Some symmetric,invariant measures of multivariate association

A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

A unified approach to multiple‐set canonical correlation analysis and principal components analysis

Multiple‐set canonical correlation analysis and principal components analysis are popular data reduction techniques in various fields, including psychology. Both techniques aim to extract a series of weighted composites or components of observed variables for the purpose of data reduction. However, their objectives of performing data reduction are different. Multiple‐set canonical correlation analysis focuses on describing the association among several sets of variables through data reduction, whereas principal components analysis concentrates on explaining the maximum variance of a single set of variables. In this paper, we provide a unified framework that combines these seemingly incompatible techniques. The proposed approach embraces the two techniques as special cases. More importantly, it permits a compromise between the techniques in yielding solutions. For instance, we may obtain components in such a way that they maximize the association among multiple data sets, while also accounting for the variance of each data set. We develop a single optimization function for parameter estimation, which is a weighted sum of two criteria for multiple‐set canonical correlation analysis and principal components analysis. We minimize this function analytically. We conduct simulation studies to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of functional neuroimaging data to illustrate its empirical usefulness.     

Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

A Comparison of Different Methods for the Elicitation of Attribute Weights: Structural Modeling,Process Tracing,and Self-Reports

This study investigates whether structural modeling, process tracing, and self-reports are able to provide similar information about attribute weights in multiattribute evaluation processes. In three experiments subjects had to evaluate a large number of profiles of fictitious persons described on a number of attributes. The experiments differed in type of judgment task, type of subjects, and number of attributes. Subject attribute weights were derived in all cases by fitting a statistical model (statistical weights), by analyzing verbal protocols (verbal protocol weights), and by directly asking the subject how important the attributes are for the judgments (subjective weights). Correspondence between the three sets of weights is examined in two ways: by computing the correlation between three sets of weights and by calculating how adequately the different sets of weights, applied in a linear model, can predict the subject′s judgments. The first method appears to be inappropriate for investigating correspondence. The correlations are rather unstable because of the small number of attributes, and apart from that, they tend to underestimate real correspondence when the weights in the respective sets are approximately equal. The second method shows that the three sets of weights are about equally adequate in predicting the actual subject judgments. It is concluded that this method convincingly demonstrates that the three different ways of eliciting attribute weights yield similar results.     

Canonical versus factor analytic perspectives on the structure of associations between the MMPI and the Buss-Durkee Hostility Inventory

A canonical correlational analysis of MMPI profiles versus Buss-Durkee hostility and aggression scales was conducted, resulting in the identification of a general hostility-aggression-maladjustment dimension and three relatively specific aggression-personality relationships. These findings were contrasted with those obtained from a factor analysis of the combination of both sets of scores. Though often used in previous studies of associations between the Buss-Durkee Hostility Inventory and other measures, factor analysis was seen to be less than optimally suited to such a purpose. It was concluded that canonical correlation is preferable to factor analysis when seeking to clarify the multivariate structure of relationships between two sets of variables.     

K-fold Crossvalidation in Canonical Analysis

A computer-assisted, K-fold crossvalidation technique is discussed within the framework of canonical correlation analysis of randomly generated data sets. Results of the analysis suggest that this technique of multi-crossvalidation can be an effective method to reduce the contamination of canonical variates and canonical correlations by sample-specific variance components.     

Multivariate relationships between measures of learning style and memory

Three self-report instruments assessing “learning style” were administered to 106 undergraduate students: the Inventory of Learning Processes, the Study Behavior Questionnaire, and the Learning Style Inventory. Performance measures on word list and prose learning tasks were also obtained using both immediate and delayed retention tests. Canonical correlation analysis of the pairs of learning style instruments indicated that there was a small to moderate amount of overlapped variance. Inspection of the canonical variates revealed that what overlap existed was generally due to a common factor related to the depth of processing conception of memory. No significant canonical correlations were obtained between any of the learning style instruments and word list retention measures. However, the Inventory of Learning Processes and the Learning Style Inventory each obtained one significant canonical correlation with the set of prose retention measures. There was little overlap between sets of retention measures derived separately from prose and word list tasks. The findings have implications concerning (a) the validity and equivalency of available self-report instruments assessing learning style, (b) the role of the learning task in studying individual differences in learning and memory processes, and (c) the assessment of “depth of processing” at the level of individual differences.     

Approximate entropy and point correlation dimension of heart rate variability in healthy subjects.

The contribution of nonlinear dynamics to heart rate variability in healthy humans was examined using surrogate data analysis. Several measures of heart rate variability were used and compared. Heart rates were recorded for three hours and original data sets of 8192 R-R intervals created. For each original data set (n = 34), three surrogate data sets were made by shuffling the order of the R-R intervals while retaining their linear correlations. The difference in heart rate variability between the original and surrogate data sets reflects the amount of nonlinear structure in the original data set. Heart rate variability was analyzed by two different nonlinear methods, point correlation dimension and approximate entropy. Nonlinearity, though under 10 percent, could be detected with both types of heart rate variability measures. More importantly, not only were the correlations between these measures and the standard deviation of the R-R intervals weak, the correlation among the nonlinear measures themselves was also weak (generally less than 0.6). This suggests that in addition to standard linear measures of heart rate variability, the use of multiple nonlinear measures of heart rate variability might be useful in monitoring heart rate dynamics.     

Approximate entropy and point correlation dimension of heart rate variability in healthy subjects

The contribution of nonlinear dynamics to heart rate variability in healthy humans was examined using surrogate data analysis. Several measures of heart rate variability were used and compared. Heart rates were recorded for three hours and original data sets of 8192 R-R intervals created. For each original data set (n=34), three surrogate data sets were made by shuffling the order of the R-R intervals while retaining their linear correlations. The difference in heart rate variability between the original and surrogate data sets reflects the amount of nonlinear structure in the original data set. Heart rate variability was analyzed by two different nonlinear methods, point correlation dimension and approximate entropy. Nonlinearity, though under 10 percent, could be detected with both types of heart rate variability measures. More importantly, not only were the correlations between these measures and the standard deviation of the R-R intervals weak, the correlation among the nonlinear measures themselves was also weak (generally less than 0.6). This suggests that in addition to standard linear measures of heart rate variability, the use of multiple nonlinear measures of heart rate variability might be useful in monitoring heart rate dynamics.     

The Relation of Self-Reported Abilities to Aptitude Test Scores: A Replication and Extension

Scales from the Self-Directed Search (SDS) were correlated with those of the Ball Aptitude Battery (BAB) for a sample of 198 high school students. The present study found larger canonical correlations between the SDS and aptitude measures than those reported by Kelso, Holland, and Gottfredson (1977). However, the variance across the data sets associated with these canonical correlations was minimal, and much of the shared variance associated with the BAB canonical variates was due tog, with specific abilities contributing only minor portions of the observed shared variance.     

Studying Canonical Analysis: A Reply To Thorndike's Comments

Robert M. Thorndike (1976) commented on the results of a Monte Carlo study on the stability of canonical correlations, canonical weights, and canonical variate-variable correlations (Barcikowski and Stevens, 1975). In this paper each of his comments are examined by the authors of the Monte Carlo Study. In addition, a possible solution to the large number of subjects necessary for stable weights and variate-variable correlations using ridge regression procedures is suggested.     

The simultaneous prediction of any number of criteria by the use of a unique set of weights

When several criteria are available, it is ordinarily necessary (1) to select one of them asthe criterion, (2) to use several and thus arrive at several different sets of weights, or (3) to combine them into a single measure. A formula is derived for the determination of a unique set of weights. The use of these weights will produce the highest possible average coefficient of correlation between the various criteria and two (or more) weighted independent variables. If desired, the criteria may be assigned any predetermined weights. The weights then derived for the independent variables are such that the weighted average of the correlation coefficients between the various criteria and the independent variable composite will be a maximum. In the use of these formulas, no assumptions are necessary regarding the interrelationships existing among the criteria and it is not necessary to compute the intercorrelations among the criteria. A numerical example is included.     

Piagetian tasks measure intelligence and intelligence tests assess cognitive development: A reanalysis

A reanalysis of the intercorrelations of Wechsler subtests, achievements tests, and Piagetian tasks originally published by Stephens et al. (1972) has corrected errors in the original design and has used models which allow both differences and similarities to appear. Unit weighted composites of the two sets of tests were formed after first partialling chronological age out of the complete table of intercorrelations. The correlation between the composites is .876 which corrects, using split-half estimates of reliabilities, to .918. Next the matrix of partial correlations was factored and the factors rotated to a hierarchical structure. Piagetian tasks contribute almost equally to the definition of the general factor in intelligence along with the Wechsler subtests and the achievement tests. The general factor, furthermore, makes the single largest contribution to the variance of most of the measures of both sets. The smaller group factors consist of an Academic Achievement factor and three Piagetian factors. Of the latter the largest and most clearly identified is the factor called Operational Thought in the earlier analysis, but is here described simply as Conservation. It is concluded that the evidence for communality in function far outweighs the differences between intelligence tests and Piagetian tasks.     

Functional Multiple-Set Canonical Correlation Analysis

We propose functional multiple-set canonical correlation analysis for exploring associations among multiple sets of functions. The proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered. As in classical multiple-set canonical correlation analysis, computationally, the method solves a matrix eigen-analysis problem through the adoption of a basis expansion approach to approximating data and weight functions. We apply the proposed method to functional magnetic resonance imaging (fMRI) data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task.     

Hardiness and sense of coherence and their relation to negative affect

This study attempted to determine the degree to which measures of health proneness and measures of negative affect represent two distinct, although related, constructs. In addition, it examined the relation between Antonovsky's salutogenic construct of sense of coherence (SOC) and Kobasa's health proneness construct of hardiness. Five health proneness and three negative affect measures were filled out by 164 male patients with coronary heart disease. The pattern of correlations between these measures and confirmatory factor analysis indicated that although the measures of health proneness are negatively related to measures of negative affect, these two sets of measures and the constructs to which they are related can be differentiated from each other. However, SOC was found to be less independent of negative affect than was hardiness.     

The comparison of interdependent correlations between optimal linear composites

A general procedure is provided for comparing correlation coefficients between optimal linear composites. The procedure allows computationally efficient significance tests on independent or dependent multiple correlations, partial correlations, and canonical correlations, with or without the assumption of multivariate normality. Evidence from some Monte Carlo studies on the effectiveness of the methods is also provided.This research was supported in part by an operating grant (#67-4640) to the first author from the National Sciences and Engineering Research Council of Canada. The authors would also like to acknowledge the helpful comments and encouragement of Alexander Shapiro, Stanley Nash, and Ingram Olkin.     

Job Experience and Big Five Personality Dimensions

Using a sample of engineers (N=80), the correlation between job experience and the personality dimensions was estimated. The results showed very small correlations. Based on these results, we suggest that composites created with job experience and personality measures can be useful for personnel selection.     

Research on individual differences requires correlational analysis,not ANOVA

The data of Welsh and Baucom (1977) have been reanalyzed using correlational analysis for the complete distributions in place of the analysis of variance of extreme groups. The analysis reveals a number of statistically significant correlations between composites of M-F scales and several measures of intelligence, but these are trival in size. Also the sign of the correlations depends directly on which M-F scales are selected to form composites. One can obtain a correlation between either masculinity and intelligence or femininity and intelligence depending on whether the supposed trait is measured from scales formed from the Adjective Check List or from a combination of the MMPI and the SVIB.