A Multivariate Associative Finite Growth Mixture Modeling Approach Examining Adolescent Alcohol and Marijuana Use

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Journal of Psychopathology and Behavioral Assessment (JOBA) pp1225-joba-487541 October 19, 2004 13:6 Style file version June 25th, 2002

Journal of Psychopathology and Behavioral Assessment, Vol. 26, No. 4, December 2004 ( C© 2004)

A Multivariate Associative Finite Growth Mixture ModelingApproach Examining Adolescent Alcohol and Marijuana Use

Hollie Hix-Small,1,2 Terry E. Duncan,1 Susan C. Duncan,1 and Hayrettin Okut1

Accepted March 11, 2004

Theoretical and empirical substance use development research suggests that adolescent populationsare not homogenous and can often be separated into subpopulations characterized by qualitatively dif-ferent patterns of substance use development. This paper demonstrates the application of a multivariateassociative finite latent growth mixture modelling approach to examine heterogeneity in patterns ofadolescent alcohol and marijuana use and the influence of age, gender, parent, and peer substanceuse. Substance use problem outcomes were also examined. Participants were male and female ado-lescents (N = 1, 044) ranging in age from 11 to 17 years at the first assessment (Mean age = 14.47;SD = 1.95). Individuals were 45% female and 82% Caucasian. Using growth mixture methodology, a7-class model captured distinct simultaneous alcohol and marijuana use patterns over a 3-year period.Findings highlight the importance of examining subgroups of adolescent substance use, rather thanfocusing only on single samples.

KEY WORDS: Growth Generalized Mixture Modeling; substance use development; associative Latent GrowthModel; Parallel-Process Growth Model.

The prevalence of adolescent substance use is welldocumented (Johnston, O’Malley, & Bachman, 1996),as are relationships among adolescent alcohol use,marijuana use, and problem behavior (Donovan, 1996;Jessor, 1987; Jessor & Jessor, 1977). Data collected bythe National Household Survey on Drug Abuse foundthat 11.4% of 12- to 17-year-olds reported currentdrug use (Substance Abuse and Mental Health ServicesAdministration [SAMHSA], 1998). Reports also indicatethat approximately 10% of American adults and 3% ofadolescents are addicted to alcohol or other drugs (U.S.Department of Health and Human Services [USDHHS],1993). Studying substance use development from an earlyage is important as several studies have found that youthwho use substances at an earlier age than most of theirpeers are more likely to develop a substance use disorder(Anthony & Petronis, 1995; Costello, Erkanli, Federman,& Angold, 1999; Hanna & Grant, 1999).

1Oregon Research Institute, Eugene, Oregon.2To whom correspondence should be addressed at Oregon ResearchInstitute, 1715 Franklin Boulevard, Eugene, Oregon 97403; e-mail:[email protected].

Problem behavior theory posits that adolescentproblem behaviors, including use of different substances,are interrelated (Donovan & Jessor, 1985; Jessor &Jessor, 1977). Numerous studies have documented therelationship between use of different substances at singlepoints in time; however, there is growing evidence thatadolescents’ development of use of various substances areintercorrelated over time (S. C. Duncan & T. E. Duncan,1994, 1996). Recent methodological advances, such asLatent Growth Modeling (LGM) methodology, enableresearchers to study multivariate growth trajectories (e.g.,S. C. Duncan & T. E. Duncan, 1994; T. E. Duncan, S. C.Duncan, & Hops, 1994; T. E. Duncan, S. C. Duncan,Strycker, Li, & Alpert, 1999; McArdle, 1988; McArdle& Epstein, 1987; McArdle & Hamagami, 1991). Withinthe LGM framework it is possible to examine whetheruse of various substances covary and whether there iscovariance in the extent of curvature in their development.Associative LGM, an extension of the general LGM,allows researchers to examine the correlations amongdevelopmental parameters for pairs of behaviors. Thus,within the associative LGM framework it is possibleto assess relationships among the individual difference

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0882-2689/04/1200-0255/0 C© 2004 Springer Science+Business Media, Inc.

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256 Hix-Small, Duncan, Duncan, and Okut

parameters for pairs of behaviors (e.g., alcohol andmarijuana use), and to estimate means, variances, andcovariances for the growth factors of each substance.

In addition to studying the interrelationships betweendifferent substances, theoretical and empirical researchindicates that development in adolescent substance useand other problem behavior may be characterized by mul-tiple developmental pathways indicative of several sub-groups within the adolescent population (Loeber, 1988;B. Muthen & L. K. Muthen, 2000; Nagin & Tremblay,1999; Shulenberg, O’Malley, Bachman, Wadsworth, &Johnston, 1996; Zucker, 1994; Zucker, Fitzgerald, &Moses, 1995). That is, adolescent populations may becomposed of subpopulations characterized by qualita-tively different patterns of substance use over time. Forexample, Schulenber, et al. (1996) identified six differentalcohol use growth trajectories and these growth patternswere related to changes in behaviors and attitudes concern-ing alcohol and other drug use. Bates and Labouvie (1997)also found distinctive subgroups of substance use amongadolescents: (1) a consistently low alcohol and drug usegroup; (2) a group with heavy alcohol and/or drug useduring adolescence followed by low use during adulthood;and (3) a group with heavy alcohol and/or drug use in ado-lescence that continued into adulthood. Adolescent sub-stance abusers and users constitute a heterogeneous popu-lation and the identification of subgroups will be importantin the development of effective treatment approaches.

One recent promising technique, the latent GrowthGeneralized Mixture Modeling (GGMM) proceduredeveloped by Muthen and colleagues, allows for the studyof growth trajectory mixtures in research samples (see B.Muthen & Shedden, 1999, for a discussion). The GGMMapproach takes into account unobserved heterogeneityin the population and assumes that different individualgrowth trajectories belong to a mixture of qualitativelydifferent subpopulations. The GGMM framework notonly extends the conventional latent growth model byconsidering heterogeneity in the population, it allowsfor the simultaneous examination of the influences ofcovariates on the subpopulations and outcome variables.Membership of subgroups is not observed in the popu-lation but must be inferred from the data. Thus, a priorihypotheses about the number of subgroups in the popu-lation cannot be formulated. Instead, the procedure takesa finite approach to the GGMM methodology, allowingmodel fit to determine the number of classes or subpopu-lations. The researcher investigates the unknown numberof latent classes needed to represent the heterogeneityof the sample by specifying the mixture models withan increasing number of latent classes, with each addedclass explaining more variation in the sample. Model fit

indices as well as substantive theory guide the researchprocess.

Recent research in this area is promising. Knowl-edge of heterogeneity is likely to be advantageous toprevention and intervention researchers in identifyingrisk and protective factors for initial and subsequentsubstance use among adolescents. To date, however,empirical growth mixture modeling studies have largelyfocused on only a single substance (e.g., T. E. Duncan,Susan, Strycker, Okert, & Li, 2002; Li, Duncan, & Hops,2001; B. Muthen, 2000; B. Muthen and L. K. Muthen,2000; B. Muthen and Shedden, 1999).

Efforts at developing and providing more effectiveprevention and intervention strategies for adolescents atrisk for problem behavior are dependent on understand-ing the relationship among use of different substancesand other problem behaviors, as well as identifying andunderstanding the subgroups of at-risk adolescents andtheir substance use developmental patterns, antecedents,and outcomes. The aim of this study was to combinemultivariate LGM and growth mixture modeling, byexpanding multivariate LGM techniques to the analysisof heterogeneity in adolescent alcohol and marijuana useover time. These two substances were chosen because oftheir reported parallel development in use over the years(Johnston, O’Malley, & Bachman, 2002), and becausethey remain two of the most widely used and abused drugsin America (Bukstein, 1995). Patterns of simultaneousalcohol and marijuana use were examined, as werehypothesized covariates of adolescent substance use. Onthe basis of prior literature highlighting the influence ofpeer and parent substance use (alcohol and marijuana usespecifically) on adolescent substance use (e.g., Bukstein,1995; T. E. Duncan, S. C. Duncan, & Hops, 1996; Elder,1980; Hansen et al., 1987; Hops, T. E. Duncan, & S. C.Duncan, 1996), these covariates were included in themodel to determine their influence on initial use andthe developmental trajectories of adolescent alcohol andmarijuana use. Age and gender were also controlled forin the analyses. Lastly, an outcome variable, substanceuse problems, was included as a covariate to determinewhether initial status and developmental trajectories ofalcohol and marijuana use among different subgroups hadvarying effects on subsequent substance use problems.

This article serves to illustrate the use of the asso-ciate finite growth mixture modeling approach to analyzemultivariate longitudinal adolescent substance use data.Although previous studies have documented the relation-ship between use of different substances at single pointsin time, few have investigated the growing evidence thatadolescents’ development of use of various substances areintercorrelated over time (S. C. Duncan & T. E. Duncan,

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Associative Latent Growth Mixture Modeling 257

1994, 1996). This study assesses relationships amongalcohol and marijuana use simultaneously and investi-gates subpopulations characterized by qualitatively differ-ent patterns of these substances over time. An overview ofthe GGMM technique is presented followed by a review ofthe results, limitations, and future utility of the technique.

METHODS

Participants

Data from the National Youth Survey (NYS; Elliott,Huizinga, & Menard, 1989) were used for this study.The NYS is a longitudinal study of 1,725 young people(targets and siblings) interviewed annually between 1976and 1980. The data for this study comes from the oldestchild in the household (N = 1, 044) who completed atleast one questionnaire during Years 1–3 and Year 5.Participants ranged in age from 11 to 17 years at the firsttime point (Mean age = 14.47, SD = 1.95). Study parti-cipants were 45% female, 82% Caucasian, 13% AfricanAmerican, 4% Latino, and 1% Asian or American Indian.Individuals were not paid for participation. Informedconsent was given by participants.

Analyses involving multivariate longitudinal data arefrequently plagued by missing values. Subjects may beunavailable during one of the data collection sessions ormay prematurely drop out of the study. Although the cur-rent analysis targeted the oldest child in the household(N = 1, 044) to avoid dependency issues, only 698 cases(67%) had complete data. The substantial reduction incases with listwise deletion was primarily due to miss-ing data on the distal outcome taken from Year 5 of thestudy. Of the 1,044 cases, 4.79% were missing data formarijuana use at Time 2, 5.65% were missing data formarijuana use at Time 3, 4.69% were missing alcohol useat Time 2, and 5.75% were missing data for alcohol useat Time 3. All cases had complete data for the precedingvariables at Time 1. Mplus (L. K. Muthen & B. Muthen,1998), the program used for the current analysis, is ableto handle missing data under the MAR assumption withMaximum Likelihood Estimation and was used to esti-mate missing values for the above continuous observedoutcomes. However, within the GGMM approach Mplusis not able to handle missing data for the covariates (peersubstance use at Time 2 with 14.46% missing and partic-ipant ethnicity with 0.10% missing) and the binary mix-ture indicator (substance use problem behavior at Time 5with 24.33% missing cases) in the model. For this rea-son, NORM (Schafer, 1994), an efficient missing dataprogram, was used to impute the values for the incom-

plete data set. A single imputation, rather than multipleimputations, was performed to estimate the values for thecovariates and binary mixture indicator because of the un-feasibility of estimating multiple fully specified mixturemodels. With multiple imputations, each imputed set ofdata is modeled and then the results are combined. TheGGMM approach presently requires precise starting val-ues for convergence and the modeling process can proveto be very difficult. It was therefore decided to use thesingle imputation approach.

Measures

Measures included (1) adolescent alcohol and mar-ijuana use, (2) peer substance use, (3) parent substanceuse, and (4) adolescent substance use problems assessedat Year 5 (T5).

Adolescent alcohol and marijuana use. Adolescentalcohol and marijuana use for the first three assessmentyears was used in the analysis. These items weremeasured via self-report items assessing use of eachsubstance in the past year. Responses were on a 9-pointscale. For alcohol and marijuana use, values included 1(Never), 2 (Once or twice per year), 3 (Once every 2–3months), 4 (Once a month), 5 (Once every 2–3 weeks),6 (Once a week), 7 (2–3 times per week), 8 (Once aday), and 9 (2–3 times per day).

Peer substance use. The measure of peer substance usewas based on the mean of two items from Year 2of the study: (1) “During the last year how many ofyour friends have used marijuana or hashish?” and(2) During the last year how many of your friendshave used alcohol?” Responses were on a 5-point scalewith values of 0 (None of them), 1 (Very few of them),2 (Some of them), 3 (Most of them), and 4 (All of them).

Parent substance use. Parent substance use was assessedusing the mean of two items from Year 2 of the study:(1) “In the past year how often have your parents usedalcohol?” and “In the past year how often have yourparents used marijuana or hashish?” Response valuesincluded 0 (Never), 1 (Once or twice), 2 (Severaltimes), and 3 (Often).

Adolescent substance use problems. The outcome mea-sure of substance use problems was created from fouritems measured during the fifth assessment (T5). Itemsincluded (1) “During the last year, have you beenarrested for any alcohol related offenses?” (2) “Duringthe last year, have you been arrested for any drug relatedoffenses?” (3) “In the past year, did you ever have todo anything illegal in order to get alcohol?” and (4) “Inthe past year, did you ever have to do anything illegal in

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Table I. Sample Descriptive Statistics for the Growth Mixture Modeling Variables

SubstancePot1 Pot2 Pot3 Alc1 Alc2 Alc3 Peer use Parent use use problems % Female Age T1 N

Mean 1.75 2.12 2.48 2.42 3.11 3.60 1.40 0.11 0.14 0.45 14.47 1,044SD 1.82 2.13 2.29 1.85 2.10 2.21 1.28 0.30 0.35Skewness 2.57 1.86 1.34 1.38 0.72 0.34 0.61 3.32 2.04Kurtosis 5.56 2.23 0.47 1.17 −0.64 −1.24 −0.63 12.59 2.14

order to get drugs?” The substance use problems scorewas coded 0 if the adolescent had not experiencedany problems with substance use at T5, or 1 if theadolescent had experienced any of the problems at T5.

Demographic Indicators

Age and gender were included in the models to con-trol for their effects. Age was a continuous variable andgender was coded (0) male and (1) female.

Descriptive statistics for the variables used in theanalyses of the multivariate growth mixture models arepresented in Table I.

Analytical Model

Recently, researchers have begun exploring newways of constructing more complex and dynamic mod-els that are better suited for assessing change (Collins &Sayer, 2001). Most recently, B. O. Muthen (2001) pro-posed an extension of current LGM methodology thatincludes relatively unexplored mixture models, such asgrowth mixture models, mixture structural equation mod-els, and models that combine latent class analysis andstructural equation modeling.

Relevant to longitudinal research is the growth mix-ture modeling approach, combining categorical and con-tinuous latent variables into the same model. Muthenand colleagues (B. Muthen, Brown, Khoo, Yang, & Jo,1998; L. K. Muthen & B. Muthen, 1998; B. Muthen &Shedden, 1999) described in detail the generalization ofLGM to finite-mixture latent trajectory models and pro-posed a GGMM framework. The GGMM approach al-lows for unobserved heterogeneity in the sample, wheredifferent individuals can belong to different subpopula-tions. The modeling approach provides for the joint esti-mation of (a) a conventional finite mixture growth modelwhere different growth trajectories can be captured byclass-varying means, and (b) a logistic regression of out-come variables on the class trajectory. The model can befurther extended to estimate varying class membership

probability as a function of a set of covariates (i.e., foreach class, the values of the latent growth parameters areallowed to be influenced by covariates), and to incorporateoutcomes of the latent class variable.

Figure 1 displays the full growth mixture model es-timated in this study within the framework of Muthen andcolleagues. This model contains a multivariate growthmodel with four continuous latent growth variables, η1,and η2 (η1 = Marijauna Intercept [F1] and η2 = Mari-juana Slope [F2]) and η3 and η4 (η3 = Alcohol Intercept[F3] and η4 = Alcohol Slope [F4]), and a latent categoricalvariable, C , with K classes, Ci = (c1, c2, . . . , cK )′, whereci = 1 if individual I belongs to class k and zero otherwise.These latent attributes are represented by circles in Fig. 1.The latent continuous growth variable portion of the modelrepresents a multivariate growth model with multiple mea-sures (marijuana and alcohol use) each with multiple indi-cators, Y , measured at three time points (Willett & Sayer,1994). The categorical latent variable is used to representlatent trajectory classes underlying the latent growth vari-ables, η. Both latent continuous and latent class variablescan be predicted from a set of background variables or co-variates, X (e.g., age, gender, peer use, parent use), sincethe model allows the mixing proportions to depend onprior information and/or subject-specific variables. Thegrowth mixture portion of the model can have mixtureoutcome indicators, U (substance use problems). In thismodel, the directional arrow from the latent trajectoryclasses to the growth factors indicates that the intercepts ofthe regressions of the growth factors on X vary across theclasses of C . The directional arrow from C to U indicatesthat the probabilities of U vary across the classes of C .

The degree to which the latent classes are clearlydistinguishable can be assessed by the estimated posteriorprobabilities for each individual. By classifying each in-dividual into their most likely class, a table can be createdwith rows corresponding to individuals classified into agiven class and column entries representing the averageconditional probabilities (Nagin, 1999). Diagonals closeto 1 and off-diagonals close to 0 represent good classi-fication rates. A summary measure of the classificationis given by the entropy measure (Ramaswamy, DeSarbo,

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Fig. 1. Graphical representation of the associative growth mixture model.

Reibstein, & Robinson, 1993) where

Ek = 1 −∑

i

∑k(− pik ln pik)

nlnK(1)

and pik denotes the estimated conditional probabilityfor individual I in class k. Entropy values range from 0to 1, with values close to 1 indicating greater clarity inclassification.

B. O. Muthen (2001) considers growth mixture mod-eling as a second generation of structural equation mod-eling. Indeed, the general framework outlined by B. O.Muthen (2001) provides new opportunities for growthmodeling. Growth mixture models are applicable to lon-gitudinal studies where individual growth trajectories areheterogeneous and belong to a finite number of unob-served groups. The application of mixtures to growth mod-eling may also be used as an alternative to cluster-analytictechniques if the posterior probability of membership ofan individual in a latent class is used to assign latent classmembership.

Model Specifications

Let πik = P(Ci = k) denote the probability that thei th response falls in the kth category. For example πi1 is the

probability that the i th response belongs to the category 1(∑K

i=k πik = 1 for I , where the probabilities add up to onefor each individual and there are K −1 parameters).

In a multinomial distribution, the probability distri-bution of the counts ci j , given the total N is

P{Ci1 = ci1, Ci2 = ci2 . . . Cik = ci K }

=(

Nci1, . . . , cik

)π ci1

i1 . . . πci Ki K . (1)

Equation 1 becomes binomial in the case of K = 2. Interms of response probabilities, the model for the multi-nomial logit is

π (ci ) = exp{�ik}∑kk=1 exp{�ik}

(2)

�ik = logπik

πi K

Let u represent a binary outcome of the latent classanalysis (LCA) model and c represent the categorical la-tent variable with K classes. The individual (marginal)probability density of UI is

P(ui = 1) =K∑

k=1

P(c = k)P(ui = 1|c = k). (3)

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If the model relates c to covariate x by means of alogit multinomial model, Eq. 3 would be

P(cik = 1|xi )exp(ack + γck xi )∑K

k=1 exp(ack + γck xi )

where

{cik = 1 if cik ∈ K

otherwise cik = 0. (4)

A multinomial logistic regression uses the standardizationof zero coefficients for ack = 0 and γk = 0. Therefore, thelogit c for the odds of class K is

log it c = log[P(ci = k|xi )/P(ci = K |xi )] = ack + γk xi .

Consider the model presented in Fig. 1 in which twooutcome variables are measured repeatedly across threetime points (y1, y2, y3, and y4, y5, y6). In addition, twolatent growth factors for each set of repeated measuresand seven latent classes C = 1, 2, 3, . . . 7) are postulated,with four covariates (X1–X4) and one mixture indicator(U1). The following equation defines the conventional la-tent growth model for the continuous observed variablesY with continuous latent variables η for individual i across3 years of data:

Y = η + , (5)

where Y is a vector (Y ′ = yi1, yi2, . . . , yit ) contain-ing scores for individual i (i = 1, 2, . . . , N ) at t (t =1, 2, . . . , m) occasions, η is defined as a p × 1 vector of in-tercept and linear factors, and is an m × p design matrix(or basis functions; Meredith & Tisak, 1990) representingspecific aspects of change. Based on information fromFig. 1, m = 6, p = 4, and and η are 6 × 4 and 4 × 1matrices, respectively. Columns 1 and 3 of are defined asthe intercept factors by fixing all loadings at 1.0. Columns2 and 4 are defined as the slopes (linear rate of change) bysetting the loadings λ12, . . . , λ32 and λ44, . . . , λ64 equalto the values of yearly measurement (t = 0, 1, 2) for in-dividual i (Note that λ12 and λ44 are set to 0 so that theintercepts can be interpreted as the predicted value of theresponse variable at the 1st year of measurement). is a6 × 1 vector of residual terms for individual i . The modelexpressed in Eq. 5 has the matrix form:

y1

y2

y3

y4

y5

y6

=

1 λ12 0 01 λ22 0 01 λ32 0 00 0 1 λ44

0 0 1 λ54

0 0 1 λ64

η0

ηs

η0

ηs

+

ε1

ε2

ε3

ε4

ε5

ε6

. (6)

In fitting this model, estimates of the factor loadings in, the variances and covariances of the latent factors

(η0, ηs) in , and the means (α0, αs) of the latent factorscores in η can be obtained.

The two different curve shapes are hypothesized tobe captured by class-varying random coefficient means foreach K -1 class, αc[η0, ηs]. When including the effect ofcovariates in the model the continuous latent variables η

for individual i are linked to the categorical latent variableC and to the observed covariant vector X (where X =x1, x2, . . . , xN ) through the following equation:

ηi = Aci + �η Xi + ζi , (7)

where ACi equals an m × 1 logit parameter vector varyingacross the K classes, �η is an m × p parameter matrixcontaining the effects of X on η given, C categories, and ζ

is an m-dimensional residual vector, normally distributed,uncorrelated with other variables, and with zero mean andcovariance matrix �k .

Binary outcome variables, U , contain both a mea-surement model and a structural model (L. K. Muthen& B. Muthen, 1998). In the measurement portion of themodel, the r binary variables Ui j are assumed to be condi-tionally independent given Ci and Xi , with the followingconditional probability decomposition:

P(Ui j |Ci , Xi ) = P(ui1, ui2, . . . , uir |Ci , Xi )

= P(ui1|Ci , Xi )P(ui2|Ci , Xi ) . . . P(uir |Ci , Xi ). (8)

We define τi j = P(ui j = 1|Ci , Xi ), the r -dimensionalvector τi = (τi1, τi2, . . . , τir )′, and the r -dimensionalvector logit (τi ) = (log[τi1/(1 − τi1)], log[τi2/(1 − τi2)],. . . , log[τir/(1 − τir )])′. The logit model is therefore

logit(τi ) = uCi + Ku X p (9)

where u is an r × K parameter matrix and Ku is anr × q parameter matrix and τi jk = P(Ui jk = 1|Cik = 1).For the model in Fig. 1, where U = (U1), X = (X1–X4),and C = 1, 2, . . . , 7, the matrices of u and Ku would be

u = [λ11,12,13,...,16]′, Ku = [k11,12,13,14].

In the structural portion of the model, the categoricallatent variables of C represent mixture components thatare related to X through a multinomial logit regressionmodel for an unordered polytomous response. Definingπik = P(Cik = 1|Xi ), the K − 1 dimensional vector πi =(πi1, πi2, . . . πik)′, and the K − 1-dimensional vectorlogit

(πi ) = (log[πi1/πik], log[πi2/πi K ], . . . ,

log[πi,K−1/πi K ])′, logit (πi ) = αc + �c Xi , (10)

where αC is a K − 1 dimensional parameter vector and�C is a (K − 1) × p parameter matrix. Again, letting

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C = 1, 2, . . . 7 and X = (X1–X4), α and � of the latentclass regression model part of the model are

α = [αc1,c2,...,c6 ]′, � =

γ11 γ12 γ13 γ14

γ21 γ22 γ23 γ24

γ31 γ32 γ33 γ34

γ41 γ42 γ43 γ44

γ51 γ52 γ53 γ54

γ61 γ62 γ63 γ64

.

where α is a vector containing the regression interceptsand � is a matrix containing the regression coefficients.

In the model of Eqs. 5 through 10, the finite mixturearises because the conditional distribution of Y and Ugiven X is governed by parameters that vary across thecategories of C ; the mean vector of Y is allowed to varybecause of the inclusion of C in Eq. 8 and the probabilitiesof U are allowed to vary because of the inclusion of C inEq. 10.

Model Estimation and Evaluation

Mplus (L. K. Muthen & B. Muthen, 1998) usesthe principle of maximum likelihood estimation and em-ploys the Expectation Maximization (EM) algorithm formaximization and bootstrapping standard errors. A thor-ough description of the procedure is given by B. Muthenand Shedden (1999). Model evaluation for growth mix-ture models proceeds much like in conventional structuralequation modeling (SEM) or latent growth models for ho-mogeneous populations. Let the density likelihood func-tion of y = {y1, y2, . . . , yN } be

n∏i=1

K∑k=1

π j f (yi/θk), (11)

which leads to K ′′ terms when the inner sums are ex-tended. To form a complete data set from the observeddata yi , covariate (xi ) and unobserved (C) data in the formof an indicator variable C = {cik(i = 1, 2, . . . . . . , N , k =1, . . . . . . , K )} are created where{

cik = 1 ifcik ∈ K

cik = 0 otherwise

and cik can be considered a classification of yi . The loglikelihood function for the complete data is

log Lc =N∑

i=1

K∑k=1

cik log f {(y|θk, xi ) +N∑

i=1

K∑k=1

cik log πik

(12)The EM algorithm is used to maximize this log likelihoodfunction. The EM algorithm is used in mixture modelestimation by treating unobserved membership of the

observations as missing data and creating a complete dataset for the model (Wang, Puterman, & Cockburn, 1998).The E step of the EM process involves obtaining the ex-pectation of this log likelihood conditioned over the unob-served data. This replaces the missing or unobserved ciksin log Lc with posterior probabilities The M step of theEM process involves maximizing the resulting conditionallog likelihood with these estimated posterior probabilitiesconsidered as known. To perform the E step, we take theexpectation (13) over cik by giving θ (0), x (0), u(0), and y.Then,

∂ Ec{log Lc}∂θk

=N∑

i=1

zik∂ log f (yi |xi , θk)

∂θ= 0, (13)

where zik represents the posterior probabilities and can becalculated as

P(class j |yi ) = f (yi class j)P(class j)k∑

k=1f (yi |class j)P(class j)

= zik (14)

The ML estimators of the class probabilities, zik , are com-puted using sample averages of the estimated weights(Wang et al., 1998),

zik =∑N

i=1 zik

N. (15)

When the prior probabilities have been used, the condi-tional log function of the M likelihood step is a weightedmultinomial logit log likelihood.

Model fit for a mixture analysis is evaluated in part bythe log likelihood value. Using chi-square-based statistics(i.e., the log likelihood ratio), the fit for nested modelscan be examined. It is, however, not appropriate to usesuch values for comparing models with different numbersof classes, which is a necessary first step in evaluatingthe number of classes needed to explain the data hetero-geneity. In these instances, Akaike Information Criterion(AIC) and Bayesian Information Criterion (BIC) can beused instead. Mplus also provides a sample-size adjustedBIC (ABIC), which has been shown to give superiorperformance in a simulation study for latent class analysismodels (Yang, 1998). The Vuong–Lo–Mendell–Rubinlikelihood ratio test also provides a standard of compar-ison for ascertaining the preferred number of classes in amodel. This test is only appropriate for means-only, non-varying variance-covariance models in which only themeans are allowed to vary (Lo, Mendell, & Rubin, 2001).The p value provided by the test indicates the probabilitythat the H0 model, the model with one less class, istenable. The test is generated by eliminating the first classin the user-specified, estimated model. A resulting p value

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greater than .05 suggests that the H0 model cannot be re-jected. Conversely, a p value less than .05 indicates that theestimated model is preferable over the reduced model. Allmodels were estimated using the Mplus (L. K. Muthen &B. Muthen, 1998) structural equation modeling program.

This study combined multivariate LGM and growthmixture modeling to determine the existence of subgroupsin developmental patterns of adolescent alcohol and mar-ijuana use measured at three yearly assessments (T1–T3).The study also examined the impact of two social contextindicators (parent and peer substance use) on (1) the initialstatus of adolescent alcohol and marijuana use correctedfor measurement error; (2) the rate of change in thesesubstances over time; and (3) latent class membership,controlling for adolescent age and gender. The influenceof latent class membership and parent and peer substanceuse on subsequent substance use problems measured atT5 were also explored. The conditional model, in whichthe covariances were freed across latent classes andallowed to vary, included the effects of several covariates(age, gender, parent and peer substance use) on the growthparameters and latent classes, and of latent class member-ship on subsequent substance use problems. Because C isan unordered categorical latent variable with seven cate-gories, the interpretation of the effect of the regression ofC on the covariates, X , is not the same as for other latentvariable models using continuous latent variables. Instead,it represents the multinomial logistic regression of C on X .Regression equations are specified for the first six classeswhereas the intercept (latent class mean) and slope (logis-tic regression coefficient) for the last class are fixed at 0 asthe default. Latent Class 7 was selected as the default be-cause it most closely represents normative growth in bothalcohol and marijuana use. In this model, U represents adistal outcome predicted by class membership. The effectof U on C indicates that the probabilities of U vary acrossthe classes of C . Input specifications for the effects of thecategorical latent class variable on the mixture indicatorinclude starting values in the logit scale for the thresholdof the latent class indicator. The thresholds define theconditional probabilities of U for each class. Startingvalues are required for the estimation of these thresholds.The regressions of U on C are specified as varyingparameters in each of the latent classes. Mplus programcode necessary to estimate the conditional associativegrowth mixture model can be found in the Appendix.

RESULTS

Model fit was first evaluated for the means-onlymodel where the covariances were fixed across the latentclasses and not allowed to vary. The ABIC was evaluated

for 6th-, 7th-, and 8th-Class means-only models to deter-mine the optimal number of classes needed to fit the data.The Vuong–Lo–Mendell–Rubin likelihood ratio test wasalso evaluated for the three models, as were model entropyvalues that represent the degree to which the latent classesare clearly distinguishable.

Entropy values for the tested 6th-, 7th-, and 8th-Classmeans-only models (those without covariates) were .925,.940, and .920, respectively, with the 7th-Class modelshowing a higher entropy value than the other two models,indicating the most optimal classification rates. ResultingABIC values were estimated at 20570.847, 20126.781,and 20228.417 for the 6th-, 7th-, and 8th-Class models,suggesting better fit for the 7th-Class model. The Vuong–Lo–Mendell–Rubin test p values for the 7th- and 8th-Classmodels indicated preferred model fit for the 7th-Classmodel with a p value of 0.018 (7th Class vs. 6th Class),while the 8 Class model test resulted in a p value of 0.105,indicating the H0 model could not be rejected and providedfurther evidence in support for the 7th-Class model.

Model fitting procedures for the 7th-class conditionalmodel depicted in Fig. 1 were evaluated with the log likeli-hood H0 value, BIC value, and entropy estimate. The finalmodel resulted in a log likelihood H0 value of −9389.868,BIC value of 20218.554, and an entropy estimate of .944.Class counts and proportion of cases out of the total sam-ple size for the conditional seven latent class model were66 (0.063) for Class 1, 122 (0.117) for Class 2, 81 (0.077)for Class 3, 47 (0.045) for Class 4, 33 (0.032) for Class5, 142 (0.136) for Class 6, and 553 (0.530) for Class 7.Figure 2 displays the trajectories of the within-class meanestimates for the seven latent classes in graphical form andindividual class descriptives are presented in Table II.

The coefficients for the regression of the growth fac-tors on X (Fig. 1) are presented in Table III. As can beseen from Table III, the participant’s age at Time 1 had asignificant effect on the alcohol intercept in Classes 2, 6,and 7, suggesting that the older members of each class hadhigher alcohol use at Time 1 compared to the rest of theirrespective class members. Age at Time 1 also had a signif-icant effect on the intercept of marijuana use for Classes 2,4, 5, and 7. These findings again suggest that older mem-bers of Classes 2, 4, 5, and 7 used more marijuana at Time1 compared to the rest of their respective class members.Age was also found to have a significant effect on the slopeof alcohol for Class 1, indicating that older children in thisclass had steeper trajectories of alcohol use.

Gender had a significant negative effect on thealcohol intercept for Class 5, suggesting that females inthis class used less alcohol at Time 1 than did males inthis class. A negative effect for gender was found on thealcohol slope of Class 4, indicating that males in this

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Fig. 2. Conditional within-class mean trajectories for the seven latent classes.

class had steeper alcohol trajectories over time. Genderhad a significant effect on the intercept of marijuanause for Classes 2 and 5, indicating that the females inthese two classes used more marijuana at Time 1 than themales in these classes. The effect of gender on the slopeof marijuana had a negative effect for Classes 1, 2, and5. These findings indicate that the males in these threeclasses had steeper marijuana trajectories over time.

A significant positive effect for peer substance usewas found for the alcohol intercept in Classes 1, 2, and 7,and for the alcohol slope in Classes 2 and 7. Peer substanceuse had a significant positive effect on the marijuana in-tercept in Classes 2, 5, and 7 and a negative effect on themarijuana intercept for Classes 1 and 4. Peer use had apositive effect on the slope for Classes 1, 4, and 7.

Parent substance use had a significant negative effecton the alcohol intercept for Class 1 and a positive effectfor Class 6. Parent substance use had a significant negativeeffect on the marijuana intercept for Classes 2, 4, and 5,whereas a positive effect was found on the marijuana slopeof Classes 4 and 5.

Table II. Class Descriptive Statistics for the Growth Mixture Modeling Variables

SubstancePot1 Pot2 Pot3 Alc1 Alc2 Alc3 Peer use Parent use use problems % Female Age T1 N

Class:1 7.46 6.49 6.25 6.06 5.66 6.09 3.30 0.28 0.35 0.36 15.51 662 1.27 3.66 6.13 2.74 4.22 5.35 2.26 0.13 0.23 0.43 14.94 1223 1.12 1.65 1.87 5.07 5.37 5.06 1.86 0.14 0.22 0.32 15.44 814 5.20 4.74 5.26 4.23 4.60 5.29 2.63 0.25 0.30 0.43 15.76 475 3.49 3.81 3.68 4.12 4.96 4.91 2.46 0.07 0.06 0.60 16.33 336 1.08 1.38 1.35 2.00 3.89 5.44 1.65 0.09 0.13 0.50 15.21 1427 1.05 1.16 1.25 1.34 1.76 1.90 0.67 0.08 0.08 0.46 13.67 553

Note. Class sample statistics are weighted by estimated class probabilities.

The coefficients for the multinomial logistic regres-sion of the latent classes on X (Fig. 1) are presented inTable IV. As can be seen from Table IV, the coefficientsshow how the odds of belonging to a particular latent class(e.g., Classes 1–6) compare to the normative class (Class7). For latent Classes 3, 4, 5, and 6, the odds of belong-ing to these latent classes compared to the normative class(Class 7) are significantly increased for those individu-als who are older and peer substance use was found tohave a significant positive effect for all latent classes com-pared to the normative class. For latent Class 1 and 4, theodds are increased for those individuals whose parents usesubstances.

The model also includes the prediction of substanceuse problems (U ) by trajectory class. Probability estimatesfor the regression of U on C are presented in Table V.While the normative class has a probability of .085 ofdeveloping problems with substance use at T5, the otherclasses, with the exception of Class 5, which has a proba-bility of .062, have substantially higher probabilities rang-ing from .126 to .348. Table V also presents the odds

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Table III. Coefficients for the Regression of the Growth Factors on X

Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7

Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic

Alc Int onAge −0.212 −1.257 0.340 3.868 0.006 0.033 0.079 0.386 −0.326 −0.622 0.163 3.000 0.056 2.791Gender −0.497 −1.153 −0.270 −0.805 0.177 0.212 −0.321 −0.647 −1.588 −2.447 −0.063 −0.172 −0.076 −0.584Peer use 0.256 3.102 0.295 2.711 0.396 1.860 0.432 1.368 −0.053 −0.140 0.200 1.847 0.256 3.102Parent use −0.780 −2.088 −0.176 −0.333 −0.167 −0.372 0.380 0.859 0.196 0.082 0.705 2.399 0.114 1.037

Alc Slp onAge 0.297 2.713 −0.099 −1.440 0.078 0.860 −0.030 −0.282 0.391 1.438 0.011 0.212 0.016 0.769Gender −0.263 −1.017 −0.066 −0.282 −0.595 −1.888 −0.674 −2.346 0.135 0.257 −0.249 −1.310 −0.032 −0.459Peer use 0.208 1.829 0.111 2.233 0.010 0.051 0.161 1.057 −0.056 −0.328 −0.095 −1.465 0.111 2.233Parent use 0.232 0.975 0.182 0.418 −0.112 −0.365 0.099 0.516 0.562 0.721 −0.173 −0.756 0.146 1.461

Pot Int onAge −0.055 −0.647 0.015 2.269 0.020 1.119 0.270 2.000 0.015 2.269 0.018 1.448 0.015 2.269Gender 0.323 1.400 0.272 2.518 −0.026 −0.314 0.727 1.557 0.272 2.518 −0.003 −0.073 −0.028 −1.450Peer use −0.275 −3.414 0.016 3.065 0.075 1.455 −0.521 −2.640 0.061 3.065 0.043 1.613 0.061 3.065Parent use −0.132 −0.652 −0.364 −2.977 −0.090 −1.658 −1.478 −6.827 −0.364 −2.977 0.130 0.962 0.007 0.172

Pot Slp onAge 0.116 0.952 −0.049 −0.786 −0.080 −0.767 −0.281 −1.896 −0.054 −0.274 −0.021 −1.019 −0.007 −0.832Gender −1.150 −3.255 −0.602 −2.989 −0.238 −0.746 −0.516 −1.535 −1.811 −6.763 −0.104 −1.498 0.018 0.599Peer use 0.704 4.429 0.139 1.616 0.275 1.375 0.565 2.752 0.043 0.407 0.061 1.092 0.147 3.986Parent use 0.209 1.053 0.123 0.456 −0.041 −0.136 0.885 3.276 1.474 2.323 −0.047 −0.363 −0.042 −0.734

Note. All t values greater than 2.00 are significantly different from 0.

of having substance use problems at T5 in each class, andthe corresponding odds ratios when comparing a classwith the normative class. As can be seen from Table V,the odds of subsequent substance use problems are great-est for Classes 1, 2, 3, and 4. Of interest are the signif-icant odds for Classes 3 and 6, where only alcohol usewas elevated compared to the other classes where bothalcohol and marijuana use were substantially elevatedby T3.

DISCUSSION

Research involving the development of substance usehas routinely applied traditional latent variable growth

Table IV. Coefficients for the Multinomial Logistic Regression of the Latent Classes on X

Class 1 Class 2 Class 3 Class 4 Class 5 Class 6

Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic Effect t Statistic

Age 0.256 1.810 0.077 1.015 0.364 3.458 0.454 3.248 0.859 5.015 0.296 4.130Gender −0.324 −0.761 −0.034 −0.103 −0.510 −0.723 −0.034 −0.084 0.704 1.592 0.259 1.052Peer use 2.246 8.547 1.374 7.959 0.933 5.211 1.538 7.652 1.364 5.784 0.813 5.363Parent use 1.005 2.235 0.204 0.448 0.581 1.163 1.135 2.517 −0.311 −0.349 −0.036 −0.081

Note. Coefficients reflect the odds of belonging to the respective class compared to the normative class (Class 7) increased/decreased as a function ofthe covariate X .

modeling where a model contains observed continuousoutcomes or latent variable indicators, latent continuousvariables, and observed background variables. Althoughuseful in examining individual differences in growth, B. O.Muthen (2001) points to a number of limitations asso-ciated with the conventional latent variable approach tomodeling growth and development.

One limitation is an underlying assumption that thedata come from a single-population growth model encom-passing all different types of trajectories (i.e., all individ-uals belong to one and the same population). In particu-lar, it is assumed that covariates have the same influenceon the growth factors for all trajectories. This assump-tion may be unrealistic in research studying adolescentsubstance use and other problem behaviors. For example,

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Table V. Probability of Substance Use ProblemsGiven Latent Class Membership

Probability Odds Odds ratio

Class 1 .348 .533 5.743Class 2 .227 .294 3.168Class 3 .219 .280 3.016Class 4 .297 .422 4.554Class 5 .062 .066 0.708Class 6 .126 .144 1.556Class 7 .085 .093 1.000

as tested in this study, variation in adolescent marijuanaand alcohol use development may differ as a function ofparent and peer influences. Moreover, adolescents withdifferent growth trajectories may not only have differ-ent antecedents but also different growth shapes, concur-rent processes, and outcomes. Earlier starters in use ofone substance may have a steeper developmental trajec-tory compared to later starters; their use may be morelikely to co-occur with the development of other substanceuse and delinquent behaviors, and thus may increase theprobability of subsequent risk behaviors. Thus, the as-sumption that a single-population model can account forall types of individual differences may be overly restric-tive for studying adolescent substance use development. Agrowth mixture modeling approach enables the researcherto study qualitatively different developmental processesacross individuals belonging to several unobservedsubgroups.

Evidence for the interrelationships among adolescentproblem behaviors is substantial. Multivariate analyseshave shown that a single common factor can accountfor the relationships among these behaviors (Donovan &Jessor, 1985; Donovan, Jessor, & Costa, 1988; Farrell,Danish, & Howard, 1992; Osgood, Johnston, O’Malley,& Bachman, 1988), and where gender differences havebeen investigated, these interrelationships have beenfound to exist for both males and females (Donovan &Jessor, 1985; Farrell et al., 1992). In the present studysignificant and positive correlations between the slopesfor marijuana and alcohol use existed for the two latentclasses characterized by elevated levels of marijuana andalcohol use and an elevated risk of subsequent problembehavior, providing support for the idea of a problembehavior syndrome among problem adolescents. Despiteevidence that various problem behaviors and use ofvarious substances are intercorrelated, however, mostgrowth mixture studies have focused on growth in a singlebehavior over time, rather than on multiple behaviors.This study extended prior research by investigating an as-sociative growth mixture model for alcohol and marijuana

use simultaneously over three points in time and examinedthe correlational structure between these two measures.

Findings from this study suggested the existence ofseven subpopulations or latent classes defined by theirvarying developmental trajectories. The labeling of theselatent classes, which correspond to “cluster” in the data,was derived from the patterns observed in the class mem-bership probabilities. In the conditional model, the sevenclasses were characterized as follows. Class 7 presentednormative substance use development with low initial sta-tus and low to moderate growth in both substances overtime. Classes 3 and 6 presented low initial marijuana useand little use over time. However, these two classes showedvery different alcohol use patterns. Class 3 had high initialstatus on alcohol with sustained use over time and Class6 reported low initial alcohol use with a significant in-crease in use over time. Conversely, Class 2 showed lowinitial status on both substances with substantial use ofboth marijuana and alcohol over time. Classes 1, 4, and5 presented significantly different moderate to high initialstatus on both substances with continued use across allthree assessments.

Results from the conditional model indicateddiffering antecedents for the latent classes. For severalof the latent classes (e.g., 3, 4, 5, and 6), the oddsof belonging to these latent classes compared to thenormative class (Class 7) were significantly increasedfor older individuals and those whose peers engagedin substance use. Findings were consistent across thesefour latent classes even though the classes varied ondevelopmental trajectories, and differed substantially onboth intercepts and slopes of alcohol and marijuana use.For latent Class 2, characterized by accelerated growthin both alcohol and marijuana use, the odds of belongingto this class were significantly increased for individualswhose peers engaged in similar substance use behavior.For latent Class 3, characterized by a nonsignificantdevelopmental trajectory for both alcohol and marijuanause, but an elevated intercept for alcohol use, the oddswere increased for older individuals whose peers engagedin substance use. The effects of parent use on latent classmembership were significant in Classes 1 and 4.

Associations with deviant and substance using peershave often been associated with an acceleration in sub-stance use development (S. C. Duncan & T. E. Duncan,1996; Hawkins, Catalano, & Miller, 1992). Involvementwith such peers is likely to exacerbate the problematicbehavior of the adolescent, leading to social conditionsthat support even more extreme forms of the behaviors(Patterson & Dishion, 1985). In the present study, peersubstance use was predictive of membership in all latentclasses when compared to the normative class (Class 7).

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However, it was only found to have a significant effect onthe growth of alcohol use for Classes 2 and 7. The growthof marijuana use over time was influenced by peer use forClasses 1, 4, and 7. Such a result might indicate differ-ing effects for various subgroups of peer substance use onadolescent substance use development. No significant ef-fect for parent use was found on the growth of alcohol use.However, parent use had a significant effect on the growthof marijuana use for Classes 4 and 5, both of which weremoderate users at initial status and had continued moder-ate use over time.

The finding that parent use had an effect on mem-bership of only two latent classes is contrary to researchfindings documenting the influence of parent substanceuse on adolescent use (e.g., Hops et al., 1996). However,most prior studies have focused on single-sample analysesand have not examined the influence of parent substanceuse on adolescents’ membership in distinct substance usesubgroups. It also has been shown that peer, as opposed toparent, influences on adolescent behavior become greaterwith age and are particularly influential in the early stagesof substance use (Coombs, Paulson, & Richardson, 1991;Kandel, 1985). Clearly, further studies are needed to fur-ther explore the role of parent and peer substance use indetermining adolescent membership in different substanceuse subgroups.

The conditional model also included the effect oftrajectory class membership on subsequent substanceuse problems at T5. Findings revealed that the odds ofdeveloping substance use problems (e.g., being arrestedfor any drug- or alcohol-related offenses and engaging inillegal behaviors to get drugs or alcohol) were substan-tially increased for Classes 1, 2, 3, and 4 compared to thenormative class (Class 7). Combined with the effects ofparent and peer use, the increased odds for Classes 1 and4 suggest that peer, and especially parent, use might be akey factor in initiation and maintenance of substance useand that these concomitant effects place the adolescent atgreater risk for subsequent drug-related problems. Suchfindings suggest that problem substance use outcomesmay be reduced by targeting specific prevention andintervention efforts toward subgroups, based on theirinitial status and developmental trajectories of alcohol andmarijuana use. The existence of these specific subgroups,however, should be replicated before any definitiveconclusions can be drawn regarding such interventions.

Study Limitations

Several cautions to the reader in the interpretation ofthese data are in order. The behavioral outcomes of in-

terest, alcohol and marijuana use, utilized in this study,were based solely on self-reports by the adolescent andthus may be partly affected by confounds because of asingle informant perspective. In addition, the study exam-ined the relationship between only two contextual factors,peer and parent substance use. It is realized that these arenot the sole mechanisms that influence adolescent sub-stance use, and that there are a plethora of other variablesof interest such as race, age of onset, family life, economicbackground, and biological tendencies, which researchersshould consider including in any substantive model. Thedata collection period from 1976 to 1980 may also limitthe applicability of findings to current substance use de-velopment. However, adolescent substance use trend datahas shown that after a general peak in marijuana use inthe 1970s, there was a decline in the 1980s, followed bya resurgence in the 1990s that brought use statistics to apoint just below those reported in the mid-1970s (Johnstonet al., 2002). Caution is also warranted in the interpreta-tion of the problem behavior outcome due to limitationsassociated with analyzing a single imputation of a binaryvariable with a fair amount of missing data.

It should also be noted that age heterogeneity couldhave differentially impacted the findings had the data beenanalyzed within a cohort-sequential, or accelerated, de-sign. Here, subjects would be grouped into cohorts onthe basis of their age at the initial assessment, and classmembership would correspond to each cohort using amultiple-group approach. Although multiple imputationmethodology is available for cohort-sequential, or accel-erated, design, questions remain as to the appropriatenessof these methods for data that are missing by design. Cur-rent missing data methods use all observations in the datato estimate the parameters in a model using a maximumlikelihood estimator. The estimation assumes that data aremissing at random (MAR) or missing completely at ran-dom (MCAR). While tests of MCAR exist for these typesof data structures, they must be carried out within the mul-tiple sample framework (see T. E. Duncan et al., 1999,chap. 8). Moreover, because the accelerated design con-founds cohort and time effects, the estimation within themultiple sample framework allows the researcher to de-termine if lack of model fit can be attributed to either anincorrect specification of the hypothesized growth func-tion, or to a lack of fit that may be attributed to differencesthat exist between age cohorts (cohort effect).

Although we could not find any examples of a cohort-sequential LGM conducted within a mixture framework(Linda Muthen, personal communication, February 19,2004), Mplus (L. K. Muthen & B. Muthen, 1998), in futureversions, will have an option (KNOWNCLASS) to treattraining data (e.g., cohort designation) within the multiple

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groups approach allowing for a “behind the scene” esti-mation of cohort sequential designs within the mixtureframework.

From a methodological perspective, several caveatsare also expressed. At present, there are no clear rulesfor assuring identification of a particular mixture model.If the model is not properly identified, it is possibleto have many different sets of parameter estimatesthat describe the same growth process equally well.Therefore, it is necessary to undergo a somewhat te-dious process of finding suitable initial starting valuesand comparing model solutions. Although this studysuccessfully estimated the hypothesized models, cautionregarding model estimation seems warranted, given thedifficulty of arriving at permissible solutions in somesituations. User-specified start values for latent means,variances/covariances of growth parameters, and uniquevariances are often required for model convergence.Researchers should routinely use differing sets of initialvalues to verify that a stable solution has been achieved.

At present there is little guidance for determin-ing whether parameters vary significantly across latentclasses. This study was successful in freeing a numberof variances and covariances across classes. However,the process of freeing parameters in current versions ofMplus is still a trial- and- error process. Without furtherrefinement of the analytical technique, and the addition ofdiagnostic information pertaining to the appropriatenessof constraints placed across latent classes, questions re-garding what constitute an acceptable model solution willremain.

Despite these limitations, growth mixture modelingrepresents an important new development in the studyof change. With new modeling opportunities comes thepotential for exploring new and more complex theoriesof development in a plethora of behavioral fields ofstudy. Prevention research has now established a numberof strategies and intervention programs that are bothefficacious and effective. Therefore, a major emphasis ofprevention research programs is identifying and deter-mining content, implementation, context, and audiencecomponents that account for intervention strategy andprogram effectiveness. Understanding these componentsis key to both tailoring programs to meet the needs ofspecific subgroups and, when appropriate, to generalizingstrategies to other settings. NIH, for example, has iden-tified the development of prevention audience profiles,including methodologies appropriate for the identificationof individuals at risk for future drug abuse and dependence,as an area of importance in their prevention researchprograms. Researchers have already begun to use thesetechniques to aid them in the identification of individuals

at risk for future problem behaviors and in the develop-ment of targeted, rather than more global, interventionstrategies.

Naturally, there is no single statistical procedure forthe analysis of longitudinal data, as different researchquestions dictate different data structures and, subse-quently, different statistical models and methods. How-ever, the field of methodology for the analysis of changehas matured sufficiently that researchers have now begunto identify larger frameworks in order to integrate knowl-edge. The growth mixture modeling approach presentedhere is strengthened by its association with the generallatent variable modeling framework and the inclusion of amultivariate design. The LGM approach allows for a morecomprehensive and flexible approach to research designand data analysis than any other single statistical model forlongitudinal data in standard use by social and behavioralresearchers. The approach makes available to a wide audi-ence of researchers an analytical framework for a varietyof analyses of growth and developmental processes. Thepotential for integrating typical causal modeling featuresfound in a majority of SEM applications, the dynamic fea-tures of latent growth curve modeling, and the ability tomodel sample heterogeneity using a combination of con-tinuous and categorical latent variables within the mixtureframework described here, makes possible a more preciseunderstanding of the influence various competing social–contextual factors have on the development of a variety ofsubstance use and other problem behaviors.

APPENDIX: MPLUS PROGRAM CODE FORTHE CONDITIONAL ASSOCIATIVEGROWTH MIXTURE MODEL

The following program specifications correspond tothose necessary to test the hypothesized model (Fig. 1)utilizing the Mplus (L. K. Muthen & B. Muthen, 1998)structural equations modeling program.

Title: NYS Marijuana & Alcohol ModelData: File is nysmix 013003.dat;Variable: Names are aggid talc1yr1 talc1yr2 talc1yr3

tpot1yr1 tpot1yr2 tpot1yr3 prob5 peeruse parusetagel tgender1 tethnic1;

Missing = All (−99);Usevariables tpot1yr1 tpot1yr2 tpot1yr3 talc1yrl talc1yr2

talc1yr3 peeruse paruse tage1 tgender1 prob5;categorical = prob5;Classes = c(7);Analysis: Estimator = MLR;

Type = Mixture Missing; miteration = 5000;

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Model: %overall%pint by tpot1yr1-tpot1yr3@1;pslp by tpot1yr1@0 tpot1yr2@1 tpot1yr3@2;[tpot1yr1-tpot1yr3@0];tpot1yr1 tpot1yr2 tpot1yr3;pint∗.547 pslp∗.419;pint with pslp∗.0235;aint by talc1yr1-talc1yr3@1;aslp by talc1yr1@0 talc1yr2@1 talc1yr3@2;[talc1yr1-talc1yr3@0];talc1yr1 talc1yr2 talc1yr3;aint∗.165 aslp∗.218;aint with aslp∗.2320;aslp with pslp∗.030;aint with pint∗.038;c#1 on tage 1 tgender 1 peeruse paruse;c#2 on tage 1 tgender 1 peeruse paruse;c#3 on tage 1 tgender 1 peeruse paruse;c#4 on tage 1 tgender 1 peeruse paruse;c#5 on tage 1 tgender 1 peeruse paruse;c#6 on tage 1 tgender 1 peeruse paruse;pslp on tagel∗.0121;pslp on tgender1∗-.002;slp on peeruse∗-.0121;slp on paruse∗-.021;int on tage1∗.0121;int on tgender1∗-.021;int on peeruse∗-.0121;pint on paruse∗.035;aslp on tagel∗.0121;aslp on tgenderl∗-.002;aslp on peeruse∗-1.087;aslp on paruse∗-.021;aint on tagel∗.0121;aint on tgender1∗.0121;aint on peeruse∗.1373;aint on paruse∗.035;%c#1%[aint∗5.647 aslp∗.250];[pint∗7.231 pslp∗-.989];[prob5$1∗0.554];aint with aslp∗.020;pint with pslp∗-.015;aslp with pslp∗-.125;aint with pint∗.0255;aint on tage1 tgender1 paruse;aslp on tage1 tgender1 peeruse paruse;pint on tagel;pint on tgender1;pint on peeruse;pint on paruse;pslp on tage1 tgender1 peeruse paruse;

%c#2%[aint∗2.810 aslp∗3.095];[pint∗1.243 pslp∗4.703];[prob5$1∗1.062];aint with aslp∗.020;aint∗.013;aslp with pslp∗-.125;aint with pint∗.0255;aint on tage1 tgender1 peeruse paruse;aslp on tage1 tgender1 paruse;pslp on tage1 tgender1 peeruse paruse;%c#3%[aint∗4.000 aslp∗.003];[pint∗1.051 pslp∗.003];[prob5$1∗1.069];aint with aslp∗.020;pint with pslp∗-.015;aslp∗.015;aslp with pslp∗-.125;aint with pint∗.0255;aint on tage1 tgender1 peeruse paruse;aslp on tage1 tgender1 peeruse paruse;pint on tage1;pint on tgender1;pint on peeruse;pint on paruse;pslp on tage1 tgender1 peeruse paruse;%c#4%[aint∗2.859 aslp∗.038];[pint∗5.158 pslp∗.471];[prob5$1∗1.786];aint with aslp∗.020;pint with pslp∗-.015;aint∗.013;aslp∗.015;pslp∗.020;aslp with pslp∗-.125;aint with pint∗.0255;aint on tage1 tgender1 peeruse paruse;aslp on tage1 tgender1 peeruse paruse;pint on tage1;pint on tgender1;pint on peeruse;pint on paruse;pslp on tage1 tgender1 peeruse paruse;%c#5%[aint∗4.390 aslp∗1.035];[pint∗3.931 pslp∗.566];[prob5$1∗1.404];aint with aslp∗.020;aslp with pslp∗-.125;aint on tage1 tgender1 peeruse paruse;

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aslp on tage1 tgender1 peeruse paruse;pslp on tage1 tgender1 peeruse paruse;%c#6%[aint∗2.586 aslp∗1.918];[pint∗1.150 pslp∗.452];[prob5$1∗2.043];aint on tage1 tgender1 paruse;aint on peeruse;aslp on tage1 tgender1 paruse;aslp on peeruse;pint on tage1 tgender1 peeruse paruse;pslp on tage1 tgender1 peeruse paruse;%c#7%[aint∗1.400 aslp∗.483];[pint∗1.351 pslp∗.273];[prob5$1∗2.861];aint on tage1;aint on tgender1;aint on paruse;aslp on tage1;pint on tgender1;pint on paruse;pslp on tgender1;pslp on peeruse;Output: tech1 tech7 tech10 tech12;

SAVEDATA:save = cprobabilities;file is c:\\cprobsav1.dat;format is free;

ACKNOWLEDGMENTS

This research was supported by Grant DA09548 fromthe National Institute on Drug Abuse. Partial support inpreparing this article was provided by Grant AA11510from the National Institute on Alcohol Abuse and Al-coholism. The National Youth Survey was supported byGrant No. MH27552 from the Center for Studies of Crimeand Delinquency, NIMH.

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