BAYESIAN METHODS IN PSYCHOLOGICAL TESTING

A Bayesian method of estimating ability is described that generalizes Kelley's regression estimate of true score based on a weighted average of observed score and the presumed known population mean true score. This Bayesian method uses all available data to estimate the population mean true score and incorporates this information into the estimate of each individual ability parameter. This method is superior to methods that do not include estimates of population mean values. A normal law error model is used to illustrate the method. Modal estimates of an intuitively attractive form from the joint posterior distribution of the ability parameters are identified as a solution to a set of nonlinear equations. The method is then illustrated in the context of a Poisson process model having both person-ability and item-difficulty parameters. The posterior marginal distributions for individual sets of ability and difficulty parameters are given and modal estimates are described. Modal estimates from the joint conditional distribution of the ability parameters given the item difficulty parameters are also given. The desirability of incorporating prior information is discussed and a method of accomplishing this is described. The application of these Bayesian methods to central prediction and sequential testing are discussed. It is suggested that the Bayesian method is uniquely appropriate to each of these problems. Finally, these ideas are also shown to be relevant to the problems of selecting predictor variables and multiple comparisons.