faX {fungible}

Factor Extraction (faX) Routines

Description

This function can be used to extract an unrotated factor structure matrix using the following algorithms: (a) unweighted least squares ("fals"); (b) maximum likelihood ("faml"); (c) iterated principal axis factoring ("fapa"); and (d) principal components analysis ("pca").

Usage

faX(R, n = NULL, numFactors = NULL, facMethod = "fals", faControl = NULL)

Arguments

R

(Matrix) A correlation matrix used for factor extraction.

n

(Numeric) Sample size associated with the correlation matrix. Defaults to n = NULL.

numFactors

(Numeric) The number of factors to extract for subsequent rotation.

facMethod

(Character) The method used for factor extraction. The supported options are "fals" for unweighted least squares, "faml" for maximum likelihood, "fapa" for iterated principal axis factoring, 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.

• "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.

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

• "pca": Principal components are extracted.

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

• Initial communality estimate: According to Widaman and Herringer (1985), the initial communality estimate does not have much bearing on the resulting solution when the a stringent convergence criterion is used. In their analyses, a convergence criterion of .001 (i.e., slightly less stringent than the default of 1e-4) is sufficiently stringent to produce virtually identical communality estimates irrespective of the initial estimate used. It should be noted that all four methods for estimating the initial communality in Widaman and Herringer (1985) are the exact same used in this function. Based on their findings, it is not recommended to use a convergence criterion lower than 1e-3.

Value

This function returns a list of output relating to the extracted factor loadings.

• loadings: (Matrix) An unrotated factor structure matrix.

• h2: (Vector) Vector of final communality estimates.

• faFit: (List) A list of additional factor extraction output.

  • facMethod: (Character) The factor extraction routine.

  • df: (Numeric) Degrees of Freedom from the maximum likelihood factor extraction routine.

  • n: (Numeric) Sample size associated with the correlation matrix.

  • objectiveFunc: (Numeric) The evaluated objective function for the maximum likelihood factor extraction routine.

  • RMSEA: (Numeric) Root mean squared error of approximation from Steiger & Lind (1980). Note that bias correction is computed if the sample size is provided.

  • testStat: (Numeric) The significance test statistic for the maximum likelihood procedure. Cannot be computed unless a sample size is provided.

  • pValue: (Numeric) The p value associated with the significance test statistic for the maximum likelihood procedure. Cannot be computed unless a sample size is provided.

  • gradient: (Matrix) The solution gradient for the least squares factor extraction routine.

  • maxAbsGradient: (Numeric) The maximum absolute value of the gradient at the least squares solution.

  • Heywood: (Logical) TRUE if a Heywood case was produced.

  • converged: (Logical) TRUE if the least squares or principal axis factor extraction routine converged.

Author(s)

• Casey Giordano ([e-mail])

• Niels G. Waller ([e-mail])

References

Jung, S. & Takane, Y. (2008). Regularized common factor analysis. New trends in psychometrics, 141-149.

Steiger, J. H., & Lind, J. (1980). Paper presented at the annual meeting of the Psychometric Society. Statistically-based tests for the number of common factors.

Widaman, K. F., & Herringer, L. G. (1985). Iterative least squares estimates of communality: Initial estimate need not affect stabilized value. Psychometrika, 50(4), 469-477.

See Also

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

Examples

## Generate an example factor structure matrix
lambda <- matrix(c(.62, .00, .00,
                   .54, .00, .00,
                   .41, .00, .00,
                   .00, .31, .00,
                   .00, .58, .00,
                   .00, .62, .00,
                   .00, .00, .38,
                   .00, .00, .43,
                   .00, .00, .37),
                 nrow = 9, ncol = 3, byrow = TRUE)

## Find the model implied correlation matrix
R <- lambda %*% t(lambda)
diag(R) <- 1

## Extract (principal axis) factors using the factExtract function
Out1 <- faX(R          = R,
            numFactors = 3,
            facMethod  = "fapa",
            faControl  = list(communality = "maxr",
                              epsilon     = 1e-4))

## Extract (least squares) factors using the factExtract function
Out2 <- faX(R          = R,
            numFactors = 3,
            facMethod  = "fals",
            faControl  = list(treatHeywood = TRUE))