Mplus Users Guide v6.. - Muthén & Muthén

Special Modeling Issuesinvariance. Some disciplines also require invariance of residualvariances. For categorical outcomes, the measurement parameters arethresholds and factor loadings. For the Delta parameterization ofweighted least squares estimation, scale factors can also be considered.For the Theta parameterization of weighted least squares estimation,residual variances can also be considered.MODELS FOR CONTINUOUS OUTCOMESFollowing is a set of models that can be considered for measurementinvariance of continuous outcomes. They are listed from least restrictiveto most restrictive.1. Intercepts, factor loadings, and residual variances free across groups;factor means fixed at zero in all groups2. Factor loadings constrained to be equal across groups; intercepts andresidual variances free; factor means fixed at zero in all groups3. Intercepts and factors loadings constrained to be equal acrossgroups; residual variances free; factor means zero in one group andfree in the others (the Mplus default)4. Intercepts, factor loadings, and residual variances constrained to beequal across groups; factor means fixed at zero in one group and freein the othersMODELS FOR CATEGORICAL OUTCOMESFollowing is a set of models that can be considered for measurementinvariance of categorical outcomes. They are listed from least restrictiveto most restrictive. For categorical outcomes, measurement invariancemodels constrain thresholds and factor loadings in tandem because theitem probability curve is influenced by both parameters.WEIGHTED LEAST SQUARES ESTIMATOR USING THEDELTA PARAMETERIZATION1. Thresholds and factor loadings free across groups; scale factorsfixed at one in all groups; factor means fixed at zero in all groups2. Thresholds and factor loadings constrained to be equal acrossgroups; scale factors fixed at one in one group and free in the others;factor means fixed at zero in one group and free in the others (theMplus default)433