Rotations
Thurstone postulated criteria for "simple structure"; that with certain rotations, factors might be
more parsimoniously interpreted.
Example: Plot 8 physical variables
Unrotated F1 & F2, orthogonally rotated F1* F2*, oblique rotation F1∆ F2∆
F*2
.6
.7
.8
F2∆
.5
1. Height
2. Arm Span
3. Length, forearm
4. Length, lower leg
5. Weight
6. Bitrochanteric diameter
7. Chest girth
8. Chest width
.4 .1
.3 .2
F1∆
F*1
F1* & F2* easier to interpret than F1 and F2. (F1* length & F2* width)
F1∆ & F2∆ even more clearly defined in space, & sensible that underlying constructs could be
correlated (nomological network...)
©Dawn Iacobucci, p.1
Rotations
Next:
Theory: Thurstone's criteria for simple structure
Practice: How to implement
Do not bother to interpret unrotated factor pattern matrix. All rotation algorithms built to
max simple structure; ease interpretation of factors.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Thurstone's 5 rules for rotating a B to A = BT to simple structure:
(from Thurstone, L.L. (1947), Multiple Factor Analysis, U of C Press.)
1) Each row (var) of the factor pattern matrix (A) should have at least one zero (or
practically: near-zero) loading. (i.e., a variable shouldn't load on all factors.) (Comment:
Probably also want at least one loading per variable to be high — otherwise variable
contributes just unique variation and perhaps should be removed from subset of variables
F analyzed.)
©Dawn Iacobucci, p.2
Rotations
2) Each column (factor) should have at least r zero elements (and the zeros for 1 factor
should be unique from the zeros for any other factor — linear independence)
E.g., okay
not okay (1st & 3rd)
O
O
O
O
O
O
O
O O
O
O
O
O
O
O
O
O
O
3) For every pair of columns (factors), there should be at least r variables with a zero
coefficient in one column and a nonzero coefficient in the other (linear indep again)
4) When r > 3, for every pair of columns (factors), there should be a large proportion of
variables with zeros in both columns.
5) For every pair of columns (factors), there should be only a small proportion of variables
with nonzeros in both columns.
Example of a factor pattern matrix that exhibits
simple structure (from McDonald, 1985, p.82,
the x's indicate nonzero loadings.):
X
X
X
0
X
0
0
0
X
0
X
0
X
X
X
0
X
0
X
0
0
X
0
0
X
X
X
©Dawn Iacobucci, p.3