# Inter-Battery Factor Analysis by the Method of Maximum Likelihood This function conducts maximum likelihood inter-battery factor analysis using procedures described by Browne (1979). The unrotated solution can be rotated (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. ```r 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 ) ``` ## 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`. - `NVarX`: (Integer) Given batteries X and Y, `NVarX` denotes the number of variables in battery X. - `numFactors`: (Numeric) The number of factors to extract for subsequent rotation. Defaults to `numFactors = NULL`. - `itemSort`: (Logical) if `itemSort = TRUE` the factor loadings will be sorted within batteries. - `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. * 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. - `Seed`: (Integer) Starting seed for the random number generator. ## Returns The `faIB` function will produce abundant output in addition to the rotated inter-battery factor pattern and factor correlation matrices. * loadings : (Matrix) The rotated inter-battery 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 : Chi-square goodness of fit value (see Browne, 1979, for details). Note that we apply Lawley's (1959) correction when computing the chi-square value. * DF : Degrees of freedom for the estimated model. * p-value : P-value associated with the above chi-square statistic. * MAD : Mean absolute difference between the model-implied and the sample across-battery 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. ## Details * 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 local-solution 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 = 1e-5.` and thus ` ` will only evaluate the complexity values to five significant digits. Any differences beyond that value will not effect the final sorting. ## Examples ```r # 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 upper-triangle 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 inter-battery factors:", jFactors,"\n") print( round(tempOutput$fit,2) ) } # END for (jFactors in 1:3) ## Compare output with Cudeck (1982) Table 2 (p. 60) BoruchOutput <- faIB(R = BoruchCorr, n = 111, NVarX = 7, numFactors = 2, rotate = "oblimin", rotateControl = list(standardize = "Kaiser")) ## Print the inter-battery factor loadings print(round(BoruchOutput$loadings, 3)) print(round(BoruchOutput$Phi, 3)) ``` ## 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. 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. Tucker, L. R. (1958). An inter-battery method of factor analysis. **Psychometrika, 23**(2), 111-136. ## See Also 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()` ## Author(s) * Niels G. Waller (nwaller@umn.edu) * Casey Giordano (Giord023@umn.edu)

faIB function