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Understanding Multivariate Analysis in Non-Experimental Designs

Explore Phenotypic Multivariate Analysis and models like Principal Components Analysis, Triangular Decomposition, Factor Analysis, and Structural Equation Models. Discover SPSS and SAS tools for data reduction and transformation. Learn about underidentification in correlation analyses.

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Understanding Multivariate Analysis in Non-Experimental Designs

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  1. Phenotypic multivariate analysis

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

  3. E C A F a e c f1 f2 f3 P P P P This lecture

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

  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 (xi) to (a smaller number of) uncorrelated (orthogonal) components (yi) • yi is a linear function of xi • Transformation of the data, not a model

  7. PCA path diagram • D • P • S = observed covariances = P D P’ y1 y2 y3 y4 y5 x4 x2 x3 x5 x1

  8. y1 y2 y3 y4 y5 x4 x2 x3 x5 x1 PCA equations • Covariance matrix qSq = qPqqDq qPq’ = 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!

  9. Correlations: satisfaction, n=100

  10. work home 0 0 ++ 0 0 ++ ++ ++ ++ ++ 0 0 Var 4 Var 1 Var 2 Var 3 Var 5 Var 6

  11. Triangular decomposition (Cholesky) 1 1 1 1 1 y1 y2 y3 y4 y5 x4 x2 x3 x5 x1 1 operationalization of all PCA outcomes Model is just identified and saturated (df=0)

  12. Triangular decomposition • S = Q * Q’ ( = P# * P# ‘) • 5Q5 = f11 0 0 0 0 f21 f22 0 0 0 f31 f32 f33 0 0 f41 f42 f43 f44 0 f51 f52 f53 f54 f55 • Q is a lower matrix • This is not a model! This is a transformation of the observed matrix S. Fully determinate!

  13. 3Q3 = f11 0 0 f21 f22 0 f31 f32 f33 Calculate Q * Q’ Var x1: f11*f11 Var x2: f21*f21+f22*f22Cov x1,x3: f31*f11Cov x2,x3: f31*f21+f32*f22 Matrix algebra, Cholesky

  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. f1 f2 f3 X4 X7 X5 X2 X1 X3 X6 e5 e7 e4 e3 e2 e1 Structural Model

  23. 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 Practice!

  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 3rd 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

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