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Toward a simple structure: a comparison of different rotation techniques

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Abstract

This paper thematise the problem of seeking and devising a simple structure, when the solution envisages the extraction of more than one component or factor. To this avail, we shall make a comparison between a number of rotation techniques, both orthogonal and oblique, to evaluate just how capable they are of delivering the highest possible semplification of the data yielded by the analysis. To evaluate the results obtained through empirical controls, we have drawn up a simple structure index. For reasons of space, we shall apply principal components analysis to our method, although the results obtained here also hold for factor analysis.

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Notes

  1. This determines the circumstance that the sum of the squared component loadings in a row of the component matrix is never equal to the communality, except by chance. Similarly, only by chance the sum of the squared component loadings in the columns is equal to the total variance of the matrix.

  2. In SPSS for Windows the number of steps of the Promax procedure can be set by means of a K parameter. The latter is set at 4 by default: by increasing that value, the number of steps is increased, and, as a result, the correlation between the components is also increased. The minimun possible value for K is 1, which entails low correlations between components. Originally, Hendrickson and White (1964) proposed a coefficient b which worked like the delta of the Direct Oblimin (see below) in that it enabled researchers to establish the desired angle between components or factors.

  3. According to Kline, “using Direct Oblimin is generally recommended. However, it is good practice to use several rotation methods since no method guarantees not being deceived by specific data patterns. In most cases differences will not be relevant. If an orthogonally simple structure is preferred, most experts indicate Varimax as the most effective method” (1994, 74).

  4. In the SPSS for Windows programme the default value for delta is zero.

  5. With the exception of the Equamax technique, seeing that, in our applications we have always asked for the extraction of the first two principal components only. In this case, and only in this case, the rotation angles of the Equamax technique are identical to those of Varimax: consequently, the two techniques produce the same results. In the case of Direct Oblimin we have used the two suggested values for delta (\(-\)1 and 0). In the case of Promax we have used three values for K: 4 (given by default in the SPSS procedure), 10 and 20.

  6. Dc (Christian Democratic Party), the Police Force, Big Corporations and even, to some extent, the Psdi (Italian Social Democratic Party).

  7. Revolutionary Groups, the feminist Movement, Student Protesters and, in a sense, the immigrants from the South of Italy.

  8. Pci (the Italian Communist Party), Psi (the Italian Socialist Party) and Trade Unions.

  9. Msi (Italian Social Movement) and Dc (the Christian Democratic Party).

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  1. University of Roma La Sapienza, Rome, Rome, Italy

    Giovanni Di Franco

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  1. Giovanni Di Franco
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Correspondence to Giovanni Di Franco.

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Di Franco, G. Toward a simple structure: a comparison of different rotation techniques. Qual Quant 48, 1785–1797 (2014). https://doi.org/10.1007/s11135-013-9874-9

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