A COMPARISON OF BAYESIAN ESTIMATES OF TRUE SCORE

Abstract

ABSTRACTTwo Bayesian estimates of true score are compared numerically to evaluate their relative usefulness in applications. The first of these, due to Box and Tiao, is the mean of the posterior distribution with a particularly convenient prior distribution. The second of these, taken from a development due to Lindley, is the relative mode based on the indifference prior distribution suggested by Novick. The first estimate is found to be relatively convenient computationally while the second estimate is attained only by an iterative procedure. These two estimates are also compared with a natural asymptotic estimator which is even more convenient computationally and with Stein's simultaneous estimator. When reliabilities are low and sample sizes and the number of replications are small the Lindley estimate exhibits a total regression to the population mean while the Box‐Tiao mean estimate exhibits only a partial regression. This same result is obtained even when the sample estimate of reliability is negative. When reliabilities, sample sizes and numbers of replications are moderate there is little difference among the first three estimators, but unless the number of persons is large the regression effect is markedly smaller in the Stein estimate. Because of the uniformly good risk properties of the Stein estimate and its substantial difference from the other three estimators it is suggested that the asymptotic regression estimate and the Bayesian estimates based on improper prior distributions may not be acceptable for small numbers of persons.