Appropriate critical values when testing for a single multivariate outlier by using the Mahalanobis distance

The Mahalanobis distance is a well-known criterion which may be used for detecting outliers in multivariate data. However, there are some discrepancies about which critical values are suitable for this purpose. Following a comparison with Wilks's method, this paper shows that the previously recommended {p(n - 1)/(n - p)}F-p,F-n-p are unsuitable, and p(n - 1)F-2(p,n-p-1)/n(n - p - 1 + pF(p,n-p-1)) are the correct critical values when searching for a single outlier. The importance of which critical values should be used is illustrated when searching for a single outlier in a clinical laboratory data set containing 10 patients and five variables. The jackknifed Mahalanobis distance is also discussed and the relevant critical values are given. Finally, upper bounds for the usual Mahalanobis distance and the jackknifed version are discussed.