On the Roles of Observations in Collinearity in the Linear Model

It is often found that individual observations play an important role in the conditioning of a linear regression problem. A state of collinearity is sometimes masked by one or two observations. Certain other observations may induce collinearity. In this article we consider a few measures of collinearity that focus on various aspects of the problem and examine the effect of case deletion on these measures. The resulting case-deletion diagnostics have the common property that they are all positive and centered around 1. A value larger than 1 corresponds to a collinearity-reducing observation, whereas a value smaller than 1 indicates a collinearityenhancing effect. The diagnostics can be used to assess the effects of multiple observations as well. Modifications of the results for column-equilibrated data are also provided. Some of the diagnostics involve extensive computation. We provide sharp and easily computable upper and lower bounds for these, complementing some results in the existing literature. We report the accuracy of the bounds for a classical dataset and show how the proposed diagnostics can contribute to data analysis. Finally, we provide a set of similar measures and inequalities for analyzing the impact of additional observations on collinearity. These results help in the choice of new design points to reduce collinearity optimally.