# Canonical Correlation

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```					Canonical Correlation

Mechanics
Data
• The input correlation setup is 
Rxx   Rxy
Ryx   Ryy
• The canonical correlation matrix is the
product of four correlation matrices,
between DVs (inverse of Ryy,), IVs
(inverse of Rxx), and between DVs
R = R-1 RyxR-1 Rxy
yy     xx
and IVs
• It also can be thought of as a product
of regression coefficients for
predicting Xs from Ys, and Ys from Xs
What does it mean?
• In this context the eigenvalues of that R matrix represent
the percentage of overlapping variance between the
canonical variate pairs
• To get the canonical correlations, you get the
eigenvalues of R and take the square root
r  i
2
ci

rci  i

• The eigenvector corresponding to each eigenvalue is
transformed into the coefficients that specify the linear
combination that will make up a canonical variate
Canonical Coefficients
• Two sets of canonical coefficients
(weights) are required
– One set to combine the Xs
– One to combine the Ys
– Same interpretation as regression coefficients
Equations
ˆ
B y = (R -1/2 )'B y
yy

Where ( Ryy1/ 2 ) 'is the transpose of the inverse of the square root of the
ˆ
correlation matrix of DVs, and By is a normalized matrix
of eigenvectors for the DVs
Bx = R -1 R xyB*y
xx

Where B* is By from above dividing each entry by their corresponding
y

canonical correlation.
Is it significant?
• Testing Canonical Correlations
– There will be as many canonical correlations as
there are variables in the smaller set
– Not all will be statistically significant
• Bartlett’s Chi Square test (Wilk’s on
printouts)
– Tests whether an eigenvalue and the ones that
follow are significantly different than zero.
           kx  k y  1 
 2    N 1                    ln  m
                  2       
Where N is number of cases, k x is number of X variables and
k y is number of Y variables
 m  (1  1 )(1  2 )...(1  m )
Lamda, Λ, is the product of differences between eigenvalues (R c s) and 1,
2

generated across m canonical correlations.

• Essentially it is an omnibus test of whether the
eigenvalues are significantly different from zero
• It is possible this test would be significant even
though a test for the correlation itself would not
be
Variate Scores
• Canonical Variate Scores
– Like factor scores (we’ll get there later)
– What a subject would score if you could measure them directly on the
canonical variate
• The values on a canonical variable for a given case, based on the canonical
coefficients for that variable.
• Canonical coefficients are multiplied by the standardized
scores of the cases and summed to yield the canonical
scores for each case in the analysis

X = ZxBx
Y = Zy By
– Key question: how well do the variate(s) on either side relate to their own set of
measured variables?
– Bivariate correlation between a variable and its respective variate
– Would equal the canonical coefficients if all variables were uncorrelated with one
another
– Its square is the proportion of variance linearly shared by a variable with the
variable’s canonical composite
•   Found by multiplying the matrix of correlations between variables in a set by the
matrix of canonical coefficients

X1 rX1                                            ry1                    A x = R xxBx
\rX2    V1                       W1         ry2
Y1
X2 r
ry3
A y = R yyBy
X3                                                   Y2
X3 rX4
Y3
X4
Redundancy Equations
•   Redundancy                            kx     2
a
pvxc  
•   Within                                       ixc
– Percentage of variance
in a set of variables
extracted by the
canonical variate
i 1 k x
ky     2
– Is the average of the                      a
pv yc  
squared correlations                       iyc

i 1   ky
•   Across
– Variance in IVs
explained by the DVs
and vice versa
– Take how much variance
is accounted for with its
Rd  ( pv)(r )       c
2

own variables and
multiply that by the
canonical correlation
squared

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 views: 24 posted: 6/6/2012 language: pages: 10
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