p. 7-9
• Sample Canonical Variables and Canonical Correlations
¾ Data:
¾ Finding canonical variables and canonical correlations
replace population distribution by empirical distribution
replace Σ by S (or Sn)
replay ρ by R
then, the rests the same as above
¾
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
• Additional Sample Descriptive Measures
¾ Matrices of Errors of Approximations

p. 7-10
p. 7-11

the matrices of error of approximation are
The approximation error matrices may be interpreted as descriptive summaries
of how well the first r sample canonical variates reproduce the sample
covariance matrices. Patterns of large entries in the rows and/or columns of the
error matrices indicate a poor “fit” to the corresponding variables
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
p. 7-12
ordinarily, the first r variates do a better job of reproducing the elements
of S12 than the elements of S11 or S22 (Q: Why?)

¾ Proportions of Explained Sample Variance

so,
p. 7-13

the contribution of the first r canonical variates to the total sample variance:

proportions of total sample variances explained by 1st r canonical variates:
they provide some indication of how well the canonical variates represent
their respective sets
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
p. 7-14
they also provide single-number descriptions of the matrices of errors,
because

• Large Sample Inferences
¾ Note: Σ12=0 ⇒ no point in pursuing a CCA. ⇒ Q: how to test H0: Σ12=0?
¾
¾ when n is large, under H0, the likelihood ratio test statistic is approximately
distributed as a chi-square random variable with pq d.f.
¾ Bartlett (1939) suggests
¾ Q: What if H0: Σ12=0 is rejected? Next step?

the issue of multiple testing should be taken into consideration
Reading: Reference, 10.4, 10.5, 10.6
NTHU STAT 5191, 2010, Lecture Notes
made by S.-W. Cheng (NTHU, Taiwan)
p. 7-15