lecture17_s10 EFA - Exploratory Factor Analysis: common...

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Exploratory Factor Analysis: common factors, principal components, and more Psychology 588: Covariance structure and factor models Apr 14, 2010
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• As learned earlier for CFA, both CFA and EFA are of (almost) the same model form, which defines q indicator variables as linear combinations of n “common” factors plus q mutually orthogonal “unique” factors as follows: Common factor model 2 ( ) () 111 ,, 0 nq i j qnq E Ei j EE E δδ × ×× × =+ = = === x Λξ δ ξδ 0 x0 ξ 0 δ 0 for Given • Difference between CFA and EFA resides in whether some selective elements of loading matrix Λ are constrained at particular constants (e.g., at 0 or by equality) --- consequently, the parameters are estimated by different methods
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• Technically, CFA and EFA differ by the degree of constraints ¾ EFA --- loading matrix has minimal constraints for a unique solution with fixed orientation of factors To fix orientation of factor axes, n × n “constraints” need be imposed, and that is done by n scaling constraints, n ( n –1)/2 elements for orthogonal factors ξ and n ( n elements for the loading matrix Λ (i.e., canonical form) ¾ CFA --- further constraints (motivated by theoretical hypotheses) imposed to see if they agree with the data • Furthermore, CFA may allow correlated measurement errors (so long as all parameters identifiable), but EFA doesn’t allow such relaxation by definition
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() ( )( ) ( ) 12 , diag , ,..., q E EE E E θθ θ ′′ ≡= + + = =+ ≡≡ Σ xx Λξ δ ξ Λ δ x0 ΛΦΛ Θ Φξ ξ Θδ δ Implied cov matrix of the common factor model 4 • It’s a convention in EFA to scale the common factors to have a variance of 1 (instead of setting their metric equal to one of their indicators) --- by this scaling, Φ becomes a correlation matrix • When considered for realized data (i.e., “subjects”), ξ is often called “factor score” matrix
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• Common factors ξ are constrained to be mutually orthogonal, and so with the unit-variance scaling, factors becomes mutually orthogonal z-scores: Orthogonal factor model 5 • Thanks to the rotational indeterminacy, orthogonal common factors are typically estimated in a canonical form for a
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This note was uploaded on 06/11/2010 for the course PSYC 588 taught by Professor Sunjinghong during the Spring '10 term at University of Illinois at Urbana–Champaign.

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lecture17_s10 EFA - Exploratory Factor Analysis: common...

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