Factor Analysis as a Classification Method - Rotating the Factor Structure
We could plot the factor loadings shown above in a scatterplot. In that plot, each variable is represented as a point. In this plot we could rotate the axes in any direction without changing the relative locations of the points to each other; however, the actual coordinates of the points, that is, the factor loadings would of course change. In this example, if you produce the plot it will be evident that if we were to rotate the axes by about 45 degrees we might attain a clear pattern of loadings identifying the work satisfaction items and the home satisfaction items.
Rotational strategies. There are various rotational strategies that have been proposed. The goal of all of these strategies is to obtain a clear pattern of loadings, that is, factors that are somehow clearly marked by high loadings for some variables and low loadings for others. This general pattern is also sometimes referred to as simple structure (a more formalized definition can be found in most standard textbooks). Typical rotational strategies are varimax, quartimax, and equamax; these are described in greater detail in the context of the Rotation dialog.
We have described the idea of the varimax rotation before (see Extracting Principal Components), and it can be applied to this problem as well. As before, we want to find a rotation that maximizes the variance on the new axes; put another way, we want to obtain a pattern of loadings on each factor that is as diverse as possible, lending itself to easier interpretation. Below is the table of rotated factor loadings.
Rotational Strategies in Factor Analysis
There are various rotational strategies that have been proposed. The goal of all of these strategies is to obtain a clear pattern of loadings, that is, factors that are somehow clearly marked by high loadings for some variables and low loadings for others. This general pattern is also sometimes referred to as simple structure (a more formalized definition can be found in most standard textbooks). Typical rotational strategies are varimax, biquartimax, quartimax, and equamax. Some authors (e.g., Catell & Khanna; Harman, 1976; Jennrich & Sampson, 1966; Clarkson & Jennrich, 1988) have discussed in some detail the concept of oblique (non-orthogonal) factors, in order to achieve more interpretable simple structure. Specifically, computer (algorithmic) strategies have been developed to rotate factors so as to best represent "clusters" of variables, without the constraint of orthogonality of factors. However, the oblique factors produced by such rotations are often not easily interpreted. Use the Hierarchical analysis of oblique factors button on the Loadings tab of the Factor Analysis Results dialog instead in order to identify (correlated, oblique) clusters of variables (see also Hierarchical factor analysis).
Note that you can use the Structural Equation Modeling (SEPATH) module to test the adequacy (goodness of fit) of specific orthogonal or oblique factor solutions.
Varimax raw. This selection of the Factor rotation box performs a varimax rotation of the factor loadings. This rotation is aimed at maximizing the variances of the squared raw factor loadings across variables for each factor; this is equivalent to maximizing the variances in the columns of the matrix of the squared raw factor loadings.
Varimax normalized. This selection of the Factor rotation box performs a varimax rotation of the normalized factor loadings (raw factor loadings divided by the square roots of the respective communalities). This rotation is aimed at maximizing the variances of the squared normalized factor loadings across variables for each factor; this is equivalent to maximizing the variances in the columns of the matrix of the squared normalized factor loadings. This is the method that is most commonly used and referred to as varimax rotation.
Biquartimax raw. This selection of the Factor rotation box performs a biquartimax rotation of the raw factor loadings. This rotation can be considered to be an "even mixture" of the varimax and quartimax rotation. Specifically, it is aimed at simultaneously maximizing the sum of variances of the squared raw factor loadings across factors and maximizing the sum of variances of the squared raw factor loadings across variables; this is equivalent to simultaneously maximizing the variances in the rows and columns of the matrix of the squared raw factor loadings.
Biquartimax normalized. This selection is equivalent to the biquartimax raw rotation, except that it is performed on normalized (standardized) factor loadings.
Quartimax raw. This selection of the Factor rotation box performs a quartimax rotation of the (raw) factor loadings. This rotation is aimed at maximizing the variances of (the squared raw) factor loadings across factors for each variable; this is equivalent to maximizing the variances in the rows of the matrix of the squared raw factor loadings.
Quartimax normalized. This selection of the Factor rotation box performs a quartimax rotation of the normalized factor loadings, that is, the raw factor loadings divided by the square roots of the respective communalities. This rotation is aimed at maximizing the variances of the squared normalized factor loadings across factors for each variable; this is equivalent to maximizing the variances in the rows of the matrix of the squared normalized factor loadings. This is the method that is commonly referred to as quartimax rotation.
Equamax raw. This selection of the Factor rotation box performs an equamax rotation of the raw factor loadings. This rotation can be considered to be a "weighted mixture" of the varimax and quartimax rotation. Specifically, it is aimed at simultaneously maximizing the sum of variances of the squared raw factor loadings across factors and maximizing the sum of variances of the squared raw factor loadings across variables; this is equivalent to simultaneously maximizing the variances in the rows and columns of the matrix of the squared raw factor loadings. However, unlike the biquartimax rotation, the relative weight assigned to the varimax criterion in the rotation is equal to the number of factors divided by 2.
Equamax normalized. This selection of the Factor rotation box performs an equamax rotation, as described for Equamax raw; however, this rotation will be performed on the normalized factor loadings.
| STATISTICA FACTOR ANALYSIS | Factor Loadings (Varimax normalized) Extraction: Principal components
| |
| Variable | Factor 1 | Factor 2 |
| WORK_1 | .862443 | .051643 |
| WORK_2 | .890267 | .110351 |
| WORK_3 | .886055 | .152603 |
| HOME_1 | .062145 | .845786 |
| HOME_2 | .107230 | .902913 |
| HOME_3 | .140876 | .869995 |
| Expl.Var | 2.356684 | 2.325629 |
| Prp.Totl | .392781 | .387605 |
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