Canonical Correlation: Equations

							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;
    also called loadings or loading
    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      