Canonical Correlation:
Equations
Psy 524
Andrew Ainsworth
Data for Canonical Correlations


CanCorr actually takes raw data and
computes a correlation matrix and
uses this as input data.
You can actually put in the
correlation matrix as data (e.g. to
check someone else’s results)
Data

The input correlation set up is:
Rxx
Rxy
Ryx
Ryy
Equations

To find the canonical correlations:

First create a canonical input matrix.
To get this the following equation is
applied:
1
yy
1
xx
R  R Ryx R Rxy
Equations

To get the canonical correlations, you
get the eigenvalues of R and take the
square root
rci  i
Equations

In this context the eigenvalues
represent percent of overlapping
variance accounted for in all of the
variables by the two canonical variates
Equations

Testing Canonical Correlations

Since there will be as many CanCorrs
as there are variables in the smaller set
not all will be meaningful.
Equations

Wilk’s Chi Square test – tests
whether a CanCorr is significantly
different than zero.

 kx  k y  1 
    N 1 
  ln  m
2



Where N is number of cases, k x is number of x var iables and
2
k y is number of y var iables
m
 m   (1  i )
i 1
Lamda, , is the product of difference between eigenvalues and 1,
generated across m canonical correlations.
Equations

From the text example - For the
first canonical correlation:
 2  (1  .84)(1  .58)  .07

 2  2  1 
   8  1  
  ln .07
2 


2
  (4.5)(2.68)  12.04
2
df  (k x )(k y )  (2)(2)  4
Equations

The second CanCorr is tested as
1  (1  .58)  .42

 2  2  1 
   8  1  
  ln .42
2 


 2  (4.5)(.87)  3.92
2
df  (k x  1)(k y  1)  (2  1)(2  1)  1
Equations

Canonical Coefficients

Two sets of Canonical Coefficients are
required

One set to combine the Xs

One to combine the Ys

Similar to regression coefficients
Equations
By  ( Ryy1/ 2 ) ' Bˆ y
Where ( Ryy1/ 2 ) ' is the transpose of the inverse of the "special" matrix
form of square root that keeps all of the eigenvalues positive and Bˆ y
is a normalized matrix of eigen vectors for yy
Bx  Rxx-1 Rxy B*y
Where B*y is By from above dividing each entry by their corresponding
canonical correlation.
Equations

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
X  Z x Bx
Y  Z y By
Equations

Matrices of Correlations between
variables and canonical variates;
also called loadings or loading
matrices
Ax  Rxx Bx
Ay  Ryy By
Equations
First Set
Second Set
TS
TC
BS
BC
Canonical Variate Pairs
First
Second
-.74
.68
.79
.62
-.44
.90
.88
.48
Equations

Redundancy

Within Percent of
variance
explained by
the
canonical
correlate on
its own side
of the
equation
kx
2
ixc
ky
2
iyc
a
pvxc  
i 1 k x
pv yc  
i 1
a
ky
(.74)  .79

 .58
2
2
pvxc1
2
Equations

Redundancy

Across - variance in Xs explained by
the Ys and vice versa
rd  ( pv)(r )
2
c
 (.74)  .79 

 (.84)  .48
2


2
rd x1  y
2