faMain: Automatic Factor Rotation from Random Configurations with...

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/faMain.R

Description

This function conducts factor rotations (using the GPArotation package) from a user-specified number of random (orthogonal) starting configurations. Based on the resulting complexity function value, the function determines the number of local minima and, among these local solutions, will find the "global minimum" (i.e., the minimized complexity value from the finite number of solutions). See Details below for an elaboration on the global minimum. This function can also return bootstrap standard errors of the factor solution.

Usage

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
faMain(
  X = NULL,
  R = NULL,
  n = NULL,
  numFactors = NULL,
  facMethod = "fals",
  urLoadings = NULL,
  rotate = "oblimin",
  targetMatrix = NULL,
  bootstrapSE = FALSE,
  numBoot = 1000,
  CILevel = 0.95,
  Seed = 1,
  digits = NULL,
  faControl = NULL,
  rotateControl = NULL,
  ...
)

Arguments

X

(Matrix) A raw data matrix (or data frame).

R

(Matrix) A correlation matrix.

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 (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.

urLoadings

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

rotate

(Character) Designate which rotation algorithm to apply. The following are available rotation options: "oblimin", "quartimin", "targetT", "targetQ", "oblimax", "entropy", "quartimax", "varimax", "simplimax", "bentlerT", "bentlerQ", "tandemI", "tandemII", "geominT", "geominQ", "cfT", "cfQ", "infomaxT", "infomaxQ", "mccammon", "bifactorT", "bifactorQ", and "none". Defaults to rotate = "oblimin". See GPArotation package for more details. Note that rotations ending in "T" and "Q" represent orthogonal and oblique rotations, respectively.

targetMatrix

(Matrix) This argument serves two functions. First, if a user has requested either a "targetT" or "targetQ' rotation, then the target matrix is used to conduct a fully or partially specified target rotation. In the latter case, freely estimated factor loadings are designated by "NA" values and rotation will be conducted using Browne's (1972a, 1972b, 2001) method for a partially-specified target rotation. Second, if any other rotation option is chosen then all rotated loadings matrices (and assorted output) will be aligned (but not rotated) with the target solution.

bootstrapSE

(Logical) Computes bootstrap standard errors. All bootstrap samples are aligned to the global minimum solution. Defaults to bootstrapSE = FALSE (no standard errors).

numBoot

(Numeric) The number bootstraps. Defaults to numBoot = 1000.

CILevel

(Numeric) The confidence level (between 0 and 1) of the bootstrap confidence interval. Defaults to CILevel = .95.

Seed

(Numeric) Starting seed for reproducible bootstrap results and factor rotations. Defaults to Seed = 1.

digits

(Numeric) Rounds the values to the specified number of decimal places. Defaults to digits = NULL (no rounding).

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.

...

Values to be passed to the cor function.

  • use: (Character) A character string giving a method for computing correlations in the presence of missing values: "everything" (the default), "all.obs", "complete.obs", "na.or.complete", or "pairwise.complete.obs".

  • method: (Character) A character string indicating which correlation coefficient is to be computed: "pearson" (the default), "kendall", or "spearman".

  • na.rm: (Logical) Should missing values be removed (TRUE) or not (FALSE)?

Details

Value

The faMain function will produce a lot of output in addition to the rotated factor pattern matrix and the factor correlations.

Author(s)

References

Browne, M. W. (1972). Oblique rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 25,(1), 207-212.

Browne, M. W. (1972b). Orthogonal rotation to a partially specifed target. British Journal of Statistical Psychology, 25,(1), 115-120.

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111-150.

Cureton, E. E., & Mulaik, S. A. (1975). The weighted varimax rotation and the promax rotation. Psychometrika, 40(2), 183-195.

Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common factor theory. British Journal of Statistical Psychology, 8(2), 65-81.

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

Mansolf, M., & Reise, S. P. (2016). Exploratory bifactor analysis: The Schmid-Leiman orthogonalization and Jennrich-Bentler analytic rotations. Multivariate Behavioral Research, 51(5), 698-717.

Rozeboom, W. W. (1992). The glory of suboptimal factor rotation: Why local minima in analytic optimization of simple structure are more blessing than curse. Multivariate Behavioral Research, 27(4), 585-599.

Zhang, G. (2014). Estimating standard errors in exploratory factor analysis. Multivariate Behavioral Research, 49(4), 339-353.

See Also

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

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
## Example 1

## Generate an orthogonal 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)

## Generate factor correlation matrix
Phi <- matrix(.50, nrow = 3, ncol = 3)
diag(Phi) <- 1

## Model-implied correlation matrix
R <- lambda %*% Phi %*% t(lambda)
diag(R) <- 1

## Load the MASS package to create multivariate normal data
library(MASS)

## Generate raw data to perfectly reproduce R
X <- mvrnorm(Sigma = R, mu = rep(0, nrow(R)), empirical = TRUE, n = 300)

## Not run: 
## Execute 50 promax rotations from a least squares factor extraction
## Compute 100 bootstrap samples to compute standard errors and 
## 80 percent confidence intervals
Out1 <- faMain(X             = X,
               numFactors    = 3,
               facMethod     = "fals",
               rotate        = "promaxQ",
               bootstrapSE   = TRUE,
               numBoot       = 100,
               CILevel       = .80,
               faControl     = list(treatHeywood = TRUE),
               rotateControl = list(numberStarts = 2,  
                                    power        = 4,
                                    standardize  = "Kaiser"),
               digits        = 2)
Out1[c("loadings", "Phi")] 

## End(Not run)

## Example 2

## Load Thurstone's (in)famous box data
data(Thurstone, package = "GPArotation")

## Execute 5 oblimin rotations with Cureton-Mulaik standardization 
Out2 <- faMain(urLoadings    = box26,
               rotate        = "oblimin",
               bootstrapSE   = FALSE,
               rotateControl = list(numberStarts = 5,
                                    standardize  = "CM",
                                    gamma        = 0,
                                    epsilon      = 1e-6),
               digits        = 2)
               
Out2[c("loadings", "Phi")]     

## Example 3

## Factor matrix from Browne 1972
lambda <- matrix(c(.664,  .322, -.075,
                   .688,  .248,  .192,
                   .492,  .304,  .224,
                   .837, -.291,  .037,
                   .705, -.314,  .155,
                   .820, -.377, -.104,
                   .661,  .397,  .077,
                   .457,  .294, -.488,
                   .765,  .428,  .009), 
                 nrow = 9, ncol = 3, byrow = TRUE)   
                 
## Create partially-specified target matrix
Targ <- matrix(c(NA, 0,  NA,
                 NA, 0,  0,
                 NA, 0,  0,
                 NA, NA, NA,
                 NA, NA, 0,
                 NA, NA, NA,
                 .7, NA, NA,
                 0,  NA, NA,
                 .7, NA, NA), 
               nrow = 9, ncol = 3, byrow = TRUE)  
               
## Perform target rotation              
Out3 <- faMain(urLoadings   = lambda,
               rotate       = "targetT",
               targetMatrix = Targ,
               digits       = 3)$loadings
Out3

fungible documentation built on Sept. 29, 2021, 1:06 a.m.

Related to faMain in fungible...