Rotation in Factor analysis

One often talks about the rotated solutions in the context of factor analysis. This is done (i.e., a factor matrix is subjected to rotation) to attain what is technically called ―simple structure‖ in data. Simple structure according to L.L. Thurstone is obtained by rotating the axes** until:

  1. Each row of the factor matrix has one zero.
  2. Each column of the factor matrix has p zeros, where p is the number of factors.
  3. For each pair of factors, there are several variables for which the loading on one is virtually zero and the loading on the other is substantial.
  4. If there are many factors, then for each pair of factors there are many variables for which both loadings are zero.
  5. For every pair of factors, the number of variables with non-vanishing loadings on both of them is small.

All these criteria simply imply that the factor analysis should reduce the complexity of all the variables. There are several methods of rotating the initial factor matrix (obtained by any of the methods of factor analysis) to attain this simple structure. Varimax rotation is one such method that maximizes (simultaneously for all factors) the variance of the loadings within each factor. The variance of a factor is largest when its smallest loadings tend towards zero and its largest loadings tend towards unity. In essence, the solution obtained through varimax rotation produces factors that are characterized by large loadings on relatively few variables. The other method of rotation is known as quartimax rotation wherein the factor loadings are transformed until the variance of the squared factor loadings throughout the matrix is maximized. As a result, the solution obtained through this method permits a general factor to emerge, whereas in case of varimax solution such a thing is not possible. But both solutions produce orthogonal factors i.e.,
uncorrelated factors. It should, however, be emphasised that right rotation must be selected for making sense of the results of factor analysis.

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