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Canonical analysis
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- 2. Purpose To analyzethe relationship between 2 sets of variables Multiple IVs Multiple DVs
- 3. Kinds of researchquestions 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
- 4. Limitations to factoranalysis 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
- 5. Fundamental equation forcanonical analysis Multiple regression When Y is more than one… ipipiii xxxy 2211 R piiii xxxy 21 piiipiii xxxyyy 2121
- 6. Fundamental equation forcanonical analysis Step 1: division of R R = R 𝑦𝑦 −1 R 𝑦𝑥R 𝑥𝑥 −1 R 𝑥𝑦 Id TS TC BS BC 1 1.0 1.0 1.0 1.0 2 7.0 1.0 7.0 1.0 3 4.6 5.6 7.0 7.0 4 1.0 6.6 1.0 5.9 5 7.0 4.9 7.0 2.9 6 7.0 7.0 6.4 3.8 7 7.0 1.0 7.0 1.0 8 7.0 1.0 2.4 1.0 TS TC BS BC TS 1.00 0 -.161 .758 -.341 TC -.161 1.00 0 .110 .857 BS .758 .110 1.00 .051 R 𝑥𝑥 R 𝑦𝑥 R 𝑥𝑦 R 𝑦𝑦
- 7. Fundamental equation forcanonical analysis Step 2: eigenvalue and eigenvector R = R 𝑦𝑦 −1 R 𝑦𝑥R 𝑥𝑥 −1 R 𝑥𝑦 R − λI K = 0 R 𝑦𝑦 −1R 𝑦𝑥R 𝑥𝑥 −1R 𝑥𝑦 − 𝑟𝑐𝑖 2 I K 𝑞 = 0 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 = Λ = 𝑟𝑐1 2 ⋯ ⋯ ⋮ ⋱ ⋮ ⋮ ⋯ 𝑟𝑐𝑖 2 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 = K = 𝑘1 ⋯ 𝑘 𝑞 Do you smell something?
- 8. Fundamental equation forcanonical analysis Step 1: division of R 1 1 N P N*P X 1 1 N Q N*Q Y 1 1 N Q N*P X Y P 1 1 Q Q (P+Q)*(P+Q) P P R 𝑥𝑥 R 𝑦𝑥 R 𝑥𝑦 R 𝑦𝑦
- 9. Fundamental equation forcanonical analysis Step 2: eigenvalue and eigenvector 1 N N*n Y 1 2 3 n… 1 N N*m X 1 2 3 m… 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 = Λ … … 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 = K
- 10. Fundamental equation forcanonical analysis χ1 χ2 χ3 χ4 X1 X2 X3 X4 X5 η1 η2 η3 η4 Y1 Y2 Y3 Y4 𝑟𝑐1 𝑟𝑐2 𝑟𝑐3 𝑟𝑐4 0 0 Canonical variate χ Canonical variate η Canonical correlation
- 11. Number of setof canonical correlation 𝜒2 = − 𝑁 − 1 − 𝑘 𝑥 + 𝑘 𝑦 + 1 2 lnΛ 𝑚 Λ 𝑚 = 1 𝑚 1 − λ𝑖 F-test Wilk’s lambda Pillai’s trace Hotelling’s trace Roy’s gcr
- 12. Canonical weight Betain regression Partialed out due to multicollinearity Instability χn X1 X2 X3 X4 X5 ηn Y1 Y2 Y3 Y4 𝑟𝑐𝑛 λ 𝑤𝑥𝑛1 λ 𝑤𝑥𝑛2 λ 𝑤𝑥𝑛3 λ 𝑤𝑥𝑛4 λ 𝑤𝑥𝑛5 λ 𝑤𝑦𝑛1 λ 𝑤𝑦𝑛2 λ 𝑤𝑦𝑛3 λ 𝑤𝑦𝑛4 χ 𝑛 = 1 𝑖 𝑋𝑖 × λ 𝑤𝑥𝑛𝑖 η 𝑛 = 1 𝑖 𝑌𝑖 × λ 𝑤𝑦𝑛𝑖
- 13. Canonical loading Structurefactor loading in FA Criterion: >.3 χn X1 X2 X3 X4 X5 ηn Y1 Y2 Y3 Y4 𝑟𝑐𝑛 λ 𝑥𝑛1 λ 𝑥𝑛2 λ 𝑥𝑛3 λ 𝑥𝑛4 λ 𝑥𝑛5 λ 𝑦𝑛1 λ 𝑦𝑛2 λ 𝑦𝑛3 λ 𝑦𝑛4
- 14. Canonical cross-loading Correlationsof each variable and other canonical variate λ 𝑥𝑛𝑖:𝑦 = 𝑟𝑐𝑛 × λ 𝑥𝑛𝑖 λ 𝑦𝑛𝑖:𝑥 = 𝑟𝑐𝑛 × λ 𝑦𝑛𝑖 χn X1 X2 X3 X4 X5 ηn Y1 Y2 Y3 Y4 𝑟𝑐𝑛 λ 𝑥𝑛1 λ 𝑥𝑛2 λ 𝑥𝑛3 λ 𝑥𝑛4 λ 𝑥𝑛5 λ 𝑦𝑛1 λ 𝑦𝑛2 λ 𝑦𝑛3 λ 𝑦𝑛4 𝑟𝑐𝑛 × λ 𝑥𝑛1 𝑟𝑐𝑛 × λ 𝑥𝑛2 𝑟𝑐𝑛 × λ 𝑥𝑛3 𝑟𝑐𝑛 × λ 𝑥𝑛4 𝑟𝑐𝑛 × λ 𝑥𝑛5 𝑟𝑐𝑛 × λ 𝑦𝑛1 𝑟𝑐𝑛 × λ 𝑦𝑛2 𝑟𝑐𝑛 × λ 𝑦𝑛3 𝑟𝑐𝑛 × λ 𝑦𝑛4
- 15. Which interpretation approachto use Priority (Hair et al., 2010) 1. Canonical cross-loading 2. Canonical loading 3. Canonical weight
- 16. Redundancy (index) Variancethe canonical variates from the IVs and extract from the DVs, and vice versa 𝑝𝑣 𝑥𝑐 = 1 𝑖 λ 𝑥𝑛𝑖 2 𝑖 𝑝𝑣 𝑦𝑐 = 1 𝑖 λ 𝑦𝑛𝑖 2 𝑖 𝑟𝑑 = 𝑝𝑣 × 𝑟𝑐𝑛 2 Adequac y coefficien t Redundanc y index
- 17. Redundancy (index) Variancethe canonical variates from the IVs and extract from the DVs, and vice versa χn X1 X2 X3 X4 X5 ηn Y1 Y2 Y3 Y4 𝑟𝑐𝑛 λ 𝑥𝑛1 λ 𝑥𝑛2 λ 𝑥𝑛3 λ 𝑥𝑛4 λ 𝑥𝑛5 λ 𝑦𝑛1 λ 𝑦𝑛2 λ 𝑦𝑛3 λ 𝑦𝑛4 𝑝𝑣 𝑥𝑐 𝑝𝑣 𝑦𝑐 𝑟𝑑η𝑛→X = 𝑝𝑣 𝑥𝑐 × 𝑟𝑐𝑛 2 𝑟𝑑χ𝑛→Y = 𝑝𝑣 𝑦𝑐 × 𝑟𝑐𝑛 2
- 18. 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
- 19. Procedure 1. Research question 2.Designing a canonical analysis 3. Check the assumptions 4. Derive canonical analysis and assess overall fit 5. Interpret the canonical variate 6. Validation and diagnosis
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