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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 one…
ipipiii xxxy 2211
Rpiiii xxxy 21
piiipiii xxxyyy 2121
Fundamental equation for canonical analysis
Step 1: division of RR=R𝑦𝑦
− 1R 𝑦𝑥R𝑥𝑥−1 R𝑥𝑦
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
R𝑥 𝑥
R𝑦 𝑥
R𝑥𝑦
R𝑦𝑦
Fundamental equation for canonical analysis
Step 2: eigenvalue and eigenvector
R=R𝑦𝑦− 1R 𝑦𝑥R𝑥𝑥
−1 R𝑥𝑦
(R− λ I )K=0
(R 𝑦𝑦−1 R𝑦𝑥 R𝑥𝑥
− 1R𝑥𝑦−𝑟 𝑐𝑖2 I )K𝑞=0
𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒=Λ=[𝑟 𝑐12 ⋯ ⋯⋮ ⋱ ⋮⋮ ⋯ 𝑟𝑐 𝑖2 ]
𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟=K=[𝑘1 ⋯ 𝑘𝑞 ]
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
PR𝑥𝑥
R𝑦 𝑥
R𝑥𝑦
R𝑦𝑦
Fundamental equation for canonical analysis
Step 2: eigenvalue and eigenvector
1
NN*n
Y
1 2 3 n…1
NN*m
X
1 2 3 m…
𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒=Λ
… …
𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟=K
Fundamental equation for canonical analysis
χ1
χ2
χ3
χ4
X1
X2
X3
X4
X5
η1
η2
η3
η4
Y1
Y2
Y3
Y4
𝑟𝑐 1❑
𝑟𝑐 2❑
𝑟𝑐 3❑
𝑟𝑐 4❑
0 0
Canonical variate χ
Canonical variate η
Canonical correlation
Number of set of canonical correlation
𝜒2=−[𝑁−1−(𝑘𝑥+𝑘𝑦+12 )] ln Λ𝑚
Λ𝑚=∏1
𝑚
(1−λ 𝑖 )
F-test Wilk’s lambda Pillai’s trace Hotelling’s trace Roy’s gcr
Canonical weight
Beta in regression Partialed out due to multicollinearity Instability
χn
X1X2X3X4X5
ηn
Y1
Y2
Y3
Y4
𝑟𝑐𝑛❑λ𝑤𝑥𝑛1
❑
λ𝑤𝑥𝑛2❑
λ𝑤𝑥𝑛3❑
λ𝑤𝑥𝑛4❑
λ𝑤𝑥𝑛5❑
λ𝑤𝑦𝑛1❑
λ𝑤𝑦𝑛2❑
λ𝑤𝑦𝑛3❑
λ𝑤𝑦𝑛4❑
χ 𝑛=∑1
𝑖
𝑋𝑖× λ𝑤𝑥𝑛𝑖 η𝑛=∑1
𝑖
𝑌 𝑖× λ𝑤𝑦𝑛𝑖
Canonical loading
Structure factor loading in FA Criterion: >.3
χn
X1X2X3X4X5
ηn
Y1
Y2
Y3
Y4
𝑟𝑐𝑛❑λ𝑥𝑛1
❑
λ𝑥𝑛 2❑
λ𝑥𝑛3❑
λ𝑥𝑛 4❑
λ𝑥𝑛 5❑
λ 𝑦𝑛1❑
λ 𝑦𝑛2❑
λ 𝑦𝑛3❑
λ 𝑦𝑛4❑
Canonical cross-loading
Correlations of each variable and other canonical variate
λ𝑥𝑛𝑖 : 𝑦❑ =𝑟 𝑐𝑛× λ𝑥𝑛𝑖
❑
λ 𝑦𝑛𝑖: 𝑥❑ =𝑟𝑐𝑛×λ 𝑦𝑛𝑖❑
χn
X1X2X3X4X5
ηn
Y1
Y2
Y3
Y4
𝑟𝑐𝑛❑λ𝑥𝑛 1
❑
λ𝑥𝑛 2❑
λ𝑥𝑛3❑
λ𝑥𝑛 4❑
λ𝑥𝑛5❑
λ 𝑦𝑛1❑
λ 𝑦𝑛2❑
λ 𝑦𝑛3❑
λ 𝑦𝑛 4❑
𝑟𝑐𝑛×λ𝑥𝑛 1❑
𝑟𝑐𝑛×λ𝑥𝑛 2❑
𝑟𝑐𝑛×λ𝑥𝑛 3❑
𝑟𝑐𝑛×λ𝑥𝑛4❑
𝑟𝑐𝑛×λ𝑥𝑛 5❑
𝑟𝑐𝑛×λ 𝑦𝑛1❑
𝑟𝑐𝑛×λ 𝑦𝑛2❑
𝑟𝑐𝑛×λ 𝑦𝑛3❑
𝑟𝑐𝑛×λ 𝑦𝑛 4❑
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
𝑝𝑣 𝑥𝑐=∑1
𝑖 λ𝑥𝑛𝑖2
𝑖
𝑝𝑣 𝑦𝑐=∑1
𝑖 λ𝑦𝑛𝑖2
𝑖
𝑟𝑑=𝑝𝑣×𝑟𝑐𝑛2
Adequacy coefficientRedundan
cy index
Redundancy (index)
Variance the canonical variates from the IVs and extract from the DVs, and vice versa
χn
X1X2X3X4X5
ηn
Y1
Y2
Y3
Y4
𝑟𝑐𝑛❑λ𝑥𝑛 1
❑
λ𝑥𝑛 2❑
λ𝑥𝑛3❑
λ𝑥𝑛 4❑
λ𝑥𝑛 5❑
λ 𝑦𝑛1❑
λ 𝑦𝑛2❑
λ 𝑦𝑛3❑
λ 𝑦𝑛 4❑
𝑝𝑣 𝑥𝑐 𝑝𝑣 𝑦𝑐
𝑟 𝑑η𝑛→ X=𝑝𝑣 𝑥𝑐×𝑟𝑐𝑛2 𝑟 𝑑χ𝑛→Y=𝑝𝑣 𝑦𝑐×𝑟𝑐𝑛2
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
過去學業表現與現在學業表現研究生焦育布想瞭解過去學業表現與現在學業表現之間的關係。他的研究問題是,大學生在高中時期的學業表現是否與現階段的學業表現有關?其中,高中學業表現包含國文、英文、三角函數與線性代數等四個科目的評量分數,大學階段的學業表現指標則包含國文、外語、微積分與統計的評量分數。請以典型相關分析解答此問題。
Canonical correlation
χ1
HS_LAN
HS_ENG
HS_TRI
HS_LIA
η1
=.994-.99
-.99
-.61
-.30
CO_LAN
CO_ENG
CO_CAL
CO_STA
-.94
-.98
-.13
.15
χ1
HS_LAN
HS_ENG
HS_TRI
HS_LIA
η1
=.965-.01
-.06
.75
.65
CO_LAN
CO_ENG
CO_CAL
CO_STA
-.27
-.17
.73
.77
𝑟 𝑑χ𝑛→Y=.58
𝑟 𝑑χ𝑛→Y=.29
𝑟 𝑑η𝑛→ X=.60
𝑟 𝑑η𝑛→ X=.23
身體活動與智能研究生游志繪依想瞭解身體活動型態對於智力的影響。他的研究問題是,青少年的身體活動與智力之間是否有關?其中,身體活動包含坐式生活、健走、中等強度以及高等強度活動量等四項指標,智力的指標則包含語文、數學邏輯、空間、音樂、肢體動覺、內省、人際與自然觀察的測驗表現。請以典型相關分析解答此問題。
Canonical correlation
χ1
Strenuous
moderate
Walk
Sedentary
η1
Language
Math
Space
Music=.351
-.98
-.74
-.13
.15
Kinesthesis
Introspection
Interpersonal
Nature science
-.43
-.06
.05
-.01
-.70
-.22
-.44
.01