CANONICAL ANALYSISWei-Jiun, Shen Ph. D.
Purpose
To analyze the relationship between 2 sets of variables
Multiple IVs Multiple DVs
Kinds of research questions
It is considered a descriptive technique or a screening procedure rather than hypothesis-testing procedure Number of canonical variate pairs interpretation of canonical variates Importance of canonical variates Canonical variate scores
Limitations to factor analysis
Theoretical issues Interpretability Linear relationship Sensitivity Causality
Practical issues Ratio of cases to IVs 10:1 Normality, linearity and homoscedasticity (not required) Missing data Absence of outliers Absence of multicollinearity and singularity
Fundamental equation for canonical analysis
Multiple regression
When Y is more than oneโฆ
ipipiii xxxy 2211
Rpiiii xxxy 21
piiipiii xxxyyy 2121
Fundamental equation for canonical analysis
Step 1: division of RR=R๐ฆ๐ฆ
โ 1R ๐ฆ๐ฅR๐ฅ๐ฅโ1 R๐ฅ๐ฆ
Id TS TC BS BC1 1.0 1.0 1.0 1.02 7.0 1.0 7.0 1.03 4.6 5.6 7.0 7.04 1.0 6.6 1.0 5.95 7.0 4.9 7.0 2.96 7.0 7.0 6.4 3.87 7.0 1.0 7.0 1.08 7.0 1.0 2.4 1.0
TS TC BS BCTS 1.00
0-.16
1.758 -.34
1TC -.16
11.00
0.110 .857
BS .758 .110 1.000
.051
BC -.341
.857 .051 1.000
R๐ฅ ๐ฅ
R๐ฆ ๐ฅ
R๐ฅ๐ฆ
R๐ฆ๐ฆ
Fundamental equation for canonical analysis
Step 2: eigenvalue and eigenvector
R=R๐ฆ๐ฆโ 1R ๐ฆ๐ฅR๐ฅ๐ฅ
โ1 R๐ฅ๐ฆ
(Rโ ฮป I )K=0
(R ๐ฆ๐ฆโ1 R๐ฆ๐ฅ R๐ฅ๐ฅ
โ 1R๐ฅ๐ฆโ๐ ๐๐2 I )K๐=0
๐๐๐๐๐๐ฃ๐๐๐ข๐=ฮ=[๐ ๐12 โฏ โฏโฎ โฑ โฎโฎ โฏ ๐๐ ๐2 ]
๐๐๐๐๐๐ฃ๐๐๐ก๐๐=K=[๐1 โฏ ๐๐ ]
Do you smell something?
Fundamental equation for canonical analysis
Step 1: division of R1
1
N
P
N*P
X
11
N
Q
N*Q
Y
11
N
Q
N*P
X Y
P1
1
Q
Q
(P+Q)*(P+Q)
P
PR๐ฅ๐ฅ
R๐ฆ ๐ฅ
R๐ฅ๐ฆ
R๐ฆ๐ฆ
Fundamental equation for canonical analysis
Step 2: eigenvalue and eigenvector
1
NN*n
Y
1 2 3 nโฆ1
NN*m
X
1 2 3 mโฆ
๐๐๐๐๐๐ฃ๐๐๐ข๐=ฮ
โฆ โฆ
๐๐๐๐๐๐ฃ๐๐๐ก๐๐=K
Fundamental equation for canonical analysis
ฯ1
ฯ2
ฯ3
ฯ4
X1
X2
X3
X4
X5
ฮท1
ฮท2
ฮท3
ฮท4
Y1
Y2
Y3
Y4
๐๐ 1โ
๐๐ 2โ
๐๐ 3โ
๐๐ 4โ
0 0
Canonical variate ฯ
Canonical variate ฮท
Canonical correlation
Number of set of canonical correlation
๐2=โ[๐โ1โ(๐๐ฅ+๐๐ฆ+12 )] ln ฮ๐
ฮ๐=โ1
๐
(1โฮป ๐ )
F-test Wilkโs lambda Pillaiโs trace Hotellingโs trace Royโs gcr
Canonical weight
Beta in regression Partialed out due to multicollinearity Instability
ฯn
X1X2X3X4X5
ฮทn
Y1
Y2
Y3
Y4
๐๐๐โฮป๐ค๐ฅ๐1
โ
ฮป๐ค๐ฅ๐2โ
ฮป๐ค๐ฅ๐3โ
ฮป๐ค๐ฅ๐4โ
ฮป๐ค๐ฅ๐5โ
ฮป๐ค๐ฆ๐1โ
ฮป๐ค๐ฆ๐2โ
ฮป๐ค๐ฆ๐3โ
ฮป๐ค๐ฆ๐4โ
ฯ ๐=โ1
๐
๐๐ร ฮป๐ค๐ฅ๐๐ ฮท๐=โ1
๐
๐ ๐ร ฮป๐ค๐ฆ๐๐
Canonical loading
Structure factor loading in FA Criterion: >.3
ฯn
X1X2X3X4X5
ฮทn
Y1
Y2
Y3
Y4
๐๐๐โฮป๐ฅ๐1
โ
ฮป๐ฅ๐ 2โ
ฮป๐ฅ๐3โ
ฮป๐ฅ๐ 4โ
ฮป๐ฅ๐ 5โ
ฮป ๐ฆ๐1โ
ฮป ๐ฆ๐2โ
ฮป ๐ฆ๐3โ
ฮป ๐ฆ๐4โ
Canonical cross-loading
Correlations of each variable and other canonical variate
ฮป๐ฅ๐๐ : ๐ฆโ =๐ ๐๐ร ฮป๐ฅ๐๐
โ
ฮป ๐ฆ๐๐: ๐ฅโ =๐๐๐รฮป ๐ฆ๐๐โ
ฯn
X1X2X3X4X5
ฮทn
Y1
Y2
Y3
Y4
๐๐๐โฮป๐ฅ๐ 1
โ
ฮป๐ฅ๐ 2โ
ฮป๐ฅ๐3โ
ฮป๐ฅ๐ 4โ
ฮป๐ฅ๐5โ
ฮป ๐ฆ๐1โ
ฮป ๐ฆ๐2โ
ฮป ๐ฆ๐3โ
ฮป ๐ฆ๐ 4โ
๐๐๐รฮป๐ฅ๐ 1โ
๐๐๐รฮป๐ฅ๐ 2โ
๐๐๐รฮป๐ฅ๐ 3โ
๐๐๐รฮป๐ฅ๐4โ
๐๐๐รฮป๐ฅ๐ 5โ
๐๐๐รฮป ๐ฆ๐1โ
๐๐๐รฮป ๐ฆ๐2โ
๐๐๐รฮป ๐ฆ๐3โ
๐๐๐รฮป ๐ฆ๐ 4โ
Which interpretation approach to use
Priority (Hair et al., 2010)1. Canonical cross-loading2. Canonical loading3. Canonical weight
Redundancy (index)
Variance the canonical variates from the IVs and extract from the DVs, and vice versa
๐๐ฃ ๐ฅ๐=โ1
๐ ฮป๐ฅ๐๐2
๐
๐๐ฃ ๐ฆ๐=โ1
๐ ฮป๐ฆ๐๐2
๐
๐๐=๐๐ฃร๐๐๐2
Adequacy coefficientRedundan
cy index
Redundancy (index)
Variance the canonical variates from the IVs and extract from the DVs, and vice versa
ฯn
X1X2X3X4X5
ฮทn
Y1
Y2
Y3
Y4
๐๐๐โฮป๐ฅ๐ 1
โ
ฮป๐ฅ๐ 2โ
ฮป๐ฅ๐3โ
ฮป๐ฅ๐ 4โ
ฮป๐ฅ๐ 5โ
ฮป ๐ฆ๐1โ
ฮป ๐ฆ๐2โ
ฮป ๐ฆ๐3โ
ฮป ๐ฆ๐ 4โ
๐๐ฃ ๐ฅ๐ ๐๐ฃ ๐ฆ๐
๐ ๐ฮท๐โ X=๐๐ฃ ๐ฅ๐ร๐๐๐2 ๐ ๐ฯ๐โY=๐๐ฃ ๐ฆ๐ร๐๐๐2
Some important issue
Importance of canonical variates Test for the significance Canonical correlation >.3 Variate and its own variables Redundancy
Interpretation of canonical variates Mathematical resolution of combining variables Loading >.3
Procedure
1. Research question2. Designing a canonical analysis3. Check the assumptions4. Derive canonical analysis and assess overall
fit5. Interpret the canonical variate6. Validation and diagnosis
PRACTICE
้ๅปๅญธๆฅญ่กจ็พ่็พๅจๅญธๆฅญ่กจ็พ็ ็ฉถ็็ฆ่ฒๅธๆณ็ญ่งฃ้ๅปๅญธๆฅญ่กจ็พ่็พๅจๅญธๆฅญ่กจ็พไน้็้ไฟใไป็็ ็ฉถๅ้กๆฏ๏ผๅคงๅญธ็ๅจ้ซไธญๆๆ็ๅญธๆฅญ่กจ็พๆฏๅฆ่็พ้ๆฎต็ๅญธๆฅญ่กจ็พๆ้๏ผๅ ถไธญ๏ผ้ซไธญๅญธๆฅญ่กจ็พๅ ๅซๅๆใ่ฑๆใไธ่งๅฝๆธ่็ทๆงไปฃๆธ็ญๅๅ็ง็ฎ็่ฉ้ๅๆธ๏ผๅคงๅญธ้ๆฎต็ๅญธๆฅญ่กจ็พๆๆจๅๅ ๅซๅๆใๅค่ชใๅพฎ็ฉๅ่็ตฑ่จ็่ฉ้ๅๆธใ่ซไปฅๅ ธๅ็ธ้ๅๆ่งฃ็ญๆญคๅ้กใ
Canonical correlation
ฯ1
HS_LAN
HS_ENG
HS_TRI
HS_LIA
ฮท1
=.994-.99
-.99
-.61
-.30
CO_LAN
CO_ENG
CO_CAL
CO_STA
-.94
-.98
-.13
.15
ฯ1
HS_LAN
HS_ENG
HS_TRI
HS_LIA
ฮท1
=.965-.01
-.06
.75
.65
CO_LAN
CO_ENG
CO_CAL
CO_STA
-.27
-.17
.73
.77
๐ ๐ฯ๐โY=.58
๐ ๐ฯ๐โY=.29
๐ ๐ฮท๐โ X=.60
๐ ๐ฮท๐โ X=.23
่บซ้ซๆดปๅ่ๆบ่ฝ็ ็ฉถ็ๆธธๅฟ็นชไพๆณ็ญ่งฃ่บซ้ซๆดปๅๅๆ ๅฐๆผๆบๅ็ๅฝฑ้ฟใไป็็ ็ฉถๅ้กๆฏ๏ผ้ๅฐๅนด็่บซ้ซๆดปๅ่ๆบๅไน้ๆฏๅฆๆ้๏ผๅ ถไธญ๏ผ่บซ้ซๆดปๅๅ ๅซๅๅผ็ๆดปใๅฅ่ตฐใไธญ็ญๅผทๅบฆไปฅๅ้ซ็ญๅผทๅบฆๆดปๅ้็ญๅ้ ๆๆจ๏ผๆบๅ็ๆๆจๅๅ ๅซ่ชๆใๆธๅญธ้่ผฏใ็ฉบ้ใ้ณๆจใ่ข้ซๅ่ฆบใๅ ง็ใไบบ้่่ช็ถ่งๅฏ็ๆธฌ้ฉ่กจ็พใ่ซไปฅๅ ธๅ็ธ้ๅๆ่งฃ็ญๆญคๅ้กใ
Canonical correlation
ฯ1
Strenuous
moderate
Walk
Sedentary
ฮท1
Language
Math
Space
Music=.351
-.98
-.74
-.13
.15
Kinesthesis
Introspection
Interpersonal
Nature science
-.43
-.06
.05
-.01
-.70
-.22
-.44
.01
Top Related