Multivariate Analysis - Research Methodology - page 345

each set of variables that maximizes the intensity of
relationship between the dependent and indepen-
dent variable sets. Actually, during the above pro-
cess, the process develops the number of
independent canonical functions that maximizes
the correlation between the canonical variates.
The characteristic feature of the canonical correla-
tion is that the canonical weights are derived in
such a way so as to maximize the correlation
between the canonical variates.
In canonical analysis, one comes across with
the following terminologies (

two optimally weighted canonical variates of
the sets of independent and dependent variables.

(e) : It is the amount of vari-

ance of a canonical variate explained by the
other canonical variate in the canonical func-
tion. For example, redundancy index of
canonical variate of dependent variables is
the amount of variance of dependent canoni-
cal variate explained by the canonical
variates of independent variates.
Canonical correlation analysis is generally
aimed at

Redundancy index

X i 's are the inde- : (1) assessing the inten-

sity of linear relationship between two sets of
variables (dependent and independent) measured
from the same elements, (2) working out the
weights of the linear combinations, that is, canoni-
cal variates in such a way that each set will have
maximum correlation, and (3) assessing the con-
tribution of each variable to the canonical function.
Sample size

three objectives

pendent variables and Y j 's are the dependent

variables):
(a) Canonical loadings : It is the simple correla-

tion between the individual independent
variables and the canonical variates made
out of

the independent variables. That

means r x i vs : x cv canonical loading measures

the simple linear correlation between an
observed variable and the sets of canonical
variates. Thus, canonical loading reflects the
variance that the variable shares with canon-
ical variate, that is, its relative importance to
the canonical variate; the larger the value, the
greater the importance.

plays a great role in canonical
analysis. Small sample size fails to represent the
correlation adequately, while very large sample
may lead to a very small correlation significant
at every instance. As a guideline, the researcher is
advised to maintain at least ten observations per
variable, though there is no or little importance of
variables to be included in the dependent or inde-
pendent sets, because of the maximization of cor-
relation during productions of canonical variates
that affect the entire process. Here lies the impor-
tance of selection of variables in the canonical
variates. It is always welcome that the researcher
should have conceptua understanding in linking
the sets for variables before canonical correlation
analysis.
Various ways of interpreting the canonical
correlations have been propounded. As most of
the multivariate analyses require computer soft-
ware and the output generated may vary among
these packages, among these the cross loading
approach is mostly preferred, if not available, either
to calculate these manually or to use canonical
loadings for interpretation.

(b) Canonical cross : As the name

suggests, it is the correlation between the
independent/dependent variables and the
opposite canonical variates, that is, canonical
variates of dependent variables/canonical
variates of independent variables, respec-
tively. That means

loading

r y j vs : x cv .As
such cross loadings provide a more direct
measure of the dependent-independent vari-
able relationship.
(c) Canonical function : The correlational
relationships between two canonical variates
are known as canonical function. The two
canonical variates are made of independent
and dependent variables separately. The
canonical functions are derived in such a
way that these are mutually independent of
each other, that is, orthogonal.
(d) C

r x i vs : y cv or

Example 12.3. Using the same data set of 37

varieties for 19 characters and taking X1-X10
as independent and X11-X19 as dependent
variables, let us see the canonical correlation
analysis through SAS.

: The square of the canonical
correlation between two canonical variates is
known as canonical root. This is also known as
eigenvalue. Actually canonical root estimates
the amount of variations shared between the

anonical root