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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 δ Λξ Λ T T ξ Λξ ± ± 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 Λ Λ T ±
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: 1 1 1 , , ′ ′ ′ = = = = = = = Σ Θ ΛΛ Λ T T ξξ T T Λ Λ TT Λ ΛΦΛ Λ Λ T Λ Λ T Φ TT ± ± ± ± ± ± 1 cos , 1,..., n ij ik jk ij k t t 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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