Complete Independence Hypothesis

Most studies involve multiple observations on multiple variables. With k variables, there are k(k − 1)/2 bivariate correlations among the measures. Each of the individual correlations could be evaluated for statistical significance. Additionally, a multiple correlation could be obtained by regressing each of the k variables on the remaining k − 1 variables. Due to the large number of parameters, trying to ascertain the joint significance of the entire set of correlations is complex. A more direct approach is to evaluate the k by k symmetric correlation matrix R for complete independence. A population in which the null hypothesis of complete independence is true is characterized by a population correlation matrix P = I, the identity matrix, in which all correlations are equal to 0. If this null hypothesis can be rejected, it may be concluded that the variables in the data set are significantly related. Two common statistical tests for assessing complete independence are denoted L1 and L2. L1 is based on Fisher's Z or tanh-1 transformation of rij, the bivariate correlation between variables i and j. Because tanh-1(r) = {log(1 + r) – log(1 − r)}/2, where log is the natural or Naperian log, has variance of 1/(N-3), the statistic L1 is distributed as a chi-square with k(k − 1)/2 degrees of freedom where k is the number of variables:

L2 is based on the distribution of the determinant of R, denoted |R|, which ranges between 0 and 1. Values closer to 0 indicate greater dependence among the measures, and values closer to 1 indicate greater independence. The statistic L2 is also distributed as a chi-square with k(k − 1)/2 degrees of freedom for k variables, where the multiplier ρ = –(N – 1 – (2k + 5)/6) and N is the sample size:

L1 and L2 have both been subjected to Monte Carlo sampling studies to evaluate Type I error rates (i.e., incorrectly rejecting the null hypothesis when it is true). A hypothesis test is considered to be biased if the estimated Type I error rate exceeds the test size α. When N is small relative to the number of variables, L2 does not perform well. L2 is biased when N is less [Page 167] than 4 times the number of variables. Even by employing finite sample correction terms, L2 is biased when N is less than twice the number of variables. In contrast, L1 is an unbiased hypothesis test, even for small N.

Power comparisons between L1 and L2 indicate that L1 is a more powerful test of P = I than is L2. In small samples, relative to the number of variables, L1 should be preferred in terms of both Type I and Type II error rates.