Data sets. Nine data sets with three to eight factors were analysed. All data sets have already been used in other contexts for the empirical illustration of factor rotation. The first ``classical'' data set is the 20-Variables Box-problem (Thurstone, 1947). Thurstone had a sample of 20 boxes, their three dimensions numerically fixed. Then he computed some relations between the edges and derived variables. The second data set is the 26-Variables Box problem (Thurstone, 1947). This data set is based on 30 boxes, whose three dimensions were measured. This data set was ``resistant'' against analytical rotation, until Cureton and Mulaik (1975) developed their weighted Varimax procedure. The weighted Varimax solution of Cureton and Mulaik (1975) was used as the prerotated starting solution for this problem. The third data set is documented in Harman (1960), and goes back to studies of Holzinger. The subjects were 145 children in Chicago, who worked on 24 tests (for the labels of the tests, see Table 4). The fourth is the Ball-problem (Cattell & Dickman, 1962). This data set is based on 32 measures, as diameter and weight, taken from a sample of 80 balls. The fifth data set is based on Overall and Kletts (1972) study of 6000 patients with 16 variables of the ``Brief Psychiatric Rating Scale''. The sixth data set is based on a study by Horn (1963), consisting of 172 male subjects. The subjects worked on items of the 16 P.F. and on some additional items. The seventh data set is a study of Baehr (1963), where 400 groups of subjects completed 30 items concerning their working situation. The eigth data set is based on Rimoldi's (1948) study of 138 subjects in 19 intelligence tests. The ninth data set is based on a study by Pemberton (1952) of 154 subjects with 25 tests.
Criteria for comparison.
For the empirical comparison of the solutions of
different methods of oblique rotation, the following criteria are proposed:
the number of main loadings is defined as about 
(Fürntratt, 1969), the hyperplane count without division of the absolute
loadings by h, and the hyperplane count with division of the absolute
loadings by h, and the number of factors with significant simple structure
according to the test of Bargmann (1955). The solutions will not be
compared with visually rotated solutions, because of the subjective
component in visual rotation.
Compared methods of factor rotation. The Trasid-rotation will be compared with Promax-, Oblimin- and Harris-Kaiser II-rotation. For the three methods, several parameters for the obliqueness of the solutions will be entered and the solution with the best hyperplane count (with division by h) will be retained. In addition, the published results for Maxplane-rotation and the few empirical results for Hyball-, PPFP-, Promaj- and Simplimax-rotation will be discussed.
Comparison of the solutions. With respect to the number of main loadings according to Fürntratt (1969) the Trasid-rotation should not be inferior. The number of main loadings are comparable for the different methods of factor rotation (see Table 1). For the Promax-solutions the mean value for the percent main loadings is slightly below the values of the remaining methods. For the mentioned new methods of factor rotation and for Maxplane, no values for the main loadings have been published.
| Data set | Factors | H-K II | Promax | Oblimin | Trasid |
| Thurstone-20 | 3 | 20 (100) | 20 (100) | 20 (100) | 20 (100) |
| Thurstone-26 | 3 | 26 (100)* | 24 (92) | 26 (100)* | 24 (92) |
| Harman | 4 | 24 (100) | 22 (92) | 22 (92) | 22 (92) |
| Cattell/Dickman | 4 | 31 (97) | 31 (97) | 31 (97) | 30 (94) |
| Overall/Klett | 4 | 16 (100) | 16 (100) | 16 (100) | 16 (100) |
| Horn | 5 | 11 (55) | 12 (60) | 13 (65) | 16 (80) |
| Baehr | 6 | 28 (93) | 28 (93) | 29 (97) | 28 (93) |
| Rimoldi | 7 | 15 (79) | 13 (68) | 15 (79) | 13 (68) |
| Pemberton | 8 | 22 (88) | 18 (72) | 20 (80) | 21 (84) |
| Mean | 21 (89) | 20 (86) | 21 (89) | 21 (89) |
With regard to the hyperplane count, first an illustration of the maximisation of the hyperplane count with division by h will be given. In Figure 3 the hyperplane count is plotted by the width W of Trasid-iterations.
Figure 3: Plot of the hyperplane count with division by h by the
width of the Trasid-iteration
One can see that, with the exception of the data sets of Horn and Pemberton, no increase of the hyperplane counts occurs for a width greater than 0.10. On the whole, the increase of hyperplane counts is not important for the widths greater than 0.07. So, in the present data sets, a width of 0.15 seems sufficient to reach an optimal simple structure. In cases with complex data and many factors (above 10) a greater width (e.g. 0.20) may be recommended.
Trasid is expected to yield superior results for the hyperplane count in
comparison to the other methods of factor rotation. The mean percentages
for the hyperplane count (last row in Table 2) will be compared. The
simple structure can be evaluated with regard to the hyperplane count
without division by h (i.e. the number of variables with 
) and
for the hyperplane count with division by h (i.e. the number of variables
with 
). Concerning Maxplane only for values for the hyperplane
count without division by h could be retained from Hakstian (1971).
The Maxplane solutions have clearly the lowest values for the hyperplane
count. Harris-Kaiser II-, Promax- and Oblimin-rotation yield comparable
hyperplane counts. However, the Oblimin-solution have slightly higher
hyperplane counts. The Trasid-solutions clearly have the highest number
of variables falling in the hyperplane. This holds for the number of
variables with as well as the number of variables with
. Not only is the mean percentage highest, but there is also no
solution where Trasid yields a lower hyperplane count than any other
method in Table 2.
The data set with the 24 tests (Harman, 1960) has been analysed via
PPFP (Katz & Rohlf, 1975). There were 46 absolute loadings 
0.10,
which is only one less than in the Trasid-solution and more than in the
Maxplane-, Harris-Kaiser II-, Promax-, and Oblimin-solution (see Table 2).
For Overall and Klett's (1972) data set, PPFP yields 31 absolute loadings
0.10 (see Katz & Rohlf, 1975), whereas Trasid yields 33 absolute
loadings 0.10 (see Table 2).
Table 2
Hyperplane count (|a|/h <= .10, behind |a| < = .10)
|
|
H-K II |
Maxplane | Promax | Oblimin | Trasid |
|
Thurstone-20 Thurstone-26 Harman Cattell/Dickman Overall/Klett Horn Baehr Rimoldi Pemberton |
27 (68)/ 27 (68) 7 (13)/ 7 (13)* 31 (43)/ 41 (57) 52 (54)/ 53 (55) 16 (33)/ 22 (46) 26 (33)/ 43 (54) 86 (57)/ 99 (66) 46 (40)/ 67 (59) 90 (51)/109 (62) |
/ 25 (63)
/ 32 (44)
/ 35 (44)
/ 45 (39) |
27 (68)/ 27 (68) 27 (52)/ 27 (52) 32 (44)/ 39 (54) 65 (68)/ 67 (70) 18 (38)/ 25 (52) 29 (36)/ 43 (54) 89 (59)/ 95 (63) 52 (46)/ 67 (59) 91 (52)/113 (65) |
27 (68)/ 27 (68) 11 (21)/ 11 (21)* 31 (43)/ 40 (56) 67 (70)/ 68 (71) 19 (40)/ 27 (56) 31 (39)/ 39 (49) 85 (57)/ 94 (63) 55 (48)/ 72 (63) 91 (52)/115 (66) |
27 (68)/ 27 (68) 27 (52)/ 27 (52) 43 (60)/ 47 (65) 77 (80)/ 78 (81) 29 (60)/ 33 (69) 48 (60)/ 52 (65) 109(73)/114 (76) 78 (68)/ 84 (74) 128(73)/129 (74) |
|
Mean |
47 (47)/ 58 (58) |
/ 34 (48) |
48 (51)/ 56 (60) |
51 (52)/ 60 (62) |
63 (66)/ 66 (69) |
Notes. The percent values are in brackets. They are refered to the maximum of variables in the hyperplane, based on the assumption of one main loading for every variable. Before the slash are the values for |a|/h <= .10, behind for |a| <=.10. In the last row are the mean values for the nine (or eight) data sets. *These solutions correspond largely to the unrotated solutions after factor extraction (i.e. all main loadings on the first factor) and are not included in the respective mean value. For Thurstones 26-Variables problem Cureton and Mulaik's (1975) weighted Varimax solution was used as starting solution for Harris-Kaiser-, Promax-, Oblimin- and Trasid-Rotation.
Harman's 24 tests have also been analysed via Simplimax-rotation
(Kiers, 1994), where 40 absolute loadings with 0.10 resulted, which is
less than with Trasid or PPFP. For the 26-variables Box-problem, there
were 27 variables in the hyperplanes. This is exactly the value of the
Promax- and Trasid-solution, but with the weighted Varimax-solution of
Cureton and Mulaik (1975) as starting position, whereas Simplimax reaches
it without this prerotation. However, Oblimin- and Harris Kaiser II-rotation
do not reach this solution, even when starting from the weighted
Varimax-solution.
A comparison with the Hyball-procedure (Rozeboom, 1991b) was possible
only for Harman's 24 tests. The best Hyball-variant (scan) reaches exactly
the same number of variables with absolute loadings 0.10 as Trasid.
Since there was no information about the absolute loadings divided by h,
a comparison could not be made for this parameter. Another comparison
with 10 factors out of a data set by Thurstone (1938) was not possible
because Rozeboom (1991b) does not mention the method of factor extraction
used. The third data set in Rozeboom (1991b) has not been published,
consequently it could not be submitted to Trasid-rotation.
The Promaj-rotation (Trendafilov, 1994) of Harman's 24 tests yields 40 variables in the hyperplane. This corresponds to the value of Simplimax and is clearly below the value of Trasid-, Hyball-, and PPFP-rotation. The Promaj-solution of the 20-variables Box-problem has 20 variables in the hyperplane, which is below the value reached with Harris-Kaiser II-, Promax-, Oblimin- and Trasid-rotation (see Table 2). From this it may be concluded, that the Promaj-rotation might be an interesting method of factor rotation, but that it probably does not maximise the hyperplane count.
While the Trasid-solutions have a higher mean hyperplane count than the
Maxplane-, Harris-Kaiser II-, Promax-, and Oblimin-solutions, the
difference is quite small compared to PPFP. With the Hyball-rotation
only one comparison resulting in an equal number of absolute loadings
0.10 was possible. In comparison to the Simplimax- and Promaj-solutions
the Trasid-solutions have clearly more absolute loadings 0.10.
The Trasid-rotation can also be compared with the Harris-Kaiser II-, Promax-, and Oblimin-rotation on the basis of the test for the significance of simple structure developed by Bargmann (1955). Because the Bargmann test is based on the absolute loadings divided by h, which are below or equal 0.10, and the number of absolute loadings divided by h is higher in the Trasid-solutions than in the Harris-Kaiser II-, Promax-, and Oblimin-solutions, a greater number of factors with significant simple structure is expected for the Trasid-solutions. In Table 3 the number of factors with significant simple structure according to Bargmann (1955) is shown for three different levels of significance.
| Data set | Harris-Kaiser II | Promax | Oblimin | Trasid | |
| Thurstone-20 | p .01 | 3 | 3 | 3 | 3 |
| p .05 | 3 | 3 | 3 | 3 | |
| p .25 | 3 | 3 | 3 | 3 | |
| Thurstone-26 | p .01 | 0 | 3 | 0 | 3 |
| p .05 | 0 | 3 | 0 | 3 | |
| p .25 | 1 | 3 | 1 | 3 | |
| Harman | p .01 | 1 | 2 | 0 | 3 |
| p .05 | 1 | 2 | 1 | 3 | |
| p .25 | 3 | 2 | 3 | 4 | |
| Cattell/Dick | p .01 | 2 | 3 | 4 | 4 |
| p .05 | 3 | 3 | 4 | 4 | |
| p .25 | 3 | 4 | 4 | 4 | |
| Overall/Klett | p .01 | 0 | 0 | 0 | 0 |
| p .05 | 0 | 0 | 0 | 1 | |
| p .25 | 0 | 1 | 2 | 4 | |
| Horn | p .01 | 0 | 0 | 0 | 2 |
| p .05 | 0 | 0 | 0 | 3 | |
| p .25 | 0 | 1 | 1 | 4 | |
| Baehr | p .01 | 3 | 3 | 3 | 5 |
| p .05 | 4 | 4 | 4 | 5 | |
| p .25 | 4 | 5 | 6 | 6 | |
| Rimoldi | p .01 | 0 | 0 | 0 | 2 |
| p .05 | 0 | 0 | 0 | 3 | |
| p .25 | 1 | 1 | 0 | 6 | |
| Pemberton | p .01 | 0 | 1 | 1 | 4 |
| p .05 | 1 | 2 | 2 | 7 | |
| p .25 | 3 | 2 | 4 | 7 | |
| Sum | p .01 | 9/ 9 | 15/12 | 11/11 | 26/23 |
| p .05 | 12/12 | 17/14 | 14/14 | 32/29 | |
| p .25 | 18/17 | 22/19 | 24/23 | 41/38 |
For all levels of significance the number of factors with significant simple structure is greater with Trasid, than with Harris-Kaiser II, Promax, and Oblimin. The differences on the 25%-level are also relevant here, because the Bargmann test is known to be conservative. The clear superiority of the Trasid-rotation in this criterion is probably due to the fact that the number of absolute loadings divided by h below or equal to 0.10 is maximised with the greatest priority with Trasid. For the new methods of factor rotation PPFP, Simplimax, Hyball, and Promaj no comparison could be drawn based on the Bargmann test, because no information on the number of absolute loadings divided by h lower than or equal to 0.10 has been reported.
Because the Trasid-solutions have generally a very high hyperplane count, they may be different from many other solutions. So, it might be interesting to give an example in order to show if the solutions are just simpler derivates of the solutions that are generally reported in the literature or if they are substantially different. Because Harman's 24 tests are one of the most cited example, the factor pattern for this data set is reported here (see Table 4).
| Test | F1 Verbal | F2 Speed | F3 Deduction | F4 Memory | |
| Visual Perception | -.02 | .07 | .70 | .04 | |
| Cubes | .01 | -.02 | .46 | .01 | |
| Paper From Board | .06 | -.09 | .59 | -.03 | |
| Flags | .10 | -.02 | .56 | -.05 | |
| General Information | .79 | .07 | .01 | -.03 | |
| Paragraph Compreh. | .79 | -.08 | .06 | .05 | |
| Sentence Completion | .89 | .02 | -.01 | -.10 | |
| Word Classification | .51 | .13 | .24 | -.03 | |
| Word Meaning | .86 | -.19 | .07 | .09 | |
| Addition | .07 | .73 | -.29 | .23 | |
| Code | .06 | .58 | -.10 | .33 | |
| Counting Dots | -.13 | .73 | .07 | .07 | |
| Straight-Curved Cap. | .04 | .58 | .27 | -.04 | |
| Word Recognition | .17 | .05 | -.03 | .48 | |
| Number Recognition | .04 | -.05 | .14 | .48 | |
| Figure Recognition | -.05 | -.05 | .45 | .37 | |
| Object Number | .05 | .06 | .02 | .64 | |
| Number-Figure | -.17 | .15 | .32 | .51 | |
| Figure-Word | .04 | .04 | .22 | .35 | |
| Deduction | .27 | -.05 | .45 | .12 | |
| Numerical Puzzles | .01 | .30 | .35 | .17 | |
| Problem Reasoning | .29 | -.13 | .40 | .24 | |
| Series Completion | .24 | .06 | .53 | .07 | |
| Arithmetic Problems | .26 | .35 | .07 | .26 | |
 | 12 | 14 | 9 | 12 | |
 | 11 | 13 | 8 | 11 | |
| Correlations | F1 | 1.00 | |||
| among factors | F2 | .41 | 1.00 | ||
| F3 | .48 | .46 | 1.00 | ||
| F4 | .35 | .27 | .26 | 1.00 |
are printed in bold face.
The solution is quite close to the Oblimin-solution reported in Harman (1976, p. 324) or the Promax-solution reported in Hendrickson and White (1964). The main difference is, that much more variables fall into the hyperplane in the Trasid-solution, but the meaning of the factors did not change. However, the intercorrelations of the factors are very different for the two solutions. This is, what was observed in general, when Trasid-solutions were compared with other methods of factor rotation: The solutions differ often with regard to the intercorrelations of the factors, but less with regard to the main loadings in the factor pattern. The differences between the intercorrelations of the factors are especially important when factor solutions of higher order are intended. But also, when the intercorrelations among factors are interpreted in terms of the relations between constructs, these differences are very important. It should be noted that different cut-off values for the hyperplane and, more generally, different criteria for simple structure may produce different solutions. Trasid only attemps to yield optimal hyperplane counts.
(Fürntratt-criterion)