SchmidLeiman {fungible}

Schmid-Leiman Orthogonalization to a (Rank-Deficient) Bifactor Structure

Description

The Schmid-Leiman (SL) procedure orthogonalizes a higher-order factor structure into a rank-deficient bifactor structure. The Schmid-Leiman method is a generalization of Thomson's orthogonalization routine.

Usage

SchmidLeiman(
  R,
  numFactors,
  facMethod = "fals",
  rotate = "oblimin",
  rescaleH2 = 0.98,
  faControl = NULL,
  rotateControl = NULL
)

Arguments

R

(Matrix) A correlation matrix.

numFactors

(Vector) The number of latent factors at each level of analysis. For example, c(3, 1) estimates three latent factors in the first-order common factor model and one latent factor in the second-order common factor model (i.e., 3 group factors and 1 general factor). This function can orthogonalize up to (and including) a three-order factor solution.

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.

rotate

(Character) Designate which rotation algorithm to apply. See the faMain function for more details about possible rotations. Defaults to rotate = "oblimin".

rescaleH2

(Numeric) If a Heywood case is detected at any level of the higher-order factor analyses, rescale the communality value to continue with the matrix algebra. When a Heywood case occurs, the uniquenesses (i.e., specific-factor variances) will be negative and the SL orthogonalization of the group factors is no longer correct.

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.

rotateControl

(List) A list of control values to pass to the factor rotation algorithms.

• numberStarts: (Numeric) The number of random (orthogonal) starting configurations for the chosen rotation method (e.g., oblimin). The first rotation will always commence from the unrotated factors orientation. Defaults to numberStarts = 10.

• gamma: (Numeric) This is a tuning parameter (between 0 and 1, inclusive) for an oblimin rotation. See the GPArotation library's oblimin documentation for more details. Defaults to gamma = 0 (i.e., a quartimin rotation).

• delta: (Numeric) This is a tuning parameter for the geomin rotation. It adds a small number (default = .01) to the squared factor loadings before computing the geometric means in the discrepancy function.

• kappa: (Numeric) The main parameterization of the Crawford-Ferguson (CF) rotations (i.e., "cfT" and "cfQ" for orthogonal and oblique CF rotation, respectively). Defaults to kappa = 0.

• k: (Numeric) A specific parameter of the simplimax rotation. Defaults to k = the number of observed variables.

• standardize: (Character) The standardization routine used on the unrotated factor structure. The three options are "none", "Kaiser", and "CM". Defaults to standardize = "none".

  • "none": No standardization is applied to the unrotated factor structure.

  • "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: (Numeric) The rotational convergence criterion to use. Defaults to epsilon = 1e-5.

• power: (Numeric) Raise factor loadings the the n-th power in the promaxQ rotation. Defaults to power = 4.

• maxItr: (Numeric) The maximum number of iterations for the rotation algorithm. Defaults to maxItr = 15000.

Details

The obtained Schmid-Leiman (SL) factor structure matrix is rescaled if its communalities differ from those of the original first-order solution (due to the presence of one or more Heywood cases in a solution of any order). Rescaling will produce SL communalities that match those of the original first-order solution.

Value

• L1: (Matrix) The first-order (oblique) factor pattern matrix.

• L2: (Matrix) The second-order (oblique) factor pattern matrix.

• L3: (Matrix, NULL) The third-order (oblique) factor pattern matrix (if applicable).

• Phi1: (Matrix) The first-order factor correlation matrix.

• Phi2: (Matrix) The second-order factor correlation matrix.

• Phi3: (Matrix, NULL) The third-order factor pattern matrix (if applicable).

• U1: (Matrix) The square root of the first-order factor uniquenesses (i.e., factor standard deviations).

• U2: (Matrix) The square root of the second-order factor uniquenesses (i.e., factor standard deviations).

• U3: (Matrix, NULL) The square root of the third-order factor uniquenesses (i.e., factor standard deviations) (if applicable).

• B: (Matrix) The resulting Schmid-Leiman transformation.

• rotateControl: (List) A list of the control parameters passed to the faMain function.

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

• HeywoodFlag(Integer) An integer indicating whether one or more Heywood cases were encountered during estimation.

Author(s)

• Casey Giordano ([e-mail])

• Niels G. Waller ([e-mail])

References

Abad, F. J., Garcia-Garzon, E., Garrido, L. E., & Barrada, J. R. (2017). Iteration of partially specified target matrices: application to the bi-factor case. Multivariate Behavioral Research, 52(4), 416-429.

Giordano, C. & Waller, N. G. (under review). Recovering bifactor models: A comparison of seven methods.

Schmid, J., & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 53-61.

See Also

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

Examples

## Dataset used in Schmid & Leiman (1957) rounded to 2 decimal places
SLdata <-
  matrix(c(1.0, .72, .31, .27, .10, .05, .13, .04, .29, .16, .06, .08,
           .72, 1.0, .35, .30, .11, .06, .15, .04, .33, .18, .07, .08,
           .31, .35, 1.0, .42, .08, .04, .10, .03, .22, .12, .05, .06,
           .27, .30, .42, 1.0, .06, .03, .08, .02, .19, .11, .04, .05,
           .10, .11, .08, .06, 1.0, .32, .13, .04, .11, .06, .02, .03,
           .05, .06, .04, .03, .32, 1.0, .07, .02, .05, .03, .01, .01,
           .13, .15, .10, .08, .13, .07, 1.0, .14, .14, .08, .03, .04,
           .04, .04, .03, .02, .04, .02, .14, 1.0, .04, .02, .01, .01,
           .29, .33, .22, .19, .11, .05, .14, .04, 1.0, .45, .15, .17,
           .16, .18, .12, .11, .06, .03, .08, .02, .45, 1.0, .08, .09,
           .06, .07, .05, .04, .02, .01, .03, .01, .15, .08, 1.0, .42,
           .08, .08, .06, .05, .03, .01, .04, .01, .17, .09, .42, 1.0),
         nrow = 12, ncol = 12, byrow = TRUE)

Out1 <- SchmidLeiman(R          = SLdata,
                     numFactors = c(6, 3, 1))$B

## An orthogonalization of a two-order structure
bifactor <- matrix(c(.46, .57, .00, .00,
                     .48, .61, .00, .00,
                     .61, .58, .00, .00,
                     .46, .00, .55, .00,
                     .51, .00, .62, .00,
                     .46, .00, .55, .00,
                     .47, .00, .00, .48,
                     .50, .00, .00, .50,
                     .49, .00, .00, .49),
                   nrow = 9, ncol = 4, byrow = TRUE)

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

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

Out2 <- SchmidLeiman(R          = R,
                     numFactors = c(3, 1),
                     rotate     = "oblimin")$B