promaxQ {fungible}

Conduct an Oblique Promax Rotation

Description

This function is an extension of the promax function. This function will extract the unrotated factor loadings (with three algorithm options, see faX) if they are not provided. The factor intercorrelations (Phi) are also computed within this function.

Usage

promaxQ(
  R = NULL,
  urLoadings = NULL,
  facMethod = "fals",
  numFactors = NULL,
  power = 4,
  standardize = "Kaiser",
  epsilon = 1e-04,
  maxItr = 15000,
  faControl = NULL
)

Arguments

R

(Matrix) A correlation matrix.

urLoadings

(Matrix) An unrotated factor-structure matrix to be rotated.

facMethod

(Character) The method used for factor extraction (faX). The supported options are "fals" for unweighted least squares, "faml" for maximum likelihood, "fapa" for iterated principal axis factoring, "faregLS" for regularized least squares, "faregML" for regularized maximum likelihood, and "pca" for principal components analysis. The default method is "fals".

• "fals": Factors are extracted using the unweighted least squares estimation procedure using the fals function.

• "faml": Factors are extracted using the maximum likelihood estimation procedure using the factanal function.

• "fapa": Factors are extracted using the iterated principal axis factoring estimation procedure using the fapa function.

• "faregLS": Factors are extracted using regularized least squares factor analysis using the fareg function.

• "faregML": Factors are extracted using regularized maximum likelihood factor using the fareg function.

• "pca": Principal components are extracted.

numFactors

(Scalar) The number of factors to extract if the lambda matrix is not provided.

power

(Scalar) The power with which to raise factor loadings for minimizing trivial loadings. The default value is 4.

standardize

(Character) Which standardization routine is applied to the unrotated factor structure. The three options are "none", "Kaiser", and "CM". The default option is "Kaiser" as is recommended by Kaiser and others. See faStandardize for more details.

• "none": Do not rotate the normalized factor structure matrix.

• "Kaiser": Use a factor structure matrix that has been normed by Kaiser's method (i.e., normalize all rows to have a unit length).

• "CM": Use a factor structure matrix that has been normed by the Cureton-Mulaik method.

epsilon

(Scalar) The convergence criterion used for evaluating the varimax rotation. The default value is 1e-4 (i.e., .0001).

maxItr

(Scalar) The maximum number of iterations allowed for computing the varimax rotation. The default value is 15,000 iterations.

faControl

(List) A list of optional parameters passed to the factor extraction (faX) function.

• treatHeywood: (Logical) In fals, if treatHeywood is true, a penalized least squares function is used to bound the communality estimates below 1.0. Defaults to treatHeywood = TRUE.

• nStart: (Numeric) The number of starting values to be tried in faml. Defaults to nStart = 10.

• start: (Matrix) NULL or a matrix of starting values, each column giving an initial set of uniquenesses. Defaults to start = NULL.

• maxCommunality: (Numeric) In faml, set the maximum communality value for the estimated solution. Defaults to maxCommunality = .995.

• epsilon: (Numeric) In fapa, the numeric threshold designating when the algorithm has converged. Defaults to epsilon = 1e-4.

• communality: (Character) The method used to estimate the initial communality values in fapa. Defaults to communality = 'SMC'.

  • "SMC": Initial communalities are estimated by taking the squared multiple correlations of each indicator after regressing the indicator on the remaining variables.

  • "maxr": Initial communalities equal the largest (absolute value) correlation in each column of the correlation matrix.

  • "unity": Initial communalities equal 1.0 for all variables.

• maxItr: (Numeric) In fapa, the maximum number of iterations to reach convergence. Defaults to maxItr = 15,000.

Details

• Varimax Standardization: When conducting the varimax rotation, it is recommended to standardize the factor loadings using Kaiser's normalization (i.e., rescaling the factor indicators [rows] so that the vectors have unit length). The standardization/normalization occurs by pre-multiplying the unrotated factor structure, A, by the inverse of H, where H^2 is a diagonal matrix with the communality estimates on the diagonal. A varimax rotation is then applied to the normalized, unrotated factor structure. Then, the varimax-rotated factor structure is rescaled to its original metric by pre-multiplying the varimax factor structure by H. For details, see Mulaik (2009).

• Oblique Procrustes Rotation of the Varimax Solution: According to Hendrickson & White (1964), an unrestricted (i.e., oblique) Procrustes rotation is applied to the orthogonal varimax solution. Specifically, a target matrix is generated by raising the varimax factor loadings to the user-specified power (typically, power = 4) (must retain the signs of the original factor loadings). This should quickly diminish trivial factor loadings while retaining larger factor loadings. The Procrustes rotation takes the varimax solution and rotates it toward the promax-generated target matrix. For a modern description of this approach, see Mulaik (2009, ch. 12, p. 342-343).

• Choice of a Power: Changing the power in which varimax factor loadings are raised will change the target matrix in the oblique Procrustes rotation. After raising factor loadings to some power, there will be a larger discrepancy between high and low loadings than before (e.g., squaring factor loadings of .6 and .7 yields loadings of .36 and .49 and cubing yields loadings of .216 and .343). Furthermore, increasing the power will increase the number of near-zero loadings, resulting in larger factor intercorrelations. Many (cf. Gorsuch, 1983; Hendrickson & White, 1964; Mulaik, 2009) advocate for raising varimax loadings to the fourth power (the default) but some (e.g., Gorsuch) advocate for trying power = 2 and power = 6 to see if there is an improvement in the simple structure without overly inflating factor correlations.

Value

A list of the following elements are produced:

• loadings: (Matrix) The oblique, promax-rotated, factor-pattern matrix.

• vmaxLoadings: (Matrix) The orthogonal, varimax-rotated, factor-structure matrix used as the input matrix for the promax rotation.

• rotMatrix: (Matrix) The (rescaled) transformation matrix used in an attempt to minimize the Euclidean distance between the varimax loadings and the generated promax target matrix (cf. Hendrickson & White, 1964; Mulaik, 2009, p. 342-343, eqn. 12.44).

• Phi: (Matrix) The factor correlation matrix associated with the promax solution. Phi is found by taking the inverse of the inner product of the (rescaled) rotation matrix (rotMatrix) with itself (i.e., solve(T' T), where T is the (rescaled) rotation matrix).

• vmaxDiscrepancy: (Scalar) The value of the minimized varimax discrepancy function. promax does not have a rotational criterion but the varimax rotation does.

• convergence: (Logical) Whether the varimax rotation congerged.

• Table: (Matrix) The table returned from GPForth from the GPArotation package.

• rotateControl: (List) A list containing (a) the power parameter used, (b) whether the varimax rotation used Kaiser normalization, (c) the varimax epsilon convergence criterion, and (d) the maximum number of iterations specified.

  • power: The power in which the varimax-rotated factor loadings are raised.

  • standardize: Which standardization routine was used.

  • epsilon: The convergence criterion set for the varimax rotation.

  • maxItr: The maximum number of iterations allowed for reaching convergence in the varimax rotation.

Author(s)

• Casey Giordano ([e-mail])

• Niels G. Waller ([e-mail])

References

Gorsuch, R. L. (1983). Factor Analysis, 2nd. Hillsdale, NJ: LEA.

Hendrickson, A. E., & White, P. O. (1964). Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17(1), 65-70.

Mulaik, S. A. (2009). Foundations of Factor Analysis. Chapman and Hall/CRC.

See Also

Other Factor Analysis Routines: BiFAD(), Box26, GenerateBoxData(), Ledermann(), SLi(), SchmidLeiman(), faAlign(), faEKC(), faIB(), faLocalMin(), faMB(), faMain(), faScores(), faSort(), faStandardize(), faX(), fals(), fapa(), fareg(), fsIndeterminacy(), orderFactors(), print.faMB(), print.faMain(), summary.faMB(), summary.faMain()

Examples

## Generate an orthgonal factor model
lambda <- matrix(c(.41, .00, .00,
                   .45, .00, .00,
                   .53, .00, .00,
                   .00, .66, .00,
                   .00, .38, .00,
                   .00, .66, .00,
                   .00, .00, .68,
                   .00, .00, .56,
                   .00, .00, .55),
                 nrow = 9, ncol = 3, byrow = TRUE)

## Model-implied correlation (covariance) matrix
R <- lambda %*% t(lambda)

## Unit diagonal elements
diag(R) <- 1

## Start from just a correlation matrix
Out1 <- promaxQ(R           = R,
                facMethod   = "fals",
                numFactors  = 3,
                power       = 4,
                standardize = "Kaiser")$loadings

## Iterate the promaxQ rotation using the rotate function
Out2 <- faMain(R             = R,
               facMethod     = "fals",
               numFactors    = 3,
               rotate        = "promaxQ",
               rotateControl = list(power       = 4,
                                    standardize = "Kaiser"))$loadings

## Align the factors to have the same orientation
Out1 <- faAlign(F1 = Out2,
                F2 = Out1)$F2

## Show the equivalence of factor solutions from promaxQ and rotate
all.equal(Out1, Out2, check.attributes = FALSE)