DOCUMENT RESUME ED 383 760 TM 023 293 AUTHOR Fan, Xitao TITLE Structural Equation Modeling and Canonical Correlation Analysis: What Do They Have in Common? PUB DATE Apr 95 NOTE 28p.; Paper presented at the Annual Meeting of the American Educational Research Association (San Francisco, CA, April 18-22, 1995). PUB TYPE Reports Evaluative/Feasibility (142) Speeches /Conference Papers (150) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Comparative Analysis; *Correlation; Mathematical Models; *Research Methodology; *Structural Equation Models ABSTRACT This paper, in a fashion easy to follow, illustrates the interesting relationship between structural equation modeling and canonical correlation analysis. Although computationally somewhat inconvenient, representing canonical correlation as a structural equation model may provide some information which is not available from conventional canonical correlation analysis. Hierarchically, with regard to the degree of generality of the techniques, it is suggested that structural equation modeling stands to be a more general approach. For researchers interested in these techniques, understanding the interrelationship among them is meaningful, since our judicious choice of these analytic techniques for our research depends on such understanding. Three tables and two figures present the data. (Contains 25 references.) (Author) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. * *********************************************************************** C) 47) U.S. DEPARTMENT OF EDUCATION "PERMISSION TO REPRODUCE THIS t". Othce of Educational Research end improvement MATERIAL HAS BEEN GRANTED BY EDUCATIONAL RESOURCES INFORMATION en CENTER (ERIC) 00 This document has been reproduced as XiV10 M colcolved from the Person Or organuation ongmatmg d C:) 0 Mmor changes have bean made to Improve 14 reproduction oualdy Points of vow or opniOnS Stetild inthisdo t mint 60 not necSaartly represent official TO THE EDUCATIONAL RESOURCES OE RI posdan or policy INFORMATION CENTER (ERIC)." Structural Equation Modeling and Cancnical Correlation Analysis: What Do They Have in Common? Xitao Fan Utah State University Department of Psychology Utah State University Logan, Utah 84322-2810 (801)797-1451 BEST COPY AVAILABLE Ii) 4N0 Paper presented at 1995 Annual Meeting of American Educational Research Association (AERA), San Francisco, California, April 18-22, E.195. Session #:52.42. ((-= 2 Abstract The paper, in a fashion easy to follow, illustrates the interesting relationship between structural equation modeling and canonical correlation analysis. Although computationally somewhat inconvenient, representing canonical correlation analysis as a structural equation model may provide some information which is not available from conventional canonical correlation analysis. Hierarchically, with regard to the degree of generality of the techniques, it is suggested that structural equation modeling stands to be a more general approach. For researchers interested in these techniques, understanding the interrelationship among them is meaningful, since our judicious choice of these analytic techniques for our research depends on such understanding.. 3 SEM & Canonical Analysis 1 INTRODUCTION The utilization of multivariate methods has been widely recognized as to be important in social and behavioral science research. The importance mainly stems from two considerations: 1) we intend to honor the complex social reality in which we operate and which we eventually want to generalize to; 2) we intend to avoid inflating experiment-wise error rate in our statistical analysis (Fish, 1988; Johnson & Wichern, 1988; SAS/STAT User's Guide, Version 6, Vol. 4, 1989; Stevens, 1986). Among the multivariate statistical techniques, canonical correlation analysis has occupied an important strategic position. It has often been conceptualized as a unified approach to many parametric statistical testing procedures, univariate and multivariate alike (Baggaley, 1981; Dunteman, 1984; Fornell, 1978; Knapp, 1978; Kshirsagar, 1972; SAS/STAT User's Guide, Version 6, Vol. 4, 1989; Thompson, 1984, 1991). Many popular parametric statistical techniques share one thing in common: they are designed to analyze linear relationships among variables. It has been shown that many of the seemingly different techniques are really not that different from one another after all (Fan, 1992; Knapp, 1978; Thompson, 1991). More specifically, "most of the practical problems arising in statistics can be translated, in some form or the other, as the problem of measurement of association between two vector variates N and y" (Kshirsagar, 1972, p. 281). Canonical correlation analysis, which summarizes the relationships between 4 SEM & Canonical Analysis 2 two groups (vectors) of variables, naturally brings order to the superficial chaos of a myriad of analytic techniques. Theoretically or empirically, the interesting relationship between canonical correlation analysis and many other statistical testing procedures ranging from simple correlation, t-test, to MANOVA, discriminant analysis has been shown many times (Knapp, 1971; Kshirsagar, 1972; Tatsuoka, 1989; Thompson, 1991). Structural equation modeling (SEM) has been heralded as a unified model which joins methods from econometrics, psychometrics, sociometrics, and multivariate statistics (Bentler, 1994). The generality and wide applicability of structural equation modeling approach has been amply demonstrated (JOreskog & Sorbom, 1989; Bentler, 1992). The statistical relationship among structural equation modeling and canonical correlation analysis, however, is less obvious. Consequently such relationship is less known by researchers, despite the insightful discussion on this topic provided by Bagozzi, Fornell and Larcker (1981). The purpose of this paper is to use concrete analysis examples to show the relationship among structural equation modeling and canonical correlation analysis. Also, this paper attempts to raise some questions about whether the structural equation modeling approach to canonical analysis will allow us to accomplish more than conventional canonical correlation approach. For researchers who are interested in these sophisticated analytical tools, to understand the underlying relationship among these seemingly different methods 5 SEM & Canonical Analysis 3 will contribute to their judicious choice among these techniques as they encounter different research situations. SEM AND CANONICAL CORRELATION ANALYSIS Canonical Correlation Analysis Pioneered by Hotelling (1935), canonical correlation analysis seeks to identify and measure the association between two sets of variables. Canonical correlation analysis can be understood as the bivariate correlation of two synthetic variables which are the linear combinations of the two sets of original variables (Johnson & Wichern, 1988; Thomson, 1984, 1991). The two sets of original variables are linearly combined to produce pairs of synthetic variables which have maximum correlation, with the restriction that each member of each subsequent set of such synthetic variables is orthogonal to all members of all other sets.- The maximum number of such pairs of synthetic variables which can be produced equals the number of variables in the smaller set of the two. In this sense, the synthetic variables in canonical correlation analysis, which are the linear combinations of the original variables, are similar to the synthetic variables produced in some other multivariate analysis techniques such-as principal component analysis, discriminant analysis, etc. The difference is that, in different statistical analysis, the original variables are linearly combined to satisfy different criteria. For example, in 6 SEM & Canonical Analysis' 4 discriminant analysis, the original variables are linearly combined to produce synthetic variables which maximizes the ratio of between-group variance to within-group variance, so that different groups can be maximally differentiated on the synthetic variables. In principal component analysis, the synthetic variables (principal components) are constructed so that the variance on the synthetic variables are maximized, and the maximum amount of variance in the original variables can be accounted for by the smallest number of these principal components. As can be expected for a multivariate statistical method, in canonical correlation analysis, eigenstructures of certain matrices play a crucial role in deriving the linear coefficients needed to produce the synthetic variables and in deriving the canonical correlation coefficients for different canonical functions. Assume two sets of variables X and Y, with X set is the smaller of the two. When combined, the two sets of variables have the following partitioned correlation matrix: Rxy R= R R yx yY The derivation of the linear coefficients for combining original variables into canonical variates is based on the following two matrices A and B, which have the same eigenvalues but have different eigenvectors ai and bi associated with the eigenvalues BEST COPYAVAILABLE 7 SEM & Canonical Analysis 5 A = Rxy Ryx B = R;y1 Ryx 122 Ray The elements in the eigenvectors ai and bi turn out to be the linear coefficients needed for the two original sets of variables X and Y respectively. These coefficients are often referred to as canonical function coefficients (Thompson, 1984) or canonical weights (Pedhazur,