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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

Factor analysis

Caroline van Baal

March 3rd 2004, Boulder

phenotypic factor analysis
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?
factor analysis and related methods
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?
data analysis in non experimental designs using latent constructs
Data analysis in non-experimental designs using latent constructs
  • Principal Components Analysis
  • Triangular Decomposition (Cholesky)
  • Exploratory Factor Analysis
  • Confirmatory Factor Analysis
  • Structural Equation Models
exploratory factor analysis
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
slide6
INF

SIM

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COM

ARI

DIG

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1

Factor 1

IQ, “g”

slide7
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1

1

Factor 1

verbal

Factor 2

performal

efa equations
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
exploratory factor analysis1
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
slide10
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1

1

Factor 1

verbal

Factor 2

performal

Fix to 0

confirmatory factor analysis
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
slide12
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1

1

Factor 1

verbal

Factor 2

performal

slide13
INF

SIM

VOC

COM

ARI

DIG

COD

BLC

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PIA

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1

1

Factor 1

verbal

Factor 2

performal

slide14
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VC

FD

PO

slide15
INF

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cfa equations
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
structural equations models
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
slide18
Second order factor model: C = P*(A*I*A’+B*B')*P' + U*U’

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2nd order Factor

“g”

F1

F2

F3

VC

FD

PO

five steps characterize structural equation models
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
practice
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!
script phenofact mx
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
slide22
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)

slide24
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
principal components analysis
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
pca path diagram
c1

c2

c3

c4

c5

x4

x2

x3

x5

x1

PCA path diagram
  • D
  • P
  • S = observed covariances = P * D * P’
pca equations
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!
slide29
work

home

0

0

++

0

0

++

++

++

++

++

0

0

Var 4

Var 1

Var 2

Var 3

Var 5

Var 6

triangular decomposition cholesky
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)
triangular decomposition
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!
saturated model latent factors script phenochol mx
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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