Based
on Cliff, N., & Krus, D.J. (1976)
Interpretation of canonical variate analysis: rotated vs. unrotated solutions. Psychometrika, 41, 35-42.
Interpretation of canonical variate
analysis:
rotated vs. unrotated solutions
Norman Cliff and
Abstract.-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
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(1) |
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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
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(2) |
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and
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(3) |
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A supermatrix containing the submatrices comprising the canonical structure can be composed as
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(4) |
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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
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(5) |
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or
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(6) |
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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
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(7) |
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and for the criterion set of variables as
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(8) |
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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
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(9) |
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and
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(10) |
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where
and
are rotated canonical weights. The canonical
correlations for the rotated solution can be computed as
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(11) |
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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. 277278,
Hall, 1969). All these procedures are, however, incidental to the canonical
analysis itself. 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.
Koons, P. B. Canonical analysis. In H. Borko (Ed.),
Computer applications in the behavioral sciences.
Krus, D. J. and Tellegen, A. Consciousness III: Fact or fiction? Psychological Reports, 1975, 36, 23-30.
Nie, N., Brent, D. H., and