Ch. 11: Canonical Correlation
I. Situation
A. Two sets of (metric) variables, Xs and Ys.
B. Want to see if two sets of variables are significantly
correlated with each other.
C. The correlation between the linear combination of Xs and the
linear combination of Ys.
D. We can compute a second canonical correlation, the
correlation between the second linear combination of Xs and
the second linear combination of Ys.
E. The second linear combination is orthogonal (independent) to
the first linear combination.
A. The maximum number of canonical correlation is min(p, q).
II. Terminology
A. Canonical variable (variate)
1. The linear combination of each set of variables which
maximizes the correlation between two sets of variables.
2. CanXi = bi1 xi1 + bi2xi2 + . . + biq xiq = b’x
CanYi = ai1yi1 + ai2 yi2 + . . + aipyip = a’y
B. Canonical loading (canonical structure)
1. Correlation between each variable and the canonical
variable.
2. RXqCanXi RYpCanYi
C. Canonical cross loading
1. Correlation between each variable of one set and the
canonical variable of the other set.
2. RXqCanYi RYpCanXi
D. Standardized canonical coefficient
1. Standardized regression coefficients of each variable for
the prediction (composition) of the canonical variable
from the same variable set.
2. Shows the relative contribution of each variable to the
canonical variable in the presence of all other variables.
E. Canonical root: squared canonical correlation (CanR2).
F. Eigenvalue (SAS)
1. λ = eigenvalue of the E-1H matrix = CanR2/(1-CanR2)
2. Relative significance of CanR2 to the coefficient of
alienation (1-CanR2).
3. Eigenvalue of (E+H)-1H matrix, ξ,
(E+H)-1H = [(Y’Y)-1(Y’X)(X’X)-1(X’Y)]
= [RYY-1RYXRXX-1RXY]
III. Test of significance
A. H0: Σyx = 0 (B1 = 0) H1: Σyx
≠
0 (B1
≠
0)
B. TS
Λ =
)
1(
|||
|
||
||||
||
||
||
2
1i
s
i
XXYYXXYY
CanR
RR
R
SS
S
HE
E−=
==
+Π
=
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