# faMB: Multiple Battery Factor Analysis by Maximum Likelihood... In fungible: Psychometric Functions from the Waller Lab

## Description

`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 `faMain` function, 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).

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```faMB( X = NULL, R = NULL, n = NULL, NB = NULL, NVB = NULL, numFactors = NULL, epsilon = 1e-06, rotate = "oblimin", rotateControl = NULL, PrintLevel = 0, Seed = 1 ) ```

## Arguments

 `X` (Matrix) A raw data matrix (or data frame) structured in a subject (row) by variable (column) format. Defaults to `X = NULL`. `R` (Matrix) A correlation matrix. Defaults to `R = NULL`. `n` (Numeric) Sample size associated with either the raw data (X) or the correlation matrix (R). Defaults to `n = NULL`. `NB` (Numeric) The number of batteries to analyze. In interbattery factor analysis NB = 2. `NVB` (Vector) The number of variables in each battery. For example, analyzing three batteries including seven, four, and five variables (respectively) would be specified as `NVB = c(7, 4, 5)`. `numFactors` (Numeric) The number of factors to extract for subsequent rotation. Defaults to `numFactors = NULL`. `epsilon` (Numeric) The convergence threshold for the Gauss-Seidel iterator when analyzing three or more batteries. Defaults to `epsilon = 1e-06`. `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. `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. `PrintLevel` (Numeric) When a value greater than zero is specified, `PrintLevel` prints the maximum change in communality estimates for each iteration of the Gauss-Seidel function. Note that Gauss-Seidel iteration is only called when three or more batteries are analyzed. Defaults to `PrintLevel = 0`. `Seed` (Integer) Starting seed for the random number generator. Defaults to `Seed = 1`.

## Value

The `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:

• 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.

## Author(s)

• Niels G. Waller (nwaller@umn.edu)

• Casey Giordano (Giord023@umn.edu)

## References

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