G89.2247 Lecture 7

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G89.2247Lecture 7

• Confirmatory Factor Analysis
• CFA as measurement models
• Illustration with POMS data
• Issues in Confirmatory Factor Analysis

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Confirmatory FAPOMS Example
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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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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 RVar(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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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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Path Formulation of Factor Analysis Model
UNIQUE LATENT VAR
MANEFEST VAR
l11
l21
UNIQUE LATENT VAR
MANEFEST VAR

• COMMON
• LATENT VAR

l31
UNIQUE LATENT VAR
MANEFEST VAR
l41
MANEFEST VAR
UNIQUE LATENT VAR
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Matrix Formulationof Factor Analysis Model

• Let the vector of manifest variables be X
• One factor model
• X L f1 d

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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' Ywhere
  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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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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Multiple Factor Analysis

• Suppose we believe there are three factors
• Ff1 , f2 , f3
• The factor model states
• X L F d
• Var(X) LFL' Ywhere Var(F) F, Var(d) Y
• There is no unique representation of F
• L F L(M -1M)F LF

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More than one factorCFA 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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CFA Issues

• Model identification
• 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 k1 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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Second order factor models

• Just as latent variables might explain
  correlation among items, second order latent
  variables might explain correlation among factors

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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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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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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. 1993 114(3) 552-566