ROTATION IN CANONICAL VARIATE ANALYSIS

David J. Krus, Thomas J. Reynolds, and Patricia H. Kennedy
University of Southern California

Abstract. Interpretations of the rotated and not-rotated canonical analysis of the Ferraro and Billings study on the drug use were compared, illustrating advantages of the rotated solution.

DORMANT for nearly half a century, Hotelling’s (1935, 1936) can­onical variate analysis has come of age. The principal reason behind its resurrection was its computerization and inclusion in major statistical packages (Cooley and Lohnes, 1971; Dixon, 1974; Nie, Hull, Jenkins, Steinbrenner and Bent, 1975). Simultaneously with its rebirth, a long­standing skepticism about its interpretability was revitalized. Thus, Warwick (1975, p. 518) commented: 

There can be no assurance that the canonical correlation analysis will find any comprehensible patterning to the data at all. Moreover, canonical correlation analysis can produce canonical variates, even clearly defined ones, that may not make any sense to the researcher.

In a similar vein, Kerlinger and Pedhazur (1973, p. 344) stressed that considerable caution and circumspection should be exercised in the interpretation of the canonical variate analysis.

As an integral part of the general linear model, canonical analysis describes formal structures in hyperspace, defined by orthogonal references axes and invariant with respect to rotation of their coordinates. Provided that certain conditions are fulfilled, rotation can greatly facilitate interpreting the obtained solution and thus can aid behavioral researchers searching for a meaningful and parsimonious description of scrutinized relationships (Cliff and Krus, 1976).

Interpretation of a not-rotated solution

To illustrate the conceptual advantages provided by the rotated solution, an example of canonical analysis recently presented by Harris, (1975, pp. 137—140) may be considered. The data analyzed were collected by Ferraro and Billings (1974) as a part of their study on drug use at the University of New Mexico. Based on the responses of 2675 undergraduates, canonical variate analysis was performed to show relationships between a set of general background variables and a number of questions pertaining to marijuana use (Table 1).

Table 1. Description of variables included in the analysis.

Background Variables	VARIABLES	QUESTIONS AND DIRECTION OF THEIR CODING
	GENDER	Male-Female: 1-0
	AGE	17, 18, 19, etc.: 0, 1, 2, etc.
	CLASS	Freshman, Sophomore, Junior, Senior, Graduate Student: 0,1,2,3,4
	INCOME	(Parents’ annual income): <$5,000; $ 5,000—10,000; ...>25,000:1,2,3,4
	RELIGION	(“How active are you in your religion?”): Very active, Active, Barely active, Inactive: 0, 1, 2, 3
	POLITICS	(“How active are you politically?”): Very active, Active, Barely active, Inactive: 0. 1,2, 3
	VIETNAM	(“Do you agree with your parents on Vietnam?”): Strongly disagree, Generally disagree, Do not know, Generally agree, Strongly agree: 0, 1,2, 3,4
	ABORTION	(“Do you agree with your parents on premarital sex and abortion?”): Strongly disagree, Generally disagree, Do not know, Generally agree, Strongly agree: 0, 1, 2,3,4
Marijuana Use	PAST USE	(“How many times have you used marijuana?”): Never, I time, 2-4 times, 5-14 times, 15-30 times, more than 30times: 0, 1,2,3,4,5
	ONSET	(‘When did you first use marijuana?”): Elementary school, Junior high, High school, Fresh Year, ... Senior year, Have never used: 1,2,3,4,... 7,8
	PRESENT USE	(“How often do you currently use marijuana?”): Do not use marijuana, Less than once per month, 1-4 times per month, 1-3 times per week, 4-6 times per week, 7 or more times per week: 0, 1,2,3,4,5,6,7
	FUTURE USE	(“If you had the opportunity in the future, would you use marijuana?”): Yes, No: 0, I
	RESTRICTION	(“If marijuana were legalized, should it be restricted to those 18 years of age or older?”): Yes, No: 0, 1
	LEGALIZATION	(“Do you feel that society should legalize marijuana at this time?”): No, Undecided, Yes: 0, 1, 2

The canonical regression weights that were not rotated were reported by Harris (1975, p.139, Table 5.2) and are reproduced here in Table 2.

Table 2. Canonical variate analysis of the Ferraro and Billings data: Unrotated solution.

	Canonical Variates	1	2	3	4a	5a	6a
	Eigenvalues	.3035	.1182	.0254	.0062	.0042	.0008
	Canonical Correlations	.551	.344	.159	.079	.065	.028
Predictor Set	Gender	-.188	.185	.719
	Age	-.309	.473	-.322
	Class	.045	.616	.155
	Income	.085	-.078	-.385
	Religion	.623	.180	-.117
	Politics	-.258	.031	.011
	Vietnam	-.194	-.123	-.023
	Abortion	-.395	.002	-.504
Criterion Set	Pastd	.295	1.815	-.581
	Onset	-.185	1.920	-.228
	Present use	.061	-.508	-.662
	Future use	-.244	-.209	-.157
	Restriction	.067	-.147	.491
	Legalization	.414	.270	.723

^a Coefficients have been omitted since these canonical correlations are not significant.
 Bold coefficients were emphasized in the interpretation of the canonical variates.

The interpretation of this not-rotated solution was offered as follows (Harris 1975, pp. 139—140):

The first pair of canonical variates seems to be identifying a tendency for young, politically active but religiously inactive respondents who disagree with their parents on the topics of pre­marital sex and abortion to have tried marijuana at an early age, to have used marijuana frequently in the past, to be in favor of legalizing its use, and to be willing to try marijuana again in the future should the opportunity present itself.

The second (canonical variate) seems to suggest primarily that older, upper-class respondents have used marijuana extensively in the past, but began this use late in their academic career.

The third (canonical variate) indicates that young females from low-income families who disagree with their parents on the issues of premarital sex and abortion tend not to have used marijuana much in the past nor to be using it at present, but nevertheless to be more in favor of legalizing marijuana for everyone, 18 or not, than most other respondents.

The grayed interpretations have a questionable face validity.

Interpretation of the rotated solution

A rotated solution of the same data is presented in Table 3.

Table 3. Canonical analysis of the Ferraro and Billings data: Rotated solution .

Canonical Variates	1	2	3	4	5	6
Eigenvalues	.1060	.1264	.0265	.0143	.0131	.0202
Canonical Coreelations	.326	.356	.163	.120	.114	.142
Past use	1.958	1.240	.370	.854	-.759	-.288
Onset	1.885	.038	.589	-,350	-.182	-.010
Class	.718	.058	-.014	.204	.200	.223
Religion	.175	.947	.219	-.044	-.097	.102
Abortion	.004	.004	1.067	-.005	.003	.003
Legalization	.379	-.108	-.884	.050	.552	.402
Future use	.156	-.089	.123	1.305	.630	.098
Vietnam	.048	.043	-.285	.940	-.195	-.087
Present use	-.454	-.469	.626	-.361	1.319	-.386
Politics	-.133	-.017	.054	-.274	-.782	-.078
Income	-.086	-.159	.106	-.182	.412	-.388
Restriction	-.125	.343	.428	-.092	.012	.847
Sex	.210	-.131	.045	-.075	-.040	.827
Age	.337	-.332	.016	-.315	-.431	-.500

Interpretation of this rotated solution is offered as follows:

The first rotated canonical variate suggests the earlier onset of the habit within the younger age group. The ambiguity of religious precepts with respect to consciousness-altering experiences is reflected in the uniqueness of the second canonical variate. The third and fourth rotated variates jointly indicate that congruence of opinions within the family results in a negative attitude toward the drug. The fifth canonical variate suggests a profile of a user as relatively affluent and politically active. The last rotated variate predictably points out that it is a group of young males advocating the age-unrestricted legalization of the drug.

The grayed narrative is in sharp contrast with the grayed narrative in the interpretation of the solution that was not rotated.

Judged as a whole, the interpretations of both rotated and not-rotated solutions are partially in agreement. However, the rotated solution is easier to interpret, as rotation tends to simplify the obtained structure of canonical loadings. Also, the rotated canonical variates tend to be more self-consistent than their not-rotated counterparts. These observations suggest that rotation enhances the capacity of the canonical variate analysis to abstract meaning underlying the scrutinized relationships.

References

Cliff, N. and Krus, D. J. Interpretation of canonical analysis: Rotated vs. not-rotated solutions. Psychometrika. 1976, 41, 35-42.

Cooley, W. W. and Lohnes, P. R. Multivariate data analysiS. New York: Wiley, 1971.

Dixon, W. J. (Ed.) BMD: Biomedical computer programs. Berkeley and Los Angeles: University of California Press, 1974.

Ferraro, D. P., and Billings, D. K. Marijuana use by college students: Three year trends, 1970-1972. International Journal of the Addic­tions, 1974, 9, 321-327.

Harris, R. J. A primer of multivariate statistics. New York: Academic Press, 1975.

Hotel ling, H. The most predictable criterion. Journal of Educational psychology, 1935, 26, 139—192.

Hotelling, H. Relations between two sets of variates. Biometrika, I 936, 28, 321—377.

Kerlinger, F. N. and Pedhazur, E. J. Multiple regression in behavioral research. New York: Holt, Rinehart and Winston, 1973.

Nie, N. H., Hull, C. H., Jenkins, J. G., Steinbrenner, K., and Bent, D. H. Statistical package for the social sciences. New York: McGraw-Hill, 1975.

Warwick, P. V. Canonical correlation analysis. In Norman Nie, et al. Statistical package for the social sciences. New York:

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