A Gibbs sampler approach to estimation of a multiple correlation coefficient in the presence of missing dat

A Bayesian approach to estimation of the multiple correlation coefficient when the data are partially missing is theoretically preferable to the commonly used ad hoc procedures or maximum likelihood estimation. However, in the presence of multiple missing data patterns, derivation of the marginal posterior density in closed form becomes analytically intractable, thus Monte Carlo methods must be used instead. The Gibbs sampler is an iterative simulation technique which avoids both complicated analytic derivation and numerical high-dimensional integration methods by generating random variables from a marginal distribution indirectly, without explicit calculation of the density. This paper demonstrates the use of the Gibbs sampler to approximate the marginal distribution of the multiple correlation coefficient conditional on the observed data, when the data are missing at random.;An executable computer program (written in FORTRAN) was developed for the PC to implement the technique for data sets containing up to 200 cases with 2 to 10 variables possessing any missing data structure (subject to certain minimal restrictions). These limits can be increased by varying the parameters at the compilation stage of the FORTRAN program.;A simple example illustrating the steps in the estimation process performed by the Gibbs sampler is presented. The procedure is carried out on simulated and real data to yield point estimates and interval estimates in terms of highest density regions. Results are compared for alternative trials employing varying Gibbs sequence lengths and different sample sizes for simulation of the marginal distribution of the multiple correlation coefficient. Estimates on all data sets are compared with those obtained from use of a standard statistical computer package (SPSS). Interval estimate for the real data sets are compared with exact Bayesian intervals obtained analytically.