Mplus Users Guide v6.. - Muthén & Muthén

Special Modeling Issuesfactor analysis, as few as three integration points per dimension may besufficient.PRACTICAL ASPECTS OF NUMERICAL INTEGRATIONFollowing is a list of suggestions for using numerical integration:• Start with a model that has a small number of latent variables,random slopes, or latent variable interactions for which numericalintegration is required and add to this number in small increments• Start with an analysis using the TECH8 and TECH1 options of theOUTPUT command in conjunction with the MITERATIONS andSTARTS options of the ANALYSIS command set to 1 and 0,respectively, to obtain information on the time required for oneiteration and to check that the model specifications are correct• With more than 3 dimensions of integration, reduce the number ofintegration points per dimension to 10 or use Monte Carlointegration with the default of 500 total integration points• If the TECH8 output shows large negative values in the columnlabeled ABS CHANGE, increase the number of integration points toimprove the precision of the numerical integration and resolveconvergence problems• Because non-identification based on a singular information matrixmay be difficult to determine when numerical integration isinvolved, it is important to check for a low condition number whichmay indicate non-identification, for example, a condition numberless than 1.0E-6MULTIPLE GROUP ANALYSISIn this section, special issues related to multiple group or multiplepopulation analysis are discussed. Multiple group analysis is used whendata from more than one population are being examined to investigatemeasurement invariance and population heterogeneity. Measurementinvariance is investigated by testing the invariance of measurementparameters across groups. Measurement parameters include interceptsor thresholds of the factor indicators, factor loadings, and residualvariances of the factor indicators. Population heterogeneity isinvestigated by testing the invariance of structural parameters acrossgroups. Structural parameters include factor means, variances, and421