lecture18_s10 EFA

lecture18_s10 EFA - Exploratory Factor Analysis: rotation...

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Exploratory Factor Analysis: rotation Psychology 588: Covariance structure and factor models Apr 16, 2010
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• Given an initial (orthogonal) solution (i.e., Φ = I ), there exist infinite pairs of “rotated” factor loading and score matrices such that all have exactly identical fit since Rotational indeterminacy 2 1 −= = = x δΛ ξΛ TT ξ ± ± subject to “rows” of T having unit-norm (sum of squares = 1) so that the total variances of ξ and are the same, i.e., --- rotation preserves the VAF collectively by the n factors, or equivalently, rotation doesn’t change communalities () tr tr ′′ = ξξ ξξ ±± ξ ± • If --- orthogonal (or rigid) rotation which preserves the angles between the initial factors, and so the initial orthogonal factors rotated rigidly to orthogonal factors , == TT I ΛΛ T ±
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• Note that the rotation matrix is defined for “factor score” matrix ξ , not for the loading matrix Λ --- makes no difference in the orthogonal case, but should be clear in the “oblique” case • Rotational indeterminacy shown in the covariance structure: 11 1 ,, ′′ −= = = = == = ΣΘ Λ Λ Λ TT ξξ ΛΛ TT Φ Λ T T Φ TT ±± ± ± 1 cos , 1,. .., n ij ik jk ij k tt i j n φ θ = = If rows of T are orthogonal, angles between rotated factors remain orthogonal; otherwise, the angle between factors i and j is defined by
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• Covariances between x and (a.k.a., “factor structure” matrix P , as compared to “factor pattern” matrix ): () ( ) ( ) EE ′′ =+ = x ξΛ ξ δ ξ Λ Φ ±± ± ξ ± • Communality is invariant over rotation and represented generally in an oblique system as: 1 22 11 2 2 nn n i k ij ik jk kj j k h λ λλφ == < = ∑∑ ± Λ ± θ π 1 ξ ± i x 1 i ± 2 i ± 1 i p 2 i p 2 ± • Could absolute factor loadings in an oblique system be greater than 1 even when a correlation matrix is analyzed (so that communality is upper bounded by 1)?
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1. Each row (variable) of the loading matrix should have at least one zero 2. Each column (factor) should have at least n zeros 3. Every pair of columns should have several
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lecture18_s10 EFA - Exploratory Factor Analysis: rotation...

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