Crawford, J. R. and Garthwaite, P. H.
(2007).
Comparison of a single case to a control or normative sample in neuropsychology: Development of a Bayesian approach.
*Cognitive Neuropsychology*, 24(4) pp. 343–372.

## Abstract

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.

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