Collapsibility hypotheses and diagnostic bounds in regression analysis

Abstract.

Deleted-case diagnostic statistics in regression analysis are based on changes in estimates due to deleting one or more cases. Bounds on these statistics, suggested in the literature for identifying influential cases, are widely used.

 In a linear regression model for Y in terms of X and Z, the model is “collapsible” with respect to Z if the Y−X relation is unchanged by deleting Z from the model. Deleted-case diagnostic statistics can be viewed as test statistics for collapsibility hypotheses in the mean shift outlier model. It follows that, for any given case, all deleted-case statistics test the same hypothesis, hence all have the same p-value, while the bounds correspond to different levels of significance among the several statistics. Furthermore, the bound for any particular deleted-case statistic gives widely varying levels of significance over the cases in the data set.