Mplus  
Monday January 20, 2020 



ChiSquare Difference Testing Using the SatorraBentler Scaled ChiSquare Chisquare testing for continuous nonnormal outcomes has been discussed in a series of papers by Satorra and Bentler. A popular test statistic is the SatorraBentler scaled (meanadjusted) chisquare, where the usual normaltheory chisquare statistic is divided by a scaling correction to better approximate chisquare under nonnormality. A littleknown fact, however, is that such a scaled chisquare cannot be used for chisquare difference testing of nested models because a difference between two scaled chisquares for nested models is not distributed as chisquare. Mplus issues a warning about this. In discussions with Albert Satorra, Bengt suggested that Albert might want to figure out how to get a chisquare difference test for the SatorraBentler scaled chisquare and he did, producing the following book chapter which can be downloaded as a working paper (in postscript format). Satorra, A. (2000). Scaled and adjusted restricted tests in multisample analysis of moment structures. In Heijmans, R.D.H., Pollock, D.S.G. & Satorra, A. (eds.), Innovations in multivariate statistical analysis. A Festschrift for Heinz Neudecker (pp.233247). London: Kluwer Academic Publishers. The formulas in the paper are, however, complex and subsequently Albert and Peter Bentler wrote a paper showing that simple hand calculations using output from nested runs can give the desired chisquare difference test of nested models using the scaled chisquare. This paper is available here: Satorra, A. & Bentler, P.M. (2010). Ensuring positiveness of the scaled difference chisquare test statistic. Psychometrika 75: 243. doi:10.1007/s113360099135y Difference Testing Using ChiSquare
Following are the steps needed to compute a chisquare difference test in Mplus using the MLM (SatorraBentler), MLR, and WLSM chisquare. DIFFTEST should be used for MLMV and WLSMV. The nested model is the more restrictive model with more degrees of freedom than the comparison model.
Difference Testing Using the Loglikelihood Following are the steps needed to compute a chisquare difference test based on loglikelihood values and scaling correction factors obtained with the MLR estimator.
Computing the Strictly Positive SatorraBentler ChiSquare Difference Test The robust chisquare difference test can sometimes produce a negative value. An alternative approach that avoids this is given in Satorra, A., & Bentler, P.M. (2010). Ensuring positiveness of the scaled difference chisquare test statistic. Psychometrika, 75, 243248. Mplus Web Note No. 12 shows how to compute this new alternative test. For an application article, see Bryant and Satorra (2011). 