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INTERPRETATION OF CANONICAL ANALYSIS: ROTATED VS. UNROTATED SOLUTIONS

Norman Cliff
University of Southern California

David J. Krus
Arizona State University

Orthogonal rotation of canonical variates is shown to preserve the major properties of the canonical solution and may increase its interpretability.

Difficulty of interpretation is widely believed to be a characteristic of Hotelling’s [1935, 1936] canonical variate analysis. Thus, e.g., Kerlinger (1973, p. 150) acknowledges the canonical correlation as a powerful method of multivariate prediction with “limitations . . . in the interpretation of the results it yields.” Similar doubts are expressed in the manual for the SPSS programs package. With respect to results provided by the program for canonical analysis, authors observe (Nie, Bent, and Hull, 1970, p. 244) that “the user has to examine two sets of coefficients simultaneously, and the result may often be very complex and difficult to interpret.”

The canonical correlation technique of multivariate prediction belongs to a family of methods which involve solving the characteristic equation for its latent roots and vectors. In this type of solution, rotation leaves many optimizing properties preserved, provided it takes place in certain ways and in a subspace. Thus rotation from the “maximum intervariate correlation structure” into a different, presumably simpler and more meaningful structure is possible. It is proposed here that this procedure is not only a legitimate, but also a desirable solution, increasing the meaningfulness, interpretability and utility of the method.

Theoretical Rationale 
 

Consider a supermatrix of correlations, incipient to canonical analysis

where X is the set of standardized predictor variables and Y is the set of standardized criterion variables. Let C signify a matrix of canonical weights for the predictor set of variables and D signify a matrix of canonical weights for the criterion set of variables. The canonical variates U and V then can be defined as

and

A supermatrix containing the submatrices comprising the canonical structure can be composed as

The identity submatrices in the above supermatrix indicate that this solution is orthogonal. The lambda submatrices with canonical correlations in the principal diagonal and zeroes in the off-diagonal elements can be defined either as

or

The intravariate submatrices of the canonical structure, containing correlations between the variables included in the analysis and the canonical variates can be defined for the predictor set of variables as

and for the criterion set of variables as

These submatrices, if stacked, are analogous to the matrix of factor loadings of the factor analysis, which contains correlations of variables with the factors. In the canonical analysis, these stacked submatrices define canonical loadings, containing correlations of variables with the canonical variates. The canonical loadings or the stacked canonical weights C and D can be rotated into the simple structure by any of the rotation methods developed within the framework of factor analysis. The canonical loadings or the canonical weights can be imagined as vectors in a hyperspace and must be stacked and rotated simultaneously, since the transformation of these structures to a more interpretable position within the hyperspace must be orthonormal. If the canonical weights were rotated, the rotated canonical loadings can be obtained as

and

where  and  are rotated canonical weights. The canonical correlations for the rotated solution can be computed as

If the original and rotated canonical weights are not equivalent, then the eigenvalues located along the principal diagonal of L will be more evenly distributed then eigenvalues of the orthogonal solution. This is in contrast with the unrotated solution, where the maximization of the canonical correlations during each successive root extraction causes the concentration of extracted variance in the first variate pairs. Also, the off-diagonal elements of  will be non-zero. For an instance of a rotated canonical analysis, see Krus and Tellegen (1975).

Discussion

The intimate relationship between factor analysis and canonical correlation has been mentioned repeatedly (Burt, 1948; Horst, 1960), and alternatives for evaluating the relationships between two sets of variables have been suggested. Alternatives include reduction of the criteria to their independent dimensions by factor analysis; making separate rotated factor analyses of predictors and criteria; and determining factor scores for both the predictors and criteria, and correlating these factor scores (cf. Koons, 1962, pp. 277—278). All these procedures are, however, incidental to the canonical analysis itself. Also, Hall (1969) suggests rotating the canonical variates of multivariate analysis of variance. His discussion, however, does not present a complete rationale for the proposal and is confined to the specific MANOVA case.

As supported by our theoretical reasoning and empirical scrutiny, rotation of the canonical variates does not seem to violate the properties of the method and the transformed variates appear to be more meaningful than the original ones

.

REFERENCES 

Burt, C. Factor analysis and canonical correlation. British Journal of Psychology, 1948, 1, 96-106.

Hall, C. E. J. Rotation of canonical variates in multivariate analysis of variance. Experimental Education, 1969, 38, 31-38.

Hotelling, H. The most predictable criterion. Journal of Educational Psychology, 1935, 26, 139-142.

Hotelling, H. Relations between two sets of variates. Biometrika, 1936, 28, 321-377.

Horst., P. Relations among m sets of variables. Psychometrika, 1961, 26, 129-149.

Kerlinger, F. N. Foundations of behavioral research. New York: Holt, 1973.

Koons, P. B. Canonical analysis. In H. Borko (Ed.), Computer applications in the behavioral sciences. Englewood Cliffs, N. J.: Prentice-Hall, 1962.

Krus, D. J. and Tellegen, A. Consciousness III: Fact or fiction? Psychological Reports, 1975, 36, 23-30.

Nie, N., Brent, D. H., and Hull, C. H. Statistical package for social sciences. New York:

McGraw-Hill, 1970.

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