Factor rotation is commonly used in order to achieve simple structure and interpretable solutions. Mostly, standard procedures are used, which are implemented in statistics-programme packages (e.g. SPSS, SAS, etc.). The standard procedure in exploratory factor analysis has been criticised by Kaiser (1970). One of the problems with standard procedures is that they are not applied with consistent stringency. Different methods of factor extraction and rotation may be used for different purposes, but the selection of specific methods in a specific context is often not justified in the case of standard procedures.
In the context of oblique factor rotation the low degree of proficiency in the application of standard procedures is especially problematic because of the relatively arbitrary default parameters in the statistics packages. Mostly the default values are not changed by the users, even when it is neccessary in order to achieve optimal simple structure. An example is the direct Oblimin rotation (Jennrich & Sampson, 1966) in SPSS (1988), where the default value is Gamma=0, or Promax-rotation (Hendrickson & White, 1964) in SAS (1982) with a default parameter of k=3. Also in the Harris-Kaiser rotation a parameter must be specified. The parameters in Oblimin-, Promax- and Harris-Kaiser-rotations have an influence on the obliqueness of the factors (see Gorsuch, 1983, p. 189-190; Wittmann & Hampel, 1976). But mostly, the aim of the rotation is not a specific degree of obliqueness but optimal simple structure. Trendafilov (1994) describes the optimal way of using Promax with different parameters, where the parameter which will give the best simple structure is finally chosen. Hendrickson and White (1964) recommend a variation of the parameter k in relation to the complexity of the data. This shows, that an optimal oblique factor rotation for simple structure with procedures, which allow variable parameters, cannot be achieved without comparison and evaluation of the simple structure of several solutions.
Which criteria for a comparison of different solutions are available? Thurstone's (1947) criteria for simple structure have been evaluated as not sufficiently objective (e.g. Kaiser, 1958; Harman, 1976). Many authors have tried to integrate Thurstone's qualitative rules into a mathematical formula in order to have a more objective basis for factor rotation. This yielded the analytical methods of factor rotation (e.g. Harman, 1976; Gorsuch, 1983). But for the purpose of oblique factor rotation, in many rotational procedures (e.g. Oblimin, Promax, Harris-Kaiser), a parameter which has an influence on the obliqueness of the solution has to be specified (see above).
But, as already mentioned, the choice of the optimal value for the parameter of a function can be done only on the basis of a comparison of different solutions with different parameters. Thus, a more general criterion for the evaluation and comparison of the simple structure of factor solutions, which is not dependent on a specific function and parameter, is needed.
In order to compare simple structure of different factor solutions, a very
simple parameter, which reflects many aspects of Thurstone's criteria, has
been proposed by Cattell (1952): The hyperplane count. The hyperplane count
is mostly defined as the number of variables with an absolute loading 
0.10. The disadvantage of this criterion is that the exact value for a
loading falling in the hyperplane seems quite arbitrary. However, Cattell
(1952) proposes some values for the absolute loadings depending on the
sample size (see also Pawlik, 1967). According to Cattell (1988) the cut-off
value should be equal to the standard error of a zero loading. So, if the
arbitrariness of the cut-off value has to be avoided, the standard-errors
for zero loadings should be calculated.
Having a cut-off value as for the hyperplane count might be considered as a problem in itself, because the loadings are not treated in a homogeneous way as, for example, in those criteria of simple structure which are based on the whole factor matrix (e.g. Varimax, Oblimin). In fact, there are two ways to treat the problem of maximisation of simple structure: One way is to maximise a criterion which is based on the whole factor matrix as for example the variance of the squared loadings. The other way is to try to define cut-off values for high versus low loadings (Kaiser & Cerny, 1978) or for zero loadings (hyperplane count; Cattell, 1952). It seems perhaps more elegant to maximise a criterion without a cut-off value, but the maximisation of the hyperplane count has some advantages too. When criteria which are based on the whole factor matrix are maximised, the maximisation is in most cases partly due to small changes of the medium loadings. But what is the exact meaning of the changes of the medium loadings? For example, why should medium loadings be reduced via rotation when they do not become virtually zero? What is the psychological or theoretical meaning of reducing a secondary-loading of e.g. 0.35 to 0.30, when it is known that the standard-error of zero loadings is about 0.10? The variable was factorially complex before rotation and is slightly less complex after rotation. Because the ``true'' complexity of the variable is not known, it seems quite arbitrary to rotate in such a way, that a medium loading becomes slightly lower and to make those small changes in medium loadings a part of the criterion for simple structure. The same holds true for the case that a medium loading after the rotation becomes slightly higher. The theoretical meaning of rotational criteria which are also based on a maximisation or minimisation of the medium loadings of a solution does not seem very clear. In contrast, when the number of virtual zero loadings is maximised via hyperplane count only rotations are accepted which give an enhanced number of virtual zero loadings. The psychological meaning of a zero loading is clear: The variable does not define the respective factor. The maximisation of the number of variables which do not define a factor enhances the simple structure and is not based on changes in the theoretically problematic area of medium loadings since they are not reduced to zero. One may argue, that also when the hyperplane count is maximised, some medium loadings will change their value. But the important thing is, that the rotational criterion will not profit from the changes in the medium loadings. So the influence on medium loadings is not direct and does only occur in cases, when the number of zero loadings is enhanced. To sum it up, the present argument in favour of maximising hyperplane counts is that the theoretical meaning of the maximised zero loadings is clear or at least transparent whereas slight changes in medium loadings are not. Of course, the present argument does not refer to all aspects of the rotational problem (as e.g. the problem of main loadings), and further aspects of the criteria which avoid cut-off values could be very helpful. But the specific advantage of the hyperplane count should be taken into account. Some further arguments in favour of the hyperplane count as criterion for factor rotation are proposed in Cattell (1988).
It is not argued that the hyperplane count is the only reasonable criterion for simple structure. But many rotational procedures have at least partly been evaluated according to this criterion (Boyle & Stanley, 1986; Gorsuch, 1970, 1983; Hakstian, 1971; Hakstian & Abell, 1973; Hakstian & Boyd, 1972; Hendrickson & White, 1964; Katz and Rohlf, 1974, 1975; Piaggio, 1972; Rozeboom, 1991a,b), and the criterion has several times been applied in temperament research (e.g. Burdsal & Bolton, 1979; Cattell, 1994). In addition, Bargmann (1955) has developed a significance test based on the hyperplane count, which permits the evaluation of the simple structure of a factor.
Cattell and Muerle (1960) claimed that analytical methods of oblique factor rotation fail to realise Thurstone's original criteria for simple structure. They developed the Maxplane-rotation, which is not based on a mathematical function, but on an iterative procedure, which maximises hyperplane count (Eber, 1966). However, in studies which compare different rotational procedures on the basis of the simple structure of the corresponding solutions, Maxplane had a lower hyperplane count than most of the analytical methods of factor rotation (Hakstian, 1971).
In recent decades new methods of factor rotation have not been used widely. Even more classical methods such as Harris-Kaiser-Rotation, for which some advantages have been demonstrated empirically (Wittmann & Hampel, 1976) have not been used often. Instead, only a limited number of procedures (Oblimin and Promax for oblique, Varimax for orthogonal rotation) with the above-mentioned problems concerning default parameters have been used widely. However, in recent years, at least three methods for oblique factor rotation have been developed.
The Hyball-Rotation (Rozeboom, 1991a,b) allows the rotation of factor subspaces, keeping parts of the factor-matrix invariant. This procedure may have some specific advantages and disadvantages of subjective, graphic rotation and may be especially recommended for the specific purpose of subspace rotation. Trendafilov (1994) presented another new method (Promaj), a further development of Promax. A target matrix is found by means of majorisation theory (Marshall & Olkin, 1979). Kiers (1994) developed another method (Simplimax), where a target-matrix is involved. Upon comparison of solutions one has to choose the preferred number of small loadings (zero-elements). The optimal position of the zero-elements is then found with Simplimax.
All these methods of factor rotation, unfortunately, have not been directly empirically compared with other methods. In general, the authors give only two examples of factor rotation. Of course, the criterion of interest in this context - the closeness to hyperplanes, which according to Cattell (1952, 1988) can be operationalised through the hyperplane count - is not always maximised by those methods. A method, which is concerned with the hyperplane count, is Primary Pattern Function Plane (PPFP) a further development of Maxplane (Katz and Rohlf, 1974, 1975). But here again, only a few empirical applications are presented. In particular Rimoldi's (1948) data - in which Maxplane was shown to have problems (Hakstian, 1971) - were not involved in the application.