Factor Analysis - Factor Rotation
Factor Analysis - Factor Rotation
Factor analysis is complicated somewhat (or made more interesting, depending on your perspective) by the fact that it is possible to generate several factor analysis solutions (loadings and factor scores) for any data set. Each solution is termed a particular factor rotation and is generated by a factor rotation scheme. Each time the factors are rotated the pattern of loadings changes, as does the interpretation of the factors. Geometrically, rotation means simply that the dimensions are rotated. (For a geometric interpretation of factor analysis see the appendix to this chapter.) Although there are many such rotation programs, varimax rotation is the most common and will be described here.
The basic "unrotated" factor analysis usually employs principal components analysis (also termed principal factor analysis) and will be introduced first. The objective of the principal components is to generate a first factor that will have the maximum explained variance. Then with the first factor and its associated loadings fixed, principal components will locate a second factor maximizing the variance explained in this second factor. The procedure continues until there are as many factors generated as variables or until the analyst concludes that the number of useful factors has been exhausted. Determination of the number of factors to include will be considered shortly.
When principal components/principal factor analysis is used, the interpretation of the factors can be difficult. The use of varimax rotation can improve greatly the interpre tab ility. A study of the perceptions of 94 consumers of a particular brand of coffee will illustrate.3 The consumers, after
sampling the coffee, rated it on 14 semantic-differential scales. The ratings were factor analyzed by the principal components method, and the results are shown on the left side of Figure 17-2. The first factor explained nearly 75 percent of the variance and seems clearly to reflect a general like-dislike dimension. This interpretation is supported by the fact that scale 14 was, in fact, an overall preference rating and had a high loading with the first factor. The remaining three factors really contain no loadings over 0.40 and are difficult to interpret. Such interpretation difficulty is not uncommon and motivates the use of varimax rotation.
Varimax rotation, probably the most widely used rotation scheme, searches for a set of factor loadings such that each factor has some loadings close to zero and some loadings close to -1 or +1. The logic is that interpretation is easiest when the variable-factor correlations are either close to +1 or — 1, indicating a clear association between the variable and the factor; or close to zero, indicating a clear lack of association.4
The right portion of Figure 17-2 shows a varimax rotated solution. Notice that, like the principal components solution, a total of 83.3 percent of the variance is explained by the four factors. Further, the communalities are the same. However, in the varimax rotation each of the four factors explains a substantial amount of the variance, whereas in the principal components solution the first factor explained nearly all of the variance. In this data set the varimax rotation was not successful in pushing some loadings to zero. Thus, the interpretation is still somewhat difficult. However, it is possible to provide an interpretation of the first four factors by considering the variables with the largest factor loadings. The factors might be labeled:
Factor 1 Mellow-comforting Factor 2 Heartiness Factor 3 Genuineness Factor 4 Freshness
The varimax rotation thus leads to quite a different perspective, which, incidentally, is not necessary a more correct perspective. A subjective judgment guided by theory is needed to determine which perspective, the principal components or the varimax rotation, is more valid.
The factor solution presented in Figure 17-1 also was a varimax solution and actually was fairly easy to interpret. Sometimes a varimax solution
(or principal components) will generate seemingly interpretable factors even when, in actuality, there is little structure present. To guard against this eventuality it is prudent to split the data in half when possible and run tad factor analyses. If the same factors emerge in each, then some confidenc-e that the factors actually exist may be warranted.
Factor analysis is complicated somewhat (or made more interesting, depending on your perspective) by the fact that it is possible to generate several factor analysis solutions (loadings and factor scores) for any data set. Each solution is termed a particular factor rotation and is generated by a factor rotation scheme. Each time the factors are rotated the pattern of loadings changes, as does the interpretation of the factors. Geometrically, rotation means simply that the dimensions are rotated. (For a geometric interpretation of factor analysis see the appendix to this chapter.) Although there are many such rotation programs, varimax rotation is the most common and will be described here.
The basic "unrotated" factor analysis usually employs principal components analysis (also termed principal factor analysis) and will be introduced first. The objective of the principal components is to generate a first factor that will have the maximum explained variance. Then with the first factor and its associated loadings fixed, principal components will locate a second factor maximizing the variance explained in this second factor. The procedure continues until there are as many factors generated as variables or until the analyst concludes that the number of useful factors has been exhausted. Determination of the number of factors to include will be considered shortly.
When principal components/principal factor analysis is used, the interpretation of the factors can be difficult. The use of varimax rotation can improve greatly the interpre tab ility. A study of the perceptions of 94 consumers of a particular brand of coffee will illustrate.3 The consumers, after
sampling the coffee, rated it on 14 semantic-differential scales. The ratings were factor analyzed by the principal components method, and the results are shown on the left side of Figure 17-2. The first factor explained nearly 75 percent of the variance and seems clearly to reflect a general like-dislike dimension. This interpretation is supported by the fact that scale 14 was, in fact, an overall preference rating and had a high loading with the first factor. The remaining three factors really contain no loadings over 0.40 and are difficult to interpret. Such interpretation difficulty is not uncommon and motivates the use of varimax rotation.
Varimax rotation, probably the most widely used rotation scheme, searches for a set of factor loadings such that each factor has some loadings close to zero and some loadings close to -1 or +1. The logic is that interpretation is easiest when the variable-factor correlations are either close to +1 or — 1, indicating a clear association between the variable and the factor; or close to zero, indicating a clear lack of association.4
The right portion of Figure 17-2 shows a varimax rotated solution. Notice that, like the principal components solution, a total of 83.3 percent of the variance is explained by the four factors. Further, the communalities are the same. However, in the varimax rotation each of the four factors explains a substantial amount of the variance, whereas in the principal components solution the first factor explained nearly all of the variance. In this data set the varimax rotation was not successful in pushing some loadings to zero. Thus, the interpretation is still somewhat difficult. However, it is possible to provide an interpretation of the first four factors by considering the variables with the largest factor loadings. The factors might be labeled:
Factor 1 Mellow-comforting Factor 2 Heartiness Factor 3 Genuineness Factor 4 Freshness
The varimax rotation thus leads to quite a different perspective, which, incidentally, is not necessary a more correct perspective. A subjective judgment guided by theory is needed to determine which perspective, the principal components or the varimax rotation, is more valid.
The factor solution presented in Figure 17-1 also was a varimax solution and actually was fairly easy to interpret. Sometimes a varimax solution
(or principal components) will generate seemingly interpretable factors even when, in actuality, there is little structure present. To guard against this eventuality it is prudent to split the data in half when possible and run tad factor analyses. If the same factors emerge in each, then some confidenc-e that the factors actually exist may be warranted.
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