# Factor Rotations in Factor Analyses.

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1 Factor Rotations in Factor Analyses. Hervé Abdi 1 The University of Texas at Dallas Introduction The different methods of factor analysis first extract a set a factors from a data set. These factors are almost always orthogonal and are ordered according to the proportion of the variance of the original data that these factors explain. In general, only a (small) subset of factors is kept for further consideration and the remaining factors are considered as either irrelevant or nonexistent (i.e., they are assumed to reflect measurement error or noise). In order to make the interpretation of the factors that are considered relevant, the first selection step is generally followed by a rotation of the factors that were retained. Two main types of rotation are used: orthogonal when the new axes are also orthogonal to each other, and oblique when the new axes are not required to be orthogonal to each other. Because the rotations are always performed in a subspace (the so-called factor space), the new axes will always explain less variance than the original factors (which are computed to be optimal), but obviously the part of variance explained by the total subspace after rotation is the same as it was before rotation (only the partition of the variance has changed). Because the rotated axes are not defined according to a statistical criterion, their raison d être is to facilitate the interpretation. In this article, I illustrate the rotation procedures using the loadings of variables analyzed with principal component analysis (the so-called R-mode), but the methods described here are valid also for other types of analysis and when analyzing the subjects scores (the so-called Q-mode). Before proceeding further, it is important to stress that because the rotations always take place in a subspace (i.e., the space of the retained factors), the choice of this subspace strongly influences the result of the rotation. Therefore, in the practice of rotation in factor analysis, it is strongly recommended to try several sizes for the subspace of the retained factors in order to assess the robustness of the interpretation of the rotation. Notations 1 In: Lewis-Beck M., Bryman, A., Futing T. (Eds.) (2003). Encyclopedia of Social Sciences Research Methods. Thousand Oaks (CA): Sage. Address correspondence to Hervé Abdi Program in Cognition and Neurosciences, MS: Gr.4.1, The University of Texas at Dallas, Richardson, TX , USA herve 1

5 Table 3: Wine example: Loadings, after varimax rotation, of the seven variables on the first two components. For For Hedonic meat dessert Price Sugar Alcohol Acidity Factor Factor interpretation is somewhat marginal, maybe because the factorial structure of such a small data set is already very simple. The first dimension remains linked to price and the second dimension now appears more clearly as the dimension of sweetness. x 2 Hedonic Acidity Alcohol For meat x 1 Price For dessert Sugar Figure 2: Pca: Original loadings of the seven variables for the wine example. Oblique Rotations In oblique rotations the new axes are free to take any position in the factor space, but the degree of correlation allowed among factors is, in general, small because two highly correlated factors are better interpreted as only one factor. Oblique rotations, therefore, relax the orthogonality constraint in order to gain simplicity in the interpretation. They were strongly recommended by Thurstone, but are used more rarely than their orthogonal counterparts. 5

6 x 2 y 2 Hedonic Acidity Alcohol For meat x 1 θ = 15 o Price y 1 For dessert Sugar Figure 3: Pca: The loading of the seven variables for the wine example showing the original axes and the new (rotated) axes derived from varimax. Table 4: Wine example. Promax oblique rotation solution: correlation of the seven variables on the first two components. For For Hedonic meat dessert Price Sugar Alcohol Acidity Factor Factor For oblique rotations, the promax rotation has the advantage of being fast and conceptually simple. Its name derives from procrustean rotation because it tries to fit a target matrix which has a simple structure. It necessitates two steps. The first step defines the target matrix, almost always obtained as the result of a varimax rotation whose entries are raised to some power (typically between 2 and 4) in order to force the structure of the loadings to become bipolar. The second step is obtained by computing a least square fit from the varimax solution to the target matrix. Even though, in principle, the results of oblique rotations could be presented graphically they are almost always interpreted by looking at the correlations between the rotated axis and the original variables. These correlations are interpreted as loadings. For the wine example, using as target the varimax solution raised to the 4th power, for a promax rotation gives the loadings shown in Table 4. From these 6

7 y 2 Hedonic Acidity Alcohol y 1 For meat Price For dessert Sugar Figure 4: Pca: The loadings after varimax rotation of the seven variables for the wine example. loadings one can find that, the first dimension once again isolates the price (positively correlated to the factor) and opposes it to the variables: alcohol, acidity, going with meat, and hedonic component. The second axis shows only a small correlation with the first one (r =.01). It corresponds to the sweetness/going with dessert factor. Once again, the interpretation is quite similar to the original pca because the small size of the data set favors a simple structure in the solution. Oblique rotations are much less popular than their orthogonal counterparts, but this trend may change as new techniques, such as independent component analysis, are developed. This recent technique, originally created in the domain of signal processing and neural networks, derives directly from the data matrix an oblique solution that maximizes statistical independence. *References [1] Hyvärinen, A., Karhunen, J., & Oja, E. (2001). Independent component analysis. New York: Wiley. [2] Kim, J.O. & Mueller, C.W. (1978). Factor analysis: statistical methods and practical issues. Thousand Oaks (CA): Sage Publications. [3] Kline, P. (1994). An easy guide to factor analysis. Thousand Oaks (CA): Sage Publications. 7

8 [4] Mulaik, S.A. (1972). The foundations of factor analysis. New York: McGraw- Hill. [5] Nunnally, J.C. & Berstein, I.H. (1994). Psychometric theory (3rd Edition). New York: McGraw-Hill. [6] Reyment, R.A. & Jöreskog, K.G. (1993). Applied factor analysis in the natural sciences. Cambridge: Cambridge University Press. [7] Rummel, R.J. (1970). Applied factor analysis. Evanston: Northwestern University Press. 8

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