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Discriminant analysis
- 2. Purpose To predict group membership from a set of predictors Level of DV > 2 What analytic strategy will you use?
- 3. Kinds of research questions Goals of discriminant analysis are to find the dimension or dimensions along which groups differ, and to find classification function to predict group membership. Significance of prediction Number of significant discriminant functions Dimensions of discrimination Classification functions Adequacy of classification Effect size Importance of predictor variables Significance of prediction with covariates Estimation of group means
- 4. Limitations to discriminant analysis Theoretical issues Random assignment Generalizability Practical issues Unequal sample size, missing data, and power Multivariate normality Absence of outliers Homogeneity of variance-covariance matrices Linearity Absence of multicollinearity and singularity
- 5. Fundamental equation for discriminant analysis Derivation and test of discriminant functions 𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆 𝑏𝑔 + 𝑆 𝑤𝑔 Λ = 𝑆 𝑤𝑔 𝑆 𝑏𝑔 + 𝑆 𝑤𝑔 Approximate F = 𝑑𝑓1, 𝑑𝑓2 = 1−𝑦 𝑦 𝑑𝑓2 𝑑𝑓1 Λ 1 2 = 𝑦;
- 6. Fundamental equation for discriminant analysis Classification Equal group size Unequal group size 𝐶𝑗 = 𝑐𝑗0 +𝑐𝑗1 𝑋1 + 𝑐𝑗2 𝑋2 + ⋯ + 𝑐𝑖𝑝 𝑋 𝑝 𝐂𝑗 = 𝐖−1 𝐌𝑗 𝑐𝑗0 = − 1 2 𝐂𝑗 ′ 𝐌𝑗 𝐂𝑗 = 𝑐𝑗0 + 𝑖=1 𝑝 𝑐𝑗𝑖 𝑋𝑖 + 𝑙𝑛 𝑛𝑗 𝑁
- 7. Fundamental equation for discriminant analysis Projection 𝐶𝑗 = 𝑐𝑗0 +𝑐𝑗1 𝑋1 + 𝑐𝑗2 𝑋2 + ⋯ + 𝑐𝑖𝑝 𝑋 𝑝 Xp Cj θ
- 8. Fundamental equation for discriminant analysis Solution B − ΛW K = 0 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 = Λ = 𝑟𝑐1 2 ⋯ ⋯ ⋮ ⋱ ⋮ ⋮ ⋯ 𝑟𝑐𝑖 2 𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 = K = 𝑘1 ⋯ 𝑘 𝑞 Do you smell something?
- 10. Discriminant function Special case of canonical analysis 1 1 N P N*P X 1 2 3 1 2 3 1 2 N*1 Y
- 11. Discriminant function Special case of canonical analysis 1 1 N P N*P X 1 N N*(L-1) …D1 D2 D3 DL-1 1 0 0 … 0 0 1 0 … 0 0 0 1 … 0 0 0 0 … 1
- 12. Discriminant function Special case of canonical analysis χ1 χ2 X1 X2 X3 X4 X5 η1 η2 D1 D2 𝑟𝑐2 𝑟𝑐3 0 0
- 13. Example: 1 predictor DV Y=0 Y=1 0 1 2 3 4 5 6 7 8 9 10 X 1 27 4 9 31013 5 6 12 11 3 14 15
- 14. Example: 2 predictors DV Y=0 Y=1 Y=2 0 1 2 3 4 5 6 7 X1 12 4 3 5 6 7 8 9 10 12 11 13 14 15 0 1 2 3 4 5 6 7 X2 14 2 5 3 6 78 9 10 12 11 13 14 15
- 15. Example: 2 predictors DV Y=0 Y=1 Y=2 0 1 2 3 4 5 6 7 1 2 4 3 5 6 7 8 9 10 1211 13 14 15 Discriminant function 2 = 0*X1+1*X2 Discriminant function 1 = 1*X1+0*X2 1 2 3 4 5 6 7 Number Min (N of IV, df for DV)
- 16. Principle of discriminant function Various linear combination (projection)
- 17. Principle of discriminant analysis Maximize difference among levels Minimize variance within level X1 X2 C1
- 18. Types of discriminant analysis Direct(standard) discriminant analysis Test for the significance Sequential discriminant analysis Mathematical resolution of combining variables Stepwise(statistical) discriminant analysis Mathematical resolution of combining variables
- 19. Criteria for overall statistical significance General Wilk’s lamda Hotelling’s trace Roy’s gcr Pillai’s criterion Distance-based (usually for stepwise) Mahalanobis’D2 Rao’s V
- 20. Interpreting discriminant functions Discriminant function plots (F1 & F2; F2 & F3;…) 0 1 2 3 4 5 6 7 G1 Discriminant function 2 = 0*X1+1*X2 Discriminant function 1 = 1*X1+0*X2 1 2 3 4 5 6 7 G2 G3
- 21. Interpreting discriminant functions Discriminant weight (coefficient) Beta in regression analysis Discriminant loading Simple linear correlation between IV and discriminant function >.33 (Tabachnick & Fidell, 2009) >.40 (Hair et al., 2010)
- 22. Evaluating predictor variables Classification function coefficient / loading Magnitude Mean difference Compare MGi and Mpooled(Gn-i) Inflation of type 1 error Adjusted alpha Potency index
- 23. Potency index χn X1 X2 X3 X4 X5 ηn D1 D2 D3 D4 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑐𝑛λ 𝑥𝑛1 λ 𝑥𝑛2 λ 𝑥𝑛3 λ 𝑥𝑛4 λ 𝑥𝑛5 λ 𝑦𝑛1 λ 𝑦𝑛2 λ 𝑦𝑛3 λ 𝑦𝑛4 Square d 𝑃𝑜𝑡𝑒𝑛𝑐𝑦 𝑖𝑛𝑑𝑒𝑥 𝑥𝑛𝑖→y = λ 𝑥𝑛𝑖 2 × 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑐𝑛 𝑡𝑜𝑡𝑎𝑙 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 2
- 24. Effect size Total Partial η2 Specific Eigenvalue (each discriminant function) Squared structure matric loading (each predictor) 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 η2 = 1 − Λ 1 3
- 25. Use of classification procedure Generalization (hit ratio) Cross-validation and new case Jackknified classification Leave one out classification Evaluating improvement in classification Sequential Early step classification Correct Incorrect Later step classification Correct A B Incorrect C D χ2 1 = 𝐵 − 𝐶 − 1 𝐵 + 𝐶
- 26. Standards of hit ratio Proportion chance criterion Press’s Q 𝑃𝑟𝑒𝑠𝑠′ 𝑠 𝑄 = 𝑁 − 𝑛𝐾 2 𝑁(𝐾 − 1) 𝐶 𝑝𝑟𝑜 = 𝑝2 + (1 − 𝑝)2 ℎ𝑖𝑡 𝑟𝑎𝑡𝑖𝑜 > 1.25 × 𝐶 𝑝𝑟𝑜 𝑝2 χ2 > 3.84 (𝛼 = .05) χ2 > 6.63 (𝛼 = .01)
- 27. 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
- 28. PRACTICE