Simplicity Functions for Rotations

The FACTOR Procedure

Simplicity Functions for Rotations

To rotate a factor pattern is to apply a non-singular linear transformation to the unrotated factor pattern matrix. To arrive at an optimal transformation you must define a so-called simplicity function for assessing the optimal point. For the promax or the Procrustean transformation, the simplicity function is defined as the sum of squared differences between the rotated factor pattern and the target matrix. Thus, the solution of the optimal transformation is easily obtained by the familiar least-squares method.

For the class of the generalized Crawford-Ferguson family (Jennrich 1973), the simplicity function being optimized is

f=k₁Z+k₂H+k₃V+k₄Q

where

$Z=(\sum_{j}\sum_{i}b_{ij}^2)^2, \ H=\sum_{i}(\sum_{j}b_{ij}^2)^2$

$V=\sum_{j}(\sum_{i}b_{ij}^2)^2, \ Q=\sum_{j}\sum_{i}b_{ij}^4$

k₁, k₂, k₃, and k₄ are constants, and b_ij represents an element of the rotated pattern matrix. Except for specialized research purposes, it is rare in practice to use this simplicity function for rotation. However, it reduces to many well-known classes and special cases of rotations. One of these is the Crawford-Ferguson family (Crawford and Ferguson 1970), which minimizes

f_cf=c₁(H-Q)+c₂(V-Q)

where c₁ and c₂ are constants and (H-Q) represents variable (row) parsimony and (V-Q) represents factor (column) parsimony. Therefore, the relative importance of the variable and the factor parsimony is adjusted via the constants c₁ and c₂. The orthomax class (Carroll, see Harman 1976) maximizes the function

$f_{or}=pQ-\gamma V$

where $\gamma$ is the orthomax weight and is usually between 0 and the number of variables p. The oblimin class minimizes the function

$f_{ob}=p(H-Q)-\tau (Z-V)$

where $\tau$ is the oblimin weight and is usually between 0 and the number of variables p.

All the above definitions are for rotations without row normalization. For rotations with Kaiser normalization the definition of b_ij is replaced by b_ij/h_i, where h_i is the communality of variable i.

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