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

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.

Notes

¹ A computer program that implements this test (dissocs.exe), and the unstandardized difference test described earlier, can be downloaded from http://www.abdn.ac.uk∼psy086/dept/SingleCaseMethodsComputerPrograms.HTM

² Crawford and Howell's (1998) test for a deficit is used for computational convenience in setting these criteria. That is, a computer program is available to test whether a case meets these criteria (see General Discussion section for details). Use of Crawford and Howell's test, rather than the equivalent Bayesian test for a deficit, means that only one set of 100,000 observations need be sorted (to obtain credible limits on the abnormality of the standardized difference) rather than three sets (two further sorts would be required to obtain the limits on the abnormality of the case's x and y scores).

³ The public domain software package R (www.cran.r-project.org) provides a very useful routine that finds the optimal Box–Cox normalizing transformation by the method of maximum likelihood. As noted, however, although it will find the best available transformation, this by no means ensures that the data will be normalized successfully.