Phenotypic multivariate analysis

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Data analysis in non-experimental designs using latent constructs

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

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

PCA path diagram

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S = observed covariances = P D P’

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PCA equations

Covariance matrix qSq = qPq qDq 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!

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Correlations: satisfaction, n=100

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Triangular decomposition (Cholesky)

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1 operationalization of all PCA outcomes

Model is just identified and saturated (df=0)

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Triangular decomposition

S = Q * Q’ ( = P# * P# ‘)

5Q5 = f11 0 0 0 0f21 f22 0 0 0f31 f32 f33 0 0f41 f42 f43 f44 0f51 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!

Matrix algebra, Cholesky

3Q3 = f11 0 0

f21 f22 0f31 f32 f33

Calculate Q * Q’

Var x1: f11*f11Var x2: f21*f21+f22*f22Cov x1,x3: f31*f11Cov x2,x3: f31*f21+f32*f22

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

EFA path diagram

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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

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

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

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

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

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

Structural Model

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Practice!

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

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

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

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

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