SPSSX Discussion

SPSS returns somewhat different from expected Bartlett's factor scores when analyzing covariance matrix

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Kirill Orlov

SPSS returns somewhat different from expected Bartlett's 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 Bartlett's (BART) factor scores somewhat incorrectly when covariances are analyzed. Why say it?
The returned scores do not coincide with the ones obtained with the (well-known in literature) formula of BART factor scores, the formula written in SPSS Algorithms. The difference is relatively small and both (returned and formula-computed) correlate very high. Still, they differ.

(When correlations are analyzed, BART 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 BART 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 BART(ALL)
/METHOD=COVARIANCE.

list FAC1_1 FAC2_1 FAC3_1 FAC4_1.

     FAC1_1      FAC2_1      FAC3_1      FAC4_1

    -.76456     -.06254    -1.07623      .46919
    1.32150     -.83911    -1.50343    -4.55797
   -1.53584      .97536    -1.37050      .66921
    -.15484     -.24267     1.20869     -.21816
   -1.45819     3.19812    -1.31374     1.81365
    -.32088      .51201      .04104    -2.15243
    2.66934     -.54847      .68039     -.84125
     .38508     1.98194     3.72881     -.70459
    -.73210    -1.31561     -.58367    -2.28756
    2.44101      .92027      .84606     1.97404
    1.77584     1.13614      .10774     -.56691
    2.25641     1.49373      .32944    -1.67253
    -.07128    -1.45765    -2.06055     -.46630
....
etc.

Now compute BART 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 P]^(-1) 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.

Note. SPSS Algorithms gives formula actually equivalent to the above. However, SPSS uses the complete residual matrix as U: R-R_hat, - in place of just its diagonal, the uniquenesses. The SPSS' approach is possible, albeit it is less cited in literature. OK, in the syntax below, will define U as SPSS Algorithms have chosen to.

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= r-a*t(a).
Comp b= t(inv(t(p)*inv(U)*p)*t(p)*inv(U)).
comp scores= x*b. /*BART factor scores
save scores /out= *. /*save as new dataset
end matrix.

list col1 to col4.

-.78062 -.06656 -1.07819   .40538
1.21840 -.92199 -1.33383 -5.05430
-1.47620 1.02746 -1.44319   .80984
-.07969 -.16349 1.07069   .07214
-1.42346 2.96132 -1.07329 1.57721
-.19991   .57980 -.05178 -1.72192
2.70254 -.48230   .50769 -.62257
.33908 1.75105 4.05108 -1.08254
-.77428 -1.23826 -.56579 -2.22770
2.41910   .97235   .82930 2.11172
1.67543 1.24914   .08409 -.60222
2.16093 1.61011   .38269 -1.33740
.05704 -1.56273 -2.04387 -.54313
.12891   .72187 -.80808 -1.18446
-1.62887 2.01300 -4.16674 -4.83334
....
etc.

The computed, according to formula, scores differ from the ones returned by SPSS FACTOR. It appears to me that SPSS does not follow its own formula claimed in Algorithms!
-------------------------------------------------------------------------------------------------------------------------------

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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