Rotation method (statistics)

As a rotating basis or rotation method is called in the multivariate statistics , a group of procedures to coordinate systems can be rotated until they meet a predefined criterion. The rooms in which these coordinate systems are located do not have any special requirements. They are arbitrarily n-dimensional, but ideally metric.

Use of rotation processes

Rotation methods are often used in conjunction with factor analysis or principal component analysis as an aid to interpretation.

The procedure is as follows: After factors that clarify the variance of the variables have been determined using the extraction method, an attempt is made to rotate the factors “counter to the data” until only a few factors with a high load are left. These can then be assigned more clearly to hypothetical laws, which is referred to as an aid to interpretation . This is necessary because the first calculated factor solution is often difficult to interpret. Usually a simple structure is sought, i. H. The rotation is done in such a way that the individual variables only upload to one factor (usually a value of 0.5) and no other factor. The simple structure, however, is only an intended goal that by no means has to be achieved.

The rotation does not increase the explained proportion of variance . It only helps to better understand the content of the factors.

Selected rotation processes

Orthogonal factor rotation.

After an oblique rotation, the axes can be at an oblique angle to one another.

Rotation methods are differentiated according to whether they are orthogonal or oblique. The most common method is Varimax . Other orthogonal rotation methods are Quartimax and Equamax (also Equimax), which is a combination of Varimax and Quartimax.

There are also oblique-angled rotation methods such as Oblimin and Promax , which loosen the assumption of the orthogonality of the factors. These methods are used more for practical reasons and are sometimes motivated by significantly improved interpretability. In the mathematical-formal sense they are z. B. is not allowed in the orthogonal factor model , since this is only unambiguous except for orthogonal transformations. In this case, even the principle of uncorrelated factors would be abandoned in favor of easy interpretability.

The figure on the right illustrates the orthogonal rotation procedure. In the first part of the figure (e.g. the first non-rotated factor solution), all variables load very similarly onto the first factor, which makes interpretation more difficult. The aim is to turn the axbox so that as many of the variables as possible have high values in one coordinate and values close to zero in the other coordinate. After the rotation, the variables form a group that loads up to factor 2 and hardly to factor 1. Another group for factor 1 was also marked in green. The observation represents an outlier. ${\ displaystyle X_ {1}, X_ {2}, X_ {3}}$ ${\ displaystyle X_ {4}}$

The common Varimax method is presented in more detail below.

Varimax

The Varimax and Oblimin methods in comparison.

As Varimax refers to a mathematical calculation method, can be rotated with which coordinate systems in n-dimensional spaces. The method developed by Henry Felix Kaiser at the end of the 1950s is mainly used in statistical processes and plays an important role as a content-related interpretation aid , particularly in factor analysis .

method

Varimax is assigned to the rotation method. When used in conjunction with the factor analysis, the factors are rotated in space in continuous steps until the variance of the squared charges per factor is maximum. This is how this process got its name. Medium-sized charges are therefore either smaller or stronger and can thus be assigned more clearly to their respective factors. An orthogonal design is used because the proponents of this method assume that the latent factors are independent of one another.

From a geometrical point of view, the (orthogonal) coordinate axes are rotated in space compared to the old axes, whereby the origin of the axes remains the same. The component transformation matrix is formed from the cosine of the angle between the factors and the original coordinate axes . By multiplying this matrix with the unrotated factor load matrix, the rotated factor loads can be calculated:

{\ displaystyle {\ underline {K}} ^ {*} = {\ underline {L}} \ cdot {\ underline {T}}}

.

${\ displaystyle K ^ {*}}$ : Matrix of rotated factor loads
${\ displaystyle L}$ : Matrix of unrotated factor loads
${\ displaystyle T}$ : Component transformation matrix

literature

HF Kaiser: The varimax criterion for analytic rotation in factor analysis. In: Psychometrika. 23, 1958, pp. 187-200. doi: 10.1007 / BF02289233
HF Kaiser: Computer program for varimax rotation in factor analysis. In: Educational and Psychological Measurement. 19, 1959, pp. 413-420. doi: 10.1177 / 001316445901900314
W. Schiller: About the meaningful effort in factor analysis. In: Archives for Psychology. 140, 1988, pp. 73-95. (e.g. comparison of different rotation processes)

Individual evidence

↑ Exploratory factor analysis - method presentation by Eric Klopp
^ ^A ^b Marcus J. Schmidt, Svend Hollensen: Marketing research: an international approach . Pearson, 2006, pp. 312 .

[1] Exploratory factor analysis - method presentation by Eric Klopp

[marketint-2] A ^b Marcus J. Schmidt, Svend Hollensen: Marketing research: an international approach . Pearson, 2006, pp. 312 .