j-th largest latent root of ~k, then we consider testing the hypothesis Hp = ) 2) against Hl : ~~1) A~2) . For this problem, Sugiyama and Ushizawa(1993) proposed the Ansari-Bradley test(A-B test) using PC-scores and showed to be effective compared to the F-test under multivariate normality by a simulation study. As the variance of PC-scores equals the corresponding latent root, they investigated applying the A-B test, which is one of the nonparametric tests for two-sample problem of variance.…”