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.     

Multivariate Relationships Between Motor and Personality Measures in Culturally Deprived High School Pupils

The relationship between personality, as measured by the IPAT-HSPQ, and certain motor performance variables was studied by utilizing the multivariate technique of canonical correlation. Ss for the study were 91 culturally deprived high school students who were achieving well below their expected capacity. A significant canonical relationship was not found between the domains of personality and motor performance. Also, no significant relationships between motor sub-domains and personality were found. It was thus concluded that for the population under study, the domains of motor performance and personality were independent Some discussion of the multivariate technique of canonical correlation was offered and compared with univariate procedures.     

A Multivariate Analysis of Psychographic Variables

This study examined the relationship between demographic and personality variables through the use of a multivariate technique, canonical analysis. The principal advantage of such a technique lies in its ability to simultaneously consider sets of variables rather than focusing narrowly in a single predictor or criterion variable. The four demographic variables examined, sex, age, education, and marital status, produced four statistically significant relationships with personality traits measured by the Edwards Personal Preference Schedule. These relations were examined and suggestions made as to how the results might be used to indicate areas for further research.     

Examining the relationship between the Eysenck personality inventory and the sixteen personality factors questionnaire using predictive configural frequency analysis

A nonparametric analogue of canonical correlation analysis, called P-CFA, was used to determine how the Eysenck Personality Inventory is related to Cattell's Sixteen Personality Factors Questionnaire. The P-CFA methods used showed that Eysenck's personality dimensions of Extraversion and Neuroticism can be predicted quite well from suitably chosen primary scales in the Cattell questionnaire. The results were consistent with those obtained in another study using canonical analysis. Beyond that, P-CFA methods were used to predict Eysenckian trait-type categories from Cattell's primary scales. The results were weak and not predictable from canonical analysis, but showed potential analytic capabilities of P-CFA that are not available in canonical analysis.     

Canonical correlations with fallible data

The technique of canonical correlation may be used to determine the extent to which two sets of measurements reflect the same underlying traits. However, if the two sets are not perfectly reliable, the unreliability may obscure the fact that they are essentially dependent on similar processes. If we attempt to determine sets of weights so as to maximize the correlation between linear composites after correction for attenuation in the composites, it turns out that the results may be obtained by determining the canonical correlations and canonical regression weights between the true score components of the measures making up the two sets. In addition, formulas are developed for calculating the correlations between the canonical variates and original measures, both corrected and uncorrected for attenuation. A numerical example relating the verbal to the performance subtests of the Wechsler Intelligence Scale for Children is presented.     

Regularized Multiple-Set Canonical Correlation Analysis

Multiple-set canonical correlation analysis (Generalized CANO or GCANO for short) is an important technique because it subsumes a number of interesting multivariate data analysis techniques as special cases. More recently, it has also been recognized as an important technique for integrating information from multiple sources. In this paper, we present a simple regularization technique for GCANO and demonstrate its usefulness. Regularization is deemed important as a way of supplementing insufficient data by prior knowledge, and/or of incorporating certain desirable properties in the estimates of parameters in the model. Implications of regularized GCANO for multiple correspondence analysis are also discussed. Examples are given to illustrate the use of the proposed technique. The work reported in this paper is supported by Grants 10630 and 290439 from the Natural Sciences and Engineering Research Council of Canada to the first and the second authors, respectively. The authors would like to thank the two editors (old and new), the associate editor, and four anonymous reviewers for their insightful comments on earlier versions of this paper. Matlab programs that carried out the computations reported in the paper are available upon request.     

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.     

Simultaneous optimization of factor assessibility and representativeness

P. M. Bentler has shown that Rao's canonical factor analysis is in effect a psychometric analysis, leading to factors that are maximally assessible from the data. He contrasts this with Kaiser and Caffrey's alpha factor analysis that leads to factors that maximally represent the true factors in the content domain. Noting the problems associated with factors that may be highly assessible, but not very representative, or vice versa, Bentler suggests the need for a technique that would, insofar as possible, be optimal with respect to both criteria. Such a technique is presented here, and is shown to resolve into a traditional scaling method, which in turn acquires a richer psychometric interpretation.The author would like to gratefully acknowledge support for this research from the Workshop in Political Theory and Policy Analysis (Indiana University) and Esco Research Associates. The paper benefitted substantially from the criticisms and suggestions of an anonymous referee.     

Revisiting Interpretation of Canonical Correlation Analysis: A Tutorial and Demonstration of Canonical Commonality Analysis

In the face of multicollinearity, researchers face challenges interpreting canonical correlation analysis (CCA) results. Although standardized function and structure coefficients provide insight into the canonical variates produced, they fall short when researchers want to fully report canonical effects. This article revisits the interpretation of CCA results, providing a tutorial and demonstrating canonical commonalty analysis. Commonality analysis fully explains the canonical effects produced by using the variables in a given canonical set to partition the variance of canonical variates produced from the other canonical set. Conducting canonical commonality analysis without the aid of software is laborious and may be untenable, depending on the number of noteworthy canonical functions and variables in either canonical set. Commonality analysis software is identified for the canonical correlation case and we demonstrate its use in facilitating model interpretation. Data from Holzinger and Swineford (1939) Holzinger, K. J. and Swineford, F. 1939. “A study in factor analysis: The stability of a bi-factor solution.”. In Supplementary Educational Monographs Vol. 48, 81–91. Chicago: Department of Education, University of Chicago.  [Google Scholar] are employed to test a hypothetical theory that problem-solving skills are predicted by fundamental math ability.     

Restricted Arrow

In this paper I present a range of substructural logics for a conditional connective ↦. This connective was original introduced semantically via restriction on the ternary accessibility relation R for a relevant conditional. I give sound and complete proof systems for a number of variations of this semantic definition. The completeness result in this paper proceeds by step-by-step improvements of models, rather than by the one-step canonical model method. This gradual technique allows for the additional control, lacking in the canonical model method, that is required.     

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.     

Reading Ability as a Predictor of Academic Procrastination Among African American Graduate Students

The present study examined the relationship between reading ability (i.e., reading comprehension and reading vocabulary) and academic procrastination among 120 African American graduate students. A canonical correlation analysis revealed statistically significant and practically significant multivariate relationships between these two reading ability variables and graduate students' levels of academic procrastination. Specifically, the first canonical correlation analysis revealed a statistically significant and practically significant multivariate relationship between reading ability and academic procrastination resulting from fear of failure. The second canonical correlation analysis revealed a statistically significant and practically significant multivariate relationship between reading ability and academic procrastination associated with writing a term paper, performing administrative tasks, attending meetings, keeping up with weekly reading assignments, and, most notably, performing academic tasks. Implications are discussed in the context of designing and implementing strategies designed to improve African American student performance and instruction in graduate-level courses.     

Forced classification: A simple application of a quantification method

This study formulates a property of a quantification method, the principle of equivalent partitioning (PEP). When the PEP is used together with Guttman's principle of internal consistency (PIC) in a simple way, the combination offers an interesting way of analyzing categorical data in terms of the variate(s) chosen by the investigator, a type of canonical analysis. The study discusses applications of the technique to multiple-choice, rank-order, and paired comparison data.This study was supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. A7942). Comments on the earlier drafts from anonymous reviewers and the editor were much appreciated.     

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.     

A unified treatment of the weighting problem

A general procedure is described for obtaining weighted linear combinations of variables. This includes as special cases, multiple regression weights, canonical variate analysis, principal components, maximizing composite reliability, canonical factor analysis, and certain other well-known methods. The general procedure is shown to yield certain desirable invariance properties, with respect to transformations of the variables.The author wishes to thank Dr. A. J. Cropley for preparing the necessary computer programs for this study.     

Graded modalities,II (canonical models)

This work intends to be a generalization and a simplification of the techniques employed in [2], by the proposal of a general strategy to prove satisfiability theorems for NLGM-s (= normal logics with graded modalities), analogously to the well known technique of the canonical models by Lemmon and Scott for classical modal logics.     

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.     

Regularized Generalized Canonical Correlation Analysis

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables. It constitutes a general framework for many multi-block data analysis methods. It combines the power of multi-block data analysis methods (maximization of well identified criteria) and the flexibility of PLS path modeling (the researcher decides which blocks are connected and which are not). Searching for a fixed point of the stationary equations related to RGCCA, a new monotonically convergent algorithm, very similar to the PLS algorithm proposed by Herman Wold, is obtained. Finally, a practical example is discussed.     

The psychophysical dimensions of similarity among random shapes

Six psychological dimensions were recovered from similarity judgments of 20 random forms using Kruskal’s nonmetric scaling technique. These were related to 10 physical factors describing the forms by using canonical correlation analysis and rotations of the resulting vectors. Four factors accounted for 2/3 of the variance of the intersection of the physical and psychological spaces for each of 11 Ss. The factors are described as Complexity, Curvature, Curvature Dispersion, and Straight-length Dispers ion. The problem of the proper psychological distance metric is discussed, and individual differences among Ss are analyzed. The results lead to the conclusion that the four major factors found may be good representations of processes occurring in the perceptual system.     

Relationships among several methods of linearly constrained correspondence analysis

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.The work reported in this paper has been supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. We wish to thank Carolyn Anderson, Ulf Böckenholt, Henk Kiers, Shizuhiko Nishisato, Jim Ramsay, Tadashi Shibayama, Cajo ter Braak, and Peter van der Heijden for their helpful comments on earlier drafts of this paper.