Mplus Users Guide v6.. - Muthén & Muthén

Examples: Multilevel Modeling With Complex Survey DataIn this example, the model with two within factors and two betweenfactors shown in the picture above is estimated. The within-level factorindicators are categorical. In the within part of the model, the filledcircles at the end of the arrows from the within factor fw1 to u1, u2, andu3 and fw2 to u4, u5, and u6 represent random intercepts that arereferred to as u1, u2, u3, u4, u5, and u6 in the between part of the model.In the between part of the model, the random intercepts are shown incircles because they are continuous latent variables that vary acrossclusters. The random intercepts are indicators of the between factor fb.This example illustrates the common finding of fewer between factorsthan within factors for the same set of factor indicators. The betweenfactor f has observed cluster-level continuous variables as factorindicators.By specifying ESTIMATOR=WLSMV, a robust weighted least squaresestimator using a diagonal weight matrix will be used. The defaultestimator for this type of analysis is maximum likelihood with robuststandard errors using a numerical integration algorithm. Note thatnumerical integration becomes increasingly more computationallydemanding as the number of factors and the sample size increase. In thisexample, three dimensions of integration would be used with a total of3,375 integration points. For models with many dimensions ofintegration and categorical outcomes, the weighted least squaresestimator may improve computational speed. The ESTIMATOR optionof the ANALYSIS command can be used to select a different estimator.In the within part of the model, the first BY statement specifies that fw1is measured by u1, u2, and u3. The second BY statement specifies thatfw2 is measured by u4, u5, and u6. The metric of the factors are setautomatically by the program by fixing the first factor loading for eachfactor to one. This option can be overridden. Residual variances of thelatent response variables of the categorical factor indicators are notparameters in the model. They are fixed at one in line with the Thetaparameterization. Residuals are not correlated as the default. The ONstatement describes the linear regressions of fw1 and fw2 on theindividual-level covariates x1 and x2. The residual variances of thefactors are estimated as the default. The residuals of the factors arecorrelated as the default because residuals are correlated for latentvariables that do not influence any other variable in the model excepttheir own indicators. The intercepts of the factors are fixed at zero asthe default.261