SPSSX Discussion

SPSS returns incorrect Anderson-Rubin factor scores when analyzing covariance matrix

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Kirill Orlov

SPSS returns incorrect Anderson-Rubin factor scores when analyzing covariance matrix

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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