faMB estimates multiple battery factor analysis using maximum
likelihood estimation procedures described by Browne (1979, 1980). Unrotated
multiple battery solutions are rotated (using the GPArotation package)
from a user-specified number of of random (orthogonal) starting configurations.
Based on procedures analogous to those in the
rotation complexity values of all solutions are ordered to determine
the number of local solutions and the "global" minimum solution (i.e., the
minimized rotation complexity value from the finite number of solutions).
faMB( X = NULL, R = NULL, n = NULL, NB = NULL, NVB = NULL, numFactors = NULL, epsilon = 1e-06, rotate = "oblimin", rotateControl = NULL, PrintLevel = 0, Seed = 1 )
(Matrix) A raw data matrix (or data frame) structured in a subject
(row) by variable (column) format. Defaults to
(Matrix) A correlation matrix. Defaults to
(Numeric) Sample size associated with either the raw data (X) or
the correlation matrix (R). Defaults to
(Numeric) The number of batteries to analyze. In interbattery factor analysis NB = 2.
(Vector) The number of variables in each battery. For example,
analyzing three batteries including seven, four, and five variables
(respectively) would be specified as
(Numeric) The number of factors to extract for subsequent
rotation. Defaults to
(Numeric) The convergence threshold for the Gauss-Seidel iterator
when analyzing three or more batteries. Defaults to
(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.
(List) A list of control values to pass to the factor rotation algorithms.
(Numeric) When a value greater than zero is specified,
(Integer) Starting seed for the random number generator.
faMB function will produce abundant output in addition
to the rotated multiple battery factor pattern and factor correlation matrices.
loadings: (Matrix) The (possibly) rotated multiple battery factor solution with the lowest evaluated complexity value 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: Chi-square goodness of fit value.
Note that, as recommended by Browne (1979), we apply Lawley's (1959) correction when computing the chi-square value when
NB = 2.
DF: Degrees of freedom for the estimated model.
pvalue: P-value associated with the above chi-square statistic.
AIC: Akaike's Information Criterion where a lower value indicates better fit.
BIC: Bayesian Information Criterion where a lower value indicates better fit.
RMSEA: Root mean squared error of approximation (Steiger & Lind, 1980).
R: (Matrix) The sample correlation matrix, useful when raw data are supplied.
Rhat: (Matrix) The reproduced correlation matrix with communalities on the diagonal.
Resid: (Matrix) A residual matrix (R - Rhat).
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 (of length equal to the
numberStarts argument within
rotateControl) containing all local solutions
in ascending order of their rotation complexity values (i.e., the first solution
is the "global" minimum). Each solution returns the following:
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) An integer indicating how many sets of local solutions with the same discrepancy value were obtained.
localSolutionSets: (List) A list (of length equal to the
numLocalSets) that contains all local solutions with the same
rotation complexity value. Note that it is not guarenteed that all
solutions with the same complexity values have equivalent factor loading patterns.
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 unspun (i.e., not multiplied by a random orthogonal transformation matrix) common factor orientation.
Call: (call) A copy of the function call.
Niels G. Waller (email@example.com)
Casey Giordano (Giord023@umn.edu)
Boruch, R. F., Larkin, J. D., Wolins, L., & MacKinney, A. C. (1970). Alternative methods of analysis: Multitrait-multimethod data. Educational and Psychological Measurement, 30(4), 833–853. https://doi.org/10.1177/0013164470030004055
Browne, M. W. (1979). The maximum-likelihood solution in inter-battery factor analysis. British Journal of Mathematical and Statistical Psychology, 32(1), 75-86.
Browne, M. W. (1980). Factor analysis of multiple batteries by maximum likelihood. British Journal of Mathematical and Statistical Psychology, 33(2), 184-199.
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111-150.
Browne, M. and Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods and Research, 21(2), 230-258.
Burnham, K. P. & Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological methods and research, 33, 261-304.
Cudeck, R. (1982). Methods for estimating between-battery factors, Multivariate Behavioral Research, 17(1), 47-68. 10.1207/s15327906mbr1701_3
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.
Steiger, J. & Lind, J. (1980). Statistically based tests for the number of common factors. In Annual meeting of the Psychometric Society, Iowa City, IA, volume 758.
Tucker, L. R. (1958). An inter-battery method of factor analysis. Psychometrika, 23(2), 111-136.
Other Factor Analysis Routines:
# These examples reproduce published multiple battery analyses. # ----EXAMPLE 1: 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. ## Load Thurstone & Thurstone's data used by Browne (1979) data(Thurstone41) Example1Output <- faMB(R = Thurstone41, n = 710, NB = 2, NVB = c(4,5), numFactors = 2, rotate = "oblimin", rotateControl = list(standardize = "Kaiser")) summary(Example1Output, PrintLevel = 2) # ----EXAMPLE 2: Browne, M. W. (1980)---- # Data originally reported in: # Jackson, D. N. & Singer, J. E. (1967). Judgments, items and # personality. Journal of Experimental Research in Personality, 20, 70-79. ## Load Jackson and Singer's dataset data(Jackson67) Example2Output <- faMB(R = Jackson67, n = 480, NB = 5, NVB = rep(4,5), numFactors = 4, rotate = "varimax", rotateControl = list(standardize = "Kaiser"), PrintLevel = 1) summary(Example2Output) # ----EXAMPLE 3: Cudeck (1982)---- # Data originally reported by: # Malmi, R. A., Underwood, B. J., & Carroll, J. B. (1979). # The interrelationships among some associative learning tasks. # Bulletin of the Psychonomic Society, 13(3), 121-123. DOI: 10.3758/BF03335032 ## Load Malmi et al.'s dataset data(Malmi79) Example3Output <- faMB(R = Malmi79, n = 97, NB = 3, NVB = c(3, 3, 6), numFactors = 2, rotate = "oblimin", rotateControl = list(standardize = "Kaiser")) summary(Example3Output) # ----Example 4: Cudeck (1982)---- # Data originally reported by: # Boruch, R. F., Larkin, J. D., Wolins, L. and MacKinney, A. C. (1970). # Alternative methods of analysis: Multitrait-multimethod data. Educational # and Psychological Measurement, 30,833-853. ## Load Boruch et al.'s dataset data(Boruch70) Example4Output <- faMB(R = Boruch70, n = 111, NB = 2, NVB = c(7,7), numFactors = 2, rotate = "oblimin", rotateControl = list(standardize = "Kaiser", numberStarts = 100)) summary(Example4Output, digits = 3)
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