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Factor analysis

Factor analysis. Caroline van Baal March 3 rd 2004, Boulder. Phenotypic Factor Analysis. (Approximate) description of the relations between different variables Compare to Cholesky decomposition

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Factor analysis

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  1. Factor analysis Caroline van Baal March 3rd 2004, Boulder

  2. Phenotypic Factor Analysis • (Approximate) description of the relations between different variables • Compare to Cholesky decomposition • Testing of hypotheses on relations between different variables by comparing different (nested) models • How many underlying factors?

  3. Factor analysis and related methods • Data reduction • Consider 6 variables: • Height, weight, arm length, leg length,verbal IQ, performal IQ • You expect the first 4 to be correlated, and the last 2 to be correlated, but do you expect high correlations between the first 4 and the last 2?

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

  5. 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 = P * f + u • x = observed variables • f = latent factors • u = unique factors • P = matrix of factor loadings

  6. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 1 Factor 1 IQ, “g”

  7. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 1 1 Factor 1 verbal Factor 2 performal

  8. EFA equations • C = P * D * P’ + U * U’ • C = observed covariance matrix • Nvar by nvar, symmetric • P = factor loadings • Nvar by nfac, full • D = correlations between factors • Nfac by nfac, standardized • U = specific influences, errors • Nvar by nvar, diagonal

  9. 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

  10. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 1 1 Factor 1 verbal Factor 2 performal Fix to 0

  11. Confirmatory factor analysis • An initial model is constructed, because: • its elements are described by a theoretical process • its elements have been obtained from a previous analysis in another sample • The model has a specific number of factors • Variables do not have to load on all factors • Measurement errors may correlate • Some latent factors may be correlated, while others are not

  12. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 1 1 Factor 1 verbal Factor 2 performal

  13. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 1 1 Factor 1 verbal Factor 2 performal

  14. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA VC FD PO

  15. INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA VC FD PO

  16. CFA equations • x = P * f + u • x = observed variables, f = latent factors • u = unique factors, P = factor loadings • C = P * D * P’ + U * U’ • C = observed covariance matrix • P = factor loadings • D = correlations between factors • U = diagonal matrix of errors

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

  18. Second order factor model: C = P*(A*I*A’+B*B')*P' + U*U’ INF SIM VOC COM ARI DIG COD BLC MAZ PIC PIA OBA 2nd order Factor “g” F1 F2 F3 VC FD PO

  19. Model specification Identification E.g., if a factor loads on 2 variables only, multiple solutions are possible, and the factor loadings have to be equated Estimation of parameters Testing of goodness of fit Respecification K.A. Bollen & J. Scott Long: Testing Structural Equation Models, 1993, Sage Publications Five steps characterize structural equation models

  20. IQ and brain volumes (MRI) 3 brain volumes Total cerebellum, Grey matter, White matter 2 IQ subtests Calculation, Letters / numbers Brain and IQ factors are correlated Datafile: mri-IQ-all-twinA-5.dat Practice!

  21. BEGIN MATRICES ; P FULL NVAR NFACT free ; ! factor loadings D STAND NFACT NFACT !free ; ! correlations between factors U DIAG NVAR NVAR free ; ! subtest specific influences M Full 1 NVAR free ; ! means END MATRICES ; BEGIN ALGEBRA; C= P*D*P' +U*U' ; ! variance covariance matrix END ALGEBRA; Means M / Covariances C / Script: phenofact.mx

  22. in exploratory factor analysis, if nfact = 2, one of the factor loadings has to be fixed to 0 to make it an identified model fix P 1 2 In confirmatory factor analysis, specify a brain and an IQ factor SPECIFY P 101 0 102 0 103 0 0 204 0 205 0 206 (if a factor loads on 2 variables only, it is not possible to estimate both factor loadings. Equate them, or fix one of them to 1)

  23. Phenotypic Correlations: MRI-IQ, Dutch twins (A), n=111/296 pairs

  24. What is the fit of a 1 factor model? • C = P * P’ + U*U’, P = 5x1 full, U = 5x5 diagonal • What is the fit of a 2 factor model? • Same, P = 5x2 full with 1 factor loading fixed to 0 • (Reducion: fix first 3 factor loadings of factor 2 to 0) • Data suggest 2 latent factors: a brain (first 3) and an IQ factor (last 2): what is the evidence for this model? • Same, P = 5x2 full with 5 factor loadings fixed to 0 • Can the 2 factor model be improved by allowing a correlation between these 2 factors? • C = P * D * P’ + U*U’, P = 5x2 full matrix (5 fixed),D = stand 2x2 matrix, U = 5x5 diagonal matrix

  25. Principal Components Analysis • SPSS, SAS, Mx (functions \eval, \evec) • Transformation of the data, not a model • Is used to reduce a large set of correlated observed variables (xi) to (a smaller number of) uncorrelated (orthogonal) components (ci) • xi is a linear function of ci

  26. c1 c2 c3 c4 c5 x4 x2 x3 x5 x1 PCA path diagram • D • P • S = observed covariances = P * D * P’

  27. c1 c2 c3 c4 c5 x4 x2 x3 x5 x1 PCA equations • Covariance matrix qSq = qPq * qDq * qPq’ • P = full q by q matrix of eigenvectors • D = diagonal matrix of eigenvalues • P is orthogonal: P * P’ = I (identity) Criteria for number of factors • Kaiser criterion, scree plot, %var • Important: models not identified!

  28. Correlations: satisfaction, n=100

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

  30. PCA: Factor loadings(eigenvalues 2.89 & 1.79)

  31. 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! Model is saturated (df=0)

  32. Triangular decomposition • S = Q * Q’ ( = P# * P# ‘, where P# is P*D) • 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!

  33. Saturated model, # latent factorsscript: phenochol.mx • BEGIN MATRICES ; • P LOWER NVAR NVAR free ; ! factor loadings • M FULL 1 NVAR free ; ! means • END MATRICES ; • BEGIN ALGEBRA; • C= Q*Q' ; ! variance covariance matrix • K=\stnd(C) ; ! correlation matrix • X=\eval(K) ; ! eigen values (i.e., variance of latent factors) • Y=\evec(K) ; ! eigenvectors (i.e., regression coefficients) • END ALGEBRA; • Means M / • Covariances C /

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