faIB  R Documentation 
This function conducts maximum likelihood interbattery factor analysis using procedures described by Browne (1979). The unrotated solution can be rotated (using the GPArotation package) from a userspecified 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.
faIB(
X = NULL,
R = NULL,
n = NULL,
NVarX = 4,
numFactors = 2,
itemSort = FALSE,
rotate = "oblimin",
bootstrapSE = FALSE,
numBoot = 1000,
CILevel = 0.95,
rotateControl = NULL,
Seed = 1
)
X 
(Matrix) A raw data matrix (or data frame) structured in a subject
(row) by variable (column) format. Defaults to 
R 
(Matrix) A correlation matrix. Defaults to 
n 
(Numeric) Sample size associated with either the raw data (X) or
the correlation matrix (R). Defaults to 
NVarX 
(Integer) Given batteries X and Y, 
numFactors 
(Numeric) The number of factors to extract for subsequent
rotation. Defaults to 
itemSort 
(Logical) if 
rotate 
(Character) Designate which rotation algorithm to apply. The following are available rotation options: "oblimin", "quartimin", "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. 
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. 
rotateControl 
(List) A list of control values to pass to the factor rotation algorithms.

Seed 
(Integer) Starting seed for the random number generator. 
Global Minimum: This function uses several random starting configurations for factor rotations in an attempt to find the global minimum solution. However, this function is not guaranteed to find the global minimum. Furthermore, the global minimum solution need not be more psychologically interpretable than any of the local solutions (cf. Rozeboom, 1992). As is recommended, our function returns all local solutions so users can make their own judgements.
Finding clusters of local minima: We find localsolution sets by sorting the rounded
rotation complexity values (to the number of digits specified in the epsilon
argument of the rotateControl
list) into sets with equivalent values. For example,
by default epsilon = 1e5.
and thus will only evaluate the complexity
values to five significant digits. Any differences beyond that value will not effect the final sorting.
The faIB
function will produce abundant output in addition
to the rotated interbattery factor pattern and factor correlation matrices.
loadings: (Matrix) The rotated interbattery factor solution with the lowest evaluated discrepancy function. This solution has the lowest discrepancy function of the examined random starting configurations. It is not guaranteed to find the "true" global minimum. Note that multiple (or even all) local solutions can have the same discrepancy functions.
Phi: (Matrix) The factor correlations of the rotated factor solution with the lowest evaluated discrepancy function (see Details).
fit: (Vector) A vector containing the following fit statistics:
chiSq: Chisquare goodness of fit value (see Browne, 1979, for details). Note that we apply Lawley's (1959) correction when computing the chisquare value.
DF: Degrees of freedom for the estimated model.
pvalue: Pvalue associated with the above chisquare statistic.
MAD: Mean absolute difference between the modelimplied and the sample acrossbattery correlation matrices. A lower value indicates better fit.
AIC: Akaike's Information Criterion where a lower value indicates better fit.
BIC: Bayesian Information Criterion where a lower value indicates better fit.
R: (Matrix) Returns the (possibly sorted) correlation matrix, useful when raw data are supplied.
If itemSort = TRUE
then the returned matrix is sorted to be consistent with the factor loading matrix.
Rhat: (Matrix) The (possibly sorted) reproduced correlation matrix.If itemSort = TRUE
then the returned matrix is sorted to be consistent with the factor loading matrix.
Resid: (Matrix) A (possibly sorted) residual matrix (R  Rhat) for the between battery correlations.
facIndeterminacy: (Vector) A vector (with length equal to the number of factors) containing Guttman's (1955) index of factor indeterminacy for each factor.
localSolutions: (List) A list containing all local solutions in ascending order of their factor loadings, rotation complexity values (i.e., the first solution is the "global" minimum). Each solution returns the
loadings: (Matrix) the factor loadings,
Phi: (Matrix) factor correlations,
RotationComplexityValue: (Numeric) the complexity value of the rotation algorithm,
facIndeterminacy: (Vector) A vector of factor indeterminacy indices for each common factor, and
RotationConverged: (Logical) convergence status of the rotation algorithm.
numLocalSets (Numeric) How many sets of local solutions with the same discrepancy value were obtained.
localSolutionSets: (List) A list containing the sets of unique local minima solutions. There is one list element for every unique local solution that includes (a) the factor loadings matrix, (b) the factor correlation matrix (if estimated), and (c) the discrepancy value of the rotation algorithm.
rotate (Character) The chosen rotation algorithm.
rotateControl: (List) A list of the control parameters passed to the rotation algorithm.
unSpunSolution: (List) A list of output parameters (e.g., loadings, Phi, etc) from the rotated solution that was obtained by rotating directly from the unrotated (i.e., unspun) common factor orientation.
Call: (call) A copy of the function call.
Niels G. Waller (nwaller@umn.edu)
Casey Giordano (Giord023@umn.edu)
Boruch, R. F., Larkin, J. D., Wolins, L., & MacKinney, A. C. (1970). Alternative methods of analysis: Multitraitmultimethod data. Educational and Psychological Measurement, 30(4), 833–853. https://doi.org/10.1177/0013164470030004055
Browne, M. W. (1979). The maximumlikelihood solution in interbattery factor analysis. British Journal of Mathematical and Statistical Psychology, 32(1), 7586.
Browne, M. W. (1980). Factor analysis of multiple batteries by maximum likelihood. British Journal of Mathematical and Statistical Psychology, 33(2), 184199.
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111150.
Burnham, K. P. & Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological methods and research, 33, 261304.
Cudeck, R. (1982). Methods for estimating betweenbattery factors, Multivariate Behavioral Research, 17(1), 4768. 10.1207/s15327906mbr1701_3
Cureton, E. E., & Mulaik, S. A. (1975). The weighted varimax rotation and the promax rotation. Psychometrika, 40(2), 183195.
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), 6581.
Tucker, L. R. (1958). An interbattery method of factor analysis. Psychometrika, 23(2), 111136.
Other Factor Analysis Routines:
BiFAD()
,
Box26
,
GenerateBoxData()
,
Ledermann()
,
SLi()
,
SchmidLeiman()
,
faAlign()
,
faEKC()
,
faLocalMin()
,
faMB()
,
faMain()
,
faScores()
,
faSort()
,
faStandardize()
,
faX()
,
fals()
,
fapa()
,
fareg()
,
fsIndeterminacy()
,
orderFactors()
,
print.faMB()
,
print.faMain()
,
promaxQ()
,
summary.faMB()
,
summary.faMain()
# Example 1:
# Example from: Browne, M. W. (1979).
#
# Data originally reported in:
# Thurstone, L. L. & Thurstone, T. G. (1941). Factorial studies
# of intelligence. Psychometric Monograph (2), Chicago: Univ.
# Chicago Press.
R.XY < matrix(c(
1.00, .554, .227, .189, .461, .506, .408, .280, .241,
.554, 1.00, .296, .219, .479, .530, .425, .311, .311,
.227, .296, 1.00, .769, .237, .243, .304, .718, .730,
.189, .219, .769, 1.00, .212, .226, .291, .681, .661,
.461, .479, .237, .212, 1.00, .520, .514, .313, .245,
.506, .530, .243, .226, .520, 1.00, .473, .348, .290,
.408, .425, .304, .291, .514, .473, 1.00, .374, .306,
.280, .311, .718, .681, .313, .348, .374, 1.00, .672,
.241, .311, .730, .661, .245, .290, .306, .672, 1.00), 9, 9)
dimnames(R.XY) < list(c( paste0("X", 1:4),
paste0("Y", 1:5)),
c( paste0("X", 1:4),
paste0("Y", 1:5)))
out < faIB(R = R.XY,
n = 710,
NVarX = 4,
numFactors = 2,
itemSort = FALSE,
rotate = "oblimin",
rotateControl = list(standardize = "Kaiser",
numberStarts = 10),
Seed = 1)
# Compare with Browne 1979 Table 2.
print(round(out$loadings, 2))
cat("\n\n")
print(round(out$Phi,2))
cat("\n\n MAD = ", round(out$fit["MAD"], 2),"\n\n")
print( round(out$facIndeterminacy,2) )
# Example 2:
## Correlation values taken from Boruch et al.(1970) Table 2 (p. 838)
## See also, Cudeck (1982) Table 1 (p. 59)
corValues < c(
1.0,
.11, 1.0,
.61, .47, 1.0,
.42, .02, .18, 1.0,
.75, .33, .58, .44, 1.0,
.82, .01, .52, .33, .68, 1.0,
.77, .32, .64, .37, .80, .65, 1.0,
.15, .02, .04, .08, .12, .11, .13, 1.0,
.04, .22, .26, .06, .07, .10, .07, .09, 1.0,
.13, .21, .23, .05, .07, .06, .12, .64, .40, 1.0,
.01, .04, .01, .16, .05, .07, .05, .41, .10, .29, 1.0,
.27, .13, .18, .17, .27, .27, .27, .68, .18, .47, .33, 1.0,
.24, .02, .12, .12, .16, .23, .18, .82, .08, .55, .35, .76, 1.0,
.20, .18, .16, .17, .22, .11, .29, .69, .20, .54, .34, .68, .68, 1.0)
## Generate empty correlation matrix
BoruchCorr < matrix(0, nrow = 14, ncol = 14)
## Add uppertriangle correlations
BoruchCorr[upper.tri(BoruchCorr, diag = TRUE)] < corValues
BoruchCorr < BoruchCorr + t(BoruchCorr)  diag(14)
## Add variable names to the correlation matrix
varNames < c("Consideration", "Structure", "Sup.Satisfaction",
"Job.Satisfaction", "Gen.Effectiveness", "Hum.Relations", "Leadership")
## Distinguish between rater X and rater Y
varNames < paste0(c(rep("X.", 7), rep("Y.", 7)), varNames)
## Add row/col names to correlation matrix
dimnames(BoruchCorr) < list(varNames, varNames)
## Estimate a model with one, two, and three factors
for (jFactors in 1:3) {
tempOutput < faIB(R = BoruchCorr,
n = 111,
NVarX = 7,
numFactors = jFactors,
rotate = "oblimin",
rotateControl = list(standardize = "Kaiser",
numberStarts = 100))
cat("\nNumber of interbattery factors:", jFactors,"\n")
print( round(tempOutput$fit,2) )
} # END for (jFactors in 1:3)
## Compare output with Cudeck (1982) Table 2 (p. 60)
BoruchOutput <
faIB(R = BoruchCorr,
n = 111,
NVarX = 7,
numFactors = 2,
rotate = "oblimin",
rotateControl = list(standardize = "Kaiser"))
## Print the interbattery factor loadings
print(round(BoruchOutput$loadings, 3))
print(round(BoruchOutput$Phi, 3))
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