3Schepers.qxd


An issue which often arises, for example in studies of the Big

Five, is whether two or more test batteries, given to the same

sample of participants, have a common factor structure.

Traditionally, researchers have simply conducted a joint factor

analysis of such batteries of tests. However, due to the effects

of differential skewness of the variables involved, the

resulting factor structures have often been distorted. Factors

of skewness rather than content have usually been 

obtained. Finch and West (1997, p.470) pointed out in this

regard that joint factor analyses confound two sources of

covariation, namely covariation within batteries and

covariation between batteries.

To overcome the confounding of the two sources of covariation

mentioned, Tucker (1958) proposed his interbattery factor

analysis. His model will now be briefly described.

Assume that two batteries of tests, with a postulated common

factor structure, have been applied to a representative sample of

participants. The variables were intercorrelated and yielded the

super matrix depicted in Figure 1.

Figure 1: Super matrix (R)

According to the fundamental theorem of factor analysis the

super matrix R can be resolved into its factors as follows:

R = FF', where

A1 = Factors of battery 1 shared in common with battery 2;

A2 = Factors of battery 2 shared in common with battery 1;

S1 = Factors specific to battery 1;

S2 = Factors specific to battery 2.

R can therefore be presented as follows:

It is therefore clear that R12 = A1A’2. 

A1A’2 contains the factors common to batteries 1 and 2.

Browne (1979) provided a maximum likelihood solution to

Tucker’s model of interbattery factor analysis. He obtained

estimates of the interbattery factor loadings by scaling the

correlations of the original variables with the canonical variates.

He subsequently extended his technique to more than two

batteries of tests (Browne, 1980).

More recently Schepers (2004b) showed that the Multiple

Battery Factor Analysis (MBFA) technique of Browne (1980) can

cope with the effects of differential skewness of variables from

two different batteries of tests.

He applied the General Scholastic Aptitude Test (GSAT) 

and Senior Ability Tests (SAT) jointly to a sample of 1598 

first-year universit y students, and subjected the inter-

correlation matrix to a principal factor analysis. The obtained

' '
1 2

11 1 '
1

21 22 2 2 '
2

12 1

'

A A
R R A S O

R S O
R R A O S

O S

F F

 
    
 = = •   
    
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1 1

2 2

A S O
F

A O S

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=  

 

JOHANN M SCHEPERS
abo@rau.ac.za

Department of Human Resource Management

University of Johannesburg

ABSTRACT
The principal objective of the study was to determine the utility of canonical correlation analysis, coupled 

with target rotation, in coping with the effects of differential skewness of variables representing two 

batteries of tests. Generally speaking joint factor analyses of two or more batteries of tests result in factors of

skewness rather than factors of content. To examine the problem, the General Scholastic Aptitude Test (GSAT)

and Senior Ability Tests (SAT) were jointly applied to a sample of 1598 first-year university students, and

subjected to both a principal factor analysis (PFA) and a canonical correlation analysis (CCA), coupled with

target rotation. Three factors were obtained in both instances. The PFA yielded factors of skewness and the CCA

factors of content. The target rotation gave a good fit with the theoretically specified values. The implications

of the findings are discussed.

Key words

Canonical correlation, Target rotation, Tarrot rotation

THE UTILITY OF CANONICAL CORRELATION ANALYSIS,

COUPLED WITH TARGET ROTATION, IN COPING WITH THE

EFFECTS OF DIFFERENTIAL SKEWNESS OF VARIABLES

19

SA Journal of Industrial Psychology, 2006, 32 (2), 19-22

SA Tydskrif vir Bedryfsielkunde, 2006, 32 (2), 19-22



SCHEPERS20

factor matrix was rotated to simple structure by means of a

Direct Oblimin rotation. 

The principal factor analysis yielded three factors, viz. a non-

verbal (spatial) factor, and two verbal factors. The verbal tests

of the GSAT loaded on one factor and the verbal tests of the SAT

on another.

Following this the intercorrelation matrix was subjected to a

multiple battery factor analysis (MBFA) and rotated to simple

structure by means of a Direct Quartimin rotation. Again a

three-factor-structure was obtained. A Tucker-Lewis reliability

coefficient of 0,967 was obtained, which is highly acceptable.

The average absolute off-diagonal residual was 0,046 which

indicates a very good fit.

Three clear-cut factors were obtained, which were identified as a

non-verbal reasoning factor, a verbal factor, and a number factor.

The three factors were strongly positively correlated, suggesting

an underlying factor of general intelligence. 

From the coefficients of skewness of the various measures of the

GSAT and SAT it would appear that the distributions of the GSAT

are quite skew. The indices range from 1,818 to -2,111. By

contrast the distributions of the SAT are moderately skew. The

indices range from 0,450 to -1,248.

From the foregoing it should be clear that even moderate

variations in degrees of skewness can distort the factor structure

of two batteries of tests if a joint factor analysis is done. By

contrast MBFA seems to cope quite well with moderate degrees

of skewness.

According to Browne (1979, p.75) the interbattery factor analysis

model is “a genuine factor analysis model in that a single set of

unobservable factor variables accounts for all correlation

coefficients between two batteries of tests”. By contrast canonical

correlation analysis “is strictly a method of component analysis

since two sets of observable linear combinations of variables are

employed to investigate relationships between the two batteries

of tests” (p.75).

Despite the fact that the rationale of the two models are quite

different, the numerical procedures of canonical correlation

analysis are very similar to that involved in obtaining

maximum-likelihood estimates of interbattery factor loadings

(Browne, 1979, p.75). It would therefore be very interesting to

examine the utility of canonical correlation analysis in coping

with the effects of differential skewness of variables.

The objective of canonical correlation analysis is to form linear

combinations of two sets of continuous variables so as to

maximise the correlation between the two composites (Cliff,

1987, p.453). According to Cliff (1987, p.455) canonical

correlation analysis can be used if “one set of variables is

dependent and the other independent or when there is no

distinction in the roles of the two sets”. It can therefore also be

applied to variables from two batteries of tests.

A statistical test is performed to determine how many significant

components there are (Bartlett, 1950; 1951). Each component

(dimension) is represented by two vectors of weights – one in

respect of the first battery of tests, and the other in respect of the

second battery of tests. The two vectors of weights representing

a component are normally referred to as a variate, and the

correlation between the two composites of a variate yields the

canonical correlation in respect of that component. Thus there

are as many canonical correlations as there are statistically

significant components.

From an interpretive point of view it is normally very difficult

to identif y the components underlying the canonical structure

matrix as it resembles an unrotated factor matrix. Rotation to

simple structure is therefore necessary. In this regard Cliff (1987,

p.456) states that the “structure correlations” between the

observed variables and the canonical variates “can be

transformed by the rotational methods of factor analysis,

although the same transformation must be applied to the

structure correlations of both batteries”. Target rotation would

seem to be ideal for this purpose.

From a theory testing point of view target rotation is more

appropriate than the usual rotations to simple structure such as

Varimax, Promax, Direct Oblimin, Quartimax, Quartimin, and

other procedures. With target rotation the common factor

structure of two batteries of tests can be specified on theoretical

grounds. This is particularly useful whenever theoretical models

are being tested.

From the foregoing it should be clear that differential skewness

of variables is very disruptive when doing joint factor analyses of

two or more batteries of tests (Ferguson, 1941; Gorsuch, 1974;

Schepers, 2004a and 2004b; Finch & West, 1997). There is thus a

real need for techniques that can cope with the effects of

differential skewness of variables of a continuous nature. 

Objectives of the study

The principal objective of the study was to determine the utility

of canonical correlation analysis, coupled with target rotation,

in coping with the effects of differential skewness of variables

from two batteries of tests.

RESEARCH DESIGN

Research approach

The primary goal of the study was to evaluate a particular

statistical technique. A cross-sectional field survey was used in

the collection of the data.

Participants

As the sample has been fully described in a previous study (cf.

Schepers, 2004b, pp.78-79) only the essential details are given here:

A representative sample of first-year university students at the

Rand Afrikaans University, during 1995, was used in the study.

Complete records in respect of 1598 participants were available

in respect of the General Scholastic Aptitude Test (GSAT) and

Senior Aptitude Tests (SAT), amongst others.

Measuring instruments

As a complete description of the measuring instruments have

been given in a previous study (cf. Schepers, 2004b, p.79) only

the essential details are given here:

The General Scholastic Aptitude Test (GSAT)

The GSAT yields a measure of academic intelligence or scholastic

aptitude. It consists of six subtests – three verbal and three non-

verbal, and measures both verbal and non-verbal intelligence

(Claassen, De Beer, Hugo & Meyer, 1998).

The Senior Aptitude Tests (SAT)

The SAT was designed for the measurement of a number 

of aptitudes of pupils in Grades 10, 11 and 12, and of adults. 

It consists of verbal, numerical, non-verbal reasoning, spatial

and memory tests. Coordination and Writing Speed were

excluded for the purposes of the present study (Fouché &

Verwey, 1991).

Procedure

For the purposes of the present study only the records of

students who had completed both the GSAT and the SAT were

used. A total of 1598 complete records were obtained.



THE UTILITY OF CANONICAL CORRELATION ANALYSIS 21

Statistical analysis

In order to attain the stated objective a Canonical Correlation

Analysis (CCA) was done (Cliff, 1987; Tabachnick & Fidell,

1983). The obtained canonical structure matrix was rotated 

to simple structure by means of a target rotation (Browne,

1972a, 1972b, 1993).

RESULTS

Principal objective: To determine the utility of canonical

correlation analysis, coupled with target rotation, in coping with

the effects of differential skewness of variables from two

batteries of tests

As a first step in the analysis, the canonical correlations of the

subtests of the GSAT with the various measures of the SAT were

computed. Bartlett’s (1950, 1951) test of significance was used to

determine the number of significant canonical correlations, and

is given in Table 1.

TABLE 1

STATISTICAL SIGNIFICANCE OF CANONICAL CORRELATIONS:

BARTLETT’S TEST IN RESPECT OF THE GSAT AND SAT

Eigenvalues Canonical Eigenavlue Significance of  eigenvalues

correlations removed remaining

�2 df p Lambda

prime

0,548444 0,740570 0 1712,319 60 <0,000001 0,340294

0,154387 0,392921 1 449,373 45 <0,000001 0,753602

0,075667 0,275076 2 182,992 32 <0,000001 0,891190

0,024308 0,155911 3 58,004 21 0,00003 0,964144

Note: N = 1598

From Table 1 it is clear that there are at least three significant

canonical correlations. Accordingly three canonical variates,

together with their associated canonical correlations, were

computed. The complete analysis is given in Table 2.

Table 2 shows that the first canonical variate yielded a

canonical correlation of 0,741, the second a canonical

correlation of 0,393 and the third a canonical correlation of

0,275. The first canonical variate suggests a general factor, with

loadings ranging from 0,421 to 0,869. The second and third

variates, however, are more difficult to interpret as no simple

structure is visible. It was therefore decided to rotate the matrix

of canonical variates to simple structure. For this purpose use

was made of a target matrix in conjunction with a Tarrot

rotation. The target matrix was specified on theoretical

grounds after studying the subtests of the GSAT and SAT. The

target matrix is given in Table 3.

The target matrix was specified with high loadings on Factor 1 in

respect of the non-verbal reasoning tests. Factor 2 was specified

with high loadings on all the verbal tests, together with the two

memory tests, and Factor 3 was specified with high loadings on

the numerical tests. 

Accordingly an oblique Tarrot rotation was performed of the

matrix of canonical variates. The rotated matrix is given in 

Table 4.

Table 4 shows that rotation of the canonical variates to simple

structure resulted in a well defined structure, yielding a good fit

with the theoretically specified target matrix. The square root of

the average squared deviation was equal to 0,144480.

TABLE 2

CANONICAL CORRELATIONS OF GSAT AND THE

RESPECTIVE MEASURES OF THE SAT

Correlations of original measures with canonical variates

Variate 1 Variate 2 Variate 3

Battery 1

GSAT 1: WORD 0,653 0,579 0,139

ANALOGIES

GSAT 2: NUMBER SERIES 0,860 -0,049 0,417

GSAT 3: VERBAL REASONING 0,869 0,216 0,075

GSAT 4: PATTERN 0,784 -0,189 -0,405

COMPLETION

GSAT 5: WORD PAIRS 0,703 0,517 -0,165

GSAT 6: FIGURE 0,823 -0,241 -0,274

ANALOGIES

Average % variance 61,79 % 12,42% 7,76% Total: 81,97%

accounted for

Average % redundancy 33,89% 1,92% 0,59% Total: 36,40%

Battery 2

SAT 1: VERBAL 0,772 0,402 0,103

COMPREHENSION

SAT 2: CALCULATIONS 0,629 0,180 0,719

SAT 3: DISGUISED WORDS 0,499 -0,660 0,161

SAT 4: COMPARISON 0,421 -0,125 0,259

SAT 5: PATTERN 0,782 -0,155 -0,155

COMPLETION

SAT 6: FIGURE SERIES 0,712 -0,132 -0,016

SAT 7: SPATIAL 2D 0,705 -0,203 -0,161

SAT 8: SPATIAL 3D 0,762 -0,255 -0,301

SAT 9: MEMORY 0,479 0,386 -0,063

(PARAGRAPH)

SAT 10: MEMORY 0,511 0,236 -0,140

(SYMBOLS)

Average % variance 40,29% 9,97% 7,85% Total: 58,11%

accounted for

Average % redundancy 22,09% 1,54% 0,59%

Canonical correlations 0,741 0,393 0,275 Total: 24,22%

Note: N = 1598

TABLE 3

TARGET MATRIX SPECIFIED FOR TARROT ROTATION

Variable Factor 1 Factor 2 Factor 3

GSAT 1 WORD ANALOGIES 0,000 9,000 0,000

GSAT 2 NUMBER SERIES 0,000 0,000 9,000

GSAT 3 VERBAL REASONING 0,000 9,000 0,000

GSAT 4 PATTERN COMPLETION 9,000 0,000 0,000

GSAT 5 WORD PAIRS 0,000 9,000 0,000

GSAT 6 FIGURE ANALOGIES 9,000 0,000 0,000

SAT 1 VERBAL COMPREHENSION 0,000 9,000 0,000

SAT 2 CALCULATIONS 0,000 0,000 9,000

SAT 3 DISGUISED WORDS 0,000 9,000 0,000

SAT 4 COMPARISON 0,000 0,000 9,000

SAT 5 PATTERN COMPLETION 9,000 0,000 0,000

SAT 6 FIGURE SERIES 9,000 0,000 0,000

SAT 7 SPATIAL 2D 9,000 0,000 0,000

SAT 8 SPATIAL 3D 9,000 0,000 0,000

SAT 9 MEMORY (PARAGRAPH) 0,000 9,000 0,000

SAT 10 MEMORY (SYMBOLS) 0,000 9,000 0,000



SCHEPERS22

TABLE 4

TARROT ROTATION OF CANONICAL FACTOR LOADINGS

(GSAT & SAT)

Variable Factor 1 Factor 2 Factor 3

BATTERY 1

GSAT 1: WORD ANALOGIES 0,013 0,937 -0,159

GSAT 2: NUMBER SERIES 0,196 0,213 0,717

GSAT 3: VERBAL REASONING 0,260 0,558 0,260

GSAT 4: PATTERN COMPLETION 0,923 0,063 -0,142

GSAT 5: WORD PAIRS 0,113 0,880 -0,151

GSAT 6: FIGURE ANALOGIES 0,883 0,006 0,028

BATTERY 2

SAT 1 VERBAL COMPREHENSION 0,003 0,732 0,183

SAT 2 CALCULATIONS -0,070 -0,043 1,020

SAT 3 DISGUISED WORDS -0,380 0,965 0,092

SAT 4 COMPARISON 0,134 -0,022 0,445

SAT 5 PATTEN COMPLETION 0,695 0,090 0,115

SAT 6 FIGURE SERIES 0,525 0,086 0,236

SAT 7 SPATIAL 2D 0,695 0,005 0,103

SAT 8 SPATIAL 3D 0,881 -0,030 -0,014

SAT 9 MEMORY (PARAGRAPH) 0,010 0,638 -0,061

SAT 10 MEMORY (SYMBOLS) 0,211 0,471 -0,083

Note: Square root of average deviation = 0,144480

FACTOR CORRELATION MATRIX

Factor 1 Factor 2 Factor 3

FACTOR 1 1,000 0,575 0,464

FACTOR 2 0,575 1,000 0,453

FACTOR 3 0,464 0,453 1,000

DISCUSSION

The principal objective of the study turned out positive:

Rotation of the canonical variates by means of a target 

rotation yielded a structure that is very similar to that obtained

with the MBFA. A CCA followed by a target rotation might 

even be preferable to a MBFA when doing confirmatory 

studies as the target matrix can be specified on theoretical

grounds prior to initiating the study. Target rotation can of

course also be used with MBFA, but then the current program

would have to be adapted.

ACKNOWLEDGEMENT

I hereby wish to thank Riëtte Eiselen and Wilhelm Koster of the

Statistical Consultation Service of the Universit y of

Johannesburg for all the hours of computational work done for

me. I value it very highly.

A special word of thanks to Annetjie Boshoff and her assistant

Afton Walters for typing the manuscript at short notice.

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