Comparison of a single case to a control or normative sample in neuropsychology: Development of a Bayesian approach

Frequentist methods are available for comparison of a patient's test score ( or score difference) to a control or normative sample; these methods also provide a point estimate of the percentage of the population that would obtain a more extreme score ( or score difference) and, for some problems, an accompanying interval estimate ( i.e., confidence limits) on this percentage. In the present paper we develop a Bayesian approach to these problems. Despite the very different approaches, the Bayesian and frequentist methods yield equivalent point and interval estimates when ( a) a case's score is compared to that of a control sample, and ( b) when the raw ( i.e., unstandardized) difference between a case's scores on two tasks are compared to the differences in controls. In contrast, the two approaches differ with regard to point estimates of the abnormality of the difference between a case's standardized scores. The Bayesian method for standardized differences has the advantages that ( a) it can directly evaluate the probability that a control will obtain a more extreme difference score, ( b) it appropriately incorporates error in estimating the standard deviations of the tasks from which the patient's difference score is derived, and ( c) it provides a credible interval for the abnormality of the difference between an individual's standardized scores; this latter problem has failed to succumb to frequentist methods. Computer programs that implement the Bayesian methods are described and made available.