A modified Aitken Pivotal Condensation Method for partial regression and multiple correlation

In this modification of the Aitken Pivotal Condensation Method for obtaining partial regression weights and multiple correlation coefficients, the unit matrices used by Aitken are eliminated, with a resultant decrease in the computational work. Through a simplified arrangement of successive matrices, each one is reduced by the same simple rule. Other modifications also contribute to ease of computation and facilitate checking.     

The evaluation of multiple and partial correlation coefficients from the factorial matrix

This paper shows how to compute multiple correlation coefficients, partial correlation coefficients, and regression coefficients from the factorial matrix. Special emphasis is given to computation technique and to approximation formulas. The method is extremely flexible in application since it may be applied to any subset of the original set of observed variables. It is also extremely useful when many of these coefficients are desired.     

A further study of partial reinforcement in the turtle

Two matched groups of mature painted turtles, Chrysemys picta picta, were trained in a simple runway, one with partial and the other with consistent reinforcement, following which both groups were extinguished. The partially reinforced animals ran more slowly in acquisition, but showed somewhat greater resistance to extinction. The results are compared with those obtained in analogous experiments with other animals.     

A cautionary tale of two statistics: partial correlation and standardized partial regression

Partial correlation and standardized partial regression are widely used to control for other variables in cross-sectional and longitudinal studies. Two aspects of these statistics can radically alter the interpretation of data and so deserve attention. First, their sign will be opposite to that of the original zero-order correlation when this correlation is of the same sign as and smaller than the product of the other 2 correlations. Second, their size will be considerably larger than a weak zero-order correlation when either that correlation is positive and the product of the other 2 correlations is negative or that correlation is negative and the product of the other 2 correlations is positive. The first aspect is illustrated with data from J. M. Gottman and L. J. Krokoff (1989) on initial conflict behavior and initial and later marital satisfaction, and the 2nd aspect is illustrated with data from C. L. Heavey, C. Layne, and A. Christensen (1993) on the same issue. Their interpretation of their findings is questioned.     

General resolution of correlation matrices into components and its utilization in multiple and partial regression

The derivation of multiple and partial regression statistics from uniqueness-augmented factor loadings, presented in the literature for orthogonal factor solutions, is generalized to oblique solutions. A mathematical rationale for the general case, without restriction to uncorrelated factors, is presented. Use of the general formulation is illustrated with a two-factor, seven-variable example.This report is based on work done under ARDC Project 7702, in support of the research and development program of the Air Force Personnel and Training Research Center, Lackland Air Force Base, Texas. Permission is granted for reproduction, translation, publication, use, and disposal in whole and in part by or for the United States Government.     

A multiple group method of factoring the correlation matrix

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be calleda multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrixF which satisfies the fundamental relationFF'=R.This study is one of a series of investigations in the development of multiple factor analysis and application to the study of primary mental abilities. We wish to acknowledge the financial assistance from the Social Science Research Committee of The University of Chicago which has made possible the work of the Psychometric Laboratory.     

A simplification of the theory of simplicity

Nelson Goodman has constructed two theories of simplicity: one of predicates; one of hypotheses. I offer a simpler theory by generalization and abstraction from his. Generalization comes by dropping special conditions Goodman imposes on which unexcluded extensions count as complicating and which excluded extensions count as simplifying. Abstraction is achieved by counting only nonisomorphic models and subinterpretations. The new theory takes into account all the hypotheses of a theory in assessing its complexity, whether they were projected prior to, or result from, projection of a given hypothesis. It assigns simplicity post-projection priority over simplicity pre-projection. It better orders compound conditionals than does the theory of simplicity of hypotheses, and it does not inherit an anomaly of the theory of simplicity of predicates — its failure to order the ordering relations. Drop Goodman's special conditions, and the problems fall away with them.     

The rowwise correlation between two proximity matrices and the partial rowwise correlation

This paper discusses rowwise matrix correlation, based on the weighted sum of correlations between all pairs of corresponding rows of two proximity matrices, which may both be square (symmetric or asymmetric) or rectangular. Using the correlation coefficients usually associated with Pearson, Spearman, and Kendall, three different rowwise test statistics and their normalized coefficients are discussed, and subsequently compared with their nonrowwise alternatives like Mantel'sZ. It is shown that the rowwise matrix correlation coefficient between two matricesX andY is the partial correlation between the entries ofX andY controlled for the nominal variable that has the row objects as categories. Given this fact, partial rowwise correlations (as well as multiple regression extensions in the case of Pearson's approach) can be easily developed.The author wishes to thank the Editor, two referees, Jan van Hooff, and Ruud Derix for their useful comments, and E. J. Dietz for a copy of the algorithm of the Mantel permutation test.     

Correlation,partial correlation,and causation

Philosophers and scientists have maintained that causation, correlation, and “partial correlation” are essentially related. These views give rise to various rules of causal inference. This essay considers the claims of several philosophers and social scientists for causal systems with dichotomous variables. In section 2 important commonalities and differences are explicated among four major conceptions of correlation. In section 3 it is argued that whether correlation can serve as a measure of A's causal influence on B depends upon the conception of causation being used and upon certain background assumptions. In section 4 five major kinds of “partial correlation” are explicated, and some of the important relations are established among two conceptions of “partial correlation”, the conception of “screening off”, the conception of “partitioning”, and the measures of causal influence which have been suggested by advocates of path analysis or structural equation methods. In section 5 it is argued that whether any of these five conceptions of “partial correlation” can serve as a measure of causal influence depends upon the conception of causation being used and upon certain background assumptions. The important conclusion is that each of the approaches (considered here) to causal inference for causal systems with dichotomous variables stands in need of important qualifications and revisions if they are to be justified.     

A note on moran's measure of multiple rank correlation

A measure of multiple rank correlation,T _y.12 ², is proposed for the situation with no tied observations in the variables. The measure is a weighted average of two squared Kendall taus. It is shown thatT _y.12 ² is equivalent to a statistic previously proposed by Moran and thus a new interpretation is given to Moran's statistic.The author wishes to thank Nancy Anderson, Willard Larkin, and Kent Norman for their helpful comments.     

Matrix formulas for part and partial correlation

General formulas for part and partial correlation of any order are derived in terms of multiple correlation coefficients, standard partial regression weights, and validities. The relationship between part correlation and the independent contribution of a predictor is discussed.     

Generalizations of the partial,part and bipartial canonical correlation analysis

Theg ₁- andg ₂-bipartial canonical correlation analyses are developed as generalizations of the partial, part, and bipartial canonical correlation analysis. Illustrative examples are provided.     

A note on multiple and canonical correlation for a singular covariance matrix

A weaker generalized inverse (Rao's g-inverse; Graybill's c-inverse) can be used in place of the Moore-Penrose generalized inverse to obtain multiple and canonical correlations from singular covariance matrices.The author expresses his gratitude to a referee for his suggestions.     

Some multiple correlation and predictor selection methods

The Doolittle, Wherry-Doolittle, and Summerfield-Lubin methods of multiple correlation are compared theoretically as well as by an application in which a set of predictors is selected. Wherry's method and the Summerfield-Lubin method are shown to be equivalent; the relationship of these methods to the Doolittle method is indicated. The Summerfield-Lubin method, because of its compactness and ease of computation, and because of the meaningfulness of the interim computational values, is recommended as a convenient least squares method of multiple correlation and predictor selection.     

Multiple bi-serial and multiple point bi-serial correlation

Normal equations, using data in various forms, are presented for securing the regression weights for prediction of a dichotomized criterion, and a simplified equation for the estimation of the multiple bi-serial or multiple point bi-serial, depending upon the proper assumption as to the nature of the distribution of the criterion, on the basis of these maximal weights is given also. The weights, unaffected by the assumption as to the nature of the criterion, are identical (or proportional) to those found by the discriminant function approach based upon analysis of variance. The author holds that the present multiple correlation approach is both easier and more informative than the discriminant function (analysis of variance) approach and suggests that the discriminant function be abandoned in favor of multiple bi-serial and/or multiple point bi-serial correlation and regression.