Today I began to wonder about this. Consider the regression Y=rX+e where X and Y are standard normal variables. Then R^2 = r^2. It was suggested that R^2 could be estimated by averaging the estimates of R^2=r^2 across multiple imputations. Yet r is estimated by averaging the estimates of r across multiple imputations. In general, these estimates will not agree: if r>0, then the estimate of R^2 will be less than the squared estimate of r. If the estimator of r is unbiased, then the proposed estimate of R^2 must be biased.
It strikes me there must be a lot of quantities for which we cannot obtain unbiased estimates using this procedure. Pertinent citations would be most appreciated.
Best wishes, Paul von Hippel Statistician Ohio State University
