Critical Values for a Sequential Test for Many Outliers

SUMMARY Tables of critical values are provided for a sequential test for detecting up to three outliers in normal samples. The test procedure is based on the joint distribution of a series of Grubbs'-type statistics applied to reducing samples. It is shown that the computations involve only the residuals in the analysis of the complete data so that reanalysis of samples of reducing size with the suspect observations removed is unnecessary. THE problem of detecting outliers in normal samples has been extensively researched in recent years and a number of test statistics are available for both the single case and the many case for testing a specified number k of outliers. In particular the maximum normed residual, or, equivalently, the Grubbs'-type statistics and adaptations and extensions of them have received considerable attention. These involve an examination of the relative size of the largest residual or the reduction in the residual sum of squares due to the elimination of one or more suspect observations. Details of these tests may be found in Grubbs (1950, 1969), Grubbs and Beck (1972), Tietjen and Moore (1972) and many others. It is well known that although these procedures have certain optimal properties when the number of outliers present is either zero or the specified number k, they may produce mis- leading results when there are fewer or more than k outliers present in the sample. The usual difficulty in practice is deciding the number of outliers for which to test. One approach to solving this problem is to use repeated applications of single procedures, deleting the outlier detected at each step and applying the test again to the reduced sample until an insignificant result is obtained. This is not recommended as the presence of two or more outliers may produce an insignificant result in the initial single test. Recently Rosner (1975) examined a number of test statistics applied sequentially to reducing samples similar to the way described above except that the test statistics are calculated for the reducing sample a predetermined number of times, k, to produce k test statistics. These are then compared, in reverse order, with critical values based on their joint distribution under the assumption of no outliers present. The procedure is designed to detect from 1 to k outliers and Rosner's investigations showed that the sequential method using the series of maximum normed residuals from samples of reducing size appears to work very well. Critical values were tabulated by Rosner for k = 2. One possible disadvantage of this approach is the need to re-estimate the mean and variance after each deletion of a suspect observation. To overcome this difficulty Rosner (1977) recently proposed a modified procedure using the trimmed mean and variance obtained by