Diagnosing collinearity-influential observations

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Abstract

It is known that the eigenstructure of a given matrix can change substantially when a row is added to or deleted from the matrix. In this article we study the effects of row deletion on the eigenstructure of a given matrix in general and on its condition index in particular. Except for very special cases, it is found that no general closed-form equation can relate the eigenstructure of a given matrix to the eigenstructure of the same matrix with one row deleted. However, we give good closed-form approximations to the relationship between the two eigenstructures. We also propose two diagnostic measures for assessing the influence of a single row on the condition index of a given matrix. Two numerical examples from regression analysis are used to illustrate the methodology. However, these methods are also relevant in other multivariate analysis problems where eigenstructure is of interest, for example, canonical correlation and principal components analyses.

Keywords

collinearity
collinearity-influential
condition index
influence
leverage
singular value decomposition.

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