Zoological Studies

Vol. 38 No. 1, 1999

Permutation Tests for Difference between Two Multivariate Allometric Patterns

Tzong-Der Tzeng and Shean-Ya Yeh*

Institute of Oceanography, National Taiwan University, Taipei, Taiwan 106

Tzong-Der Tzeng and Shean-Ya Yeh (1999) Studies that include comparisons of multivariate allometric patterns between sexes, species, discrete growth stages, or geographic populations have gradually increased. Some statistical methods assume that compared groups share the same multivariate allometric pattern, so comparisons of multivariate allometric patterns also have to be performed before using these methods. Several methods have been used to detect the difference between 2 multivariate allometric patterns, but these methods lack an objective guide to test whether the 2 multivariate allometric patterns are the same or not. In this study, a permutation test was used to determine whether the difference of 2 patterns was significant or not. Four examples were used to explain and verify this test. The multivariate allometric pattern was estimated by the 1st eigenvector of the sample covariance matrix of the logarithmic measurement. The angle between the 2 first eigenvectors was taken as the test statistic. For each example, 5000 permutations were performed to assess the significance level. Finally, the effect of sample size difference on the permutation test was also examined. We found that all 1st eigenvalues explained the largest part of total variance and all 1st eigenvectors can satisfactorily interpret the multivariate allometric patterns. These tests can successfully detect the relationship between 2 multivariate allometric patterns in each example, so they can be a tool to test whether the difference of 2 multivariate allometric patterns is significant or not. Although this method is not sensitive to sarnple size differences, we still suggest that the sample size difference be as small as possible when using permutation tests to address this question.

Key words: First eigenvector, Angle, Allometry, Reorder.

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