Robert F. Ling

Cluster Analysis Algorithms for Data Reduction and Classification of Objects

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Some Cautionary Notes on the Use of Principal Components Regression

Abstract Many textbooks on regression analysis include the methodology of principal components regression (PCR) as a way of treating multicollinearity problems. Although we have not encountered any strong justification of the methodology, we have encountered, through carrying out the methodology in well-known data sets with severe multicollinearity, serious actual and potential pitfalls in the methodology. We address these pitfalls as cautionary notes, numerical examples that use well-known data sets. We also illustrate by theory and example that it is possible for the PCR to fail miserably in the sense that when the response variable is regressed on all of the p principal components (PCs), the first (p − 1) PCs contribute nothing toward the reduction of the residual sum of squares, yet the last PC alone (the one that is always discarded according to PCR methodology) contributes everything. We then give conditions under which the PCR totally fails in the above sense.

K-Clustering as a Detection Tool for Influential Subsets in Regression

This article describes a new methodology for the detection of influential subsets in regression. The method is based on an adaptation of computational and graphical techniques used in cluster analysis and makes use of some general properties of influential subsets, but it is independent of any specific measure of influence. For small to moderate data sets the proposed method is computationally efficient, compared to existing search methods, and it identifies subset candidates that merit attention according to some or all measures of joint influence that have appeared in the literature to date. Examples are given illustrating the method applied to two data sets previously analyzed in published studies.

General Classes of Influence Measures for Multivariate Regression

Abstract Many of the existing measures for influential subsets in univariate ordinary least squares (OLS) regression analysis have natural extensions to the multivariate regression setting. Such measures may be characterized by functions of the submatrices H I of the hat matrix H, where I is an index set of deleted cases, and Q I , the submatrix of Q = E(E T E)−1 E T , where E is the matrix of ordinary residuals. Two classes of measures are considered: f(·)tr[H I Q I (I − H I − Q I ) a (I − H I ) b ] and f(·)det[(I − H I − Q I ) a (I − H I ) b ], where f is a scalar function of the dimensions of matrices and a and b are integers. These characterizations motivate us to consider separable leverage and residual components for multiple-case influence and are shown to have advantages in computing influence measures for subsets. In the recent statistical literature on regression analysis, much attention has been given to problems of detecting observations that, individually or jointly, exert a disproportionate ...

Probability Tables for Cluster Analysis Based on a Theory of Random Graphs

Abstract Statistics based on a theory of random graphs have been proposed as an analytic aid to assess the randomness of a clustered structure. Probability tables for two such statistics are tabulated. Exact values of Pn,v , the cumulative probabilities of the minimum number of edges needed to connect a random graph, are tabulated for n = 10(1)30(5)80-(10)100. Exact and approximate values of En,v, the expected number of components in a random graph with n vertices and v edges, are tabulated for n = 10(1)30(5)100.

A Study of the Accuracy of Some Approximations for t, χ2, and F Tail Probabilities

Abstract This article summarizes an empirical study on the numerical accuracy of some computational formulas for the tail probabilities of t, χ2, and F distributions, in terms of the maximum absolute error for all p, 0 < p ≤ 1. The results are intended to fill a conspicuous gap in the statistical literature concerning the empirical quality of the approximations, and they are useful for designing efficient and accurate computing algorithms for such probabilities.

Correlation and Causation.

The ideal method of science is the study of the direct influence of one condition on another in experiments in which all other possible causes of variation are eliminated. Unfortunately, causes of variation often seem to be beyond control. In the biological sciences, especially, one often has to deal with a group of characteristics or conditions which are correlated because of a complex of interacting, uncontrollable, and often obscure causes. The degree of correlation between two variables can be calculated by well-known methods, but when it is found it gives merely the resultant of all connecting paths of influence. The present paper is an attempt to present a method of measuring the direct influence along each separate path in such a system and thus of finding the degree to which variation of a given effect is determined by each particular cause. The method depends on the combination of knowledge of the degrees of correlation among the variables in a system with such knowledge as may be possessed of the causal relations. In cases in which the causal relations are uncertain the method can be used to find the logical consequences of any particular hypothesis in regard to them. CORRELATION

General considerations on the design of an interactive system for data analysis

Among the most important criteria in the design and implementation of an interactive system for data analysis are: data structure, control language, user interface, sytem versatility, extensibility, and portability. The design of an interactive system, viewed as a set of constrained decisions based on these criteria, will be discussed. The concepts and considerations discussed in this article about the design of interactive systems are general in nature and are neither problem-specific nor discipline-specific. Specific examples from statistical packages and their designs are cited for illustration purposes only.

The Accuracy of Peizer Approximations to the Hypergeometric Distribution, with Comparisons to Some other Approximations

Abstract Results of an extensive empirical study of the accuracy of 12 normal and 3 binomial approximations to the hypergeometric distribution are presented in terms of maximum absolute error under various conditions on the variables. The most useful conditions employ the minimum cell in the given or complementary 2 × 2 table and the tail probability itself. Of the normal approximations, the best by far are of a heretofore unpublished type originated by Peizer. Especially detailed results on both absolute and relative errors are given for one Peizer approximation. Its absolute error is at most .0001, for example, if the minimum cell is at least 4.

A unifying representation of some case-deletion influence measures in univariate and multivariate linear regression

Abstract In recent years numerous measures have been proposed for assessing the influence of observations on least-squares regression results. Various influence measures were introduced based on different motivational arguments and each was designed to measure the influence of observations on different aspects of various regression results. The chief motivation and result of this paper is the determination of the main mathematical components common to many multiple-case regression diagnostics, and to make use of these components in formulating an algebraic representation which unifies the existing measures. The representation, which we refer to as the JI-class, is based on a few common interpretable components. These components are functions of the elements of two orthogonal projector matrices; the projector matrix for the column space of the explanatory variables, and the projector matrix for the residuals. The proposed representation enables one to perceive such existing measures not only in their original diversified formulations, but also in a manner that allows associations or similarities to be recognized among them in a clear, concise fashion. The class possesses several properties that can be used to study the relationships among the existing influence measures, especially those with seemingly different motivations and initial characterizations. The JI-class, which we use for a unifying representation of some existing influence measures, can also be used to generate new influence measures.