Two tests of complete independence were compared in terms of relative power efficiency on 50 equi-correlation matrices and 26 published data sets. Λ1 a test based on Fisher's tanh-1 (r) transformation had greater power efficiency than, Λ0 a test based on -log|R| with finite sample corrections. Consequently, sample size requirements for rejecting P=I should be determined from Λ1 instead of Λ0. In addition, Baggaley's estimate of Q=-log|R|/k from the average absolute off-diagonal correlation was compared with a root mean square estimate of Q. The root mean square estimate was preferable but also demonstrated considerable bias. Because of the bias in estimating Q from the off-diagonal correlations, because Λ1 has greater relative power efficiency for P=I than ?0, and because the test statistic for Λ1 can be approximated from the root mean square off-diagonal correlation, it is preferable to tabulate power results for Λ1 as a function of the average correlation instead of Λo.