  Cook's distance [Technometrics 19 (1977) 15-18] is one of the most important
diagnostic tools for detecting influential individual or subsets of
observations in linear regression for cross-sectional data. However, for many
complex data structures (e.g., longitudinal data), no rigorous approach has
been developed to address a fundamental issue: deleting subsets with different
numbers of observations introduces different degrees of perturbation to the
current model fitted to the data, and the magnitude of Cook's distance is
associated with the degree of the perturbation. The aim of this paper is to
address this issue in general parametric models with complex data structures.
We propose a new quantity for measuring the degree of the perturbation
introduced by deleting a subset. We use stochastic ordering to quantify the
stochastic relationship between the degree of the perturbation and the
magnitude of Cook's distance. We develop several scaled Cook's distances to
resolve the comparison of Cook's distance for different subset deletions.
Theoretical and numerical examples are examined to highlight the broad spectrum
of applications of these scaled Cook's distances in a formal influence
analysis.
