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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