Canonical Correlation: Equations

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```							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          1
R  R Ryx R Rxy
yy          xx
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 
2
  ln  m
             2       
Where N is number of cases, k x is number of x var iables and
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  
2
  ln .07
            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  
2
  ln .42
           2 
 2  (4.5)(.87)  3.92
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

          ˆ
By  ( Ryy1/ 2 ) ' By

Where ( Ryy1/ 2 ) ' is the transpose of the inverse of the "special" matrix
ˆ
form of square root that keeps all of the eigenvalues positive and By
is a normalized matrix of eigen vectors for yy
Bx  Rxx Rxy B*
-1
y

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

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;
matrices

Ax  Rxx Bx
Ay  Ryy By
Equations

Canonical Variate Pairs
First      Second
First Set   TS     -.74          .68
TC      .79          .62
Second Set   BS     -.44          .90
BC      .88          .48
Equations

Redundancy                   kx
a   2
pvxc  

ixc
   Within -
Percent of            i 1 k x
variance                 ky    2
a
explained by
pv yc  
iyc
the
canonical               i 1   ky
correlate on
its own side             (.74)  .792   2

of the         pvxc1                  .58
equation                       2
Equations

   Redundancy
   Across - variance in Xs explained by
the Ys and vice versa

rd  ( pv)(r )  c
2

 (.74)  .79 
2       2
rd x1  y                  (.84)  .48
       2      

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