canonical correlation analysis

1. Canonical Correlation

2. Introduction
 If we have two sets of variables, x1,...., xn and y1,….., ym,
and there are correlations among the variables, then
canonical correlation analysis will enable us to find linear
combinations of the x's and the y's which have maximum
correlation with each other.
 Canonical correlation begin with the observed values of
two sets of variables relating to the same set of areas, and
a theory or hypothesis that suggests that the two are
interrelated.
 The overriding concern is with the structural relationship
between the two sets of data as a whole, rather than the
associations between individual variables

3.  Canonical correlation is the most general form of
correlation.
 Multiple regression analysis is a more specific case in
which one of the sets of data contains only one variable,
while product moment correlation is the most specific
case in that both sets of data contain only one variable.
 Canonical correlation analysis is not related to
factor/principal components analysis despite certain
conceptual and terminological similarities. Canonical
correlation analysis is used to investigate the inter-
correlation between two sets of variables, whereas
factor/principal components analysis identifies the
patterns of relationship within one set of data.

4. Difficulties in Canonical Correlation
 Canonical correlation is not the easiest of techniques to
follow, though the problems of comprehension are
conceptual rather than mathematical.
 Unlike multiple regression and principal components
analysis, we cannot provide a graphic device to illustrate
even the simplest form. For with canonical correlation
analysis we are dealing with two sets of data. Even the
most elementary example must, therefore, have at least
two variables on each side and so we require 2 + 2 = 4
dimensions. Tied as we are, however, to a three
dimensional world, a true understanding of the
technique in the conventional cognitive/visual sense of
the term, is beyond our grasp.

5. Conceptual Overview
 Data Input
i. The size of the matrices : There is no requirement in canonical
analysis that there must be the same number of variables
(columns) in each matrix, though there must be the same number
of areas (rows). (There must of course be more than one variable
in each set otherwise we would be dealing with multiple regression
analysis)
ii. The order of the matrices : Neither set of data is given
priority in the analysis so it does not matter which we term the
criteria and which the predictors. Unlike simple linear regression
there is no concept of a 'dependent' set or an 'independent' set. But
in practice the smaller set is always taken second as this simplifies
the calculation enormously

6. Advantages
 Useful and powerful technique for exploring the relationships among
multiple dependent and independent variables. Results obtained from a
canonical analysis should suggest answers to questions concerning the
number of ways in which the two sets of multiple variables are related, the
strengths of the relationships.
 Multiple regressions are used for many-to-one relationships, canonical
correlation is used for many-to-many relationships.
Canonical Correlation- More than one such linear correlation
relating the two sets of variables, with each
such correlation representing a different
dimension by which the independent set of
variables is related to the dependent set.

7.  Interpretability:
Although mathematically elegant, canonical solutions are often un-
interpretable. Furthermore, the rotation of canonical variates to
improve interpretability is not a common practice in research, even
though it is commonplace to do this for factor analysis and principle
components analysis.
 Linear relationship:
Another problem using canonical correlation for research is that
the algorithm used emphasizes the linear relationship between
two sets of variables. If the relationship between variables is not
linear, then using a canonical correlation for the analysis may
miss some or most of the relationship between variables.

8.  The Canonical Problem
 Latent Roots and weights
 Canonical Scores
 Results and Interpretation
i. Latent Roots
ii. Canonical Weights
iii. Canonical Scores

9. Mathematical Model
 The partitioned intercorrelation matrix
where
R11 is the matrix of intercorrelations among the p criteria
variables
R22 is the matrix of intercorrelations among the q predictor
variables
R12 is the matrix of intercorrelations of the p criteria with
the q predictors
R21 is the transpose of R12

10.  The Canonical Equation
i. The product matrix

11. ii. The canonical roots
• The significance of the roots:
Wilk’s Lambda (ᴧ) :
Bartlett’s chi squared:

12. • The canonical vectors
Weights B for the predictor variables are given by :
Weights A for the criteria variables are given by :

13.  The canonical scores
The scores Sa for the criteria are given by
Sa = Zp A
The scores Sb for the predictors are given by
Sb = Zq B
where Zp and Zq are the standardized raw data

14. RESEARCHERS-A. O. UNEGBU &
JAMES J. ADEFILA
Canonical correlation analysis-promotion
bias scoring detector
(a case study of American university of
Nigeria(AUN))
`

15. Introduction
 Problem: AUN bids to keep with her value statement
i.e. highest standards of integrity,
transparency and academic honest.
 Solution: Appraise & select Faculties for promotion
based on various promotion committees’
scores.
 Issues : Dwindling funding,
need for a bias free selection technique,

16. Research Hypotheses
 H01 : CCA cannot detect bias scoring for any of the
candidates from any of the named
committees with 90% confidence level.
 H02: CCA cannot detect significantly whether or
not score-weights of each of the Promotion
Assessors have over bearing influence on the
promotability of candidates.
 H03: CCA cannot at 90% level of certainity
discriminate between candidates that have
earned promotion scores and those that could not
from various promotion committees of the
university.

17. Research objectives
 To test the efficacy of Canonical Correlation Analysis
as a relevant statistical tool for adaption in bias free
promotion score processing and promotion bias
scoring detector so as to ensure fairness, integrity,
transparency and academic honest in analysis of
applicants’ score and in reaching Faculties’
promotion decision.

18. Steps of the Research
1) Data collection
2) Manual computations
3) SPSS analysis
4) Test the Hypothesis

19. AUN promotion procedure
Weights:
Dean of the School 7.5%
School Promotion Committee 7.5%
The Academic Vice President 10%
External Assessor/Reviewer 10%
University Wide Promotion Committee 15%
The Senate Committee 20%
President of the University 30%
Total 100%
The benchmark for promotion is securing a weighted
average score should be more than 65%age.

20. Each of the Committee’s point allocation will
be based on the below criteria
Teaching Effectiveness 40 %
Scholarship, research & creative works 40 %
Service to the University & to Community 20 %

21. Supporting documents for Teaching Effectiveness
 Peer evaluation
 Student evaluation
 Course Syllabi
 Record of participation in teaching seminars, workshops,
etc
 Contributions to the development of new academic
programs
 Faculty awards for excellence in teaching

22. Scholarship, Research and Creative Works
 Terminal degrees/Professional qualifications
 At least Five publications, three of which shall be
journal articles
 Computer Software and Program development
 Creative work in the areas of advertising, public
relations, layout design, photography and graphics, visual
arts etc.

23. Service to the University, Profession and
Community
 Membership/leadership in departmental, school-wide
or university-wide committees
 Planning or participation in workshops, conferences,
seminars .
 Evidence of participation in mentoring or career
counseling of students.
 Membership in Civil Society organizations
 Evidence of service as external assessor or
external examiner on examination committees

24. Raw Scores of Candidates

25. Processed scores of the Candidates

26. Scores of Promotable and Non-promotable Candidates

27. Data Input
 The data input view containing the three groups of
assessors and individual assessors

28. SPSS Results
 Analyze ⇒General Linear
Model⇒Multivariate
 SPSS classified candidates into two groups of
promotable and non promotable of 5 and 9
respectively.
 The result leads to the rejection of Null hypothesis
Ho3 which states that Canonical Correlation Analysis
cannot with 90% confidence level discriminate
between promotable and non promotable candidates

30. Multivariate Test
 The Multivariate tests indicate the effect of scores of the
group and individual assessors both on status
determination and bias impact on such status. The figure
shows that the computed values and critical table values
differences are very insignificant.
 Candidate’s status determination resulting from scores
across the assessors and those that might result from bias
scoring are very insignificant(Wilk’s lambda value
=0.041)
 There is no between-status differences in the scores
between assessors of both group and individuals
 Rejection of Null hypothesis (Ho1) which states that
Canonical Correlation Analysis cannot detect bias

32.  The results of the table show that the scores of each
assessor had a significant effect on the determination
of each Candidate Status as the significance is 0.135.

33. Test for homogeneity of variance
 Overbearing score weight influence test hypothesis is
aimed at detecting across the individual assessors’
mark allocations and weights assigned to each.
 In this test, the assessors having low significance
value mean that there is homogeneity of variance.

35.  This Leads to rejection of null hypothesis (Ho2)
which states that Canonical Correlation Analysis
cannot detect significantly whether or not score-
weights of each of the promotion assessors has
overbearing influence on the promotability of
candidates.

36. Shortcomings and limitations of the process
 Procedures that maximize correlation between canonical
variate pairs do not necessarily lead to solutions that make
logical sense. it is the canonical variates that are actually
being interpreted and they are interpreted in pairs. a variate
is interpreted by considering the pattern of variables that are
highly correlated (loaded) with it. variables in one set of the
solution can be very sensitive to the identity of the variables
in the other set.
 The pairings of canonical variates must be independent of
all other pairs.

37. Conclusion from research analysis:
 From Table it can be seen that the order of promotable rankings but application
of Canonical Correlation Analysis results produced different ranking of
candidates.
Rejection of Null Hypothesis(H03):The results as shown in tables
indicate the Canonical Correlation Analysis status discriminatory ability of
grouping Candidates into promotable and Non-promotable status. The result
leads to the rejection of Null hypothesis Ho3 which states that Canonical
Correlation Analysis cannot with 90% confidence level discriminate between
promotable and nonpromotable candidates based on their earned scores.

38. Continued………….
Rejection of Null Hypothesis(Ho1):Pillar’s trace of 0.041, Wilk’s
Lambda of 0.041, Hotelling’s trace of 0.041 and Roy’s Largest Root of 0.041 -
all of them showed that p<0.05, it means that there is no between-status
differences in the scores between assessors of both group and individuals,
thereby leading to the rejection of Null hypothesis (Ho1) which states that
Canonical Correlation Analysis cannot detect bias.
Rejection of Null Hypothesis(Ho2):For Group Assessors - Internal
Assessors with p=0.096, External Academic Assessors with p=0.526 and The
President’s Assessment with p=0.0001, shows that except that of the President,
the weight assigned to scores of other two are group assessors are
insignificant- lead us to reject the Null hypothesis (Ho2) which states that
Canonical Correlation Analysis cannot detect significantly whether or not
score-weights of each of the promotion assessors has overbearing influence on
the promotability of candidates.

39. Thank You !!