Mixture Modeling

Mixture ModelingChongming YangResearch Support CenterFHSS CollegeMixture of Distributions

Mixture of Distributions

Classification TechniquesLatent Class Analysis (categorical indicators)Latent Profile Analysis (continuous Indicators)Finite Mixture Modeling (multivariate normal variables)

Integrate Classification Models into Other ModelsMixture Factor AnalysisMixture RegressionsMixture Structural Equation ModelingGrowth Mixture ModelingMultilevel Mixture Modeling Disadvantages of Multi-steps PracticeMultistep practiceRun classification model Save membership VariableModel membership variable and other variables DisadvantagesBiases in parameter estimatesBiases in standard errors SignificanceConfidence IntervalsLatent Class Analysis (LCA)SettingLatent trait assumed to be categoricalTrait measured with multiple categorical indicatorsExample: drug addiction, SchizophreniaAimIdentify heterogeneous classes/groups Estimate class probabilitiesIdentify good indicators of classesRelate covariates to Classes Graphic LCA ModelCategorical Indicators u: u1, u2,u3, urCategorical Latent Variable C: C =1, 2, , or K

Probabilistic Model Assumption: Conditional independence of u so that interdependence is explained by C like factor analysis modelAn item probability

Joint Probability of all indicators

LCA ParametersNumber of Classes -1Item Probabilities -1Class Means (Logit)Latent Class Analysis with Covariates

Posterior Probability(membership/classification of cases)

EstimationMaximum Likelihood estimation via Expectation-Maximization algorithmE (expectation) step: compute average posterior probabilities for each class and itemM (maximization) step: estimate class and item parametersIterate EM to maximize the likelihood of the parameters Test against DataO = observed number of response patternsE = model estimated number of response patternsPearson

Chi-square based on likelihood ratio

Determine Number of Classes Substantive theory (parsimonious, interpretable)Predictive validityAuxiliary variables / covariatesStatistical information and testsBayesian Information Criterion (BIC)EntropyTesting K against K-1 ClassesVuong-Lo-Mendell-Rubin likelihood-ratio testBootstrapped likelihood ratio test

Bayesian Information Criterion (BIC)

L = likelihoodh = number of parametersN = sample sizeChoose model with smallest BICBIC Difference > 4 appreciable

Quality of ClassificationTesting K against K-1 Classes Bootstrapped likelihood ratio test LRT = 2[logL(model 1)- logL(model2)], where model 2 is nested in model 1.Bootstrap Steps:Estimate LRT for both modelsUse bootstrapped samples to obtain distributions for LRT of both modelsCompare LRT and get p values

Testing K against K-1 Classes Vuong-Lo-Mendell-Rubin likelihood-ratio test

Determine Quality of IndicatorsGood indicatorsItem response probability is close to 0 or 1 in each classBad indicatorsItem response probability is high in more than one classes, like cross-loading in factor analysisItem response probability is low in all classes like low-loading in factor analysisLCA ExamplesLCALCA with covariatesClass predicts a categorical outcome Save Membership VariableVariable: idvar = id;

Output:Savedata: File = cmmber.txt; Save = cprob; Latent Profile AnalysisFinite Mixture Modeling(multivariate normal variables)Finite = finite number of subgroups/classesVariables are normally distributed in each classMeans differ across classes Variances are the same across Covariances can differ without restrictions or equal with restrictions across classesLatent profile can be special case with covariances fixed at zero. Mixture Factor AnalysisAllow one to examine measurement properties of items in heterogeneous subgroups / classesMeasurement invariance is not required assuming heterogeneityFactor structure can changeSee Mplus outputsFactor Mixture AnalysisParental Control

Parental Acceptance

Feel people in your family understand youFeel you want to leave homeFeel you and your family have fun togetherFeel that your family pay attention to youFeel your parents care about youFeel close to your motherFeel close to your fatherParents let you make your own decisions about the time you must be home on weekend nightsParents let you make your own decisions about the people you hang around withParents let you make your own decisions about what you wearParents let you make your own decisions about which television programs you watchParents let you make your own decisions about which television programs you watchParents let you make your own decisions about what time you go to bed on week nightsParents let you make your own decisions about what you eatTwo dimensions of Parenting

Mixture SEMSee mixture growth modelingMixture Modeling with Known ClassesIdentify hidden classes within known groupsUnder nonrandomized experiments Impose equality constraints on covariates to identify similar classes from known groups Compare classes that differ in covariates