Testing for a single outlier from a general linear regression

Several authors have considered the problem of detection of outliers from the general linear model Y = Xbeta + mu. Ellenberg [1973] among others, has advocated use of a detection method which involves examination of the set of internally standardized least squares residuals. Mickey [1974] and Snedecor and Cochran [1968], apparently concerned about the usefulness of an outlier detection method which is based on residual estimates that themselves are biassed by the presence of the outlier, have proposed two other alternatives. It is shown that the three approaches are exactly equivalent. A detection procedure is described which uses as its test statistic the maximum of the internally standardized least squares residuals, and upper and lower bounds for the percentage points of the test statistic are given by Bonferroni inequalities. The computations required to obtain these approximate percentage points are illustrated in a numerical example. Finally, a brief simulation study of the performance of the procedure illustrates that the power of the test can be influenced by the position of the outlier vis-a-vis the structure of the design matrix X.