A class of permutation tests of bivariate interchangeability

A set of permutation tests that are both exact and distribution-free are pr oposed to simultaneously detect differences in marginal locations and/or sc ales in bivariate data. The tests take advantage of the fact that when the marginal means and variances are equal, the pairwise differences are symmet rically distributed about 0 and are uncorrelated with the pairwise Sums. Tw o statistics for detecting the marginal location and scale differences are combined in a quadratic form. A permutation distribution for this quadratic form follows from considering all 2(n) conditionally equally likely sign c hanges on the differences. Several methods of estimating the covariance mat rix of the quadratic form are examined, including conditional and unconditi onal (plug-in) approaches. These new tests are compared with the standard t ests in the literature and, through simulation for several families of biva riate distributions, are found to compare quite favorably. This article als o brings to light the largely overlooked likelihood ratio test for equal me ans and variances in the bivariate normal and shows its relationship to mor e recent approaches, including those presented here.