Report

Mixture Modeling Chongming Yang Research Support Center FHSS College Mixture of Distributions Mixture of Distributions Classification Techniques • Latent Class Analysis (categorical indicators) • Latent Profile Analysis (continuous Indicators) • Finite Mixture Modeling (multivariate normal variables) • … Integrate Classification Models into Other Models • • • • • Mixture Factor Analysis Mixture Regressions Mixture Structural Equation Modeling Growth Mixture Modeling Multilevel Mixture Modeling Disadvantages of Multi-steps Practice • Multistep practice – Run classification model – Save membership Variable – Model membership variable and other variables • Disadvantages – Biases in parameter estimates – Biases in standard errors • Significance • Confidence Intervals Latent Class Analysis (LCA) • Setting – Latent trait assumed to be categorical – Trait measured with multiple categorical indicators – Example: drug addiction, Schizophrenia • Aim – Identify heterogeneous classes/groups – Estimate class probabilities – Identify good indicators of classes – Relate covariates to Classes Graphic LCA Model • Categorical Indicators u: u1, u2,u3, …ur • Categorical 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 model • An item probability ( = 1) = ( = )( = 1| = =1 • Joint Probability of all indicators P(u1 , u2 , u3 ...ur ) k P(c k ) P(u k 1 1 | c k ) P(u2 | c k )...P(ur | c k ) LCA Parameters • Number of Classes -1 • Item Probabilities -1 Class Means (Logit) • Probability Scale 1 ( = 1|) = 1 + exp(−) (logistic Regression without any Covariates x) • Logit Scale = [ ( 1 − ) • Mean (highest number of Class) = 0 Latent Class Analysis with Covariates • Covariates are related to Class Probability with multinomial logistic regression P(cik 1| xi ) ck ck x e K e J 1 cj cj x Posterior Probability (membership/classification of cases) P(c k )P(u1 | c k )P(u2 | c k )...P(ur | c k ) P(c k | u1, u2 ,...ur ) P(u1, u2 ,...ur ) Estimation • Maximum Likelihood estimation via • Expectation-Maximization algorithm – E (expectation) step: compute average posterior probabilities for each class and item – M (maximization) step: estimate class and item parameters – Iterate EM to maximize the likelihood of the parameters Test against Data • O = observed number of response patterns • E = model estimated number of response patterns 2 ( o e) 2 • Pearson e • Chi-square based on likelihood ratio 2 LR 2 o log(o / e) Determine Number of Classes • • • • Substantive theory (parsimonious, interpretable) Predictive validity Auxiliary variables / covariates Statistical information and tests – Bayesian Information Criterion (BIC) – Entropy – Testing K against K-1 Classes • Vuong-Lo-Mendell-Rubin likelihood-ratio test • Bootstrapped likelihood ratio test Bayesian Information Criterion (BIC) BIC 2log(L) (h)ln(N) L = likelihood h = number of parameters N = sample size Choose model with smallest BIC BIC Difference > 4 appreciable Quality of Classification • Entropy (− ln ) = 1 − ln – = average of highest class probability of individuals – A value of close to 1 indicates good classification – No clear cutting point for acceptance or rejection Testing 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: 1. Estimate LRT for both models 2. Use bootstrapped samples to obtain distributions for LRT of both models 3. Compare LRT and get p values Testing K against K-1 Classes • Vuong-Lo-Mendell-Rubin likelihood-ratio test Determine Quality of Indicators • Good indicators – Item response probability is close to 0 or 1 in each class • Bad indicators – Item response probability is high in more than one classes, like cross-loading in factor analysis – Item response probability is low in all classes like low-loading in factor analysis LCA Examples • LCA • LCA with covariates • Class predicts a categorical outcome Save Membership Variable Variable: idvar = id; Output: Savedata: File = cmmber.txt; Save = cprob; Latent Profile Analysis • Covariance of continuous variables are dependent on class K and fixed at zero • Variances of continuous variables are constrained to be equal across classes and minimized • Mean differences are maximized across classes 11 0 • = = 0 0 22 0 0 0 Finite Mixture Modeling (multivariate normal variables) • • • • • Finite = finite number of subgroups/classes Variables are normally distributed in each class Means differ across classes Variances are the same across Covariances can differ without restrictions or equal with restrictions across classes • Latent profile can be special case with covariances fixed at zero. Mixture Factor Analysis • Allow one to examine measurement properties of items in heterogeneous subgroups / classes • Measurement invariance is not required assuming heterogeneity • Factor structure can change • See Mplus outputs Factor Mixture Analysis • Parental Control Parents let you make your own decisions about the time you must be home on weekend nights Parents let you make your own decisions about the people you hang around with Parents let you make your own decisions about what you wear Parents let you make your own decisions about which television programs you watch Parents let you make your own decisions about which television programs you watch Parents let you make your own decisions about what time you go to bed on week nights Parents let you make your own decisions about what you eat • Parental Acceptance Feel people in your family understand you Feel you want to leave home Feel you and your family have fun together Feel that your family pay attention to you Feel your parents care about you Feel close to your mother Feel close to your father Two dimensions of Parenting Mixture SEM • See mixture growth modeling Mixture Modeling with Known Classes • Identify hidden classes within known groups • Under nonrandomized experiments – Impose equality constraints on covariates to identify similar classes from known groups – Compare classes that differ in covariates