1. G89.2247 Lecture 7 Confirmatory Factor Analysis
CFA as measurement models
Illustration with POMS data
Issues in Confirmatory Factor Analysis
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2. Confirmatory FA POMS Example
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3. Psychometric Perspective on CFA
Suppose X1 is a measure of a psychological construct
The measure might not be perfect
Subject Carelessness and error
Transient variation in construct
Special features or bias of specific measure
Error is often addressed using multiple measures of the same construct (items, raters)
Supplement X1, with X2, X3
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4. A Parallel Measure Model
Suppose the average X is E(X)=T
If X1, X2, X3 are "parallel" measures
Then X1 = T + E1 X2 = T + E2 X3 = T + E3
The reliability of any X is R=[Var(T)/Var(X)]
Any X will be a "fallible" measure of T
True correlations of some Y with T will be underestimated by observed correlations between Y and X.
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5. Logic of Latent Variable (Factor) Model
Any set of variables might be highly correlated because they have a common cause
Examples
A series of items related to distress
A series of behaviors that suggest impulsive tendencies
A common cause would leave a pattern in the data, even if the cause itself is not measured
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6. Path Formulation of Factor Analysis Model
UNIQUE LATENT VAR
MANEFEST VAR
l11
l21
MANEFEST VAR
UNIQUE LATENT VAR
COMMON
LATENT VAR
l31
MANEFEST VAR
UNIQUE LATENT VAR
l41
MANEFEST VAR
UNIQUE LATENT VAR
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7. Matrix Formulation of Factor Analysis Model
Let the vector of manifest variables be X
One factor model:
X = L f1 + d
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8. Expected Variance Implied by Factor Model
Let's assume E(X)=0
Then Var(X) = E(XX')
E(XX')=E[(L f1 + d)(L f1 + d)'] = LFL' + Y where Var(f1) = F, Var(d) = Y
We can look at the implications of this pattern in a general covariance structure fitting context.
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9. Notes on Variance Patterns Shown in Excel Program
When only one factor is considered, the fitted covariance matrix S(q) has rows and columns that are proportional to each other (except on diagonal).
We can estimate the coefficients relating the factor to the manefest variables, and these are unique up to their sign.
The approach generalizes to more than one factor.
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10. Multiple Factor Analysis
Suppose we believe there are three factors
F=[f1 , f2 , f3]
The factor model states
X = L F + d
Var(X) = LFL' + Y where Var(F) = F, Var(d) = Y
There is no unique representation of F
L F = L(M -1M)F = L*F *
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11. More than one factor: CFA vs EFA
When more than one factor is considered, it is impossible to specify a unique coordinate system for the latent variable space
Factor rotations are arbitrarily chosen representations that aid interpretations
Orthogonal vs oblique rotations
CFA can specify a unique representation of the latent space based on theory
Model identification still an issue
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12. CFA Issues Model identification Number of factors
To avoid interdermanency issues, one must fix several loadings for each variable and factor.
Number of factors
For CFA usually we compare the fit of models with k and k+1 factors (competing theories)
Interpretation issues
Scaling of latent variables
Naming of latent variables
Reification of latent variables
Overly simple factor models (e.g. McCrae et al, 1996)
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13. Second order factor models
Just as latent variables might explain correlation among items, second order latent variables might explain correlation among factors 1 FA FB FC A1 A2 A3 B1 B2 B3 C1 C2 C3 F2
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14. Other Issues Influence of parts of model on overall fit
The measurement model can be influenced by structural aspects of the hybred model.
Recommend that the measurement model be examined in isolation
Items, Item parcels
Averages of items can often be useful
Domain representative vs factor representative
Kishton and Widaman (1994)
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15. Covariance Structure Models in Multiple Groups
Covariance structure model fitting is well adapted to exploration of multiple groups
Each group generates a separate sample covariance matrix to be fit
The groups can be estimated in a joint estimation phase
If we minimize the sum of LR criteria across the groups, the minimum is the solution that minimizes each group
Estimates can be constrained to be the same within as well as between groups
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16. Utility of Multiple Group Analyses
For structural models, tests of categorical variables and moderation of structure according to group
For measurement models, tests of measurement equivalence across groups
See:
Reise,SP; Widaman,KF; Pugh,RH (1993) Confirmatory factor analysis and item response theory: Two approaches for exploring measurement invariance. Psychological Bulletin (3):
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