lecture8_s10 CFA

lecture8_s10 CFA - Confirmatory Factor Analysis:...

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Confirmatory Factor Analysis: Identification and estimation Psychology 588: Covariance structure and factor models Feb 19, 2010
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Identification 2 • Covariance structure of measurement model: () xx δ =+ Σθ ΛΦΛ Θ where we can impose selective zero and equality constraints on Λ x and Φ ; and free selective off-diagonal elements in Θ δ , provided that the resulting model is identifiable • Though CFA has different parameter sets than the path model only with observed variables, the basic ideas hold: ¾ A model is identified if and only if every single free parameter has a unique solution ¾ A parameter is identified if it can be written as a function (or functions) of the data (variances/covariances)
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Single-factor model 3 • As a simplistic case, consider only one factor measured by 2 or 3 indicators, with no correlated errors (see detailed algebraic derivation in pp. 240-242) ¾ 2 indicators --- one more constraint (in addition to the scaling constraint) needed for just identification; e.g., known reliability, tau-equivalent measures ¾ 3 indicators --- just identifiable with the scaling constraint ¾ Multiple orthogonal factors with a uni-factorial loading pattern can also be considered as separate single-factor models for identification, except that any non-zero covariance between indicators of different factors will also contribute to misfit
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t-rule 4 • With no knowledge whatsoever about parameters to constrain, the factor model (for both EFA and CFA) has free parameters as many as: ¾ q × n factor loadings in Λ x ¾ n ( n +1) / 2 nonredundant elements in Φ ¾ q ( q + 1) / 2 nonredundant elements in Θ δ • Thus, constraints (including the scaling constraints) needed so as to satisfy ( ) 12 tq q + • As before, t-rule is necessary, not sufficient
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Three indicators per factor 5
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lecture8_s10 CFA - Confirmatory Factor Analysis:...

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