model_comparison(..., correction = c("swain", "bartlett", "none"),
conf.level = .9, nsim = 1001)
paired_comparison(M_0, M_1)paired_comparison produces an object of S3 class "htest"; model_comparison
produces a list with the following elements:restrictions object for the modelpaired_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 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 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 McDonald McDonald's (1989) Centrality Index
GFI AGFI 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
}
Bentler, P.M., & Bonett, D.G. (1980),
Browne, M.W. (1984),
Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.
Herzog, W., and Boomsma, A. (forthcoming),
Herzog, W., Boomsma, A., and Reinecke, S. (2007),
Hotelling, H. (1931),
McDonald, R.P. (1989),
Preacher, K.J. (2006),
Preacher, K.J., Cai, L., and MacCallum, R.C. (2007),
Satorra, A and Bentler, P.M. (2001),
Steiger, J.H. and Lind, J.C. (1980),
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),
Factanal