An Approximation to the Distribution of the Product of Two Dependent Correlation Coefficients

1. INTRODUCTION Many methodological studies depend on the product of two dependent correlation coefficients. For example, the product of two dependent correlation coefficients is needed in assessing the impact of a confounding variable on a regression coefficient (Frank, 2000). In the literature, there is an extensive research on the distribution of a single correla-but little on the distribution of the product of two correlation coefficients. Thus, the behavior of the distribution of the product of two dependent correlation coefficients is little known. There are two plausible approaches to obtain the distribution of the product of two correlation coefficients. Mathai and Saxena (1969) express the product of two correlation coefficients as a special case of the product of two generalized Mellin-Barnes functions or H-functions (Mathai & Saxena, 1978). However, the expression obtained for the distribution function is quite unwieldy. In addition, in their study, the two correlation coefficients are assumed independent , while we are interested in two dependent correlation coefficients. The other approach is described in Frank (2000). Frank transforms the two correlation coefficients to two asymptotically, normally distributed Fisher z's, then uses Aroian and colleagues' findings about the distribution of the product of two normal variables to obtain the distribution of the product of two Fisher z's, instead of the two original correlation coefficients. A problem with this approach, however, is that the approach is relying on asymptotic theory for Fisher's z, compounded with the approximation error associated with Aroian's approach, which results in very slow convergence. By following Hotelling's (1940) and Ghosh's (1966) approximations to the moments of the distribution of correlation coefficient, the present study derives a more accurate approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I (Beta) distribution. A simulation study is also conducted to assess the accuracy of the approximation .