Chi | chi square difference test
Chi-squaretestingforcontinuousnon-normaloutcomeshasbeendiscussedinaseriesofpapersbySatorraandBentler.ApopularteststatisticistheSatorra-Bentlerscaled(mean-adjusted)chi-square,wheretheusualnormal-theorychi-squarestatisticisdividedbyascalingcorrectiontobetterapproximatechi-squareundernon-normality.Alittle-knownfact,however,isthatsuchascaledchi-squarecannotbeusedforchi-squaredifferencetestingofnestedmodelsbecauseadifferencebetweentwoscaledchi-squaresfornestedmodelsisnotdistributedaschi-square.Mp...
Chi-square testing for continuous non-normal outcomes has been discussed in a series of papers by Satorra and Bentler. A popular test statistic is the Satorra-Bentler scaled (mean-adjusted) chi-square, where the usual normal-theory chi-square statistic is divided by a scaling correction to better approximate chi-square under non-normality.
A little-known fact, however, is that such a scaled chi-square cannot be used for chi-square difference testing of nested models because a difference between two scaled chi-squares for nested models is not distributed as chi-square. Mplus issues a warning about this.
In discussions with Albert Satorra, Bengt suggested that Albert might want to figure out how to get a chi-square difference test for the Satorra-Bentler scaled chi-square and he did, producing the following book chapter which can be downloaded as a working paper[1] (in postscript format).
Satorra, A. (2000). Scaled and adjusted restricted tests in m...