A Technique of Nonparametric Multivariate Analysis

Abstract

On each of n individuals p + q variables are observed. A nonnegative distance or closeness measure between any two individuals on any one variable can be based on ranks or tied ranks for orderable variables (continuous, discrete, or categorical); for nonorderable categorical variables the distance measure reflects whether the two individuals belong to the same category. Let Xi and Yi, represent weighted sums over the p and the q variables, respectively, of the distance measures between individuals i and j; there will be 'n(n + 1) such pairs of weighted sulms. A test statistic for judging whether closeness in the set of p variables is related to closeness in the set of q variables is given by Z = E E Xt iYi. The permutational distribution of the statistic is defined by the random pairing of p-variable and q-variable observation vectors. Being a U statistic, this measuLre is asymptotically normally distributed. A computing procedure is given for obtaining permutational expectations and variances so that departure from the permutational distribuLtion can be judged.