Orthogonal inter-battery factor analysis

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.This investigation has been supported by the U.S. Office of Naval Research under Contract Nonr-2752(00).     

Assessment of Developed General Intellectual Ability with Nonverbal Measures

A canonical correlation analysis was performed between two measures of general intellectual ability: The D. 48 Test and the Standard Progressive Matrices were taken as one set of variables, and general scholastic ability, verbal and quantitative ability, English and mathematics achievement, time spent in school, and ethnic/cultural group membership were used as the second set of variables. Subjects were 180 girls attending a large urban parochial high school. The D. 48 Test performances made a substantial contribution to the first canonical variate associated with the only significant canonical correlation; quantitative ability performances made a substantial contribution to the second variate. Matrices performances and performances in mathematics achievement and general scholastic ability made lesser contributions to their respective variates.     

Canonical correlation and chi-square: relationships and interpretation

A 2 x 2 chi-square can be computed from a phi coefficient, which is the Pearson correlation between two binomial variables. Similarly, chi-square for larger contingency tables can be computed from canonical correlation coefficients. The authors address the following series of issues involving this relationship: (a) how to represent a contingency table in terms of a correlation matrix involving r - 1 row and c - 1 column dummy predictors; (b) how to compute chi-square from canonical correlations solved from this matrix; (c) how to compute loadings for the omitted row and column variables; and (d) the possible interpretive advantage of describing canonical relationships that comprise chi-square, together with some examples. The proposed procedures integrate chi-square analysis of contingency tables with general correlational theory and serve as an introduction to some recent methods of analysis more widely known by sociologists.     

Canonical Correlation and Chi-Square: Relationships and Interpretation

A 2 × 2 chi-square can be computed from a phi coefficient, which is the Pearson correlation between two binomial variables. Similarly, chi-square for larger contingency tables can be computed from canonical correlation coefficients. The authors address the following series of issues involving this relationship: (a) how to represent a contingency table in terms of a correlation matrix involving r - 1 row and c - 1 column dummy predictors; (b) how to compute chi-square from canonical correlations solved from this matrix; (c) how to compute loadings for the omitted row and column variables; and (d) the possible interpretive advantage of describing canonical relationships that comprise chi-square, together with some examples. The proposed procedures integrate chi-square analysis of contingency tables with general correlational theory and serve as an introduction to some recent methods of analysis more widely known by sociologists.     

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.     

Instrumental variables in factor analysis

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.     

An Empirical Investigation Of Step-Down Canonical Correlation With Cross-Validation

Three potential applications of stepwise procedures in canonical analysis and several alternative stepping decision rules are described. A stepdown procedure using smallest interest multiple correlation as the stepping criterion was applied to data on the Minnesota Importance Questionnaire and the Minnesota Vocational Interest Inventory for two random halves of a group of 500 males in a double cross-validation design. Results indicated that up to 75 percent of variables could be dropped from either set with little drop in the R[SUBc]. Cross-validation coefficients were usually higher after dropping several variables than for the full sets.     

Predicting Academic Achievement with the High School Placement Test

The study reported here investigated the use of the High School Placement Test as a predictor of 9th grade academic achievement as measured by school grades. The sample was composed of 3,194 boys who were enrolled in an academic curriculum at 1 of the 10 randomly selected Diocesan high schools in Philadelphia. A multiple regression analysis and a canonical correlation analysis were performed using the subtest scores of the High School Placement Test as independent variables, and final 9th grade average, grades in English, social studies, Latin, general science, and algebra as the dependent variables. It was found that the Composite score of the High School Placement Test individually would have been as useful a predictor of final grades as either of the empirically determined predictor composites computed in performing the multiple regression analysis and the canonical correlation analysis.     

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.     

Measures of reliability for profiles and test batteries

A general index of reliability, termed “canonical reliability,” is developed for use with profiles, or more generally, for use with vectors of random variables. Canonical reliability is defined as the ratio of the average squared distance among true scores to the average squared distance among observed scores. Based on Mahalonobis distances, canonical reliability is shown to be consistent with multivariate analogues of parallel form correlations, squared correlation between true and observed scores, and an analysis of variance formulation. The index of reliability based on Cronbach and Gleser'sD ² is also derived from the general formulation. A comparison of the Mahalonobis andD ² approaches indicates that score vectors usingD ² distances are more reliable; however, both methods of comparing profiles are useful depending on the nature of the information that is desired. Transforming the observed variables to independent canonical variates provides a basis for comparing profiles on maximally reliable profile dimensions. For illustrative purposes, profile reliability is calculated and interpreted for the WISC subscales for a 7 1/2 year age group.     

Robust canonical discriminant analysis

A method for robust canonical discriminant analysis via two robust objective loss functions is discussed. These functions are useful to reduce the influence of outliers in the data. Majorization is used at several stages of the minimization procedure to obtain a monotonically convergent algorithm. An advantage of the proposed method is that it allows for optimal scaling of the variables. In a simulation study it is shown that under the presence of outliers the robust functions outperform the ordinary least squares function, both when the underlying structure is linear in the variables as when it is nonlinear. Furthermore, the method is illustrated with empirical data.The research of the first author was supported by the Netherlands Organization of Scientific Research (NWO grant 560-267-029).     

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.     

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.     

Tandem criteria for analytic rotation in factor analysis

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.     

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.     

Identity and counsellor preference†

This study investigated the relationship of identity development, measured by the Self-Identity Inventory (SII), universal-diverse orientation (UDO), measured by the Miville-Guzman Universality-Diversity Scale- Short (M-GUDS-S), to counsellor preference in a sample of 365 university students. After controlling for personal identity demographic variables, multiple regression analysis indicated SII and UDO accounted for significant additional variance in counsellor preference with Immersion (SII Scale 3), Relativistic Appreciation (MGUDS-S Subscale 2) and Comfort with Differences (MGUDS-S Subscale 3), identified as significant unique predictors. A canonical correlation analysis performed with 11 individual counsellor preference variables as criterion variables and 9 identity salience variables as predictor variables to explore the relationships between identity salience and counsellor preference yielded three significant canonical functions.     

An investigation of the relationship between general belief patterns and well-being

One hundred and thirty-three employees (58 males, 75 females) of the Department of Veterans Affairs Medical Center (DVAMC), Coatesville, PA anonymously volunteered data on a Demographic Survey, The Brief Symptom Inventory (BSI, Derogatis), and The Irrational Beliefs Test (IBT, Jones). A canonical correlation analysis was used to investigate associations between general patterns of beliefs and symptomatology. Two significant pairs of variates were revealed. The first pair provided associations between Demand for Approval, High Self-Expectations, Frustration Reactive, Anxious Overconcern, and Perfectionism and psychological symptomatology. The second pair provided associations between Demand for Approval, Blame Proneness, Frustration Reactive, and Emotional Irresponsibility and somatic symptoms. A MANOVA was used to assess for gender differences. There was no significant general gender effect. However, univariate analyses revealed significant effects for seven BSI variables. This study supported the belief that one's pattern of thinking is related to one's pattern of symptomatology. This has implications for use in health psychology and behavioral medicine.     

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.     

External Single-Set Components Analysis Of Multiple Criterion/Multiple Predictor Variables

Although much progress has been made in clarifying the properties of canonical correlation analysis in order to enhance its applicability, there are several remaining problems. Canonical variates do not always represent the observed variables even though the canonical correlation is high. In addition, canonical solutions are often difficult to interpret.

This paper presents a method designed to deal with these two problems. Instead of maximizing the correlation between unobserved variates, the sum of squared inter-set loadings is maximized. Contrary to the canonical correlation solution, this method ensures that the shared variance between predictor variates and criterion variables is maximal. Instead of extracting variates from both criterion and predictor variables, only one set of components (from the predictor variables) is constructed. Without loss of common variance, an orthogonal rotation is applied to the resulting loadings in order to simplify structure.     

The effects of knowledge of child development and social-emotional maturity on adolescent attitudes toward parenting

This study investigated the relationship between the combined effect of knowledge of child development and level of social-emotional maturity, and the extent to which this relationship affects adolescent attitudes toward parenting. The analysis of the data (multiple regression and canonical analysis) suggested that there were significant relationships among these variables. In general, the relationships indicated that subjects' negative attitudes toward parenting were associated with lack of knowledge of child development and low levels of social-emotional maturity, while subjects' positive attitudes toward parenting were associated with knowledge of child development and high levels of social-emotional maturity. The joint impact of knowledge of child development and social-emotional maturity factors on attitudes toward parenting accounted for 51% of the variation among the variables.