Advances in Longitudinal Data Analysis - Society of Behavioral ...

Katie Witkiewitz, PhDDepartment of PsychologyWashington State UniversityApril 7 th , 2009Workshop at the 31 st Annual Meeting of the Society of Behavioral Medicine

Goal of seminar is to introduce basicconcepts, terminology and potentialapplications of longitudinal mixture models2

What will not be covered?Level 1: y it = η 0i + η 1i x t + ε it ,Level 2: η 0i = α 0 + ζ 0iη 1i = α 1 + ζ 1iOr…3

Why learn about longitudinal mixturemodeling?Depends on your research question…5

Question 1: How doindividuals change theirdrinking behavior followingtreatment?6

Individual drinking patterns (n(= 395)following an initial post-treatmenttreatmentlapse7

Question 2: Does the numberof cigarettes smoked per daychange during telephonetobacco counseling?8

Cigarettes per day during telephone smokingcessation calls (n = 11,927)9

Question 3: Are therechanges in quit status aftertelephone tobaccocounseling?10

Quit status 6- and 12-months following telephonetobacco counseling (n = 7,624)11

Question 4: How do weexamine the effects ofweight loss interventionwhen some individuals in thetreatment group are nonresponders?12

Response to weight-loss interventionWeight loss interventionHealthy eating behaviorControlInterventionTimeThe non-responders in the intervention group lessen the effect size of theintervention in comparison to the control group13

Examples of Longitudinal Mixture ModelingBooth, K. M. (2003). Evaluating academic outcomes of head start: An application of general growth mixture modeling. EarlyChildhood Research Quarterly, 18(2), 238-254.Colder, C. R., Campbell, R. T., Ruel, E., Richardson, J. L., Flay, B. R. (2002). A finite mixture model of growth trajectories ofadolescent alcohol use: Predictors and consequences. Journal of Consulting & Clinical Psychology, 70(4), 976-985.Ellickson, P. L., Martino, S. C., Collins, R. L. (2004). Marijuana use from adolescence to young adulthood: Multipledevelopmental trajectories and their associated outcomes. Health Psychology, 23(3), 299-307.Muthén, B., et al., (2002). General growth mixture modeling for randomized preventive interventions. Biostatistics, 3, 459-475.Muthén, B. & Muthén, L. (2000). Integrating person-centered and variable-centered analyses: Growth mixture modeling withlatent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882-891.Nagin, D.S., & Tremblay, R.E. (1999). Trajectories of boys' physical aggression, opposition, and hyperactivity on the path tophysically violent and non-violent juvenile delinquency. Child Development, 70, 1181-1196.Orlando, M., Tucker, J. S., Ellickson, P. L., Klein, D. J. (2004). Developmental trajectories of cigarette smoking and theircorrelates from early adolescence to young adulthood. Journal of Consulting & Clinical Psychology, 72(3), 400-410.Orcutt, H. K., Erickson, D. J., Wolfe, J. (2004). The course of PTSD symptoms among Gulf War Veterans: A growth mixturemodeling approach. Journal of Traumatic Stress, 17(3), 195-202.Schaeffer, C. M; Petras, H., Ialongo, N., Poduska, J., Kellam, S. (2003). Modeling growth in boys' aggressive behavior acrosselementary school: Links to later criminal involvement, conduct disorder, and antisocial personality disorder. DevelopmentalPsychology, 39(6), 1020-1035.Wiesner, M., Silbereisen, R. K. (2003). Trajectories of delinquent behaviour in adolescence and their covariates: Relations withinitial and time-averaged factors. Journal of Adolescence, 26(6), 753-771.Witkiewitz, K. & Masyn, K.M. (2008)Drinking trajectories following an initial lapse. Psychology of Addictive Behaviors, 22, 157-167.Witkiewitz, K., van der Maas, H. J., Hufford, M. R., & Marlatt, G. A. (2007). Non-Normality and Divergence in Post-TreatmentAlcohol Use: Re-Examining the Project MATCH Data “Another Way.” Journal of Abnormal Psychology, 116, 378-394.Witkiewitz, K., & Villarroel, N. (2009). Dynamic association between negative affect and alcohol lapses following alcoholtreatment. Journal of Consulting and Clinical Psychology, 77, 633-644.14

Latent Variableunobserved; unmeasured; based onrelationships between observed variablesobservedvariable (x1)observedvariable (x2)observedvariable (x3)observed; measuredvariablesE1E2E3error or “residual,” notexplained by sharedvariance16

YYLatent variables can be continuous or categorical; tworepresentations of the same realityXXContinuous latent variable – correlationexplained by underlying factorEx. structural equation models, factormodels, growth curve models, multilevelmodelsCategorical latent variable – correlationreflects difference between discretegroups on mean levels of observedvariablesEx. latent class analysis, mixtureanalyses, latent transition analysis,latent profile analysis17

Types of modelsCategorical &Categorical latent variable continuous latentvariableCross-sectional Latent class analysis Factor mixture analysisLatent profile analysisLongitudinal Latent transition analysis Growth mixture modelingLatent Markov modelLatent class growth analysis18

Finite Mixture Modeling19

Direct and Indirect Applications1.Direct: identify distinct subpopulations of individualsObtain estimates of the conditional distributionsCalculate posterior probabilities that individual n camefrom subpopulation k2. Indirect: approximate intractable distribution with asmall number of simpler component distributions20

latent profile analysislatent class analysis

Latent profile analysis = categorical latent variableobserved continuous variables“Profiles”% heavydrinking daysEWitkiewitz et al. (2007). J. of Abnormal Psych. 22

Latent profile analysis = categorical latent variableobserved continuous variablesClass 1= infrequent drinking (78%)Class 2 = occasional drinking (12%)Class 3 = frequent drinking (10%)“Profiles”% heavydrinkingdaysE23

Latent class analysis = categorical latent variableobserved categorical variablesResponses Joe Jane Bob BettyAny smoking Yes Yes No NoAny smoking“CLASS”Cigarettes perdayStage ofchangeE E ECigarettes perday = 0No No Yes YesCigarettes perday

Latent class analysis = categorical latent variableobserved categorical variablesClass 1= non smoking, action stage (10%)Class 2 = smoking, 12-20 CPD, preparation stage (13%)Class 3 = smoking, 20+ CPD, preparation stage (77%)Any smoking“CLASS”Cigarettes perdayStage ofchangeE E EProbability ofendorsingC1 C2 C3Any smoking 0.00 1.00 1.000 CPD 1.00 0.00 0.0012-20 CPD 0.00 1.00 0.3420+ CPD 0.00 0.00 0.38Contemplation 0.00 0.00 0.07Preparation 0.11 0.99 0.93Action 0.89 0.01 0.0025

Latent Growth Mixture ModelLatent Markov ModelLatent Transition AnalysisRepeated Measures Latent Class Analysis

yLatent Growth Curve = longitudinal continuous latent variableBillJaneJoey it = η 0i + η 1ix t + ε itGordanSueInterceptη oSlopeη 11 2 3timeTime 1 Time 2 Time 3X 1 X 2X 3ε 1 ε 2 ε 327

Latent growth model28

Latent Growth Mixture Model: Combined continuousand categorical latent variable modelCLASS(“mixture”)• Growth factors (ie. random effects) areindicators of latent trajectory “classes”InterceptSlopePost-treatmentWeek 10Week26Week52ε 1 ε 2 ε 3 ε 429

Latent Growth Mixture Model30

Latent Class Growth Analysis (i.e., semi-parametricgroup-based modeling): Combined continuous andcategorical latent variable modelCLASS(“mixture”)InterceptSlopeAll variances in trajectories explained byclassification! With no variation aroundtrajectories…Post-treatmentWeek 10Week26Week52ε 1 ε 2 ε 3 ε 431

Latent Class Growth Analysis32

Latent Markov model = estimate probability oftransitioning over timeex. Percent heavy drinking days (%HD)%HDPost-treatment%HD10 weekspost%HD26 weekspost%HD52 weekspost%HD post-treatment%HD 10weeks post%HD 26weeks post%HD 52weeks postEEEE33

Class 1= infrequent drinking (78%)Class 2 = occasional drinking (12%)Class 3 = frequent drinking (10%)“Profiles”% heavydrinkingdaysE34

Latent Markov model = estimate probability oftransitioning over timeex. Percent heavy drinking days (%HD)%HDPost-treatment%HD10 weekspost%HD26 weekspost%HD52 weekspost%HD post-treatment%HD 10weeks post%HD 26weeks post%HD 52weeks postEEEE35

Latent Markov ModelWeek 10 Week 26 Week 52Infreq Occas. Freq. Infreq. Occas. Freq.Week 0 Week 10 Week 26Infreq.n≈1006Occas.n≈162Freq.n≈1280.84 0.13 0.030.18 0.58 0.240.08 0.13 0.79Infreq.n≈887Occas.n≈238Freq.n≈1710.85 0.11 0.040.17 0.59 0.240.11 0.13 0.76Infreq.n≈822Occas.n≈256Freq.n≈218Infreq.n≈831Occas.n≈239Freq.n≈2260.90 0.09 0.010.16 0.54 0.300.08 0.15 0.7736

Latent transition analysis = estimate probability oftransitioning from one class to another over timeSmokingClassCall 1SmokingClassCall 2Any smokingcall 1Cigarettesper day call 1Stage ofchange call 1Any smokingcall 2Cigarettesper day call 2Stage ofchange call 2E E EE E E37

Class 1= non smoking, action stage (10%)Class 2 = smoking, 12-20 CPD, preparation stage (13%)Class 3 = smoking, 20+ CPD, preparation stage (77%)Any smoking“CLASS”Cigarettes perdayStage ofchangeE E EProbability ofendorsingC1 C2 C3Any smoking 0.00 1.00 1.000 CPD 1.00 0.00 0.0012-20 CPD 0.00 1.00 0.3420+ CPD 0.00 0.00 0.38Contemplation 0.00 0.00 0.07Preparation 0.11 0.99 0.93Action 0.89 0.01 0.0038

Latent transition analysisProbability ofendorsingC1 C2 C3Any smoking 0.00 1.00 1.000 CPD 1.00 0.00 0.0012-20 CPD 0.00 1.00 0.3420+ CPD 0.00 0.00 0.38Contemplation 0.00 0.00 0.07Preparation 0.11 0.99 0.93Action 0.89 0.01 0.00Call 1C1(33%)Call 2C2(10%)C3(57%)C1 (10%) 0.43 0.00 0.57C2 (13%) 0.39 0.51 0.11C3 (77%) 0.00 0.13 0.8739

Repeated measures latent class analysis = estimate %of individuals who follow similar pattern over timeLatent classAny smokingcall 1EAny smokingcall 2EAny smokingcall 3EAnysmokingcall 4EAnysmokingcall 5E40

Repeated measures latent classanalysis41

Prior to running any analyses…Is your data appropriate? Check your distributions Sample size and power Time points and time scale43

Check your distributionsWitkiewitz et al. (2007). J. of Abnormal Psych.44

Sample size and power Most LMMs are estimated with large samples No set guidelines for parameter-to-n ratio Monte Carlo study to determine sample size for givenmodel (Muthen & Muthen, (2002) Structural Equation Modeling )45

Time, it’s on your side… # of time points? Growth mixture models - 3 is enough, but not ideal Latent Markov/transition analysis - 2 is enough Time scale? Age Calendar time Clinically meaningful scaling46

Model selection & estimation Class enumeration and model fit evaluation Type of mixture model? Parameter restrictions Starting values47

Class Enumeration???48

Fit Indices Information Criteria – based on loglikelihood and # ofparameters Akaike’s Information Criteria (AIC) Bayes Information Criteria (BIC) Adjusted BIC Lo-Mendell-Rubin likelihood ratio test (LMR) andbootstrapped likelihood ratio test (BLRT) k-1 vs. k class model49

Classification precision Model entropy criteria – range from 0 to 1 Average Latent Class Probabilities for Most LikelyLatent Class Membership by Latent ClassExample.Latent class assignment1 2 3Most likely latent class 1 0.875 0.106 0.0192 0.060 0.913 0.0283 0.021 0.051 0.92850

Assessing Model Fit in PracticeExample : Drinking trajectories following an initial lapse Type: Growth mixture model Class enumeration: 1-5 classes estimated Sample-size adjusted BIC (aBIC) prefer lower values Bootstrapped likelihood ratio test (k vs. k-1)51

Example 1: Observed data…(n = 395; % days abstinent per month, during themonths following an initial post-treatment lapse)52

Two Class Growth Mixture Model % Days AbstinentEntropy = .95aBIC = 30510.36How does itcompare to a oneclass Model?aBIC = 30707.75BLRT = 200.24 (p

Three Class Growth Mixture Model: % Days AbstinentEntropy = .91aBIC = 30458.312-Class ModelaBIC = 30510.36BLRT = 70.23 (p = .20)54

Four Class Growth Mixture Model: % Days AbstinentEntropy = .92aBIC = 30316.963-Class ModelaBIC = 30458.31BLRT = 147.64 (p < .0005)BLRT of 5 class model = 25.76 (p = .85)55

Type of mixture model???56

Example 2: Observed data…(n = 1,383; % heavy drinking days post-treatment)57

Comparison of latent growth mixture andMarkov models Growth mixture: 3-class model based on LMR Entropy = 0.98 BIC (3-class GMM) = 43886.02 Markov model: 3-class model based on LMR Entropy = 0.95 BIC (3-class LMM) = 41045.22Witkiewitz, Maisto, & Donovan (under review)58

Latent Growth Mixture Model59

Growth mixture model:“Infrequent heavy drinking class”60

Growth mixture model:“Occasional heavy drinking class”61

Growth mixture model:“Frequent heavy drinking class”62

Latent Markov model:“Infrequent heavy drinking class pattern”63

Latent Markov model:“Infrequent to occasional heavy drinkingclass pattern”64

Latent Markov model:“Frequent heavy drinking class pattern”65

Cross-classification of models66

A few more sticky issues Parameter restrictions Equality constraints Setting parameters to specific values (0, 1, known value) Starting values Local solutions - erratic likelihood functions Substantively different results Multiple random starting values can help examinelikelihood space (Hipp & Bauer, 2006)67

Common research questionsDo different levels of x correspond to agreater likelihood of being in class y?MotivationMotivationWhat is the relationship between x andwithin class change over time?CLASSInterceptDoes class membershippredict outcome y?Slope3 year drinkingTime 1Time 2ε 1ε 2ε 3ε 4Time 3Time 469

Growth Mixture Model with CovariatesCopingCognitionAffectDistal RiskIntercept"Class"Simplified Growth Mixture ModelPDA = % days abstinentMo = follow-up monthSlopeMo.1PDAE12Mo.2PDAMo.3PDAMo. 4PDAMo. 5PDAMo. 6PDAMo. 7PDAMo. 8PDAMo. 9PDAMo.10PDAE3Mo.11PDAE2Mo.12PDA1 1 1 1 11 1 1 1 111E11E10E9E8E7E6E5E4E170

Growth Mixture Model with Covariatespredicting class membershipDrinking Class Coping AffectDistalriskCognitiveriskInfrequent 1.84* -.03 -.05* .06*Occasional 1.45* .005 -.01 .02Prolapsers 1.24* .06 -.08* .05*Frequent ** ** ** **Frequent drinkers vs. other classes significantly lower coping scores vs. all classes* = (p

Growth Mixture Model with covariatespredicting within class growthcovariateCLASSInterceptSlopeTime1Time2Time3Time4ε 1ε 2ε 3ε 472

Covariates predicting within class growthWitkiewitz & Masyn (2008). Psych Addictive Behaviors73

“Predicting” a distal outcomeDoes class membershippredict outcome y?Latent class12-month pointprevalenceAny smokingcall 1EAny smokingcall 2EAny smokingcall 3EAnysmokingcall 4EAnysmokingcall 5E74

Predict 12-month point prevalenceEquality tests of mean 12-month point prevalence acrossclasses using posterior probability-based multiple imputationsClass 1 (mean = .23) vs. Class 2 (mean = .46)Class 1 vs. Class 3 (mean = .52)χ2 , p-value72.54, p

Latent transition analysis with timeinvariantcovariate predicting classes andtransitions between classesPDAPost-TxDDDPost-TxDrInCPost-TxPDAMo.6DDDMo.6DrIncMo.6PDAMo.12DDDMo.12DrIncMo.12C1C2C3Alcoholdependence76

Latent transition analysis with timeinvariantcovariate predicting classes andtransitions between classesWitkiewitz (2008). J Stud Alcohol Drugs.77

Limitations: Longitudinal Mixture Models User specified number of latent classes Classes are not real and true # of classes never known All sorts of modeling issues See Bauer (2007). Multivariate Behav ResearchBauer & Curran (2003/2004). Psych MethodsHipp & Bauer (2006). Psych MethodsSampson & Laub (2005). Criminology Adding covariates Non-normality and nonlinearity can result in spuriousclass-by-predictor relationships Can change underlying latent classes79

Summary of modeling issues Over-extraction of latent classes (Bauer & Curran, 2003) Reification of groups that do not exist (Raudenbush, 2005) Loss of power - artificially dividing growth trajectories intoclasses Spurious covariate by class interactions that do not exist(Bauer & Curran, 2003)80

Bauer and Curran (2003):“All of our models are wrong, and it is quite possiblethat there is no “right” model to discern whatsoever.The real task at hand is to decide which model is mostuseful” (p. 388).81

Software Mplus – www.statmodel.com Mx - http://www.vcu.edu/mx/ SAS ProcTraj - http://www.andrew.cmu.edu/user/bjones/ SAS ProcLCA/ProcLTA -http://methodology.psu.edu/index.php/downloads/proclcalta Latent Gold - http://www.statisticalinnovations.com/ Open Mx - http://openmx.psyc.virginia.edu/ Others?83

Screen shot of mplus webpage84

Screen shot of LGMMcommands85

WARNING: WHEN ESTIMATING A MODEL WITH MORE THAN TWO CLASSES,IT MAY BE NECESSARY TO INCREASE THE NUMBER OF RANDOM STARTSUSING THE STARTS OPTION TO AVOID LOCAL MAXIMA.THE MODEL ESTIMATION TERMINATED NORMALLYTESTS OF MODEL FITLoglikelihoodH0 Value -21907.093H0 Scaling Correction Factor 1.949for MLRInformation CriteriaNumber of Free Parameters 18Akaike (AIC) 43850.186Bayesian (BIC) 43943.193Sample-Size Adjusted BIC 43886.015(n* = (n + 2) / 24)86

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENTCLASSES BASED ON THE ESTIMATED MODELLatent Classes1 158.22360 0.122092 1009.38814 0.778853 128.38827 0.09907FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENTCLASS PATTERNS BASED ON ESTIMATED POSTERIORPROBABILITIESLatent Classes1 158.22360 0.122092 1009.38813 0.778853 128.38827 0.09907CLASSIFICATION QUALITYEntropy 0.97987

CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOSTLIKELY LATENT CLASS MEMBERSHIPClass Counts and ProportionsLatent Classes1 156 0.120372 1011 0.780093 129 0.09954Average Latent Class Probabilities for Most LikelyLatent Class Membership (Row)by Latent Class (Column)1 2 31 0.970 0.027 0.0022 0.006 0.994 0.0003 0.008 0.000 0.99288

MODEL RESULTSTwo-TailedEstimate S.E. Est./S.E. P-ValueLatent Class 1I |T4_PHD 1.000 0.000 999.000 999.000T26_PHD 1.000 0.000 999.000 999.000T52_PHD 1.000 0.000 999.000 999.000T68_PHD 1.000 0.000 999.000 999.000S |T4_PHD -1.000 0.000 999.000 999.000T26_PHD 0.000 0.000 999.000 999.000T52_PHD 2.600 0.000 999.000 999.000T68_PHD 4.200 0.000 999.000 999.000Q |T4_PHD 1.000 0.000 999.000 999.000T26_PHD 0.000 0.000 999.000 999.000T52_PHD 6.760 0.000 999.000 999.000T68_PHD 17.640 0.000 999.000 999.00089

Two-TailedEstimate S.E. Est./S.E. P-ValueLatent Class 1S WITHI 32.714 2.262 14.460 0.000MeansI 46.136 2.268 20.339 0.000S 3.534 1.426 2.479 0.013Q -0.718 0.310 -2.320 0.020InterceptsT4_PHD 0.000 0.000 999.000 999.000T26_PHD 0.000 0.000 999.000 999.000T52_PHD 0.000 0.000 999.000 999.000T68_PHD 0.000 0.000 999.000 999.000VariancesI 86.389 6.205 13.923 0.000S 26.446 1.933 13.680 0.000Q 0.000 0.000 999.000 999.000Residual VariancesT4_PHD 10.086 4.484 2.249 0.024T26_PHD 342.876 25.345 13.528 0.000T52_PHD 310.972 21.594 14.401 0.000T68_PHD 123.691 27.015 4.579 0.000 90

Estimate S.E. Est./S.E. P-ValueLatent Class 2I |T4_PHD 1.000 0.000 999.000 999.000T26_PHD 1.000 0.000 999.000 999.000T52_PHD 1.000 0.000 999.000 999.000T68_PHD 1.000 0.000 999.000 999.000S |T4_PHD -1.000 0.000 999.000 999.000T26_PHD 0.000 0.000 999.000 999.000T52_PHD 2.600 0.000 999.000 999.000T68_PHD 4.200 0.000 999.000 999.000Q |T4_PHD 1.000 0.000 999.000 999.000T26_PHD 0.000 0.000 999.000 999.000T52_PHD 6.760 0.000 999.000 999.000T68_PHD 17.640 0.000 999.000 999.000S WITHI 32.714 2.262 14.460 0.000MeansI 9.608 0.554 17.328 0.000S 5.368 0.310 17.313 0.000Q -0.825 0.078 -10.584 0.000 91

TECHNICAL 11 OUTPUTRandom Starts Specifications for the k-1 Class Analysis ModelNumber of initial stage random starts 1000Number of final stage optimizations 200VUONG-LO-MENDELL-RUBIN LIKELIHOOD RATIO TEST FOR 2 (H0) VERSUS 3CLASSESH0 Loglikelihood Value -22251.8162 Times the Loglikelihood Difference 689.446Difference in the Number of Parameters 4Mean 3.678Standard Deviation 162.623P-Value 0.0005LO-MENDELL-RUBIN ADJUSTED LRT TESTValue 666.207P-Value 0.000692

Screen shot of LMM commands93

Latent Markov model continued94

LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODELC16 Classes (Rows) by C26 Classes (Columns)1 2 31 0.578 0.242 0.1802 0.135 0.788 0.0773 0.131 0.029 0.840C26 Classes (Rows) by C52 Classes (Columns)1 2 31 0.587 0.238 0.1742 0.128 0.764 0.1083 0.113 0.036 0.851C52 Classes (Rows) by C68 Classes (Columns)1 2 31 0.537 0.162 0.3012 0.154 0.765 0.0813 0.086 0.017 0.89795

Screen shot of LTA commands96

Latent transition analysiscontinued97

Latent transition analysiscontinued98

Screen shot of RMLCAcommands99

Repeated measures latentclass analysis continued100

Covariates in MplusDo different levels of x correspond to agreater likelihood of being in class y?What is the relationship between x andchange over time across classes?What is the relationship between x andchange over time within classes?Does class membership predict outcome?C#1 C#2 on motiv;icept slope on tx;%C#1%icept slope on tx;%C#2%icept slope on tx;%C#1%[drink_3]*;%C#2%[drink_3]*;ORAUXILIARY = drink_3(e)101

Common Mplus error messagesErrorWARNING: WHEN ESTIMATING A MODEL WITH MORE THAN TWO CLASSES,IT MAY BE NECESSARY TO INCREASE THE NUMBER OF RANDOM STARTSUSING THE STARTS OPTION TO AVOID LOCAL MAXIMA.SolutionMake sure you are using STARTS =*Note, you will still get this warning with STARTS commandErrorWARNING: THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED. THESOLUTION MAY NOT BE TRUSTWORTHY DUE TO LOCAL MAXIMA.INCREASE THE NUMBER OF RANDOM STARTS.SolutionIncrease the number of random starts.102

Common Mplus error messagesErrorWARNING: THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IN CLASS 1 ISNOT POSITIVE DEFINITE. THIS COULD INDICATE A NEGATIVE VARIANCE/RESIDUAL VARIANCE FOR A LATENT VARIABLE, A CORRELATION GREATEROR EQUAL TO ONE BETWEEN TWO LATENT VARIABLES, OR A LINEARDEPENDENCY AMONG MORE THAN TWO LATENT VARIABLES. CHECK THETECH4 OUTPUT FOR MORE INFORMATION. PROBLEM INVOLVINGVARIABLE S.SolutionCheck the residual variances under Model ResultsVariancesEstimate S.E. Est./S.E. P-ValueI 7.326 3.033 2.416 0.016S -0.709 0.510 -1.392 0.164103

Common Mplus error messagesError(s)THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES MAY NOT BETRUSTWORTHY FOR SOME PARAMETERS DUE TO A NON-POSITIVE DEFINITEFIRST-ORDER DERIVATIVE PRODUCT MATRIX. THIS MAY BE DUE TO THESTARTING VALUES BUT MAY ALSO BE AN INDICATION OF MODELNONIDENTIFICATION. THE CONDITION NUMBER IS 0.186D-16. PROBLEMINVOLVING PARAMETER 18.ONE OR MORE MULTINOMIAL LOGIT PARAMETERS WERE FIXED TO AVOIDSINGULARITY OF THE INFORMATION MATRIX. THE SINGULARITY IS MOSTLIKELY BECAUSE THE MODEL IS NOT IDENTIFIED, OR BECAUSE OF EMPTY CELLSIN THE JOINT DISTRIBUTION OF THE CATEGORICAL LATENT VARIABLES ANDANY INDEPENDENT VARIABLES. THE FOLLOWING PARAMETERS WERE FIXED: 23Potential Solutions-Look at TECH1 output to determine what parameter is causing the problem-Increase number of random starts-Reduce complexity of model via parameter constraints104

Advanced longitudinal mixture models Discrete time survival mixture analysis Multilevel mixture modeling Associative latent transition analysis Markov switching models Alternative model specifications Two-part Piecewise Zero-inflated Poisson106

BASIC SETUP Programs can be simple (taking advantage of defaultsettings) or complex (by making specific modelstatements) .dat data files .inp input files .out output files .gph graphic files107

COMMAND LINESTITLE:DATA:VARIABLE:DEFINE:ANALYSIS:MODEL:PLOT:OUTPUT:SAVEDATA:• All lines end with ;• Only 80 characters per line“-” can shorten variable lists (y1-y6)!comments can be used anywhere;108

EXAMPLE PROGRAMTITLE: Drinking example 1: 2 classes - only linear slope - No covariatesDATA: FILE IS lgmm.example.dat;VARIABLE:NAMES ARE pda_gm1 pda_gm2 pda_gm3 pda_gm4 pda_gm5;USEVAR ARE pda_gm1-pda_gm5;CLASSES = c(2);ANALYSIS: TYPE = MIXTURE;STARTS = 500 250;MODEL:%OVERALL%i s | pda_gm1@0 pda_gm2@1 pda_gm3@2 pda_gm4@3pda_gm5@4;i* s*;i with s*;%C#1%[i* s*];%C#2%[i* s*];PLOT: TYPE IS plot3;SERIES IS pda_gm1(0) pda_gm2(1) pda_gm3(2) pda_gm4(3)pda_gm5(4);OUTPUT: standardized sampstat tech11 tech14; 109

VARIABLE COMMANDNAMES AREnames of variables in the data set;USEOBSERVATIONS AREUSEVARIABLES AREconditional statement to select observations;names of analysis variables;MISSING ARE variable (#);GROUPING ISIDVARIABLE ISKNOWNCLASS =CLASSES =name of grouping variable (labels);name of ID variable;name of categorical latent variable with knownclass membership (labels);names of categorical latent variables (number oflatent classes);110

ANALYSIS COMMAND Many optionsTYPE IS mixture, meanstructure, complex, twolevelESTIMATOR = ML, GLS, WLS, MLR, WLSMV Each question may require a different analysis typeand estimator111

MODEL COMMANDPART OF THE MODEL%OVERALL% - indicates the overall model%C#1% - indicates class #1 model%C#2% - indicates class #2 modelDEFINING GROWTH FACTORSi s | y1@0 y2@1 y3@2 y4@3i = intercepts = slope| = names and defines growth factorsMODEL:%OVERALL%i s | pda_gm1@0 pda_gm2@1 pda_gm3@2;%C#1%[i* s*];i* s@0;i with s@0;i on s*;GROWTH FACTOR MEANS[i* s*] [means] i* = intercept mean is estimated[i@0 s@1] i@0 = intercept mean is fixed at 0GROWTH FACTOR VARIANCESi* s* no brackets=variances, i* = intercept variance is estimatedi@0 s@1 no brackets=variances, i@0 = intercept variance is fixed at 0COVARIANCESi WITH s WITH=covariance of i and s%C#2%[i* s*];i* s*;i with s*;i on s*;REGRESSIONi ON s ON=regression of i on s112

OUTPUT COMMAND Many options for output, including: confidenceintervals, standard errors, modification indices,standardized coefficients, residuals, “tech1-tech15” Tech11 = Lo-Mendell-Rubin Likelihood ratio test Tech13 = tests for assessing multivariate skew andkurtosis Tech14 = bootstrapped likelihood ratio test113