Location of Several Outliers in Multiple-Regression Data Using Elemental Sets

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.The outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model. Each such subset yields an elemental residual for the case in question, and a suitable summary statistic of them can be used as an estimate of the case's outlying tendency. We propose two such summary statistics: an unweighted median, which is of bounded influence, and a weighted median, which is more efficient but less robust. The computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p. As a byproduct the method yields useful information on the influence (or leverage) of cases and the mutual masking of high leverage points.

Two-Step Robust Diagnostic Method for Identification of Multiple High Leverage Points

“…Hawkins et al [9] constructed an artificial three-predictor data set containing 75 observations with 10 outliers (cases 1-10) and 14 high leverage points (cases [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Most of the previous single case deletion identification methods fail to identify all of these influential observations.…”

Two-Step Robust Diagnostic Method for Identification of Multiple High Leverage Points

“…Hawkins-Bradu-Kass data: Hawkins et al [9] constructed an artificial three-predictor data set containing 75 observations with 10 outliers (cases 1-10) and 14 high leverage points (cases [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Most of the previous single case deletion identification methods fail to identify all of these influential observations.…”

Delete‐2 and Delete‐3 Jackknife Procedures for Unmasking in Regression

“…Let p be the rank of the fitted regression model. Hawkins et al (1984) used the idea of ‘elemental sets’, subsets of the data of size p from the n data points, to reveal unusual data points by repeatedly sampling subsets of size p , fitting the resultant exact model, and continuing to sample until a clear pattern of unusual data points is evident. Atkinson (1986) developed a two‐stage procedure in which a highly robust fit is used in the first stage, followed by a second, confirmatory stage using least squares‐based single‐case deletion diagnostics.…”

“…Atkinson & Riani (2002) further refined the forward search method, using an added‐variable approach to deal with ordering issues that arise from the forward search technique. A common theme to the approaches of Hawkins et al (1984) and Atkinson & Riani (2000, 2002) is the use of models fitted to subsets of the original data to discover unusual observations. Our approach also uses data subsets, those defined by a delete‐ d jackknife operation, and so shares this common theme.…”

“…However, robust methods generally do not provide explicit information about the extent and type of masking present, a key feature of our proposed technique. Our proposal adopts the spirit of Hawkins et al 's (1984) idea of using subsets of the data for fitting and assessment, but uses delete‐ d jackknife replications as the basis for generating appropriate subsets.…”