A likelihood-ratio-based normal approximation for the non-null distribution of the multiple correlation coefficient

Let X1,X2, …, Xp be jointly distributed according to a multivariate normal distribution, and let ? denote the multiple correlation coefficient between X1 and X2, X3,…, Xp Let Xli,…, Xpi, i =1, … N, be a random sample from the distribution. The logarithm of the likelihood ratio statistic for testing the hypothesis that ρ is zero is −(N/2)log(l−R2), where R is the sample multiple correlation coefficient. A Gaussian approximation to the non-null (ρ≠0) distribution of R is developed using the transformation (T/E(T))hwhere T =−log(l−R2), and h is determined from the first three cumulants of T. The approximation is simple and accurate over a wide range of the parameters p, N, and ρ.