Departure from normality in multivariate normative comparison: the Cramér alternative for Hotelling's T2

Crawford and Howell (1998) have pointed out that the common practice of z-score inference on cognitive disability is inappropriate if a patient's performance on a task is compared with relatively few typical control individuals. Appropriate univariate and multivariate statistical tests have been proposed for these studies, but these are only valid if the data are Gaussian (normal) distributed. Previous studies have investigated the consequences for Type I error rates of using the univariate test when data are not Gaussian. In this paper we examine the effects of violation of the Gaussian assumption on nominal Type I error rates for the multivariate test. We also consider a new test that has been devised recently, called Cramér's test, as a viable alternative for the multivariate normative comparison. In simulations we show that the new test not only provides a distribution free alternative for existing methods, but also has the advantage that it is substantially more powerful in most common research settings. We demonstrate the use of the new test with an application to two individuals diagnosed with autism.