model_comparison: Compare Factor Analysis Models
In FAiR: Factor Analysis in R

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

These functions produce the usual model comparison statistics for factor analysis models.

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model_comparison(..., correction = c("swain", "bartlett", "none"), 
                 conf.level = .9, nsim = 1001)
paired_comparison(M_0, M_1)

`...`	objects of `FA-class` produced by `Factanal`
`correction`	character string indicating what correction to use
`conf.level`	confidence interval for the RMSEA statistic
`nsim`	number of simulations for the nonparametric tests, see Details
`M\_0`	object of `FA-class` produced by `Factanal` that nests `M_1`
`M\_1`	object of `FA-class` produced by `Factanal` that is nested within `M_0`

For exactly two nested models, paired_comparison performs the simple version of the test recommened in Satorra and Bentler (2000); however, it is up to the user to verify that M_1 is nested within M_0.

Any number of objects of FA-class that are produced by Factanal can be passed to model_comparison and a wide variety of statistic tests and fit indices will be calculated. The exact behavior heavily depends on how the model was estimated and in the case of traditional maximum likelihood estimation also depends on the correction argument.

If correction = "swain" (the default), the maximum likelihood test statistic is scaled by one of the correction factors in Swain (1975) that has been recommended in Herzog, Boomsma, and Reinecke (2007) and in Herzog and Boomsma (forthcoming) and is based on http://www.ppsw.rug.nl/~boomsma/swain.R Users should refer to these works for details, simulation results, and in publications making use of this Swain correction. If correction = "bartlett", the correction factor recommended in Bartlett (1950), which is only strictly appropriate for exploratory factor analysis and has been implemented in factanal for a long time. If correction = "none", then no correction factor is utilized, which is also the behavior for models that do not use the traditional maximum likelihood discrepancy function. If the ADF discrepancy function is used (or one of its special cases), the primary test statistic is that advocated in Yuan and Bentler (1998) but the test in equation 2.20b of Browne (1984) is also calculated.

The (primary) test statistic is then used in the root mean squared error of approximation (RMSEA) (see Steiger and Lind 1980) to conduct a test of “close fit”, namely that the true RMSEA is less than 0.05. Confidence intervals are also reported and depend on the value of conf.level. The RMSEA is in turn used to calculate Steiger's (1989) gamma index. In the maximum likelihood case, both of these are affected by the correction factor.

If the traditional maximum likelihood discrepancy function is used, then the BIC and SIC (Stochastic Information Criterion, see Preacher 2006 and Preacher, Cai, and MacCallum 2007) are calculated. These information criteria can be used to compare nonnested models and in both cases, smaller is better.

Finally, several model comparison statistics are calculated, largely based on the summary.sem function in the sem package. Most of these statistics are discussed in Bollen (1989). These are

List element	Reference
`McDonald`	McDonald's (1989) Centrality Index
`GFI`	J<c3><b6>reskog's and Sorb<c3><b6>m's (1981) Goodness of Fit Index
`AGFI`	J<c3><b6>reskog's and Sorb<c3><b6>m's (1981) Adjusted Goodness of Fit Index
`SRMR`	Bentler's (1995) Standardized Root Mean-squared Residual
`TLI`	Tucker and Lewis (1973) Index
`CFI`	Bentler's (1995) Comparative Fit Index
`NFI`	Bentler and Bonett's (1980) Normalized Fit Index
`NNFI`	Bentler and Bonett's (1980) Nonnormalized Fit Index

paired_comparison produces an object of S3 class "htest"; model_comparison produces a list with the following elements:

`restrictions`	the `restrictions` object for the model
`exact\_fit`	a list of one or more objects, usually of S3 class "htest", indicating the results of the associated test(s) of exact fit
`close\_fit`	a list of one or more objects, usually of S3 class "htest", indicating the results of the associated test or measure of “close” fit
`fit\_indices`	a list of numeric fit indices
`infocriteria`	in the case of maximum likelihood estimation, a list of the information criteria that were calculated

If Yates' weighted least squares discrepancy function is used, the test statistic is not strictly valid.

Ben Goodrich with major contributions by Anne Boomsma, John Fox, and Walter Herzog

Bentler, P.M. (1995), EQS structural equations program manual. Encino, CA: Multivariate Software.

Bentler, P.M., & Bonett, D.G. (1980), “Significance tests and goodness of fit in the analysis of covariance structures”. Psychological Bulletin, 88, 588–606.

Browne, M.W. (1984), “Asymptotically distribution-free methods for the analysis of covariance structures”, British Journal of Mathematical and Statistical Psychology, 37, 62–83.

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

Herzog, W., and Boomsma, A. (forthcoming), “Small-Sample Robust Estimators of Noncentrality-Based and Incremental Model Fit” Structural Equation Modeling.

Herzog, W., Boomsma, A., and Reinecke, S. (2007), “The model-size effect on traditional and modified tests of covariance structures”. Structural Equation Modeling, 14, 361–390.

Hotelling, H. (1931), “The generalization of Student's ratio”, Annals of Mathematical Statistics, 2, 360–378.

J<c3><b6>reskog, K. G., and Sorb<c3><b6>m, D. (1981). LISREL V: Analysis of linear structural relations by the method of maximum likelihood. Chicago: Internationa Educational Services.

McDonald, R.P. (1989), “An index of goodness-of-fit based on noncentrality”, Journal of Classification, 6, 97–103.

Preacher, K.J. (2006), “Quantifying Parsimony in Structural Equation Modeling”, Multivariate Behavioral Research 41, 227–259.

Preacher, K.J., Cai, L., and MacCallum, R.C. (2007), “Alternatives to traditional model comparison strategies for covariance structure models.” in Modeling Contextual Effects in Longitudinal Studies, eds. Little, T.D., Bovaird, J.A., and Card, N.A. Psychology Press.

Satorra, A and Bentler, P.M. (2001), “A scaled difference chi-square test statistic for moment structure analysis,” Psychometrika, 66, 507–514.

Steiger, J.H. and Lind, J.C. (1980), “Statistically based tests for the number of common factors” Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.

Steiger, J.H. (1989), EzPATH: A supplementary module for SYSTAT and SYGRAPH. Evanston, IL: SYSTAT.

Swain, A.J. (1975). Analysis of parametric structures for variance matrices. Unpublished doctoral dissertation, Department of Statistics, University of Adelaide, Australia.

Tucker, L. R, and Lewis, C. (1973), “A reliability coefficient for maximum likelihood factor analysis”. Psychometrika, 38, 1–10.

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