On testing equality of related correlation coefficients

Abstract

SUMMARY Let ( X 1 , X 2 , X 3 ) have a trivariate distribution with mean vector µ and covariance matrix Σ having ( i, j )th element σ ij = ρ ij σ i σ j with ρ ij = 1( i, j =1, 2, 3). If we are interested in testing the hypothesis H :ρ 12 =ρ 13 against appropriate alternatives under a variety of assumptions about the form of the trivariate distribution, but with no further restrictions on Σ, we could use one of the many asymptotic or conditional procedures that are available; see, for example, Aitkin, Nelson & Reinfurt (1968), Corsten (1970) and Dunn & Clark (1969). On the other hand, an exact, small-sample, unconditional test of H does not appear to be tractable for this general Σ. We show that by placing a mild restriction on Σ, such a small-sample test of H can be based on a simple, single product-moment correlation coefficient between the variables X 1 and X 3 − X 2 provided that the conditional distribution of X 1 given X 3 − X 2 , is normal.