Posted by
Kirill Orlov on
Nov 23, 2016; 1:38pm
URL: http://spssx-discussion.165.s1.nabble.com/SPSS-returns-incorrect-Anderson-Rubin-factor-scores-when-analyzing-covariance-matrix-tp5733504.html
Problem observed on SPSS Statistics 22 (and earlier). I don't know
at this time how it is on later versions.
Problem: FACTOR apparently computes Anderson-Rubin (AR) factor
scores incorrectly when
covariances are analyzed. Why say
it?
1) The returned scores correlate (though factor solution is
orthogonal). AR scores must not correlate.
2) The returned scores do not coincide with the ones obtained with
the (well-known in literature) formula of AR factor scores.
(When correlations are analyzed, AR returned by FACTOR are correct.)
Example.
data list list /v1 v2
v3 v4 v5 v6 v7 v8 v9 v10 v11 v12.
begin data
3 6 5 3 4 4 4 5 5 4 5 4
7 4 6 6 3 6 2 7 6 6 6 3
3 7 4 2 6 2 5 4 5 3 5 2
4 6 4 5 6 4 6 5 3 3 6 4
1 5 7 2 7 1 3 6 7 2 7 5
6 7 4 6 7 2 5 1 6 2 6 3
6 7 3 7 7 7 6 6 5 5 7 5
2 2 7 7 7 4 6 6 6 2 7 5
6 5 4 5 3 4 3 6 3 3 5 4
4 7 6 6 5 6 7 7 7 3 7 6
5 7 4 6 6 6 6 7 7 4 4 3
6 7 7 4 7 7 7 7 6 2 3 7
5 6 4 1 6 3 6 3 4 7 7 3
3 6 7 3 5 5 5 4 6 4 7 2
6 7 6 1 7 3 1 3 6 1 6 1
4 7 7 6 6 4 6 6 6 5 6 4
6 5 4 6 6 4 6 4 4 4 7 3
3 6 6 4 6 3 4 5 7 3 6 5
4 7 5 5 6 6 6 6 5 4 7 6
4 6 5 4 6 3 6 6 6 3 6 6
3 5 6 5 6 2 6 6 4 2 6 5
6 7 4 2 3 1 3 7 4 5 7 3
2 6 7 6 6 4 7 7 6 4 5 2
4 6 7 4 6 6 4 6 7 6 7 7
3 6 6 5 6 6 5 7 6 3 6 4
4 6 4 5 5 3 7 5 7 6 6 6
3 7 7 7 7 3 7 7 6 3 6 6
5 7 1 2 4 3 5 6 5 6 7 3
3 6 6 4 4 5 7 4 4 5 5 3
4 5 4 4 2 5 2 5 3 3 7 2
3 6 7 5 6 3 3 1 3 2 4 4
3 5 6 5 6 4 5 4 5 3 7 4
3 7 7 5 6 5 6 4 5 6 7 3
3 5 4 3 4 2 6 5 3 4 5 5
1 4 5 4 3 3 6 6 7 2 7 2
3 6 2 6 6 4 7 6 3 3 5 3
5 7 5 5 7 5 5 6 4 3 4 5
6 7 2 5 3 4 5 7 4 4 7 5
2 7 5 7 4 3 3 6 4 3 4 4
4 7 4 4 4 7 4 4 7 1 5 7
6 7 7 7 4 2 7 3 3 7 5 2
5 7 5 5 5 5 7 7 7 5 7 6
3 5 4 3 4 3 4 5 5 4 5 4
4 6 3 7 5 3 6 4 7 2 5 3
3 6 5 7 2 4 7 6 6 4 7 4
1 7 4 6 5 5 6 3 4 4 5 5
1 5 5 4 6 3 6 6 4 4 5 3
2 7 7 2 6 4 4 7 7 5 6 3
4 6 3 6 6 2 5 5 6 5 7 5
4 5 6 5 3 5 4 4 5 5 5 6
4 7 6 3 7 3 7 6 7 4 6 6
6 1 7 7 2 6 7 1 1 1 7 3
4 5 6 6 7 4 6 7 6 3 7 4
3 6 4 7 6 5 6 5 5 2 6 6
3 7 5 4 7 3 4 6 3 5 5 6
5 6 7 6 7 5 7 6 7 6 6 5
3 7 1 4 4 7 4 6 6 6 7 3
4 6 6 4 3 1 5 3 6 1 4 3
5 6 6 5 7 4 5 3 5 4 5 5
2 4 7 4 6 6 6 6 5 1 7 5
3 6 7 4 4 7 7 7 7 4 5 5
4 3 4 2 2 3 3 7 5 3 4 2
2 6 5 4 5 2 4 4 4 4 5 5
3 4 3 4 5 3 5 5 5 4 4 3
4 7 4 3 2 3 5 1 5 5 7 3
3 6 5 5 5 3 3 4 4 2 7 5
4 6 6 6 7 7 6 7 6 6 7 4
4 6 3 5 5 6 4 4 3 3 4 6
2 6 7 7 3 2 7 2 6 6 7 6
5 6 7 4 5 2 5 5 4 5 3 4
4 7 4 4 6 2 4 6 6 1 5 4
4 5 3 4 3 3 4 3 3 4 5 4
5 6 4 5 6 4 6 4 4 6 6 4
2 5 5 5 5 3 2 6 5 3 6 3
5 7 5 1 5 6 6 7 7 4 7 3
2 7 4 2 3 7 4 3 5 4 6 6
2 6 5 2 6 5 5 5 3 4 2 4
1 4 4 5 4 5 7 5 4 3 7 4
3 6 5 5 4 6 6 5 3 4 5 5
5 5 3 4 4 4 5 4 5 5 5 5
2 6 5 4 5 5 6 6 1 2 5 6
2 1 3 6 4 4 7 4 3 4 7 7
3 6 6 5 4 6 6 7 7 6 6 3
5 7 2 5 4 2 6 5 6 5 4 3
1 6 1 7 1 3 7 1 1 2 7 6
4 4 5 6 3 5 5 5 4 5 5 4
5 7 5 4 7 6 7 7 5 7 7 2
3 7 5 3 5 7 5 4 6 5 6 5
1 6 4 5 4 3 2 6 6 3 4 6
3 6 4 4 3 6 4 5 5 4 5 6
3 6 6 7 3 3 7 6 5 3 6 5
6 6 6 2 5 6 6 7 2 7 7 6
6 6 6 5 4 7 4 6 4 6 7 6
3 2 5 4 4 5 5 6 3 4 6 4
3 4 5 4 1 5 5 7 6 5 5 6
3 5 4 5 6 4 3 6 3 6 5 6
5 7 5 4 5 5 4 6 6 4 6 6
3 1 6 5 2 5 6 6 5 3 4 3
6 7 3 3 3 3 4 4 2 6 7 6
1 4 2 6 4 3 6 2 1 1 3 6
6 7 2 4 6 7 5 5 6 5 7 5
5 6 7 3 5 3 3 7 6 6 7 6
3 6 3 6 4 5 4 5 2 5 6 5
6 6 3 3 4 2 4 6 3 4 4 5
3 5 4 3 5 3 4 5 6 3 5 4
3 6 6 5 6 5 6 6 5 3 5 5
5 7 5 4 5 5 5 6 5 3 6 5
5 7 3 6 6 7 7 4 6 5 6 6
1 7 2 3 6 3 4 7 7 4 4 6
6 6 7 6 3 6 7 4 6 6 7 5
6 7 3 1 3 2 4 6 6 3 7 3
5 7 6 2 6 3 5 6 3 5 6 4
5 7 4 5 4 5 5 5 7 3 6 2
6 7 6 3 1 6 5 6 2 6 6 1
5 6 6 6 5 6 7 7 6 5 7 6
5 5 5 4 5 6 7 4 5 4 5 5
3 5 4 4 2 3 3 3 2 5 6 5
4 6 6 4 2 5 3 6 5 5 3 3
1 7 4 4 5 3 5 6 3 4 7 4
3 7 6 6 5 2 6 7 7 4 6 4
6 6 6 6 5 6 7 7 4 6 3 6
1 7 7 6 1 3 5 7 3 4 3 5
5 7 4 5 6 3 6 6 5 6 7 4
5 7 4 5 7 7 6 5 7 4 7 4
2 6 7 7 5 2 6 7 5 4 5 4
1 5 2 2 4 1 2 4 2 4 4 3
3 6 6 4 5 1 6 4 7 6 7 2
5 7 6 3 4 6 6 6 5 5 6 4
2 7 4 5 4 2 5 7 4 3 5 4
3 7 6 2 5 3 2 6 7 6 6 4
2 4 4 5 2 1 7 6 7 3 7 5
4 7 5 7 4 2 4 7 1 6 6 4
3 6 4 2 4 3 6 6 7 2 7 3
6 5 6 5 4 7 7 3 2 6 6 5
5 7 5 4 3 5 4 3 5 3 6 4
7 4 7 5 2 5 6 6 7 6 6 4
4 6 2 5 2 7 6 6 5 4 5 6
6 7 5 5 6 5 4 4 4 4 7 6
5 5 4 4 4 6 6 5 4 5 6 6
4 7 1 1 1 1 4 7 6 5 5 4
7 7 6 2 7 5 5 7 6 3 6 4
2 5 7 5 2 5 7 5 4 2 7 3
5 7 6 6 2 5 7 7 7 6 7 7
3 6 2 5 4 3 5 4 3 4 4 7
4 7 6 3 7 3 5 7 6 3 7 6
5 6 5 3 5 4 5 5 5 3 5 4
3 6 4 6 2 6 6 3 2 6 5 5
5 7 5 4 3 4 6 7 6 6 6 5
5 5 6 7 5 2 5 5 3 5 6 3
4 6 4 4 6 5 3 4 6 5 6 3
3 7 4 2 7 1 1 7 7 1 7 6
7 6 6 3 6 6 5 6 6 4 7 4
5 6 6 3 4 3 3 6 4 2 5 2
2 3 7 6 7 5 5 7 2 1 6 4
3 7 6 4 6 5 4 5 6 5 6 5
3 4 7 4 6 2 4 5 2 3 6 5
1 3 3 4 4 5 6 4 7 5 7 4
2 6 7 5 6 4 7 6 6 2 6 1
1 6 7 7 7 2 5 6 4 1 4 4
4 5 6 3 5 5 4 6 7 6 7 4
2 3 6 5 7 3 7 6 2 4 5 2
4 6 6 4 6 6 6 6 5 6 4 3
4 6 3 7 7 5 5 2 4 6 7 7
3 4 6 6 5 3 6 7 2 2 2 2
2 4 5 5 5 4 2 5 5 2 3 5
7 6 2 6 6 4 4 7 7 2 7 2
3 6 6 7 7 7 6 7 4 5 7 4
2 6 5 5 4 4 5 7 2 5 4 6
4 7 3 6 7 4 6 6 4 2 4 6
4 6 6 5 6 6 4 2 6 3 4 3
3 5 4 6 6 4 5 7 6 4 4 4
6 6 2 2 5 3 2 5 4 3 4 3
3 6 6 6 6 6 5 3 3 4 5 3
3 6 3 1 6 5 2 3 5 2 7 3
3 6 4 2 3 4 4 6 3 5 7 5
2 1 1 4 3 3 5 4 2 3 6 4
3 5 5 5 6 5 6 6 7 4 7 6
3 7 5 6 6 5 6 4 6 5 7 5
1 7 4 4 6 5 3 4 4 3 3 4
3 5 3 3 2 5 6 5 3 3 4 5
5 6 6 4 4 7 5 1 1 1 7 7
4 5 3 5 6 6 3 4 6 4 6 5
3 7 5 4 6 3 4 6 6 3 4 4
3 6 6 4 7 6 5 3 6 4 7 2
4 6 6 4 7 6 5 6 7 4 3 3
3 6 3 6 6 6 4 5 6 3 5 5
3 4 5 6 6 2 6 4 4 4 6 3
3 7 7 2 6 5 3 4 5 6 6 5
6 3 6 6 6 2 3 7 6 2 7 6
1 7 7 2 7 7 7 7 7 1 7 4
4 5 5 4 5 3 6 5 3 3 5 3
3 5 5 4 4 4 4 4 5 4 6 3
3 4 2 4 6 3 4 7 4 3 6 4
2 6 6 3 6 3 5 5 6 5 6 3
4 5 6 4 6 3 4 5 6 3 4 2
4 4 4 6 6 6 7 6 4 3 7 4
4 5 6 4 5 5 4 5 5 2 2 3
6 6 6 7 2 6 6 5 7 6 4 2
2 7 6 6 2 1 4 1 5 2 4 3
3 6 7 2 6 4 3 6 6 2 6 3
3 6 5 5 4 5 7 5 1 5 7 5
5 7 7 6 7 2 4 6 6 4 5 4
6 7 4 5 5 3 3 6 7 4 3 2
3 7 6 4 7 4 7 6 7 3 5 4
3 5 4 4 5 4 4 5 5 5 5 5
6 6 3 4 5 2 4 6 6 3 4 3
7 4 5 3 5 6 6 4 4 5 6 4
6 7 6 2 4 7 1 4 4 2 7 4
3 6 5 6 6 4 4 2 7 3 6 4
2 6 6 5 7 1 4 5 5 2 6 2
3 3 5 5 6 4 3 4 4 3 4 2
4 5 5 3 5 4 4 4 3 4 4 3
7 7 4 4 6 7 7 4 7 4 7 6
5 5 5 4 5 4 3 5 5 5 4 4
2 5 4 5 5 2 1 4 3 3 4 2
3 7 4 5 5 3 5 7 6 3 3 3
4 7 4 5 4 4 6 6 6 3 6 4
2 6 5 7 7 7 7 7 7 4 5 5
1 4 4 6 7 2 5 5 6 3 5 3
3 6 3 6 4 5 5 4 3 4 6 5
3 7 6 7 4 5 4 7 3 6 6 5
3 5 3 5 7 4 3 7 5 2 5 1
4 7 2 4 2 7 6 5 6 4 7 6
7 5 5 4 1 7 4 4 6 4 6 4
4 7 7 5 7 3 6 7 5 3 6 6
5 5 4 4 5 4 5 4 4 4 6 4
3 4 5 4 5 3 4 5 4 3 5 5
1 7 6 4 6 4 4 4 4 2 2 1
3 6 6 4 4 3 4 3 6 4 3 3
6 6 6 4 4 4 4 4 4 3 5 4
1 6 6 5 7 1 6 5 6 3 3 3
3 4 5 4 4 4 4 3 5 3 3 4
3 5 4 4 4 6 5 6 5 2 3 3
1 7 6 5 6 2 6 6 6 4 6 5
4 5 6 4 6 4 5 6 6 2 5 4
2 6 7 5 6 7 5 6 7 2 5 1
4 6 5 4 5 4 4 5 5 3 5 3
4 6 4 4 4 5 6 6 4 4 5 4
4 6 2 5 5 5 6 3 4 3 3 5
4 5 4 5 5 3 5 4 4 2 4 4
4 6 4 5 7 4 4 2 2 3 7 4
1 6 6 5 7 5 6 6 6 2 4 5
3 7 5 3 4 3 3 4 6 4 6 3
4 7 7 6 7 4 7 6 7 2 5 3
1 4 4 6 7 7 7 5 6 4 3 4
3 4 6 3 4 5 4 3 3 6 6 5
2 5 6 7 6 3 6 6 6 3 6 3
5 6 3 4 4 5 7 6 5 2 4 3
6 6 6 7 3 4 7 7 6 5 4 4
1 7 7 1 7 1 5 7 3 1 1 7
7 7 6 3 6 7 5 6 7 5 7 6
3 5 4 5 6 4 6 6 5 5 5 4
1 5 6 2 6 1 2 4 5 2 4 3
4 6 3 4 2 5 4 6 6 5 6 4
3 6 5 6 6 5 3 2 5 4 6 4
5 6 5 1 5 2 6 1 2 1 7 4
2 6 2 3 7 3 7 2 5 2 6 4
3 6 5 5 6 4 3 4 5 4 4 4
6 7 7 4 5 6 5 7 6 3 5 6
4 5 6 3 5 4 5 4 4 5 6 4
2 5 6 7 5 3 4 3 1 4 6 4
3 6 5 6 6 7 7 7 7 4 6 5
5 7 7 3 7 4 4 6 7 3 6 2
3 6 6 6 6 5 5 2 7 2 7 2
3 5 7 4 6 3 6 6 6 4 6 4
2 4 4 5 7 4 4 6 5 2 4 3
7 7 7 7 5 7 1 4 7 7 6 2
4 7 5 6 6 3 7 5 4 3 5 5
5 7 7 2 7 4 3 5 6 3 6 4
3 6 3 7 6 4 6 7 7 3 5 3
5 4 6 4 5 4 5 6 5 3 4 5
2 6 7 5 7 6 7 6 1 6 6 1
5 7 6 6 6 2 3 6 4 5 7 4
4 6 7 5 6 6 4 5 7 4 5 5
6 6 6 6 5 6 3 4 4 6 7 5
5 5 5 6 6 6 6 5 5 6 6 6
4 3 5 3 4 3 5 4 4 4 5 4
3 6 6 6 6 5 4 4 5 4 5 4
1 7 4 3 6 7 4 7 6 4 7 6
2 7 5 3 6 3 2 7 4 4 4 4
1 6 7 3 2 4 4 4 6 3 5 6
7 2 4 3 7 7 6 4 1 7 3 2
1 7 5 5 5 2 3 5 2 3 6 4
5 7 1 1 5 3 6 7 4 5 7 4
3 5 6 3 6 2 5 5 6 2 4 2
6 7 5 4 7 6 4 4 4 4 4 7
3 7 7 5 7 6 4 7 6 5 7 6
4 5 4 5 3 3 4 2 5 3 2 6
5 6 4 6 4 5 5 4 6 5 7 6
3 6 4 5 5 5 6 7 3 3 4 2
6 7 6 6 7 4 6 7 7 3 5 2
5 5 3 4 5 5 6 5 5 4 7 4
5 6 4 5 4 7 5 4 5 5 5 5
4 5 6 3 5 5 4 6 6 4 6 6
6 6 5 4 6 4 4 6 5 6 6 5
4 5 4 3 3 5 4 5 6 6 6 6
3 6 5 3 5 4 3 5 5 2 5 3
5 5 5 2 7 2 6 6 3 3 6 3
3 6 4 4 5 5 4 5 5 4 5 4
3 5 4 2 4 4 4 4 3 4 5 3
5 7 3 3 4 5 7 6 6 5 6 5
4 6 5 5 5 3 5 6 6 3 5 4
5 7 7 4 4 5 4 6 7 3 6 5
3 6 5 4 4 4 5 6 4 3 4 3
3 5 5 3 7 3 3 5 4 3 5 5
3 7 3 3 2 2 7 7 5 4 6 5
4 7 2 3 5 2 4 5 3 3 5 3
3 6 2 5 3 6 7 6 2 5 5 6
6 6 6 5 3 7 5 6 6 5 4 5
5 5 3 5 6 5 4 5 5 3 5 5
3 5 2 4 4 5 4 5 5 4 5 6
4 6 6 5 6 4 4 7 5 3 4 5
1 7 4 2 4 3 3 7 6 3 5 5
3 5 6 4 4 3 5 5 5 5 6 5
2 5 5 4 4 3 4 6 6 2 4 3
6 7 5 4 7 6 4 5 5 6 7 6
3 4 4 6 6 6 2 6 6 3 6 1
2 7 5 3 4 2 3 4 5 3 4 3
4 6 4 3 4 7 5 3 4 4 4 5
6 3 6 4 6 3 5 6 6 2 4 6
6 6 2 2 6 7 6 7 7 4 7 6
3 6 4 5 5 3 6 5 6 4 6 5
2 6 4 3 6 5 3 6 6 2 4 3
4 6 3 5 5 3 5 4 5 4 7 6
6 7 4 7 6 3 3 6 7 3 6 6
4 5 4 4 6 5 4 3 3 3 5 4
3 5 6 4 4 4 4 3 6 4 6 3
5 6 5 5 6 7 6 3 6 5 6 5
4 7 2 3 5 3 4 4 6 6 6 5
5 5 6 3 6 4 4 5 3 4 5 4
2 4 3 3 6 3 6 3 2 3 3 2
6 7 5 6 5 4 7 7 6 6 6 3
1 5 2 3 3 4 3 2 5 3 4 3
4 6 4 3 4 5 5 4 4 3 5 4
6 7 4 4 5 4 6 4 4 4 7 4
5 6 2 3 2 6 7 3 6 3 7 7
5 6 4 3 2 3 3 1 2 6 4 6
4 5 2 3 5 4 5 6 6 5 6 4
3 7 4 2 4 3 4 4 4 4 6 5
2 6 4 4 4 4 5 5 6 4 5 4
2 4 4 5 3 3 4 5 4 4 5 4
1 6 6 3 7 3 4 7 7 7 6 2
6 6 5 5 6 4 7 7 4 5 5 4
2 6 5 5 5 6 5 7 2 3 5 3
3 4 4 4 3 4 3 4 4 2 4 4
7 4 7 2 3 2 5 3 7 4 5 1
3 5 4 3 5 5 7 6 4 4 5 5
2 7 3 4 7 3 5 7 7 1 3 5
5 3 4 2 3 4 5 5 2 4 5 5
3 5 5 5 4 5 5 5 5 3 5 4
2 5 4 5 5 4 4 7 4 4 6 4
3 7 4 1 4 2 5 6 4 6 4 6
5 6 5 2 4 6 2 6 2 5 2 5
3 5 2 2 5 4 5 2 3 3 4 4
2 7 7 3 7 5 5 7 6 3 6 7
2 6 6 4 6 4 5 5 4 5 5 4
4 7 4 2 6 6 6 6 4 2 4 3
3 7 6 4 5 4 5 5 5 5 6 5
4 1 4 6 2 5 7 2 2 6 7 6
5 6 6 5 7 5 6 6 6 2 6 5
6 6 4 4 2 5 5 3 4 4 2 6
5 5 5 4 2 4 3 2 4 7 7 6
5 4 3 5 5 5 6 2 5 4 6 5
2 6 6 6 6 4 4 5 6 4 6 5
1 7 7 3 4 3 6 4 7 6 7 6
6 6 6 2 6 3 6 7 5 5 3 4
4 6 6 4 6 6 5 5 5 3 3 4
6 6 6 5 6 6 3 7 6 4 6 4
7 7 5 7 4 6 7 7 7 4 7 7
7 7 5 4 4 4 6 7 7 7 7 5
5 5 6 6 5 5 6 6 6 4 6 6
5 6 4 4 4 4 5 6 4 5 5 4
3 5 4 5 6 7 4 4 4 4 4 5
6 6 2 5 5 4 6 6 4 3 7 4
3 6 6 6 6 3 3 5 4 2 4 6
5 6 5 3 6 5 5 7 7 2 5 3
2 7 6 5 6 6 5 6 6 3 7 7
5 5 5 6 6 6 6 7 4 5 5 5
4 6 6 5 6 6 7 7 3 4 7 5
2 6 5 3 6 2 3 5 2 2 4 2
1 7 1 4 3 6 7 4 6 7 7 4
4 7 4 4 4 5 4 6 7 5 5 5
6 3 1 6 6 6 5 4 7 3 6 7
5 7 4 4 3 3 7 3 4 4 6 6
3 6 5 4 3 3 2 4 5 3 5 4
6 7 4 6 5 6 7 6 6 4 5 5
6 6 4 5 4 3 6 6 5 5 6 6
4 7 7 6 6 6 7 6 6 3 6 5
1 7 2 2 1 2 3 6 5 2 7 4
4 6 5 6 7 5 7 6 6 3 1 5
2 6 5 6 5 7 5 6 4 6 4 4
4 5 5 5 5 3 5 6 4 5 6 3
4 7 3 5 6 3 6 6 5 6 5 5
6 6 4 4 5 7 7 4 2 4 7 5
4 4 5 4 5 3 5 5 4 4 6 4
2 3 4 3 7 5 6 3 6 4 6 3
4 6 6 5 7 5 5 7 6 5 6 6
2 7 5 2 5 4 3 3 4 4 4 4
1 7 4 4 3 3 7 3 5 6 7 6
3 5 6 6 7 3 6 7 7 5 5 5
end data.
*Do PAF factor analysis of covariances, extracting 4 orthogonal
factors, no rotations. Request AR scores.
FACTOR
/VARIABLES v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12
/MISSING LISTWISE
/ANALYSIS v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12
/PRINT INITIAL EXTRACTION FSCORE
/CRITERIA FACTORS(4) ITERATE(25)
/EXTRACTION PAF
/ROTATION NOROTATE
/SAVE AR(ALL)
/METHOD=COVARIANCE.
list FAC1_1 FAC2_1 FAC3_1 FAC4_1.
FAC1_1 FAC2_1 FAC3_1 FAC4_1
-.91595 -.07371 -1.06787 .36761
1.61923 -.99050 -1.56172 -3.56638
-1.82922 1.08925 -1.35554 .56965
-.17795 -.27227 1.19770 -.13385
-1.74295 3.60161 -1.26990 1.47338
-.34911 .54956 .01071 -1.60304
3.17461 -.60602 .67348 -.76505
.48553 2.23518 3.71303 -.43101
-.84430 -1.51140 -.62468 -1.73198
2.87296 1.07566 .88654 1.40775
2.11740 1.28308 .11283 -.49539
2.70397 1.67667 .32133 -1.34489
-.09064 -1.65124 -2.06570 -.42316
.09580 .80882 -.71860 -.88397
....
etc.
These factor scores variables are not uncorrelated!
Now compute AR scores manually according to a well-known formula
Scores = X B, where X are the centered variables (not
standardized but centered - because factor analysis was based on
covariances, not correlations).
B' = [P' U_1 R U_1 P]^(-1/2) P' U_1
where R is the covariance (in this instance of analyzing
covariances) matrix; P is the loading matrix; U_1 is the diagonal
matrix with reciproced uniquenesses.
matrix.
get x /vari= v1 to v12.
!center(x%x). /*center variables [pick function below]
!cov(x%r). /*covariance matrix [pick function below]
comp p=
{.630426033687571, -.381178993709790, -.379685752308936,
-.397888690325225;
.318764722990882, .214932831882101, -.438228894531366,
.237998627435543;
.267827629824463, .511222792329309, .112420662122596,
-.277058174614929;
.413135062223572, .042735859375348, .699697012798330,
-.031379281815216;
.232671633794717, .760892733785184, .126228350650247,
-.184917192286093;
.722696748518335, -.276205244962143, .104386477478958,
-.219443846007899;
.602353573453240, -.122277799742283, .534170992444325,
.222283299283007;
.451686809782317, .530323329528760, -.103776107047512,
.160937075055084;
.564654060259637, .588321828137206, -.319822169164320,
.139544744387092;
.525274197515675, -.490466656598620, -.199250950564010,
.020814398575153;
.462160616318379, -.188528939630216, -.120813664339187,
.136115852441336;
.375559104122328, -.302302348283490, .056628961131190,
.278348795947127}.
/*Loadings taken from the factor analysis output
comp u= diag(r)-rssq(p). /*Uniquenesses = variances minus
communalities
comp u_1= 1/u. /*their reciprocals; column vector
comp tpu= t(p&*(u_1*make(1,ncol(p),1))). /*this is P' U_1
call eigen(tpu*r*t(tpu),eivec,eival).
comp b= t(inv(eivec*mdiag(sqrt(eival))*t(eivec))*tpu).
/*Coefficients B
comp scores= x*b. /*AR factor scores
save scores /out= *. /*save as new dataset
end matrix.
list col1 to col4.
COL1 COL2
COL3 COL4
-.62418 -.04586 -.75380 .20201
.92587 -.72020 -.88088 -2.61157
-1.18063 .78365 -1.01403 .40801
-.05966 -.12862 .75539 .02627
-1.13172 2.24808 -.78258 .82521
-.18272 .44658 -.04315 -.88592
2.16938 -.37027 .36430 -.28479
.25735 1.30393 2.82856 -.56813
-.64477 -.94118 -.35982 -1.17992
1.96592 .72451 .58036 1.10209
1.33209 .93540 .06903 -.29559
1.70601 1.22868 .24723 -.66597
.04962 -1.17158 -1.43631 -.25379
.08921 .52829 -.54218 -.63200
-1.37543 1.52686 -2.88269 -2.51932
....
etc.
The computed AR scores are uncorrelated, as expected. Also,
these scores correlate with
regression-method factor scores
very high (regression-method scores are the ones with highest
"validity").
-------------------------------------------------------------------------------------------------------------------------------
define !cov(!pos=
!token(1) /!pos= !charend('%') /!pos= !charend(')'))
comp !3= !2.
comp @sum= csum(!3).
comp !3= (sscp(!3)-t(@sum)*@sum/nrow(!3))/(nrow(!3)-1).
release @sum.
!enddefine.
define !center(!pos= !token(1) /!pos= !charend('%') /!pos=
!charend(')'))
comp !3= !2.
comp !3= !3-make(nrow(!3),1,1)*(csum(!3)/nrow(!3)).
!enddefine.
(See these and many other handy MATRIX functions of mine at
http://www.spsstools.net/en/KO-spssmacros)
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