As with extraction, there are many choices of rotation method. In PROC FACTOR, 24 methods are available. Each method uses a slightly different algorithm to achieve the same broad goal—simplification of the factor structure. In general, the rotation methods fall into two broad categories: orthogonal and oblique (referring to the angle maintained between the X- and Y-axes).

Orthogonal rotations produce factors that are uncorrelated (i.e., maintain a 90° angle between axes), and oblique methods allow the factors to correlate (i.e., allow the X- and Y-axes to assume a different angle than 90°). Traditionally, researchers have been guided to orthogonal rotation because (the argument went) uncorrelated factors are more easily interpretable. In addition, the mathematics is simpler,[3] a fact that made a significant difference during much of the 20th century when EFA was performed by hand calculations or on early computers. Orthogonal rotations (which include quartimax, equamax, and varimax) are generally the default setting in other statistical software. However, in PROC FACTOR, there is no default method of rotation—the researcher must specify the method in order for it to occur.

Let us review some of the more common methods of rotation. The first three methods are orthogonal methods (i.e., produce uncorrelated factors) and the latter two are oblique methods (i.e., produce correlated factors).

Varimax rotation seeks to maximize the variance within a factor (within a column of factor loadings) such that larger loadings are increased and smaller are minimized. In this way, this method simplifies the factors by identifying a set of variables that have high loadings on one and another set that have high loadings on another. However, some variables might not load strongly on any of the factors.

Quartimax tends to focus on rows, maximizing the differences between loadings across factors for a particular variable—increasing high loadings and minimizing small loadings. This method seeks to simplify interpretation at the variable level—it wants each variable to load on one factor, and only one.

Equamax is considered a compromise between varimax and quartimax, in that it seeks to clarify loadings in both directions. It attempts to find a solution where factor loadings clearly distinguish the factors, and the variables load on one primary factor.

Promax is recommended by Thompson (2004) as the more desirable oblique rotation choice.[4] This method starts by actually conducting an orthogonal rotation to clarify the pattern of loadings. It then slowly starts to allow the factors to become correlated through a procrustean rotation (which is less common and not discussed here). The initial method of rotation can be specified through the PREROTATE option. If no option is specified, PROC FACTOR will default to using a varimax rotation.

Direct oblimin rotation is another oblique rotation that can sometimes be problematic but often gives very similar results to promax.

The two oblique methods, promax and oblimin, have parameters that allow the researcher to limit how correlated factors can be. The POWER option raises the factor loadings to a specified power, effectively increasing the correlation of the factors as the power becomes larger. The default power for these methods is 3, but users can adjust the value to any value greater than or equal to 1. A low correlation can be specified as POWER=1, but a high can be specified as POWER=5. Even though researchers can adjust the possible correlation between factors, they cannot force the factors to be correlated if they are not—in other words, you can limit how strongly correlated the factors are, but not the minimum correlation.

Each of the techniques described above is specified on the ROTATE option in the FACTOR procedure. The different rotation methods summarized above can be requested using the following keywords.

rotate = 
   VARIMAX /*Varimax rotation*/
   QUARTIMAX /*Quartimax rotation*/
   EQUAMAX /*Equamax rotation*/
   PROMAX /*Promax rotation*/
   OBLIMIN /*Direct Oblimin rotation*/

Please note that the mathematical algorithms for each rotation discussed above are different, and beyond the scope of this brief introduction. However, the overall goal of each method is the same: simplicity and clarity of factor loadings. For details about how they achieve these goals, see SAS (2015). For a good overview of the technical details of different versions of varimax rotation, see Forina, Armanino, Lanteri, and Leardi (1989).