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Examples: Multilevel Mixture Modeling

395

CHAPTER 10

EXAMPLES: MULTILEVEL

MIXTURE MODELING

Multilevel mixture modeling (Asparouhov & Muthén, 2008a) combines

the multilevel and mixture models by allowing not only the modeling of

multilevel data but also the modeling of subpopulations where

population membership is not known but is inferred from the data.

Mixture modeling can be combined with the multilevel analyses

discussed in Chapter 9. Observed outcome variables can be continuous,

censored, binary, ordered categorical (ordinal), unordered categorical

(nominal), counts, or combinations of these variable types.

With cross-sectional data, the number of levels in Mplus is the same as

the number of levels in conventional multilevel modeling programs.

Mplus allows two-level modeling. With longitudinal data, the number of

levels in Mplus is one less than the number of levels in conventional

multilevel modeling programs because Mplus takes a multivariate

approach to repeated measures analysis. Longitudinal models are two-

level models in conventional multilevel programs, whereas they are one-

level models in Mplus. Single-level longitudinal models are discussed in

Chapter 6, and single-level longitudinal mixture models are discussed in

Chapter 8. Three-level longitudinal analysis where time is the first level,

individual is the second level, and cluster is the third level is handled by

two-level growth modeling in Mplus as discussed in Chapter 9.

Multilevel mixture models can include regression analysis, path analysis,

confirmatory factor analysis (CFA), item response theory (IRT) analysis,

structural equation modeling (SEM), latent class analysis (LCA), latent

transition analysis (LTA), latent class growth analysis (LCGA), growth

mixture modeling (GMM), discrete-time survival analysis, continuous-

time survival analysis, and combinations of these models.

All multilevel mixture models can be estimated using the following

special features:

 Single or multiple group analysis

 Missing data

CHAPTER 10

396

 Complex survey data

 Latent variable interactions and non-linear factor analysis using

maximum likelihood

 Random slopes

 Individually-varying times of observations

 Linear and non-linear parameter constraints

 Maximum likelihood estimation for all outcome types

 Wald chi-square test of parameter equalities

 Analysis with between-level categorical latent variables

 Test of equality of means across latent classes using posterior

probability-based multiple imputations

For TYPE=MIXTURE, multiple group analysis is specified by using the

KNOWNCLASS option of the VARIABLE command. The default is to

estimate the model under missing data theory using all available data.

The LISTWISE option of the DATA command can be used to delete all

observations from the analysis that have missing values on one or more

of the analysis variables. Corrections to the standard errors and chi-

square test of model fit that take into account stratification, non-

independence of observations, and unequal probability of selection are

obtained by using the TYPE=COMPLEX option of the ANALYSIS

command in conjunction with the STRATIFICATION, CLUSTER,

WEIGHT, WTSCALE, BWEIGHT, and BWTSCALE options of the

VARIABLE command. Latent variable interactions are specified by

using the | symbol of the MODEL command in conjunction with the

XWITH option of the MODEL command. Random slopes are specified

by using the | symbol of the MODEL command in conjunction with the

ON option of the MODEL command. Individually-varying times of

observations are specified by using the | symbol of the MODEL

command in conjunction with the AT option of the MODEL command

and the TSCORES option of the VARIABLE command. Linear and

non-linear parameter constraints are specified by using the MODEL

CONSTRAINT command. Maximum likelihood estimation is specified

by using the ESTIMATOR option of the ANALYSIS command. The

MODEL TEST command is used to test linear restrictions on the

parameters in the MODEL and MODEL CONSTRAINT commands

using the Wald chi-square test. Between-level categorical latent

variables are specified using the CLASSES and BETWEEN options of

the VARIABLE command. The AUXILIARY option is used to test the

equality of means across latent classes using posterior probability-based

multiple imputations.

Examples: Multilevel Mixture Modeling

397

Graphical displays of observed data and analysis results can be obtained

using the PLOT command in conjunction with a post-processing

graphics module. The PLOT command provides histograms,

scatterplots, plots of individual observed and estimated values, and plots

of sample and estimated means and proportions/probabilities. These are

available for the total sample, by group, by class, and adjusted for

covariates. The PLOT command includes a display showing a set of

descriptive statistics for each variable. The graphical displays can be

edited and exported as a DIB, EMF, or JPEG file. In addition, the data

for each graphical display can be saved in an external file for use by

another graphics program.

Following is the set of cross-sectional examples included in this chapter:

 10.1: Two-level mixture regression for a continuous dependent

variable*

 10.2: Two-level mixture regression for a continuous dependent

variable with a between-level categorical latent variable*

 10.3: Two-level mixture regression for a continuous dependent

variable with between-level categorical latent class indicators for a

between-level categorical latent variable*

 10.4: Two-level CFA mixture model with continuous factor

indicators*

 10.5: Two-level IRT mixture analysis with binary factor indicators

and a between-level categorical latent variable*

 10.6: Two-level LCA with categorical latent class indicators with

covariates*

 10.7: Two-level LCA with categorical latent class indicators and a

between-level categorical latent variable

Following is the set of longitudinal examples included in this chapter:

 10.8: Two-level growth model for a continuous outcome (three-

level analysis) with a between-level categorical latent variable*

 10.9: Two-level GMM for a continuous outcome (three-level

analysis)*

 10.10: Two-level GMM for a continuous outcome (three-level

analysis) with a between-level categorical latent variable*

 10.11: Two-level LCGA for a three-category outcome*

 10.12: Two-level LTA with a covariate*

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