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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

- 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

- Principal Components Analysis
- Triangular Decomposition (Cholesky)
- Exploratory Factor Analysis
- Confirmatory Factor Analysis
- Structural Equation Models

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

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 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

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

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

- 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

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

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 modelsIQ 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!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.mxin 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)

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

- 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

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!

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

- 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 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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