0.10 only
Results with Trasid.
The possibility of maximising only the number of
absolute loadings 0.10 without division by h has already been
mentioned.
First, an example is given to demonstrate that a much higher number of
absolute loadings 0.10 can be attained, if the number of absolute
loadings divided by h lower than or equal to 0.10 is not maximised
(see Table 5).
| a .10/ a/h .10 | Bargmann-Test | |
| Thurstone-20 | 27 (68)/ 27 (68) | 3 / 3 / 3 |
| Thurstone-26 | 26 (50)/ 27 (52) | 2 / 3 / 3 |
| Harman | 21 (29)/ 51 (70) | 0 / 0 / 2 |
| Cattell/Dickman | 70 (73)/ 79 (82) | 4 / 4 / 4 |
| Overall/Klett | 13 (33)/ 36 (75) | 0 / 0 / 1 |
| Horn | 24 (30)/ 59 (74) | 0 / 0 / 1 |
| Baehr | 85 (57)/119 (79) | 3 / 5 / 5 |
| Rimoldi | 43 (37)/ 88 (77) | 0 / 0 / 0 |
| Pemberton | 94 (54)/138 (79) | 1 / 2 / 4 |
| Mean | 45 (48)/ 69 (73) | 13 / 17 / 23 |
,
behind are the values for 
. The percent values are in brackets.
They are refered to an optimum of one main loading for every variable. The
third column contains the number of factors with significant simple
structure. Before the first slash are the values for 
, after the
first slash for 
and after the second slash for 
. In the
last row are the mean values for the nine (or eight) data sets.
The mean percentage hyperplane count is about 73% and higher than for the Trasid-rotation with priority in maximising the hyperplane count with division by h, where it was about 69%. The difference between this and the other methods of factor rotation including Hyball-rotation is clear. However, also the number of absolute loadings divided by h falling in the hyperplane is still comparable with the values of the Harris-Kaiser II-rotation (see Table 2). So, when there is no restriction concerning the division of the loadings by h, the results are also very convincing.
But the reduction of the number of factors with significant simple
structure according to the Bargmann-test is very important. While the
Trasid-rotation - which maximises the number of absolute loadings divided
by h lower than or equal to 0.10 - yields 32 factors with significant
simple structure on the 5%-level (see Table 3), there are only 17 factors
with significant simple structure on the 5%-level, if maximisation is not
based on the absolute loadings divided by h (see Table 5). This shows that
the simple maximisation of the hyperplane count, without taking into account
the communalities of the variables, produces less convincing results in the
Bargmann test. In addition, this example shows that a high number of
absolute loadings lower than or equal to 0.10 does not guarantee a
significant simple-structure according to the Bargmann test. This is a
strong argument for the maximisation as it is proposed with Trasid-rotation,
and it is also an argument against the tendency in the literature to report
only the number of absolute loadings lower than or equal to 0.10 as an
indicator for simple structure without taking into account the communality
of the variables.
Example: Trasid- and Oblimin-rotation with variables of low communality. Moreover, the question arises, to what extent do variables with low communality influence the rotation, if not the hyperplane count based on the loadings divided by h but the simple hyperplane count is maximised. In order to examine this question, a matrix with 9 variables of moderate communality (.36-.40) and 11 variables of very low communality (.08) was constructed (see Table 6). The variables with moderate communality all clearly load on one of the three factors. Some of them have secondary loadings, which may fall into the hyperplane after oblique rotation. But also the loadings of variables with low communality may fall into the hyperplane after oblique rotation.
| Variable | F1 | F2 | F3 |  |
| V1 | .60 | .00 | .00 | .36 |
| V2 | .60 | .00 | .00 | .36 |
| V3 | .60 | .00 | .00 | .36 |
| V4 | .00 | .60 | .00 | .36 |
| V5 | .00 | .60 | .00 | .36 |
| V6 | -.20 | .00 | .60 | .40 |
| V7 | -.20 | .00 | .60 | .40 |
| V8 | -.20 | .00 | .60 | .40 |
| V9 | -.20 | .00 | .60 | .40 |
| V10 | -.20 | .00 | -.20 | .08 |
| V11 | -.20 | .00 | -.20 | .08 |
| V12 | -.20 | .00 | -.20 | .08 |
| V13 | -.20 | .00 | -.20 | .08 |
| V14 | -.20 | .00 | -.20 | .08 |
| V15 | -.20 | .00 | -.20 | .08 |
| V16 | -.20 | .00 | -.20 | .08 |
| V17 | -.20 | .00 | -.20 | .08 |
| V18 | -.20 | .00 | -.20 | .08 |
| V19 | -.20 | .00 | -.20 | .08 |
| V20 | -.20 | .00 | -.20 | .08 |
 | 2 | 18 | 5 |  |
 | 2 | 18 | 5 | |
The number of absolute loadings 0.10 and absolute loadings divided by
is equal in this matrix. Now, it can be examined how the
orientation on the loadings with or without division by h, influences the
rotation. First, a direct Oblimin solution (with delta=0) is reported
(see Table 7).
| Variable | F1 | F2 | F3 | |
| V1 | .52 | .00 | -.37 | .36 |
| V2 | .52 | .00 | -.37 | .36 |
| V3 | .52 | .00 | -.37 | .36 |
| V4 | .00 | .60 | .00 | .36 |
| V5 | .00 | .60 | .00 | .36 |
| V6 | .13 | .00 | .60 | .40 |
| V7 | .13 | .00 | .60 | .40 |
| V8 | .13 | .00 | .60 | .40 |
| V9 | .13 | .00 | .60 | .40 |
| V10 | -.28 | .00 | -.04 | .08 |
| V11 | -.28 | .00 | -.04 | .08 |
| V12 | -.28 | .00 | -.04 | .08 |
| V13 | -.28 | .00 | -.04 | .08 |
| V14 | -.28 | .00 | -.04 | .08 |
| V15 | -.28 | .00 | -.04 | .08 |
| V16 | -.28 | .00 | -.04 | .08 |
| V17 | -.28 | .00 | -.04 | .08 |
| V18 | -.28 | .00 | -.04 | .08 |
| V19 | -.28 | .00 | -.04 | .08 |
| V20 | -.28 | .00 | -.04 | .08 |
| | 2 | 18 | 13 |  |
| | 2 | 18 | 2 |  |
In the starting solution the number of absolute loadings 0.10 was 25.
In the Oblimin-solution the number of absolute loadings 0.10 is 33.
But the number of loadings divided by h fell from 25 to 22. The most
important change is in factor F3. The secondary loadings of the variables
V10 to V18 are now -.04. When this loadings are divided by h they are .14
and do not fall into the hyperplane. Note that in factor F1 four loadings
of the variables V6 to V9 (which have the highest communality in this
example) do not fall into the hyperplane.
In Table 8 two Trasid-solutions (after Varimax-prerotation, as described
above) are presented, the first (Trasid1, left side) is based on the
above-mentioned algorithm with highest priority for the maximisation of
the absolute loadings divided by h lower than or equal to 0.10. For the
second solution (Trasid2, right side) the Trasid-algorithm only maximises
the number of absolute loadings 0.10.
| Trasid1 | Trasid2 | ||||||
| Variable | F1 | F2 | F3 | F1 | F2 | F3 | |
| V1 | .59 | .04 | -.41 | .56 | .09 | -.28 | .36 |
| V2 | .59 | .04 | -.41 | .56 | .09 | -.28 | .36 |
| V3 | .59 | .04 | -.41 | .56 | .09 | -.28 | .36 |
| V4 | .03 | .60 | .03 | .08 | .63 | -.07 | .36 |
| V5 | .03 | .60 | .03 | .08 | .63 | -.07 | .36 |
| V6 | .04 | -.05 | .62 | .05 | -.09 | .64 | .40 |
| V7 | .04 | -.05 | .62 | .05 | -.09 | .64 | .40 |
| V8 | .04 | -.05 | .62 | .05 | -.09 | .64 | .40 |
| V9 | .04 | -.05 | .62 | .05 | -.09 | .64 | .40 |
| V10 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V11 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V12 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V13 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V14 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V15 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V16 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V17 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V18 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V19 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| V20 | -.28 | .00 | -.02 | -.27 | -.01 | -.09 | .08 |
| | 6 | 18 | 13 | 6 | 18 | 13 | |
| | 6 | 18 | 13 | 4 | 11 | 0 | |
In the second Trasid-solution maximising the absolute loadings 0.10
(without division by h) with highest priority, there are 37 absolute
loadings 0.10, which is more than in the Oblimin-solution. However,
there are only 15 absolute loadings divided by h, which are lower than
or equal to 0.10, compared to 22 in the Oblimin solution. In both solutions
the number of absolute loadings 0.10 has been maximised, whereas the
number of absolute loadings divided by h lower than or equal to 0.10 was
reduced. Because the number of loadings 0.10 was also maximised in the
variables with low communality (.08), both solutions have been
influenced by those variables.
The first Trasid-solution (Trasid1, see Table 8) differs clearly from the Oblimin-solution and the second Trasid-solution. The number of absolute loadings (both divided by h and not divided by h) which are lower than or equal to 0.10 is 37 in both cases and has been maximised. The variables with low communality have also a little influence on the solution, which can be seen in the fact that the absolute loadings of variables with low communality (V1 to V11), which were above 0.10 in the starting solution, are now below 0.10 for the factor F3. But the most important fact is, that the hyperplane count divided by h is maximal for all the variables with high communality (V1 to V9), which is not the case in the Oblimin- and in the second Trasid-solution. In fact, in Trasid2 only four loadings of the variables with high communality (V6 to V9 on F1) fall into the hyperplane with division by h, in Oblimin the loadings of V6 to V9 on F1 do not fall into the hyperplane at all.
However, both in the Trasid1-solution and in the Oblimin-solution the secondary loadings of the variables V1 to V3 are high, which is not so much the case in the Trasid2-solution. This may be an interesting aspect of the Trasid2-solution. But, as mentioned above, the ``true'' complexity of the variables V1 to V3 is not known, so that this point needs further exploration. But since it is related to the medium loadings, it is not directly related to the hyperplane count and is therefore not treated here. All in all, it should not be excluded that Trasid2 yields interesting solutions, but the hyperplane count with division by h will in most cases probably not be maximal for the variables with high communality.
The differences in the maximisation of the absolute loadings 0.10 with
division by h leads also to very different intercorrelations of the
factors (see Table 9).
| Trasid1 | Trasid2 | Oblimin | |||
| -.08 | -.21 | .00 | |||
| .32 | .01 | .03 | .25 | .12 | .00 |
The fact that the intercorrelations between the Oblimin-factors are very
different from those of the other solutions is not very surprising, because
the Oblimin-solution differs from the other two solutions with regard to
the hyperplane count (both with and without division by h). But it is
interesting to note, that in the second Trasid-solution, there are 37
absolute loadings 0.10 (without division by h), whereas in the Oblimin
solution there are 33. The difference of 4 variables in the hyperplane count
(without division by h) is accompanied by important differences in the
intercorrelations of the factors. This corresponds to Cattell's (1988)
opinion that slight differences in the hyperplane count may lead to
important differences between factor solutions. However, the second
Trasid-solution has the same number of absolute loadings 0.10 without
division by h as the first Trasid-solution (i.e. 37, see Table 8). But
the intercorrelations between the factors are very different for the two
Trasid-solutions. Thus, the difference between the solutions with regard
to the number of absolute loadings 0.10 with division by h
(15 versus 37, see Table 8) is important with regard to the
intercorrelations of factors.
Maximising the number of absolute loadings 0.10 without division by h
may lead to factor intercorrelations very different from those, which are
obtained, when the number of absolute loadings 0.10 with division by h
is maximised with the highest priority. In the case that there are many
variables with low communality and if only the number of absolute
loadings 0.10 without division by h is maximised, the maximisation of
the hyperplane count may be primarily due to variables with low
communality. In this sense, an influence of variables with low
communality on the Oblimin- and Trasid2-solutions could be shown,
which means that Oblimin- and Trasid2-solutions have to be regarded
with caution, when they are based on data sets with many variables
with low communality.