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# Factor analysis - PowerPoint PPT Presentation

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

Caroline van Baal

March 3rd 2004, Boulder

• (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?

• 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?

• Principal Components Analysis

• Triangular Decomposition (Cholesky)

• Exploratory Factor Analysis

• Confirmatory Factor Analysis

• Structural Equation Models

Exploratory Factor Analysis constructs

• 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

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

1

Factor 1

IQ, “g”

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

1

1

Factor 1

verbal

Factor 2

performal

EFA equations constructs

• C = P * D * P’ + U * U’

• C = observed covariance matrix

• Nvar by nvar, symmetric

• Nvar by nfac, full

• D = correlations between factors

• Nfac by nfac, standardized

• U = specific influences, errors

• Nvar by nvar, diagonal

• Exploratory factor analysis constructs

• 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

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

1

1

Factor 1

verbal

Factor 2

performal

Fix to 0

Confirmatory factor analysis constructs

• 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

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

1

1

Factor 1

verbal

Factor 2

performal

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

1

1

Factor 1

verbal

Factor 2

performal

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

VC

FD

PO

INF constructs

SIM

VOC

COM

ARI

DIG

COD

BLC

MAZ

PIC

PIA

OBA

VC

FD

PO

CFA equations constructs

• x = P * f + u

• x = observed variables, f = latent factors

• C = P * D * P’ + U * U’

• C = observed covariance matrix

• D = correlations between factors

• U = diagonal matrix of errors

Structural equations models constructs

• 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: constructsC = 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 constructs

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

IQ and brain volumes (MRI) constructs

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 ; constructs

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

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)

Phenotypic Correlations: MRI-IQ, factor loadings has to be fixed to 0 to make it an identified modelDutch twins (A), n=111/296 pairs

• What is the fit of a 1 factor model? factor loadings has to be fixed to 0 to make it an identified 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 factor loadings has to be fixed to 0 to make it an identified model

• 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 factor loadings has to be fixed to 0 to make it an identified model

c2

c3

c4

c5

x4

x2

x3

x5

x1

PCA path diagram

• D

• P

• S = observed covariances = P * D * P’

c1 factor loadings has to be fixed to 0 to make it an identified model

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!

Correlations: satisfaction, n=100 factor loadings has to be fixed to 0 to make it an identified model

work factor loadings has to be fixed to 0 to make it an identified model

home

0

0

++

0

0

++

++

++

++

++

0

0

Var 4

Var 1

Var 2

Var 3

Var 5

Var 6

Triangular decomposition (Cholesky) factor loadings has to be fixed to 0 to make it an identified model

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 factor loadings has to be fixed to 0 to make it an identified model

• 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 factor loadings has to be fixed to 0 to make it an identified modelscript: phenochol.mx

• BEGIN MATRICES ;

• 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 /