Computes direct oblique rotation according to a generalized fourth‑degree polynomial criterion.

Routine FGCRF performs direct oblique factor rotation for an arbitrary fourth‑degree polynomial criterion function. Let p = NVAR denote the number of variables, and let k = NF denote the number of factors. Then, the criterion function

is minimized by finding a rotation matrix T such that (λij) = Λ = AT and T−1 (T−1)T is a correlation matrix. Here, ωi = W(i), i = 1, …, 4 are user specified constants. The rotation is said to be direct because it minimizes Q with respect to the factor loadings directly, ignoring the reference structure (see, e.g., Harman, 1976).

Kaiser normalization (Harman, 1976) is specified when option parameter NRM = 1. When Kaiser normalization is performed, the rows of A are first “normalized” by dividing each row by the square root of the sum of its squared elements. The rotation is then performed. The rows of B are then “denormalized” by multiplying each row by the initial row normalizing constant.

The criterion function Q was first proposed by Jennrich (1973). It generalizes the oblimin criterion function and the criterion function proposed by Crawford and Ferguson (1970) to an arbitrary fourth degree criterion. Q is optimized by accumulating simple rotations where a simple rotation is defined to be an optimal factor rotation (with respect to Q) for two columns of Λ, and for which the requirement that T −1 (T −1)T be a correlation matrix is satisfied. FGCRF determines the optimal simple rotation by finding the roots of a cubic polynomial equation. The details are contained in Clarkson and Jennrich (1988).

An iteration is complete after all possible k(k ‑ 1) simple rotations have been performed. When the relative change in Q from one iteration to the next is less than EPS, the algorithm stops. EPS = .0001 is usually sufficient. Alternatively, the algorithm stops when the user specified maximum number of iterations, MAXIT, is reached. MAXIT = 30 is typical.

The parameters in the rotation, ω1, provide for a two‑dimensional family of rotations. When ω1 = ‑γ/p, ω2 = 1, ω3 = γ/p, and ω4 = ‑1, then a direct oblimin rotation with parameter γ is performed. Direct oblimin rotations are also performed by routine FDOBL, which is somewhat faster. For ω1 = 0, ω2 = K1, ω3 = K2, and ω4 = ‑ (K1 + K2) direct Crawford‑Ferguson rotation with parameters K1 and K2 results (see Crawford and Ferguson 1970, or Clarkson and Jennrich 1988). Other values of ω yield other rotations. Common values for ω are as in Table 1.

The example is a continuation of the example in routine FACTR. It involves nine variables. A Crawford‑Ferguson rotation with row normalization and 3 factors is performed.