Mplus Users Guide v6.. - Muthén & Muthén

CHAPTER 14WEIGHTED LEAST SQUARES ESTIMATOR USING THETHETA PARAMETERIZATION1. Thresholds and factor loadings free across groups; residual variancesfixed at one in all groups; factor means fixed at zero in all groups2. Thresholds and factor loadings constrained to be equal acrossgroups; residual variances fixed at one in one group and free in theothers; factor means fixed at zero in one group and free in the others(the Mplus default)MAXIMUM LIKELIHOOD ESTIMATOR WITHCATEGORICAL OUTCOMES1. Thresholds and factor loadings free across groups; factor meansfixed at zero in all groups2. Thresholds and factor loadings constrained to be equal acrossgroups; factor means fixed at zero in one group and free in the others(the Mplus default)PARTIAL MEASUREMENT INVARIANCEWhen full measurement invariance does not hold, partial measurementinvariance can be considered. This involves relaxing some equalityconstraints on the measurement parameters. For continuous outcomes,equality constraints can be relaxed for the intercepts, factor loadings,and residual variances. This is shown in Example 5.15. For categoricaloutcomes, equality constraints for thresholds and factor loadings for avariable should be relaxed in tandem. In addition, for the Deltaparameterization, the scale factor must be fixed at one for that variable.This is shown in Example 5.16. For the Theta parameterization, theresidual variance must be fixed at one for that variable. This is shown inExample 5.17.MODEL DIFFERENCE TESTINGIn chi-square difference testing of measurement invariance, the chisquarevalue and degrees of freedom of the less restrictive model aresubtracted from the chi-square value and degrees of freedom of thenested, more restrictive model. The chi-square difference value iscompared to the chi-square value in a chi-square table using thedifference in degrees of freedom between the more restrictive and less434