Bivariate boxplots, multiple outliers, multivariate transformations and discriminant analysis: The 1997 Hunter Lecture

Abstract

Outliers can have a large in¯uence on the model ®tted to data. The models we consider are the transformation of data to approximate normality and also discriminant analysis, perhaps on transformed observations. If there are only one or a few outliers, they may often be detected by the deletion methods associated with regression diagnostics. These can be thought of as`backwards' methods, as they start from a model ®tted to all the data. However such methods become cumbersome, and may fail, in the presence of multiple outliers. We instead consider a`forward' procedure in which very robust methods, such as least median of squares, are used to select a small, outlier free, subset of the data. This subset is increased in size using a search which avoids the inclusion of outliers. During the forward search we monitor quantities of interest, such as score statistics for transformation or, in discriminant analysis, misclassi®cation probabilities. Examples demonstrate how the method very clearly reveals structure in the data and ®nds in¯uential observations, which appear towards the end of the search. In our examples these in¯uential observations can readily be related to patterns in the original data, perhaps after transformation.

Some Algebra and Geometry for Hierarchical Models, Applied to Diagnostics

“…Similar insight into the structure of the data can be obtained for the discriminant analysis of multivariate data (Atkinson and Riani, 1997). Fig.…”

New methods for ordering multivariate data: an application to the performance of investment funds

“…The construction of bivariate boxplots for each pair of variables for data coming from di!erent populations, can be a very useful instrument for extracting information about spread, location, and separation among the groups. These tools not only enable us to highlight potential outliers, but can help to select the variables in discriminant analysis and can provide useful information about the choice of multivariate transformations [11]. Furthermore, bivariate boxplots can also be drawn on derived variables such as principal components and canonical variates, provided that the projection gives a good approximation to the original space.…”

Fitting Straight Lines with Replicated Observations by Linear Regression. IV. Transforming Data

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