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 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 |
Jöreskog's and Sorböm's (1981) Goodness of Fit Index |
AGFI |
Jöreskog's and Sorböm'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 |
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.
Jöreskog, K. G., and Sorböm, 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.
Factanal