SJoOnCesI OetL aOl.G / ISCAASL P MROETCHEDOUDRS E& T RREASJEARCH ThisarticleintroducesanewSASprocedurewrittenbytheauthorsthatanalyzeslongitudinaldata(developmentaltrajectories)byfittingamixturemodel.TheTRAJprocedure fitssemiparametric(discrete)mixturesofcensorednormal,Poisson,zero-inflatedPoisson,andBernoullidistributionstolongitudinaldata.Applicationstopsychometricscale data,offensecounts,andadichotomousprevalencemeasureinviolenceresearchareillustrated.Inaddition,theuseoftheBayesianinformationcriteriontoaddresstheproblemofmodelselection,includingtheestimationofthenumberofcomponentsinthemixture, is demonstrated. A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories BOBBY L. JONES DANIEL S. NAGIN KATHRYN ROEDER Carnegie Mellon University T hestudyofdevelopmentaltrajectoriesisacentralthemeof developmental and abnormal psychology and of life coursestudiesinsociologyandcriminology(Fergusson,Lynskey,and Horwood 1996; Loeber and LeBlanc 1990; Moffitt 1993; Patterson 1996;Patterson,DeBaryshe,andRamsey1989;Pattersonetal.1998; Patterson and Yoerger 1997; Sampson and Laub 1993). This article demonstrates a new SAS procedure, called TRAJ, developed by the authors for estimating developmental trajectories. The procedure is based on a semiparametric, group-based modeling strategy. Technically,themodelisamixtureofprobabilitydistributionsthataresuitablyspecifiedtodescribethedatatobeanalyzed.Theapproachisintended to complement two well-established methods for analyzing developmental trajectories—hierarchical modeling (Bryk and Raudenbush 1987, 1992; Goldstein 1995) and latent growth curve modeling(MeredithandTisak1990;Muthen1989;WillettandSayer 1994). In hierarchical modeling, individual variation in developmental trajectories, which are commonly called growth curves, are capturedbyarandomcoefficientsmodelingstrategy.Latentgrowthcurve SOCIOLOGICAL METHODS & RESEARCH, Vol. 29 No. 3, February 2001 374-393 © 2001 Sage Publications, Inc. 374 modeling uses covariance structure methods. These methods model variationintheparametersofdevelopmentaltrajectoriesusingcontinuous multivariate density functions. The group-based approach employsamultinomialmodelingstrategy.Thestatisticaltheoryunderlying the method has been developed in detail elsewhere (Nagin and Land 1993; Land, McCall, and Nagin 1996; Roeder, Lynch, and Nagin 1999; Nagin and Tremblay 1999; Nagin 1999), so our focus hereisonthesoftwareitselfanditsfunctionalcapabilities.However, we begin with a brief overview of the underlying statistical theory. BRIEF OVERVIEW: DERIVATION OF THE LIKELIHOOD Mixturemodelsareusefulformodelingunobservedheterogeneity in a population. An appropriate parametric model f(y, λ) is assumed forthephenomenontobestudied,wherey=(y1,y2,...,yT)denotesthe longitudinal sequence of an individual’s behavioral measurements overtheTperiodsofmeasurement.However,incontrasttothehomogeneous case, it is believed that there are unobserved subpopulations differing in their parameter values. In this case, the marginal density for the data y can be written, f C k C k p f k K k k K k ( ) ( ) ( | ) ( , ) y Y y y = = = = = == ∑∑ Pr Pr 11 λ . (1) Here pk is the probability of belonging to class k with corresponding parameter(s)λk.Thelongitudinalnatureofthedataismodeledbyhaving the parameter(s) λk depend on time. Time-stable covariates (risk factors) are incorporated into the model by assuming they influence the probability of belonging to a particular group. Time-dependent covariatescanalsodirectlyaffecttheobservedbehavior,asillustrated in Figure 1. Theriskfactorsaffectthelikelihoodofaparticulardatatrajectory, but it is assumed that nothing more can be learned about the data (Y) fromriskfactors(Z)givengroup(C).Thus,weassumetheriskfactors forsubject i,Zi =(Zi1,...,ZiR),andthedatatrajectoryforthesubject consistingoftherepeatedmeasurementsoverTmeasurementperiods, Yi =(Yi1, . . . ,YiT),areindependentgiventhegroup,Ci.Giventhatthere Jones et al. / SAS PROCEDURE TRAJ 375 are K groups, we can write the conditional distribution of the observable data for subject i, given risk factors and a time-dependent covariate, Wi = (wi1 . . . , wiT), f C k C k i i i k K i i i i i i i i ( | , ) ( | ) ( | , y z w Z z Y y W w = = = = = = = ∑Pr Pr 1 ). (2) The time-stable covariate effect on group membership is modeled with a generalized logit function (θ1 and λ1 are taken to be zero for identifiability), Pr( | ) exp( ) exp( ) Ck i i i k k i l K l l i = = = + + ′ = ′∑ Zz z z θ θ

1 (3) TRAJprovidestheoptionofmodelingthreedifferentdistributions for Pr(Yi = yi | Ci = k, Wi = wi) to analyze count, psychometric scale, anddichotomousdata.Thezero-inflatedPoisson(ZIP)modelisusefulformodelingtheconditionaldistributionofcountdatagivengroup membership when there are more zeros than under the Poisson assumption(Lambert1992).Thisiscommoninantisocialandabnormal behavior that is typically concentrated in a small fraction of the 376 SOCIOLOGICAL METHODS & RESEARCH Figure 1: Directed Acyclic Graph Representing the Independence Assumptions population. For the ZIP model, the probability of observing the data trajectory yi given membership in group k is, Pr( | , ) [ ( ) ] ( ) Y y W w i i i i i ijk ijk ijk Ck e ijk === = + − − − ρ ρ ρ λ 11 y ijk ijk y ijy ij ij ij y=> ∏∏ − 00 exp( ) ! . λλ (4) Notethatρijkistheextra-Poissonprobabilityofazero.Letageijdenote subjecti’sageinperiodj,andwij subjecti’stime-dependentcovariate value in period j. The (optional) time-dependent covariate is related linearly to log(λijk). In addition, a polynomial relationship is used to model the link between age and the model’s parameters: log(λijk) = β0k + ageijβ1k + age2ijβ2k + . . . + wijδk and log(ρijk/(1 – ρijk)) = α0k + ageijα1k + age2ijα2k + . . . . Thesoftwareallowsforspecificationofuptoathird-orderpolynomial in age. It also allows the user to specify different order polynomials acrossthektrajectorygroups.Equations(3)and(4)incorporatedinto equation (2) give the likelihood of observing the data trajectory of a subject, given his covariate values. The complete likelihood for all subjects is the product of these individual likelihood values. Thecensorednormal(CNORM)modelisusefulformodelingthe conditional distribution of psychometric scale data, given group membership (Nagin and Tremblay 1999). A distribution allowing forcensoringisusedbecausethedatatendtoclusterattheminimum ofthescale(Min)andatthescalemaximum(Max).Hence,thelikelihoodofobservingthedatatrajectoryforsubjecti,givenhebelongs to group k, is Pr(Yi = yi | Ci = k, Wi = wi) = Φ y ijk Min y Max ij ijk ij ij Min y = < < ∏∏ −        − min µ σσ ϕ µ σ 1         − −               = ∏ 1 Φ Max ijk y Max ij µ σ , where µijk = β0k + ageijβ1k +ageij k 2 2β + . . . + wijδk. (5) Jones et al. / SAS PROCEDURE TRAJ 377 The censored normal model is also appropriate for continuous data that are approximately normally distributed, with or without censoring. The uncensored case is handled by specifying a minimum and maximum that lie outside the range of the observed data values. Finally, the logistic (LOGIT) model is used to model the conditionaldistributionofdichotomousdata,givengroupmembership.The likelihoodofobservingthetrajectoryforsubjecti,givenhebelongsto group k, is Pr( | , ) ( ) Y y W w i i i i i ijk yy ijk C k p p ij ij = = = = − ∏∏ =0 1 with p age age w ijk k ij k ij k ij k k = + + + + ++ exp( ) exp( β β β δ β 01 2 2 01
age k age w ij ij k ij k β β δ 1 2 2 + + +
) . (6) Maximumlikelihoodisusedtoestimatethemodelparameters.The maximization is performed using a general quasi-Newton procedure (Dennis,Gay,andWelsch1981;DennisandMei1979)obtainedfrom Netlib. Standard error estimates are calculated by inverting the observed information matrix. Subjects with some missing longitudinaldatavaluesortime-dependentcovariatevaluesareincludedinthe analysis. However, subjects with any missing risk factor (time-stable covariate) data are excluded from the analysis. OVERVIEW OF SOFTWARE ManyresearchersarefamiliarwiththeSASpreprogrammedstatisticalprocedurestoanalyzedata.Inaddition,SAScanbeprogrammed throughstatementsinthedatastepthroughmacrosorthroughtheSAS interactivematrixlanguage.Alesser-knownfourthoptionistodevelop a customized SAS procedure using a SAS product: SAS/TOOLKIT. Our custom SAS procedure (available for the PC platform only) is a program written in the C programming language that interfaces with theSASsystemtoperformthemodelfitting.Theexecutabledynamic linklibraryisdistributedtootheruserswhoafterinstallationuseitjust astheywoulduseanypreprogrammedSASprocedure.Thefollowing 378 SOCIOLOGICAL METHODS & RESEARCH introductoryexampleillustratestheapplicationofthemethodandthe use of the SAS procedure TRAJ. EXAMPLE 1: MONTREAL LONGITUDINAL STUDY The data consist of 1,037 boys assessed annually by their teachers atage6(spring1984)andatages10through15onscalesofphysical aggression,opposition,andhyperactivity.The53participatingschools were located in low socioeconomic areas of Montreal (Canada). Time-stable covariates were recorded, including age of mother and fatheratthebirthoftheirfirstchild,yearsofschoolingforthemother and father, a home adversity index, and psychometric scale data on inattention,anxiety,andprosocialbehaviorofeachboyatage6.Consider the opposition score, which ranges from 0 to 10 and measures five items: does not share, irritable, disobedient, blames others, and inconsiderate. Figure 2 shows sample opposition data for nine subjects, illustrating the variability in the trajectory shapes. Some never exhibit difficulties; others have difficulties and then seem to learn moreadaptivecopingstrategies,asevidencedbytheirdropinoppositionscores.Alsopresentaresubjectswhocontinuetoshowhighlevels ofoppositionalbehaviorthroughage15.Figure3showsthedistributionoftheoppositionscoresforeachyeartheywererecorded.Scores of zero are most frequent. Note also that the opposition scores decreaseinfrequencyasthescoreincreases.Hence,thecensorednormal distribution seems a sensible choice for modeling these data. Thefollowingstatementsfitafive-groupmodeltotheoppositional behaviordataandplottheresults(seeFigure4).Thejustificationforthe choice of five groups is discussed in the fourth section of this article. PROC TRAJ DATA=MONTREAL OUT=OF OUTPLOT=OP OUTSTAT=OS; VAR O1-O7; /* Opposition Variables */ INDEP T1-T7; /* Age Variables */ MODEL CNORM; /* Censored Normal Model */ MIN 0; /* Lower Censoring Point */ MAX 10; /* Upper Censoring Point */ NGROUPS 5; /* Fit 5 Groups */ ORDER 3 3 3 3 3; /* Cubic Trajectory for Each Group */ RUN; %TRAJPLOT (OP, OS,“Opposition Trajectories”,,“Opposition”,“Scaled Age”); Jones et al. / SAS PROCEDURE TRAJ 379 Twenty-twopercentofthesubjectsareclassifiedasexhibitinglittle ornooppositionalbehavior(group1);thelargestpercentage,42percent, exhibit low and somewhat decreasing levels of oppositional behavior (group 2); 18 percent of the subjects show moderate levels of oppositional behavior (group 3); 7 percent of the subjects start out withhighlevelsofoppositionalbehaviorthatdropssteadilywithage (group 4); while the remaining 10 percent exhibit chronic problems with oppositional behavior (group 5). 380 SOCIOLOGICAL METHODS & RESEARCH Figure 2: Sample Data (oppositional behavior) The next examples illustrate analyses of dichotomous data and Poissondatawithextrazeros.Itisimportanttorealizethatsomemodels are difficult to fit and that there is no guarantee that the procedure will be able to fit the model successfully. In particular, the procedure may find only a local minimum; hence, the process of determining startingvaluesiscritical.Iftheuserdoesnotspecifystartingvalues(as in the introductory example), the procedure provides default starting valuesbyassumingintercept-onlytrajectoriesevenlyspacedthrough the range of the dependent variable. The next example includes the specification of starting values. EXAMPLE 2: CAMBRIDGE STUDY OF DELINQUENT DEVELOPMENT Thedataconsistof411subjectsfromaprospectivelongitudinalsurvey conducted in a working-class section of London. Farrington and Jones et al. / SAS PROCEDURE TRAJ 381 Figure 3: Distribution of Opposition Scores by Age West (1990) provide a detailed discussion of the study. The numbers of criminal offense convictions were recorded annually beginning when the boys were age 10 and continuing through age 32. Because we are dealing with count data, the Poisson model is potentially appropriate here; however, more zeros are present than would be expectedinthepurelyPoissonmodel,soweusetheZIPmodel.Thefollowing statements fit a four-group model to the offense counts data and plot the results (see Figure 5). The starting values were obtained from an analysis (Roeder et al. 1999) that used cubic trajectories for the four groups. PROC TRAJ DATA=CAMBRDGE OUT=OF OUTPLOT=OP OUTSTAT=OS; VAR C1-C23; /* Offense Count Variables */ INDEP T1-T23; /* Age Variables */ MODEL ZIP; /* Zero Inflated Poisson Model */ NGROUPS 4; /* Fit 4 Groups */ ORDER 0 2 0 2; /* Two Linear and Two Quadratic Groups */ IORDER 1; /* Linear Zero Inflation */ START –4.8 /* Group 1 - Intercept Only */ –15.5 16.2 -4.5 /* Group 2 - Quadratic Trajectory */ –1.1 /* Group 3 - Intercept Only */ –4.5 5.1 –1.3 /* Group 4 - Quadratic Trajectory */ 382 SOCIOLOGICAL METHODS & RESEARCH Figure 4: Expected (dashed lines) Versus Observed (solid line) Trajectories –0.2 0.0 /* Linear Zero Inflation */ –1.2 –2.1 –2.1; /* Group Proportion Parameters */ RUN; %TRAJPLOT (OP, OS,“Offense Counts”,,“Offense Counts”,“Scaled Age”); Sixty-six percent of the subjects are classified as never convicted (group 1), 19 percent exhibit low conviction rates limited to adolescence(group2),7percentofthesubjectsshowlowbutpersistingconvictionrates(group3),whiletheremaining8percentexhibitthehighest conviction rates (group 4). EXAMPLE 3: CAMBRIDGE DATA PREVALENCE MEASURE It is common in research on criminal careers to analyze both the frequencyofoffendingmeasuredbyoffensecountsandtheabsence or presence of offenses (a dichotomous prevalence measure). The analysisontheCambridgedataisrepeated,convertingthenumbers of criminal offense convictions to a dichotomous prevalence measure.Thelogisticmodelwillbeusedfortheprevalencedata.Thefollowing statements fit a three-group model to the prevalence measure data and plot the results (see Figure 6). Jones et al. / SAS PROCEDURE TRAJ 383 Figure 5: Expected (dashed lines) Versus Observed (solid line) Trajectories PROC TRAJ DATA=CAMBRDGE OUT=OF OUTPLOT=OP OUTSTAT=OS; VAR C1-C23; /* Prevalence Variables */ INDEP T1-T23; /* Age Variables */ MODEL LOGIT; /* Logistic Model */ NGROUPS 3; /* Fit 3 Groups */ ORDER 3 3 3; /* Cubic Trajectories */ RUN; %TRAJPLOT (OP, OS,“Prevalence Measure”,,“Prevalence”,“Scaled Age”); Fifty-eightpercentofthesubjectsareclassifiedasneverconvicted, 34 percent have a low prevalence rate that peaks during adolescence, and the remaining 8 percent exhibit the highest prevalence rate. EXAMPLE 4: INTRODUCING TIME-STABLE COVARIATES INTO THE MODEL A common objective of social science research is to establish whetheratrait(e.g.,beingpronetooppositionalbehavior)islinkedto measured covariates (e.g., risk factors). Previous applications of the semiparametricapproachcategorizedsubjectsbylatenttraitfromobservablebehavior(Nagin,Farrington,andMoffitt1995;Laub,Nagin, and Sampson 1998). The group assignments were then fit to the co 384 SOCIOLOGICAL METHODS & RESEARCH Figure 6: Expected (dashed lines) Versus Observed (solid line) Trajectories variates with standard linear models. However, this classify-analyze procedure does not account for the uncertainty in group assignment and can lead to bias (Clogg 1995; Roeder et al. 1999). This finalexampleillustratestheinclusionofriskfactorsdirectlyinto themodel.Insodoing,thisapproachaccountsforassignmentuncertainty automatically. Suppose we were interested in investigating whether and to what degreeinattention,verbalIQ,andanadversehomelifeareriskfactors for elevated levels of opposition. Figure 7 shows the distribution of measures of each of these factors for the subjects in the Montreal study. The procedure automatically drops observations with missing dataintheriskfactorvariables.Ofthesubjects,174havemissingvaluesintheriskfactorsandareomittedfromtheanalysis.Thefollowing statements perform the risk analysis on the remaining 863 subjects. PROC TRAJ DATA=MONTREAL OUT=OF OUTPLOT=OP OUTSTAT=OS; VAR O1-O7; /* Opposition Variables */ INDEP T1-T7; /* Age Variables */ MODEL CNORM; /* Censored Normal Model */ MIN 0; /* Lower Censoring Point */ MAX 6; /* Upper Censoring Point */ NGROUPS 5; /* Fit 5 Groups */ ORDER 3 3 3 3 3; /* Cubic Trajectory for Each Group */ RISK VERBALIQ, /* Risk Factors */ INATTENT,ADVERSTY; RUN; InTable1,wepresenttheriskfactorparameterestimates,standard errors, tests for the hypothesis that the parameter equals zero, and p values for the tests. Figure 8 illustrates the marginal relationships of the risk factors—inattention, adversity, and verbal IQ—to the likelihoodofbelongingtothehighestoppositioncategoryversusthelowest oppositioncategory.Includedintheplotsarethesamplevalues(asmall amount of noise has been added to the plot points to separate them): lowoppositiongrouponthebottomandhighoppositiongrouponthe topofeachgraph.Asadversityinthehomeandinattentionscoresincrease,sodoesthelikelihoodofproblemswithhighoppositionalbehavior.However,asverbalIQincreases,thelikelihoodofbelongingto the high opposition group decreases. Jones et al. / SAS PROCEDURE TRAJ 385 EXAMPLE 5: MONTREAL LONGITUDINAL STUDY WITH A TIME-VARYING COVARIATE A trajectory defines the developmental course of a behavior over age(ortime).Trajectories,however,arenotdeterministicfunctionsof age.Externaleventsmaydeflectatrajectory.Forexample,Laubetal. (1998) examine the impact of marriage on deflecting trajectories of offending from high levels of criminality toward desistance. Life events may also have transitory affects on enduring trajectories of behavior. For example, spells of mental illness may temporarily alter trajectories of high-level productivity. In this example, we extend the basic model presented in example1byintroducingatime-varyingcovariateintothetrajectorymodel. Specifically, we add to the base model relating opposition to age a binary variable equal to 2 if by the age t the individual had been held back in school, 1 if the individual has not been held back. The objec 386 SOCIOLOGICAL METHODS & RESEARCH Figure 7: Distribution of Verbal IQ, Adversity, and Inattention Index tiveistotestwhetherforsometrajectorygroupsschoolfailureisassociated with an increase in opposition. Note that the structure of the modelallowsforthepossibilitythattheimpactmayvarybytrajectory group. The number of students held back ranges from 51 at age 6 to 516 at age 15. Thefollowingstatementsfitafive-groupmodeltotheoppositional behavior data. PROC TRAJ DATA=MONTREAL OUT=OF OUTPLOT=OP OUTSTAT=OS; VAR O1-O7; /* Opposition Variables */ INDEP T1-T7; /* Age Variables */ MODEL CNORM; /* Censored Normal Model */ MIN 0; /* Lower Censoring Point */ MAX 10; /* Upper Censoring Point */ NGROUPS 5; /* Fit 5 Groups */ Jones et al. / SAS PROCEDURE TRAJ 387 Figure 8: Probability of Belonging to Group 5 (high opposition) Versus Group 1 (low opposition) as a Function of Risk Factor 388 SOCIOLOGICAL METHODS & RESEARCH ORDER 3 3 3 3 3; /* Cubic Trajectory for Each Group */ TCOV C1-C7; /* Time Varying Covariate (Held Back) */ RUN; Expected opposition trajectories for subjects never held back and always behind are given in Figure 9. Note that this was done as one way to illustrate the effect of the time-varying covariate. Other plots arepossiblebychangingwhensubjectsbegintobebehindgrade.We see that there is an increase in opposition for those behind grade in groups2,3,and5.Thereislittleeffectinthelowestoppositiongroup (group1)andinthesteadilydecreasinggroup(group4).Thosebehind grade in group 4 showed lower opposition in the first period. This is explainedbecauseofthe55subjectsclassifiedtogroup4(thesmallest group),4werebehindgradeinthefirstperiodandallhadlowopposition scores relative to the rest of the group. USING THE BAYESIAN INFORMATION CRITERION (BIC) FOR MODEL SELECTION One possible choice for testing the hypothesis of the number of componentsinamixtureisthelikelihoodratiotest.However,thenull TABLE 1: Risk Factor Parameter Estimates, Errors, Tests, and p Values Group Parameter Estimate Error Test p Value 2 Constant 1.96 0.79 2.49 .013 Inattention 0.26 0.15 1.82 .069 Adversity 2.83 0.63 4.46 .000 Verbal IQ –0.41 0.08 5.19 .000 3 Constant 0.80 0.72 1.11 .268 Inattention 0.48 0.12 3.98 .000 Adversity 0.98 0.49 1.99 .046 Verbal IQ –0.10 0.07 –1.48 .140 4 Constant –4.61 1.37 –3.36 .001 Inattention 1.21 0.16 7.72 .000 Adversity 2.92 0.79 3.71 .000 Verbal IQ 0.11 0.12 0.90 .366 5 Constant –2.46 1.30 –1.90 .058 Inattention 1.18 0.20 5.91 .000 Adversity 4.27 0.99 4.33 .000 Verbal IQ –0.28 0.11 –2.60 .009 hypothesis (i.e., three components versus more than three components)isontheboundaryoftheparameterspace,andhencetheclassicalasymptoticresultsdonothold(GhoshandSen1985).Tocircumventthisproblem,wefollowtheleadofD’Ungeretal.(1998)anduse thechangeintheBICbetweenmodelsasanapproximationtothelog oftheBayesfactor(KassandWasserman1995).Keribin(1997)demonstratedthat,undercertainconditions,thisapproximationisvalidfor testing the number of components in a mixture. Raftery (1995) and KassandRaftery(1995)aregoodreferencesforBayesfactors.Also, Jones et al. / SAS PROCEDURE TRAJ 389 TABLE 2: Interpretation of 2loge(B10) 2loge(B10) (B 10) Evidence Against H0 0 to 2 1 to 3 Not worth mentioning 2 to 6 3 to 20 Positive 6 to 10 20 to 150 Strong > 10 > 150 Very strong Figure 9: Expected Opposition Trajectories for Subjects Who Have Never Been Held Back (solid lines) Versus Subjects Who Have Always Been Behind Grade (dashed line) Fraley and Raftery (1998) address the use of Bayes factors in model-based clustering. The Bayes factor (B10) gives the posterior oddsthatthealternativehypothesisiscorrectwhenthepriorprobability that the alternative hypothesis is correct equals one-half. TheBIC(Schwarz1978),thelog-likelihoodevaluatedatthemaximumlikelihoodestimatelessone-halfthenumberofparametersinthe modeltimesthelogofthesamplesize,tendstofavormoreparsimoniousmodelsthanlikelihoodratiotestswhenusedformodelselection. TomaintainconsistentusagewiththatofJeffreys(1961)andKassand Raftery (1995), we use the BIC log Bayes factor approximation, 2loge(B10) ≈ 2(∆BIC), (7) where∆BIC is the BIC of the alternative (more complex) model less theBICofthenull(simpler)model.ThelogformoftheBayesfactor isinterpretedasthedegreeofevidencefavoringthealternativemodel (see Table 2). Table3tabulatestheBICformodelfitstotheoppositionalbehavior data. Based on the results, the five-group model is favored. CONCLUSION WedemonstratedtheuseofanewSASprocedurethatwewroteto analyze longitudinal data by fitting a mixture model. We illustrated the use of the TRAJ procedure through applications to psychometric scaledata(oppositionalbehavior)usingthecensorednormalmixture, offense counts using the ZIP mixture, and an offense prevalence 390 SOCIOLOGICAL METHODS & RESEARCH TABLE 3: Tabulated Bayesian Information Criterion (BIC) and 2loge(B10) (opposition data) Number of Groups BIC Null Model 2loge(B10) 1 –12,524.06 2 –11,818.92 1 1,410.28 3 –11,685.81 2 266.22 4 –11,683.27 3 5.08 5 –11,669.70 4 27.14 6 –11,678.51 5 –17.62 measure using the logistic mixture. Time-stable covariates (risk factors) were incorporated into the model by assuming that the risk factors are independent of the developmental trajectories, given group membership. A time-dependent covariate can also directly affect the observedbehaviortrajectory.Inaddition,theuseoftheBICtoaddress the problem of model selection, including the estimation of the number of components in the mixture, was demonstrated. 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Cambridge, MA: Blackwell. 392 SOCIOLOGICAL METHODS & RESEARCH Roeder,Kathryn,KevinG.Lynch,andDanielS.Nagin.1999.“ModelingUncertaintyinLatent ClassMembership:ACaseStudyinCriminology.” Journal of the American Statistical Association 94:766-76. Sampson,RobertJ.andJohnH.Laub.1993.CrimeintheMaking:PathwaysandTurningPoints Through Life. Cambridge, MA: Harvard University Press. Schwarz,Gideon1978.“EstimatingtheDimensionofaModel.”AnnalsofStatistics6:461-64. Willett,JohnB.andAlineG.Sayer.1994.“UsingCovarianceStructureAnalysistoDetectCorrelates and Predictors of Individual Change Over Time.” Psychological Bulletin 116(2):363-81. BobbyL.JonesisaPh.D.candidateintheDepartmentofStatisticsatCarnegieMellon University. He is currently working on his dissertation, “Analyzing Longitudinal Data WithLatentClassModels.”Heisthecoauthor(withShohiniGhose,JamesP.Clemens, PerryR.Rice,andLenoM.Pedrotti)of“PhotonStatisticsofaSingleAtomLaser,”which appeared in Physics Review A (1999). DanielS.NaginistheTeresaandH.JohnHeinzIIIProfessorofPublicPolicyattheH. John Heinz III School of Public Policy and Management, Carnegie Mellon University. He has written widely on deterrence, developmental trajectories and criminal careers, taxcompliance,andstatisticalmethodology.Hisrecentpublicationsinclude“Analyzing Developmental Trajectories: A Semi-Parametric, Group-Based Approach” in Psychological Methods (1999) and “Trajectories of Boys’ Physical Aggression, Opposition, and Hyperactivity on the Path to Physically Violent and Nonviolent Juvenile Delinquency” (with Richard E. Tremblay) in Child Development (1999). KathrynRoederisprofessorofstatisticsattheCarnegieMellonUniversity.Herresearch has focused on the development of statistical methodology for the analysis of heterogeneousdatausingmixturemodelsandsemiparametricmethods.Sheisinterestedincriminology and the genetic basis of psychiatric disorders. Recent publications include “Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology” (withKevinG.LynchandDanielS.Nagin)intheJournaloftheAmericanStatisticalAssociation(1999)and“GenomicControlforAssociationStudies”(withBernieDevlin)in Biometrics (1999). Jones et al. / SAS PROCEDURE TRAJ 393