R. Dennis Cook

Detection of influential observation in linear regression

A new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models. It is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values. Two examples are presented.

Cross-Validation of Regression Models

Abstract A methodolgy for assessment of the predictive ability of regression models is presented. Attention is given to models obtained via subset selection procedures, which are extremely difficult to evaluate by standard techniques. Cross-validatory assessments of predictive ability are obtained and their use illustrated in examples.

Influential Observations in Linear Regression

Abstract Characteristics of observations which cause them to be influential in a least squares analysis are investigated and related to residual variances, residual correlations, and the convex hull of the observed values of the independent variables. It is shown how deleting an observation can substantially alter an analysis by changing the partial F-tests, the studentized residuals, the residual variances, the convex hull of the independent variables, and the estimated parameter vector. Outliers are discussed briefly, and an example is presented.

A Comparison of Algorithms for Constructing Exact D-Optimal Designs

An empirical comparison of existing algorithms for the computer generation of exact D-optimal experimental designs is carried out. Among algorithms considered were those due to Wyrnn, Mitchell, Fedorov, and Van Schalkwyk. A procedure for rounding off approximate designs as suggested by Kiefer is also evaluated. A modification of the Fedorov algorithm is given and shown to effect substantial decreases in the computer time required for design generation.

Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach

A family of dimension-reduction methods, the inverse regression (IR) family, is developed by minimizing a quadratic objective function. An optimal member of this family, the inverse regression estimator (IRE), is proposed, along with inference methods and a computational algorithm. The IRE has at least three desirable properties: (1) Its estimated basis of the central dimension reduction subspace is asymptotically efficient, (2) its test statistic for dimension has an asymptotic chi-squared distribution, and (3) it provides a chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors. Current methods like sliced inverse regression belong to a suboptimal class of the IR family. Comparisons of these methods are reported through simulation studies. The approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in sliced average variance estimation.

Characterizations of an Empirical Influence Function for Detecting Influential Cases in Regression

Traditionally, most of the effort in fitting full rank linear regression models has centered on the study of the presence, strength and form of relationships between the measured variables. As is now well known, least squares regression computations can be strongly influenced by a few cases, and a fitted model may more accurately reflect unusual features of those cases than the overall relationships between the variables. It is of interest, therefore, for an analyst to be able to find influential cases and, based on them, make decisions concerning their usefulness in a problem at hand. Based on an empirical influence function, we discuss methodologies for assessing the influence of individual or groups of cases on a regression problem. We conclude with an example using data from the Florida Area Cumulus Experiments (FACE) on cloud seeding.

On the Interpretation of Regression Plots

Abstract A framework is developed for the interpretation of regression plots, including plots of the response against selected covariates, residual plots, added-variable plots, and detrended added-variable plots. It is shown that many of the common interpretations associated with these plots can be misleading. The framework also allows for the generalization of standard plots and the development of new plotting paradigms. A paradigm for graphical exploration of regression problems is sketched.

Fisher Lecture: Dimension Reduction in Regression

Beginning with a discussion of R. A. Fishers early written remarks that relate to dimension reduction, this article revisits principal components as a reductive method in regression, develops several model-based extensions and ends with descriptions of general approaches to model-based and model-free dimension reduction in regression. It is argued that the role for principal components and related methodology may be broader than previously seen and that the common practice of conditioning on observed values of the predictors may unnecessarily limit the choice of regression methodology.

Graphics for regressions with a binary response

Abstract Central dimension-reduction subspaces, which characterize the dependence of a response variable on one or more predictors, are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable. Graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension-reduction subspace.

Diagnostics for mixed-model analysis of variance

We describe a new method for assessment of model inadequacy in maximum-likelihood mixed-model analysis of variance. In particular, we discuss its use in diagnosing perturbations from the usual assumption of constant error variance and from the assumption that each realization of a given random factor has been drawn from the same normal population. Computer implementation of the procedure is described, and an example is presented, involving the analysis of filter cartridges used with commercial respirators.