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Abstract:
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The sample squared multiple correlation coefficient is well-known to be an upwardly biased estimator of its population parameter, especially when the sample size is small and the number of predictors approaches the sample size. A small cottage industry has undertaken to correct its bias and has produced a number of adjusted estimators, the most popular being Fisher's. Curiosity prompts the question whether a Bayesian estimator, namely the posterior expectation, suffers from an analogous defect. Finding out involves the hypergeometric and the generalized hypergeometric functions. As it turns out, the Bayesian estimator is automatically 'adjusted' for small samples with many predictors. Performance comparisons between the frequentist and Bayesian estimators are presented.
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