Mplus Users Guide v6.. - Muthén & Muthén

CHAPTER 9as y1, y2, y3, and y4 in the between part of the model. In the betweenpart of the model, the random intercepts are shown in circles becausethey are continuous latent variables that vary across classes. The brokenarrows from s to the arrows from a1 to y1, a2 to y2, a3 to y3, and a4 toy4 indicate that the slopes in these regressions are random. The s isshown in a circle in both the within and between parts of the model torepresent a decomposition of the random slope into its within andbetween components.By specifying TYPE=TWOLEVEL RANDOM in the ANALYSIScommand, a multilevel model with random intercepts and random slopeswill be estimated. By specifying ALGORITHM=INTEGRATION, amaximum likelihood estimator with robust standard errors using anumerical integration algorithm will be used. Note that numericalintegration becomes increasingly more computationally demanding asthe number of factors and the sample size increase. In this example, fourdimensions of integration are used with a total of 10,000 integrationpoints. The INTEGRATION option of the ANALYSIS command isused to change the number of integration points per dimension from thedefault of 15 to 10. The ESTIMATOR option of the ANALYSIScommand can be used to select a different estimator.The | symbol is used in conjunction with TYPE=RANDOM to name anddefine the random slope variables in the model. The name on the lefthandside of the | symbol names the random slope variable. Thestatement on the right-hand side of the | symbol defines the random slopevariable. The random slope s is defined by the linear regressions of y1on a1, y2 on a2, y3 on a3, and y4 on a4. Random slopes with the samename are treated as one variable during model estimation. The randomintercepts for these regressions are referred to by using the name of thedependent variables in the regressions, that is, y1, y2, y3, and y4. Theasterisk (*) following the s specifies that s will have variation on boththe within and between levels. Without the asterisk (*), s would havevariation on only the between level. An explanation of the othercommands can be found in Examples 9.1 and 9.12.276