Permutation tests for difference between two multivariate allometric patterns

Studies that include comparisons of multivariate allometric patterns betwee n sexes, species, discrete growth stages, or geographic populations have gr adually increased. Some statistical methods assume that compared groups sha re 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 multivari ate 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 v erify this test. The multivariate allometric pattern was estimated by the 1 st eigenvector of the sample covariance matrix of the logarithmic measureme nt. The angle between the 2 first eigenvectors was taken as the test statis tic. For each example, 5000 permutations were performed to assess the signi ficance level. Finally, the effect of sample size difference on the permuta tion test was also examined. We found that all 1st eigenvalues explained th e largest part of total variance and all 1st eigenvectors can satisfactoril y interpret the multivariate allometric patterns. These tests can successfu lly detect the relationship between 2 multivariate allometric patterns in e ach example, so they can be a tool to test whether the difference of 2 mult ivariate allometric patterns is significant or not. Although this method is not sensitive to sample size differences, we still suggest that the sample size difference be as small as possible when using permutation tests to ad dress this question.