A traditional LCA of the six smoking indicators was first examined. These initial analyses ignored the clustering of students in communities. presents the class solutions for one to six latent classes (see Model 1). The BIC drastically declines (i.e., improves) from 1 to 3 classes and then begins to level off. Entropy is also best with the 3-class model. Moreover, the 4-class solution separates one of the classes from the 3-class solution into two smaller groups, but the posterior probabilities indicates that there is substantial misclassification between these two smaller classes. For example, the posterior probabilities for the 3-class solution are .99, .93, .98 and the posterior probabilities for the 4-class solution are .79, .93, .98, and .79, with the first and fourth classes representing the separated classes from the 3-class model. The low posterior probabilities for these two classes indicate that the model has difficulty distinguishing between people in the first and fourth class. Most importantly, the substantive interpretation of the 3-class solution (as described in the next section) is theoretically meaningful, useful, and parsimonious. As such, we chose the 3-class solution as the best model. The results are presented in .
Fit Criteria for Each Model Specification
Latent Class Solution for Three-Class Model (No Level 2 Model)
In this 3-class solution, the largest class represents non-smokers and comprises 61.3% of the sample. While some of these students had smoked a cigarette in their lifetime, none of them were current smokers or thought of themselves as a smoker. Moreover, they tended to associate with non-smoking peers, believe that their parents would stop them from smoking, and perceive that smoking is harmful. The smallest class, described as the heavy smokers, represents 14.6% of the sample. Girls in this class tended to be regular smokers and viewed themselves as a heavier smoker. They also tended to associate mostly with other peers who smoked cigarettes and were less likely than other girls to perceive that their parents would stop them from smoking and that smoking is harmful. The remaining students were classified as moderate or occasional cigarette smokers. Comprising 24.1% of the sample, these students were most likely to report occasional cigarette smoking and viewed themselves as light smokers. Just over half of them reported that most of their friends smoke. Nearly all believed that their parents would try to stop them from smoking and about three-quarters believed that smoking is harmful to one's health.
Building on this 3-class, Level 1 solution, we next specified a model that utilized the parametric approach to account for the nested structure of the data. The results of the model are presented in , Model 2. The BIC improves with the addition of the random effects and the entropy remains the same as for the fixed effects model. The estimated mean of the random effect (or random mean) for the heavy smoker class indicates that, for communities at the average random mean for both heavy smoking and moderate smoking, the average probability that a student would be classified as a heavy smoker is .13. The variance of the random mean describes the variation in the probability that a student will belong to the heavy smoking class across communities (i.e., in some communities the probability is quite high, in others it is quite low). This variance is statistically significant, V(U0j
)=.61, se=.10, and indicates that communities did indeed vary significantly in their probability that a female would be a heavy smoker. Specifically, holding the probability of membership in the moderate smoker class constant, the probability that a female is a heavy smoker in a community that is 1 standard deviation below the mean of the random mean is .06 and 1 standard deviation above the mean of the random mean is .25. Larsen and Merlo (2005)
offer an alternative measure to quantify the between community variation, the Median Odds Ratio (MOR). The MOR is an estimate of the difference in the probability of the outcome for two randomly chosen people from two different randomly chosen Level 2 units. The MOR between a student in a community with the higher propensity to be a heavy smoker and a student with a lower propensity to be a heavy smoker is 2.10. This is a moderate odds ratio and also indicates substantial community-level variability in the probability of heavy smoking.
The estimated mean of the random effect (or random mean) for the moderate smoker class indicates that, for communities at the average random mean for both heavy smoking and moderate smoking, the average probability that a student would be classified as a moderate smoker is .26. The variance of this random mean describes the variation in the probability that a student would be classified as a moderate smoker across communities. This variance is also statistically significant, V(U0j)=.23, se=.05. Holding the probability of membership in the heavy smoker class constant, the probability that a female is a moderate smoker in a community that is 1 standard deviation below the mean of the random mean is .17 and 1 standard deviation above the mean of the random mean is .38. The MOR is 1.57. This is a small odds ratio, and indicates that there is some variability across communities in the probability of being a moderate smoker, but considerable less than for heavy smoking.
We also estimated the neighboring two and four class parametric random effects models. With the addition of the random effects, the three class model still appears to be the best model. The BIC shows a large decline from two classes to three, and a much smaller decline from three classes to four. Moreover, entropy is maximized with three classes and the fourth class produced in the 4-class solution is not well distinguished from one of the other classes. It should be noted that more research is needed to understand the performance of BIC in multilevel latent class models.
We extended this model by including a common factor on the Level 2 random means for the 3-class and 4-class solutions (Model 3). This dramatically reduced computation time and resulted in a reasonably small increase in BIC. The substantive interpretation of the Level 1 latent class solutions remained the same, including the qualitative typologies defined by each class and the proportion of individuals in each class.
In the next set of models we utilized the non-parametric approach. In this case, a Level 2 latent class model was added based on the random means from the Level 1 latent class solution. As presented in (Model 4a), the BIC significantly improves over the fixed effects 3-class model with the addition of two, Level 2 latent class; however, the BIC is not better than the BIC for the parametric model. Adding a third class only slightly improves the BIC, but it still does not show improvement over the parametric approach. A fourth Level 2 class was also assessed, but this model resulted in a very small number of individuals in the fourth class and the best log-likelihood failed to replicate. The results of the Level 2 latent class solution for the CB=2 and CB=3 solutions are presented in and .
Non-Parametric Multilevel Latent Class Solution, C=3, CB=2
Non-Parametric Multilevel Latent Class Solution, C=3, CB=3
With two Level 2 latent classes ( – Model 4a in ), one Level 2 latent class is comprised of communities with a relatively large number of non-smokers (i.e., 68% of the females are non-smokers). This class represents nearly 73% of the students. The second Level 2 latent class is comprised of communities with more heavy and moderate smokers. This class represents about 27% of students.
With three Level 2 latent classes ( – Model 4b in ), a low use community, moderate use community, and heavy use community emerges. Most students lived in one of the moderate use communities (65%), in these communities, about 64% of the females were non-smokers. demonstrates the nonparametric characterization of the random logit means. Here, the distribution of the random means is not assumed or represented to be normal, as is the case in the parametric ML LCA model. Rather, the histogram captures the discrete distribution of the random means. For example, in , the random mean distribution for the Level 1 heavy smoker class is represented by three bars showing a skewed distribution across the Level 2 latent classes.
The Discretized, Non-Normal Distribution of the Non-Parametric Approach. The bars represent the percent of students in the heavy user class in each between communities latent class from the C=3, CB=3 solution.
We also estimated the neighboring Level 1 two and four class nonparametric random effects models. The model with three Level 1 classes appears to be superior, showing a substantial decline over the model with two Level 1 classes, maintaining a high entropy value, and providing the most substantively interesting solution.
As a final step, we examined the inclusion of a common Level 2 factor for the individual indicators making up the Level 1 latent class model. For both the parametric (Models 5 and 6) and non-parametric approach (Model 7a and 7b), these models represented a marked improvement in log-likelihood and BIC. For example, when comparing Model 3 (the parametric model with Level 2 factor for random means) to Model 6 (the parametric model with Level 2 factor for random means and a factor for the Level 1 latent class indicators) for the 3-class solution, BIC improves from 58093 to 57827, with a difference of seven parameters. Similar improvements are observed for the non-parametric approach (Model set 4 compared to Model set 7). These improvements indicate that communities have a substantial influence on the Level 1 indicators of individual smoking typologies.
Synthesizing the information from all three multilevel models presented in , we find that the parametric approach with the inclusion of a common factor on the latent class indicators provides the best BIC for these data. Moreover, adding a second common factor on the Level 2 random means greatly decreases computation time and complexity, with minimal increase in BIC. As such, we selected this 3-class parametric random effects model with a factor on the latent class indicators and a factor on the Level 2 random means (Model 6) for further examination. We extended this model by including predictors at Level 1 (i.e., individual characteristics) and Level 2 (community characteristics). This was accomplished by regressing latent class membership on the Level 1 predictors via a multinomial logistic regression, and regressing both the random means and the latent class indicators on the Level 2 predictors via linear regression. Because of the common factor for the Level 2 random means and the common factor for the Level 1 latent class indicators, the Level 2 covariate effects must be carefully specified and interpreted. The sum of the indirect and direct effect of a particular covariate onto a random intercept C# is calculated as follows using the example of the covariate effect when comparing Class 2 to Class 3. The Class 2 factor loading λ2 is multiplied by the coefficient γ for the regression of the common factor on the covariate, and the direct effect γ2 of the covariate on the random intercept is added – that is, λ2*γ+γ2. For C#1 λ = 1 and since C#1 is identical to the common factor, the regression coefficient for the common factor regressed on the covariate is the total effect. presents the results of these conditional models.
Results of the Conditional Multilevel Latent Class Analysis: Effects on Level 1 Latent Class Solution
At Level 1, latent class membership was regressed on the student level predictors in a multinomial logistic regression. In this model, the non-smoker latent class served as the reference group. The first set of columns in presents the results that compare the moderate smokers to the non-smokers. The odds that a girl would be a moderate smoker (compared to a non-smoker) were significantly higher as her school bonding decreased, school performance decreased, parent's expectations for academic achievement decreased, parent's involvement in school decreased, involvement with friend's who were well bonded to school decreased (although this is a marginally significant effect), and if she associated with friends who had dropped out of school As is the case in any regression model, the effect of each covariate represents its unique effect after adjusting for all other variables in the model. School bonding, school performance, parental expectations for academic achievement, parental involvement in school, and friend's school bonding were all standardized to a mean of 0 and a standard deviation of one. Therefore we can interpret the regression coefficients as follows: for each one standard deviation increase in school bonding, the odds of being a moderate smoker as compared to a non-smoker decreased by about 13%, with similar interpretations for all other continuous school-related covariates. Since involvement with friends who dropped out of school is binary, the odds ratio indicates that the odds of being a moderate smoker as compared to a non-smoker were about 2.4 times higher if a girl associated with friends who had dropped out of school.
The second set of columns in presents the results that compare the heavy smokers to the non-smokers. The odds that a girl would be a heavy smoker (compared to a non-smoker) were significantly higher if she were older, as her school bonding decreased, school performance decreased, academic aspirations decreased, parent's expectations for academic achievement decreased, parent's involvement in school decreased, involvement with friend's who were well bonded to school decreased, and if she associated with friends who had dropped out of school. While all covariates are robust predictors of heavy smoking, involvement with friends who had dropped out of school is particularly strong. The odds of being a heavy smoker were 6.3 times higher if a girl associated with high school dropouts.
At Level 2, the random means from the Level 1 LCA were regressed on three community level predictors. The non-smoker class was used as the reference group in this model as well. When comparing moderate smokers to non-smokers, we find that proportion of youth living in poverty is the only significant predictor. This indicates that, holding all other predictors constant, as the proportion of young people living in poverty in the community increased, more girls indicated moderate smoking as compared to no smoking. When comparing heavy smokers to non-smokers we find that living in a tobacco growing state is the only significant predictor (p=.05). Communities in a tobacco growing state had more girls who were heavy smokers than non-smokers. Specifically, the odds of being a heavy smoker is 39% higher if the community is located in a tobacco growing state.
Finally, several Level 2 covariate effects on the Level 1 latent class indicators were observed. These results are reported in . Tobacco growing state had an effect on friend's smoking, in that the probability of endorsing the friend's smoke indicator was higher if the respondent lived in a tobacco growing state. Level of poverty in the community affected several of the latent class indicators. Poverty was significantly associated with all indicators except parental sanctions against smoking.
Results of the Conditional Multilevel Latent Class Analysis: Effects on Level 1 Latent Class Indicator Intercepts