Interval Estimation of Bivariate Correlations With Missing Data on Both Variables: A Bayesian Approach

Abstract

The posterior distribution of the bivariate correlation (ρ_xy ) is analytically derived given a data set consisting N ₁ cases measured on both x and y, N ₂ cases measured only on x, and N ₃ cases measured only ony. The posterior distribution is shown to be a function of the subsample sizes, the sample correlation (r_xy ) computed from the N ₁ complete cases, a set of four statistics which measure the extent to which the missing data are not missing completely at random, and the specified prior distribution for ρ_xy . A sampling study suggests that in small (N = 20) and moderate (N = 50) sized samples, posterior Bayesian interval estimates will dominate maximum likelihood based estimates in terms of coverage probability and expected interval widths when the prior distribution for ρ_xy is simply uniform on (0, 1). The advantage of the Bayesian method when more informative priors based on beta densities are employed is not as consistent.