EVALUATION OF THE EFFECTS OF THE ABAN AYA YOUTH PROJECT IN REDUCING VIOLENCE AMONG AFRICAN AMERICAN ADOLESCENT MALES USING LATENT CLASS GROWTH MIXTURE MODELING TECHNIQUES

Abstract

This study employs growth mixture modeling techniques to evaluate the preventive effects of the Aban Aya Youth Project in reducing the rate of growth of violence among African American adolescent males (N = 552). Results suggest three distinct classes of participants: high risk (34%), medium risk (54%), and low risk (12%) based on both the participants’ initial violence scores and their growth of violence over time. Results further show significant effects (almost 3 times as large as the effect found in the regular one-class analysis) for the high-risk class but not for the medium- or low-risk classes.

Evaluation of program effects for longitudinal randomized trials (LRTs) are sometimes fraught with methodological problems and data analytical challenges (August et al. 2003). The mixed effect model is suited to evaluate LRTs because it can properly analyze data with nested structure (e.g., observations nested within a subject) and missing observations (Raudenbush and Bryk 2002). Therefore, the model is widely used to evaluate LRTs. However, it assumes that a response variable is distributed normally over subjects. This assumption is often violated in the study of problem behaviors, such as violence. Problem behaviors are often prevalent in only a small proportion of a target population. Therefore, its distribution is often quite different from the bell-shaped normal distribution and may be highly skewed and/or multimodal. Such a distribution might be better represented by a latent class model.

We present a simple example to explain how the nonnormal distribution is approximated in a latent class model. Suppose that 1,000 subjects are sampled and their violence score is measured by a sum of 10 binary items (total score ranges from 0 to 10). Sixty percent of subjects score 0, 30% score 10, and the remaining 10% are evenly spread between 0 and 10. Such a distribution can be well approximated by a histogram with two bars at 0 and 10 whose proportion is two thirds and one third respectively (discriterization). If the scores spread more in both lower and higher groups, say, two thirds of subjects are in lower part spreading from 0 to 5 with its mode at 3 and one third of subjects are in higher part spreading from 6 to 10 with its mode at 8, then the distribution might be well approximated by two normal distributions with means at 3 and 8 (i.e., a mixture of normals).

The advantages of latent class models are not limited to the statistical (i.e., good approximation) but also are of substantive interest because we can often interpret the nonnormal distribution meaningfully and the representation gives us insight into the problem. For example, it is more informative and correct to state that the target population consists of 33% and 66% of violent engaging and nonengaging subpopulations whose means are 8 and 3 respectively rather than average of the score of 4.6 (= 8 × 0.33 + 3 × 0.66). Such models are called latent class (Hagenaars and McCutcheon 2002), semi-parametric (Nagin 1999), or finite mixture (Bauer and Curran 2003) models. We continue to use “latent class model” in the rest of this article.

Recent methodological advances combine the advantages of the mixed effect and the latent class models (D’Unger et al. 1998; Muthen et al. 2002; Muthen and Muthen 2000). We call the combined model the latent class growth mixture model following Muthen and Muthen (2001). Because it is a hybrid of the two models, it inherits an advantage of each model; that is, it can properly (a) approximate a nonnormal distribution of response variables over subjects and (b) analyze nested structured data with missing observations.

The Aban Aya Youth Project (AAYP) was a longitudinal efficacy trial designed to compare three interventions: School-Community (SC), Social Development Curriculum (SDC), and Health Enhancement Curriculum (HEC). AAYP was implemented in Grades 5 through 8 in 12 elementary schools in Chicago and the surrounding suburbs. The two experimental interventions, SDC and SC, focused on reducing health-compromising behaviors (violence, substance use, and unsafe sexual practices), whereas the control intervention, HEC, focused on promoting health-enhancing behaviors (physical activity, nutrition, and self-care). For more detailed descriptions of these research conditions including the essential differences between (and among) the research conditions, refer to previous analyses of the AAYP (Flay et al. 2004).

The present study employs the latent class growth mixture modeling techniques modeled after the work of Bengt Muthen and his colleagues (2002) to assess intervention effects on subgroups of the participants. When data are collected on many individuals over several observations, this methodology is capable of assessing individual growth or development curves, then sorting participants into their most likely subgroups or classes, and finally accounting for program effects on each subgroup. This important feature is considered especially relevant in analyzing longitudinal growth data (Muthen et al. 2002; Muthen and Muthen 2000; D’Unger et al. 1998).

There is evidence to show that cross-time intraindividual trajectories may be better measured (approximated) as subgroups of growth curves rather than as a single population representing the overall parameters of the entire population. In looking at drinking behavior among young adults, Muthen and Muthen (2000) identified three major trajectories of alcohol drinking behavior: low use class, early onset class, and escalating class. Colder and colleagues (2002, 2001) found five trajectories of alcohol use and five classes of tobacco use onset among adolescents.

Applications of the latent class growth mixture model to LRTs are limited. Muthen et al. (2002) applied it to evaluate the Good Behavior Game (GBG) and found the largest effect size in the highest risk class. However, the effect was not statistically significant, and Muthen et al. attributed this to the small number of subjects in the highest class. Our study is one of the earliest applications of the model to LRTs with statistically significant program effects.

Two underlying assumptions are made in the present study. First, we assume that the population of youth who participated in the AAYP possibly consists of a mixture of distinct subgroups or classes within the larger population; and second, we assume that the significant preventive effects of AAYP, in terms of reduction in the rate of growth of violence found in previous analyses (Flay et al. 2004; Ngwe et al. 2004), may have impacted these distinct classes or subgroups differently.

Careful and elaborated analysis is necessary to apply the model to LRTs because additional assumptions have to be checked. They are as follows: (a) The latent classes found in control and treatment groups are similar in terms of numbers, means, proportions, and so on; and (b) the class membership is not changed by the program.

This study employs a rigorous and state-of-the-art evaluation methodology (growth mixture modeling techniques) to test the above assumptions and particularly to address the issue of possibly missed preventive effects in previous analyses of the AAYP data. Specifically, the present study intends to (a) ascertain the number of distinct classes or subgroups of male participants that may be present in the AAYP data set, (b) determine the proportion of participants that are most likely to belong to each class or subgroup, and (c) determine the preventive effects of the AAYP intervention in reducing violent behaviors among these distinct subgroups as evidenced by their violence growth patterns.

METHOD

SCHOOL SELECTION AND RANDOMIZATION

The longitudinal trial of 3 interventions was conducted in a high-risk sample of 12 poor, African American metropolitan Chicago schools (9 inner-city and 3 near-suburban) between 1994 and 1998. School inclusion criteria included enrollment of >80% African American and <10% Latino/Hispanic students; Grades K-8 (or K-6 if students were tracked to 1 middle school); enrollment >500; not on probation or slated for reorganization; not a special designated school (i.e., magnet, academic center); and moderate mobility (<50% annual turnover, meaning approximately <25% transferred in and <25% transferred out). Eligible schools (N = 141 inner-city and 14 suburban) were stratified into 4 quartiles of “risk” based on a score that combined proxy risk variables using the procedures described by Graham et al. (1984). The proxies of risk came from school report card data (1991–1992) and included enrollment, attendance/truancy, mobility, family income, and achievement scores. Using a randomized block design, we assigned to each condition 2 inner-city schools from the middle of the highest risk quartile, 1 inner-city school from the middle of the second risk quartile and 1 suburban school (also from the second quartile) per condition. One inner-city school refused to participate and was replaced with 1 from the same risk level. Schools signed an agreement to participate in the study for 4 years and agreed to not participate in another prevention initiative during that time. Study schools were 91% African American. Each school received the intervention free of charge (provided to all students in the appropriate grade levels) plus $250 for each participating classroom up to a maximum of $1,000 each year of the study.

PARTICIPANTS

A total of 552 African American boys who participated in the AAYP between 1994 and 1998 provided the data included in this analysis. At the study onset in 1994, the participants were in fifth grade and the mean age of the male participants was 10.9 years with a standard deviation of 0.6 years. For more detailed descriptions of these participants, for example, the average household income of these students, please refer to Flay et al. (2004).

DATA COLLECTION

Self-report data were collected from the participating students at the beginning of fifth grade (pretest, fall 1994), and posttests at the end of Grades 5, 6, 7, and 8 in the springs of 1995 to 1998, respectively. Only students who were in project schools or transferred into project schools during the study period were included in our analysis. Students who transferred out of project schools were not followed. Trained data collectors administered surveys in classrooms during school hours using a standardized protocol. The surveys were read aloud to students to accommodate different reading levels and to ensure uniform completion of the survey.

MEASURES

Questions concerning the students’ self-reported violent behaviors over their lifetime and in the previous 3 months (90 days) to the survey comprised the measures employed in this study. These measures were used to create a violence behavior scale. Specifically, the violence scale consisted of seven violence-related items: (a) carrying a gun; (b) carrying a knife; (c) threatening to beat up siblings; (d) threatening to beat up someone else; (e) threatening to cut, stab, or shoot people; (f) cutting or stabbing someone, and (g) shooting someone. All of the violence outcomes were collected longitudinally. The seven items were scored on a scale of 0 to 3 (0 = never; 1 = yes for lifetime, but not recently; 2 = once recently; 3 = more than once recently). The resulting violence score ranged from 0 to 21. Higher average scores represent higher levels of violence. The Cronbach’s αs over the five waves of data are .60, .75, .75, .77, and .76, respectively.

MODEL SPECIFICATION

For model specifications, we apply both the conventional mixed effect model for growth, that is, the latent growth curve model (Duncan et al. 1999) or hierarchical linear model (Raudenbush and Bryk 2002) and the growth mixture model (Hedeker 2000; Muthen et al. 2002; Nagin 1999).

The growth mixture model stochastically classifies subjects to a (pre-specified) number of unobserved clusters (classes) according to repeatedly measured responses and covariate(s) (such as a treatment indicator variable). Simultaneously, different growth curves are fitted in different classes. Therefore, the growth mixture model is a combination of a classification model, that is, latent class model (Hagenaars and McCutcheon 2002) and a growth model, that is, mixed- and/or fixed-effect models for growth. The conventional mixed effect model for growth can be considered a special, one-class, growth mixture model. Because there is only one class, the probability that a participant belongs to the class is 1 for all participants. We assume that class membership does not change over time and that the program does not affect the classification. The latter assumption was examined and tested. Both the conventional mixed effect model and the growth mixture model were fitted using Mplus software (Muthen and Muthen 2001).

In this section and the rest of this article, we combine the two treatment groups, SDC and SC, and consider them as one treatment group. We do this because, first, although the reduction of growth of violence relative to control (HEC) were statistically significant in both SDC and SC, the difference of the reductions between SC and SDC were not statistically significant in the previous analysis (Flay et al. 2004) using the conventional mixed-effect model for growth. Second, we wanted to simplify the rather complex growth mixture model.

We did not include the school-level cluster effect in our models. Results from the analysis by conventional (mixed) model with the school cluster (i.e., three-level model) were almost identical to results from the analysis without it (i.e., two-level model) in terms of evaluating the program effect (Flay et al. 2004). A second reason concerns technical difficulties. Ideally, a discrete distribution would be assumed at the second (student) level within each school and a normal distribution would be assumed in the third (school) level. As far as we know, software to apply such a model is not available. In summary, we believe that the effect of not including the school level would increase standard errors of estimates only slightly—and assessing the effect properly is beyond our capability. Thus, we believe that the effect would not alter our conclusion substantially given that the program effect is highly significant in the highest class (see Results section) and the third-level cluster effect was negligible in the conventional three-level model.

MATHEMATICAL NOTATIONS

We describe the mathematical notations for our model in two parts: the classification part of the model and the growth part of the model. For the classification part, the probability that subject i belongs to class k given the program is

$$P(c_{ik}=1\mid I_i)=\frac{e^{{\alpha }_{ck}+{\gamma }_{ck}{I}_i}}{{\sum }_{k=1}^K{e}^{{\alpha }_{ck}+{\gamma }_{ck}{I}_i}},$$ (1)

where c_ik (i = 1 … 552 and k = 1 … K) is an index variable of the class membership of subject i. If then a subject i belongs to class k, c_ik = 0 otherwise. I_i is a program indicator variable, I_i = 1 if subject i is in the prevention program, I_i = 0 otherwise. α_ck is a logit intercept for class k. γ_ck is a coefficient of an effect of the program to classification to k. The last class is a reference class with coefficients standardized to zero, α_ck = 0 and γ_ck = 0. The assumption that the treatment does not affect the classification is γ_ck = 0 for all k. The assumption is tested later.

For the growth model part, we present the fixed (but not the mixed-effect) model because it fits the data better. That is, the variation of data within a class is represented only by measurement error and not by variation of regression coefficients over subjects within class. The variation of data between classes is represented by different regression coefficients over classes. Responses of subjects are modeled as

$$y_{ij}={\alpha }_{0k}+({\alpha }_{1k}+{\gamma }_{1k}{I}_i)t_j+({\alpha }_{2k}+{\gamma }_{2k}{I}_i)t_j^2+e_{ij},$$ (2)

where $e_{ij}\sim N(0,{\sigma }_k^2)$ and Cov (e_ij, e_ij′) = σ_jj′k, j=1…5, j′ = 1, j≠j′We specify the covariance of adjacent errors in class k as being constant over time and nonadjacent errors as 0. That is,σ _{jj′ k} = σ _k if |j − j′| = 1 and σ_{jj′ k} = 0 otherwise. y_ij is violence scale score of subject i at jth wave and t_j is a time point at jth wave. Time points are 0, 0.5, 1.5, 2.5, and 3.25 for Waves 1, 2, 3, 4, and 5, respectively. α_0k, α_1k, and α_2k are coefficients of intercept, slope and quadratic terms, respectively, for a class k in the control condition. γ_1k and γ_2k are coefficients for the difference between control and treatment conditions in linear and quadratic terms, respectively, in class k. There is no coefficient for the difference for the intercept because we assume equality because of the randomization. e_ij is an error of subject i at jth wave. Errors are independent over subjects. ${\sigma }_{jk}^2$ is an error variance at jth wave in class k, and σ_{jj′ k} is a covariance between errors jth and j′th waves in class k. Using vectors and matrices, the above growth model is presented as

$$f_k({\mathbf{y}}_i\mid {\mathbf{X}}_i,c_{ik}=1){\mathbf{X}}_l{\mathbf{\beta }}_k+{\mathbf{e}}_i,$$ (3)

where e_i ~ N(0, R_ik) and y_i is an n_i × 1 response vector for subject i (n_i is number of observations for subject i). X_i is an n_i × 5 design matrix for subject i, whose columns are vectors of intercept, slope, quadratic, slope by program indicator, and quadratic by program indicator. β_k is a 5 × 1 parameter vector for class k and β_k = [α_0k α_1k α_2k γ_1k γ_2k]^T. e_i is an n_i × 1 error vector for subject i. e_is are independent over subjects. R_ik is an n_i × n_i covariance matrix for the error vector for the class k. The model is estimated by the maximum likelihood method. Its observed-data log likelihood is

$$logL={\sum }_{i=1}^{N}logf({\mathbf{y}}_i\mid {\mathbf{X}}_i),$$ (4)

where f(y_i | X_i) is a mixture distribution defined with (1) and (3) as

$$f({\mathbf{y}}_i\mid {\mathbf{X}}_i)={\sum }_{k=1}^{K}P(c_k=1)f_k({\mathbf{y}}_i\mid {\mathbf{X}}_i,c_{ik}=1).$$ (5)

For the conventional growth model, Equation 5 is not necessary because there is only one class.

DATA ANALYSIS

To determine the best growth mixture model, we selected a number of classes and specified growth models in each class. Using the final model, we investigated whether the program effect, that is, the reduction in the growth of violence in the treatment group relative to the control group, was significant in any class. Because the growth mixture model is very complex, the model selection and assessment are quite complex as well. Therefore, we simplified the complexity by conducting the data analysis in three stages following Muthen et al. (2002): (a) an initial exploratory analysis separating control and treatment groups, (b) a combined analyses to select the final model, and (c) an assessment of the final model.

The separate analyses are important in checking the assumption that the treatment does not affect the classification. If the assumption is correct, we should get similar classifications in the analyses of treatment and control conditions. Moreover, the similarity of classification over conditions ensures that the program effect found in the combined step is not artificial due to differential classification of participants between control and treatment conditions. The assumption is then formally tested using Equation 1 as well. The separate analyses are also quite helpful in fitting the very complex growth mixture models because it is much easier to fit a simpler model for subgroups and then combine the simpler models to build the final complex model.

The model selection in the combined analyses is in two steps. First, we selected a best-fitted model within a specified number of classes. For example, suppose the numbers of classes we investigate are from one to four; then there are four best models. Second, we choose the overall best among the four models. The choice of models is based on Bayesian information criteria (BIC) (Schwartz 1978) because the models are not nested. The lower the BIC value, the better the model fits.

We report selection of the final model, its estimates, and its assessment. The assessment includes a goodness of fit index (BIC), classification table and index (entropy), and fit plots. High diagonal probability values indicate high quality of classification in the classification table (Muthen et al. 2002). The entropy measure (Muthen et al. 2002) is the summary of fit of classification. Its value ranges from 0 to 1, and the higher the value, the better the classification. We also present a model-fitting plot for each class (Muthen et al. 2002). We assign participants to classes randomly proportionately to the estimated class probability given on their responses. Then we plot the mean curve and observed data of the assigned participants in each class. We also report detailed estimates, such as the proportion of each class, the growth trajectories in each class, and the sizes of program effect in each class. We select the final model based on all the results of assessment, estimates, and substantive knowledge. We also briefly compare results of the conventional and growth mixture approaches. Because the growth mixture model is sensitive to starting value and sometimes converges to a local maximum, we used different starting points to avoid the problem.

RESULTS

DESCRIPTIVE DATA

The means and standard deviations of violence measures for the treatment and controls groups and by study conditions over the five time points are shown in Table 1. At the pretest period, the treatment group (a combination of SC and SDC) has a slightly (nonsignificant) higher level of violence, but at the last time point, the control group has a substantially higher level of violence.

TABLE 1.

Means (Standard Deviations) for Violence Measures By Treatment Condition and Time

	Time (in Years)
	0		0.5		1.5		2.5		3.25
Group	Treat	Control	Treat	Control	Treat	Control	Treat	Control	Treat	Control
Number of participants	219	89	209	86	212	101	170	90	171	87
Mean	3.49	3.28	5.04	5.41	5.15	4.83	6.11	5.71	5.92	7.60
Standard deviation	3.21	3.49	4.50	4.01	4.48	3.74	4.78	4.50	4.08	5.43

NOTE: Social Development Curriculum (SDC) and School-Community (SC) were combined as one treatment group.

SEPARATE ANALYSES

We fit two-class, three-class and four-class models to both control and treatment data. BIC shows that the three-class model fits best in both control and treatment (Figure 1). In Figure 1, we note that the percentage in each class is similar between the control group and the treatment group in the fitted class models. Also, the growth curves are similar between the control group and the treatment group except for the highest class. These results indicate that the classification does not depend on the treatment and that the treatment only affects the participants in the highest class in each model.

Figure 1. Separately Estimated Mean Growth Curves for Two-, Three-, and Four-Class Models for Control and Treatment Groups.

NOTE: BIC = Bayesian information criteria.

COMBINED ANALYSES

The mean curves of a best-fitted model within one-class to four-class models are shown in Figure 2. The figure also includes each model’s proportion of classes, BIC, and entropy. Nagin (1999) provided probabilistic reasoning to determine that the three-class model is preferred. Let p_j denote the posterior probability that model j is the correct model. Schwarz (1978) and Kass and Wasserman (1995) showed that p_j is reasonably approximated by the following:

$$p_j=e^{-2({\text{BIC}}_j-{\text{BIC}}_{max})}/\sum _j{e}^{-2({\text{BIC}}_j-{\text{BIC}}_{max})},$$

where BIC_max is maximum BIC score of the models under consideration. Table 2 shows the probabilities as computed using the above equation and shows that the models with varying numbers of groups are the true models. With this BIC-based probability approximation, the probability of the three-class model is near 1. Because the three-class model fits best according to BIC, and is substantively meaningful as described below, we chose the three-class model as our final model.

Figure 2. Estimated Mean Growth Curves for One-, Two-, Three-, and Four-Class Models.

NOTE: BIC = Bayesian information criteria.

TABLE 2.

Bayesian Information Criteria (BIC) and Probabilities of Correct Model Among the Four Final Models

	Probability
Number of Classes	BIC	Correct Model
1	8631	.0000
2	7659	.0000
3	7601	.9997
4	7617	.0003

Before describing the details of the final model, we compare and describe the results in Figure 2 to simplify the understanding of the final model. All four models show that the growth rate in the highest class gradually decreases in the treatment group but remains almost constant in the control group. This observation creates the program effects in the highest class in all models (see Figure 2 and Table 3). Table 3 shows the estimates of the program effect in the highest class in each model, which are all significant. We note that the larger the number of classes, the larger the program effect and significance level.

TABLE 3.

The Program Effect in the Highest Class in the Best Model in Each Class of Models

Number of Classes	Estimate	SE	p
1	−1.17	0.63	.031
2	−2.06	1.14	.036
3	−3.18	1.18	.004
4	−3.01	1.24	.008

Both the program effect and the significance level are about the same between three-class and four-class models (Table 3). However, we prefer the three-class to the four-class model because the three-class model fits better statistically (see above), is simpler, and because the proportion of the additional class in the four-class model is very small (3%, Figure 2) and not substantively important.

THREE-CLASS (FINAL) MODEL

We describe the three-class model in detail. We call the three classes, from lower to higher violence level, low, medium, and high class, respectively. Because estimates for quadratic terms were very close to 0 and not significant in the low and middle classes, they were fixed to 0, that is, α₂₁ = α₂₂ = 0 (and, consequently, coefficients for the difference between control and treatment for quadratic terms are fixed to 0, i.e., γ₂₁ = γ₂₂ = 0) (see Table 4). The proportion of control and treatment group subjects in each class is almost the same as the overall proportion, .32 to .68. This observation suggests that classification does not depend on treatment. A formal test also supports the assumption. That is, γ_c1 and γ_c2 parameters in Equation (1) are not significant jointly (the chi-square difference is 0.432 with 2 degree of freedom and corresponding p < .8). The numbers of subjects in each class are, from low to high, 68 (12%), 298 (54%), and 186 (34%). The estimates and standard errors of growth curves in each class are in Table 4. The error variances are different over classes but constant over time. The covariances of adjacent errors are not 0 and constant but nonadjacent errors are 0 over time in all classes. The higher the class, the larger the error variances, and covariances (Table 4 and Figure 3). Both the control and treatment conditions start close to 0 and remain close to 0 during the study in the low class (Figure 2). Both control and treatment conditions start at about 3 and increase about 0.7 per year in the middle class (Figure 2). Little program effect is found in both the lower and middle classes, that is, both γ₁₁ and γ₁₂ are close to 0 and not significant (Table 4).

TABLE 4.

Estimates of Parameters in Violence Growth Model in the Final Three-Class Model

	High Class (k = 3)	Medium Class (k = 2)	Low Class (k = 1)
	Estimate (SE)	Estimate (SE)	Estimate (SE)
α0k	5.954 (0.452)*	2.925 (0.296)*	0.774 (0.264)*
α1k	1.107 (1.119)	0.659 (0.208)*	0.047 (0.208)
α2k	0.308 (0.358)	0.000 (fixed)	0.000 (fixed)
γ1k	3.589 (1.305)*	−0.066 (0.177)	0.014 (0.198)
γ2k	−1.405 (0.438)*	0.000 (fixed)	0.000 (fixed)
${\alpha }_k^{\text{2}}$	19.101 (1.446)*	6.250 (0.852)*	0.914 (0.395)*
covk	5.071 (1.381)*	1.732 (0.393)*	0.356 (0.131)*

NOTE: cov_k is covariance of adjacent errors in class k that is constant over time.

Covariance of nonadjacent errors are 0.

^*p < .05.

Figure 3.

Estimated Mean Growth Curves and Observed Trajectories for the Final Three-Class Model

Both treatment and control started high, around 6, and increased substantially during the study in the high class. The score increased more than 1 point per year and the rate of growth increased over time in the control condition (α₂ is positive [Table 4]). Although the rate of growth was higher (γ₁ is positive [Table 4]) initially, it decreased over time (γ₂ is negative [Table 4]) and is much lower near the end of the intervention in the treatment condition. The decreasing rate (negative γ₂) eventually created the significant program effect of about 3.2 scale points (p < .005 [Table 3]). This estimate is about 3 times larger than the estimate from the regular one-class model (Table 3). The program effects for all three classes are jointly significant—in a comparison between the final model and a model excluding the four parameters for the program effect (γ₁₃, γ₂₃, γ₁₂, and γ₁₁), from the final model the chi-square difference is 15.1 with 4 degree of freedom and corresponding p < .005. The classification table (Table 5) and entropy value (0.659 [Figure 2]) show that the classification is reasonable. Figure 3 shows that the model fits the data well in each class and that the higher the class the larger the error variances.

TABLE 5.

Classification Table for Three-Class Model

	Average Posterior Probabilities
Most Likely Class	High Class	Medium Class	Low Class
High	.883	.117	.000
Medium	.127	.822	.050
Low	.016	.203	.781

DISCUSSION

The present study is designed on the premise that the AAYP may have produced differential preventive effects on different groups of participants. That is, the program may have produced different effects on different clusters of individuals within the same population, and these differential effects may be missed through conventional group data analysis that considers all participants belonging to one homogeneous group. As Muthen and Muthen (2000) pointed out, conventional group data analysis methods may overlook program effects for distinctive subgroups in the population.

At least three distinct subgroups of participants were found within the larger matrix of the total AAYP sample. From an intervention point of view, this is important because some researchers (D’Unger et al. 1998) have noted that subgroups of individuals with similar violence trajectories (e.g., individuals who engage in mild forms of the behavior versus those who engage in severe forms of the behavior) need to be identified and intervention programs may need to be differently structured for each subgroup. In the same manner, program evaluation techniques must be able to sort these participants into their respective subgroups and apply appropriate evaluation methodologies to assess program effects for each subgroup. As mentioned in the Results section, this identification of subgroups facilitated the uncovering of bigger program effects of the AAYP for the high-risk group.

We found no program effects in the low or middle classes mainly because there was no (for lower class) or little (for middle class) room for improvement for those participants. The lower class stayed close to 0 and the middle class started from around 3 and ended around 5 during the intervention period. Substantively, scores around 3 to 5 represent very mild levels of violence. The initial steeper growth of violence in the treatment group in the high class is puzzling to us. It is also found in the analysis of other outcome variables, such as substance use, in this sample using the conventional mixed-effect model. Therefore, our explanation is that the result of our randomization was not ideal and more participants who had the potential to engage in violent behavior (or substance use) in future were included in the treatment than the control. Because it took time for the program to be effective, the positive linear term is found. However, the program offsets the difference and further improves the behavior of the treatment group relative to the control group toward the end of intervention.

It is certainly more informative to say that the program effect is 3.2 scale units in the highest class that consists of about one third of participants and that there is little program effect in the lower classes (result of the growth mixture analysis). It may even be considered somewhat misleading to say that the program effect is about 1.2 units (result of the conventional analysis), because about two thirds of participants were not affected by the program at all. Furthermore, the growth mixture model identified the participants for whom the program was effective.

We presented two ways of modeling the variation of growth over participants. The conventional mixed model explains the variation as random effects, that is, individual intercepts and slopes are distributed normally over participants. The growth mixture model explains the variation as discrete groups. For our data the latter fit much better as we showed in the comparison of one-class and three-class models. One of the main reasons is that problem behaviors, such as our violence scale score, usually do not distribute normally among subjects but typically are found only a among subset of subjects.

A considerable variation exists between the public (media) perception and empirical evidence regarding the number of adolescents who are actually involved in violent behaviors. Whereas the media, and by extension the general public opinion, makes it seems like the majority of African American youngsters are violent, in actuality a large proportion of these violent behaviors involve a relatively small number of youths (Moffitt 1993). If this assertion is true, then there is a need to relate intervention effects to the affected subgroups.

One final point regarding the variations in analysis methods is that latent class modeling methodology is not entirely a new technique. It has been widely used in the area of criminology and public health to (a) study how juveniles and adults enter a pattern of delinquent/criminal offending or desist from such offending (D’Unger et al. 1998; Nagin, Farrington, and Moffitt 1995; Nagin and Land 1993); (b) study the trajectories of boys’ physical aggression, opposition, and hyperactivity on the path to violence and delinquency (Nagin and Tremblay 1999); (c) examine the developmental trajectories of physical aggression from school entry to late adolescence (Brame, Nagin, and Tremblay 2001); and (d) examine the different routes to delinquency, for example, early starters and late starters (Simons et al. 1994). What appears to be more recent, according to some researchers (Muthen et al. 2002), is the application of these techniques in determining and, indeed, disentangling preventive intervention effects in longitudinal randomized trials. Our illustration suggests that it is a promising method.