1. Phenotypic multivariate analysis

2. Last 2 days……. 1 P A A C C E E 1/.5 a cecae P

3. This lecture P AC E a ce P F f1 PP f2 f3

4. This lecture P F f5 P F f1 P F f2 P F f3 P F f4 C c1 c3 c4 c5 c2

5. Data analysis in non-experimental designs using latent constructs Principal Components Analysis Triangular Decomposition (Cholesky) Exploratory Factor Analysis Confirmatory Factor Analysis Structural Equation Models

6. Principal Components Analysis SPSS, SAS Is used to reduce a large set of correlated observed variables (x i ) to (a smaller number of) uncorrelated (orthogonal) components (y i ) y i is a linear function of x i Transformation of the data, not a model

7. PCA path diagram D P S = observed covariances = P D P’ x1 x2x3 x4 x5 y1y2y3y4y5

8. PCA equations Covariance matrix q S q = q P q q D q q P q ’ = P # P # ’ P = orthogonal matrix of eigenvectors D = diagonal matrix with eigenvalues P’P = I and P # = P  D Criteria for number of factors Kaiser criterion, scree plot, %var Important: models not identified! x1 x2x3 x4 x5 y1y2y3y4y5

9. Correlations: satisfaction, n=100 Var 1 work Var 2 work Var 3 work Var 4 home Var 5 home Var 6 home Var 11 Var 2.651 Var 3.65.731 Var 4.14.161 Var 5.15.18.24.661 Var 6.14.24.25.59.731

10. ++ 0 0 0 0 0 0 workhome Var 1Var 2Var 3 Var 4 Var 5Var 6

11. Triangular decomposition (Cholesky) x1 x2x3 x4 x5 y1y2y3y4y5 1 operationalization of all PCA outcomes Model is just identified and saturated (df=0) 1 1 111

12. Triangular decomposition S = Q * Q’ ( = P # * P # ‘) 5 Q 5 = f110000 f21f22000 f31f32f3300 f41f42f43f440 f51f52f53f54f55 Q is a lower matrix This is not a model! This is a transformation of the observed matrix S. Fully determinate!

13. Matrix algebra, Cholesky 3 Q 3 = f1100 f21f220 f31f32f33 Calculate Q * Q’ Var x1: f11*f11 Var x2: f21*f21+f22*f22 Cov x1,x3: f31*f11 Cov x2,x3: f31*f21+f32*f22

14. Exploratory Factor Analysis Account for covariances among observed variables in terms of a smaller number of latent, common factors Includes error components for each variable x = L * f + u x = observed variables f = latent factors u = unique factors L = matrix of factor loadings

15. EFA path diagram C L U

16. EFA equations S = L * C * L’ + U * U’ S = observed covariance matrix L = factor loadings C = correlations between factors U = diagonal matrix of errors Correlations between latent factors are allowed

17. Exploratory factor analysis No prior assumption on number of factors All variables load on all latent factors Factors are either all correlated or all uncorrelated Unique factors are uncorrelated Underidentification

18. Confirmatory factor analysis A model is constructed in advance The model has a specific number of factors Variables do not have to load on all factors Measurement errors may correlate Latent factors may be correlated

19. CFA An initial model (i.e., a matrix of factor loadings) may be specified, because: its elements have been obtained from a previous analysis in another sample its elements are described by a theoretical process

20. CFA equations x = L * f + u x = observed variables, f = latent factors u = unique factors, L = factor loadings S = L * C * L’ + U * U’ S = observed covariance matrix L = factor loadings C = correlations between factors U = diagonal matrix of errors

21. Structural equations models The factor model x = L * f + u is sometimes refered to as the measurement model The relations between latent factors can also be modelled This is done in the covariance structure model, or the structural equations model Higher order factor models

22. Structural Model X1X1 e1e1 X2X2 e2e2 X3X3 e3e3 X4X4 e4e4 X5X5 e5e5 X6X6 X7X7 e7e7 f1f1 f2f2 f3f3

23. Practice! Problem behavior in children (CBCL at age 3) 7 syndromes (aggression, oppositional, withdrawn/depressed, anxious, overactive, sleep and somatic problems Syndromes are correlated Datafile: cbcl1all.cov

24. Observed correlations (2683 subj.) Opp w/d agg anx act sleep Withdrawn.41 Aggression.63.35 Anxious.45.47.27 Overactive.53.34.52.29 Sleep.32.24.28.26.23 Somatic.21.22.18.17.15.23

25. Cholesky: How many underlying factors? – S = Q * Q’, Q is 7x7 lower – Fact7.mx What is the fit of a 1 factor model? – S = F * F’ + U, F = 7x1 full, U = 7x7 diagonal – Fact1.mx

26. What is the fit of a 2 factor model? – Same model,but F = 7x2 full with loading of aggression fixed – Fact2.mx Achenbach suggests 2 factors: externalizing and internalizing: what is the evidence for this model? – Same model, F = 7x2 full, internalizing factor and externalizing factor – Fact2a.mx

27. Can the 2 factor model be improved by adding a 3 rd general problem factor or by having a correlation between the 2 factors? – Same model, F = 7x3 full with general factor, internalizing factor and externalizing factor, Fact3.mx – S = F * C * F’ + U, F = 7x2 full matrix, C = stand 2x2 matrix (correlation), U = 7x7 diagonal matrix, Fact2b.mx