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

The posterior distribution of the bivariate correlation (ρ xy) is analytically derived given a data set consisting of N1 cases measured on both x and y, N2 cases measured only on x, and N3 cases measured only on y. The posterior distribution is shown to be a function of the subsample sizes, the sample correlation (rxy) computed from the N1 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.