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Analyzing Criminal Trajectory Profiles: Bridging Multilevel and Group-based Approaches Using Growth Mixture Modeling

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Abstract

Over the last 25 years, a life-course perspective on criminal behavior has assumed increasing prominence in the literature. This theoretical development has been accompanied by changes in the statistical models used to analyze criminological data. There are two main statistical modeling techniques currently used to model longitudinal data. These are growth curve models and latent class growth models, also known as group-based trajectory models. Using the well known Cambridge data and the Philadelphia cohort study, this article compares the two “classical” models—conventional growth curve model and group-based trajectory models. In addition, two growth mixture models are introduced that bridge the gap between conventional growth models and group-based trajectory models. For the Cambridge data, the different mixture models yield quite consistent inferences regarding the nature of the underlying trajectories of convictions. For the Philadelphia cohort study, the statistical indicators give stronger guidance on relative model fit. The main goals of this article are to contribute to the discussion about different modeling techniques for analyzing data from a life-course perspective and to provide a concrete step-by-step illustration of such an analysis and model checking.

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Notes

  1. This approach is often referred to under its technical SAS procedure name PROC TRAJ (Jones et al. 2001).

  2. ICPSR study number 8488. http://www.icpsr.umich.edu/ includes the Cambridge data until age 24. David Farrington kindly provided us with the outcome variables until age 40 for the Cambridge sample. A summary of the non-public data is given in Table 1a.

  3. Some of these boys died within the observation period. For the sake of simplicity, data for these boys are not included in our analytical illustration. The variable of interest in this context is the number of convictions per year for each of the 404 males, for whom data are available until age 31.

  4. For an overview of this modeling framework, see Muthén (2002). For a step-by-step introduction to applying latent variable models to longitudinal data, see Muthén (2004). For a recent technical presentation see Muthén and Asparouhov (in press).

  5. For an in depth discussion of growth mixture models see Muthén (2001a).

  6. In addition to the BIC, the Akaike Information Criteria (AIC) is sometimes used for model comparison. However, for finite mixture models, the AIC has been shown to overestimate the correct number of components (Soromenho 1994; Celeux and Soromenho 1996). The BIC on the other hand has been reported to perform well (Roeder and Wasserman 1997) and most consistently (Jedidi et al. 1997). For further details and comparisons see McLachlan and Peel (2000) as well as Nylund et al. (in press).

  7. http://www.statmodel.com

  8. The quadratic growth function is applied to the count part as well as the zero-inflation part of the growth model. We estimated all models with and without growth structure on the zero-inflation part. The results for the models with unstructured zero-inflation part do not differ from those presented here and are therefore not listed in addition. A cubic growth parameter was specified for the conventional growth model with random intercept, but had not significant contribution. The cubic function was then not pursued any further.

  9. Among the seven estimated parameters, three are for the means of the Poisson growth factors, one for the variance for the intercept of the Poisson growth model and three parameters for the quadratic growth model for the inflation part.

  10. The two additional parameters are the variance of the slope and the covariance between slope and intercept.

  11. Note that the three inflation parameters are held equal across classes.

  12. We used an Epanechnikov kernel for the nonparametric smoothing function (Silverman, 1986).

  13. The inflation part was modeled in the same way as it was for the GMM. That is, three parameters are estimated and set equal across the classes.

  14. What again becomes obvious here, is the small number of patterns represented by more than one person in the Cambridge data. It could be that in the case of outliers, the GMM model might have an advantage in as much as allowing for random effects can lower the effects of single influential cases. In a model that allows for variance around the growth factors, a few outliers will increase variance substantially. If the growth factors are not allowed to vary, those cases would be more likely to form a new class. Thus, we were concerned about the effect of single influential cases to the model comparison performed here. We computed the influence statistic for each observation (Cook 1986; Liski 1991). The results of the model comparison did not change after excluding influential cases (patterns).

  15. A similar strategy was employed by Loughran and Nagin (2006) in their analysis of the full data set.

  16. Unlike in the Cambridge analysis, the model here has one inflation parameter per time point, held equal across classes.

  17. For LCGA and the non-parametric GMM those can be computed by hand. For GMM with random effects, numerical integration was used.

  18. Going through the different modeling steps, the non-parametric GMM nicely bridged the results from LCGA and GMM for the Cambridge data. Without having yet looked at covariates or predictive power, the case can be made that there are only three overall patterns in the Cambridge data with high and low variations on the themes.

  19. BIC with random effect for the quadratic slope factor was 2951.3 compared to 2936.5 with just random effects on the intercept and linear slope.

  20. The non-parametric GMM in which the three sub-classes are constrained to have equal probability for later conviction shows a slight decrease in fit. The log-likelihood value for the restricted model is −1467.8 with 19 parameters compared to −1462.0 with 21 parameters in the unrestricted model.

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Acknowledgments

We like to acknowledge everyone who discussed this article with us during the last 5 years at various meetings and conferences. In particular we thank Tihomir Asparouhov for ongoing discussions that shaped our perspective on this article. Shawn Bushway, Booil Jo, John Laub, Katherine Masyn, Daniel Nagin, Paul Nieuwbeerta and three anonymous reviewers provided critical comments to earlier versions of this manuscript that we gladly took into account. Michael Lemay was of great help in data preparation and analysis. The work on this article was partially supported by grant K02 AA 00230 from National Institute on Alcohol Abuse and Alcoholism.

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Authors and Affiliations

  1. Joint Program in Survey Methodology, University of Maryland, 1218 Lefrak Hall, College Park, MD, 20742, USA

    Frauke Kreuter

  2. University of California, Los Angeles, CA, USA

    Bengt Muthén

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  1. Frauke Kreuter
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Correspondence to Frauke Kreuter.

Appendix

Appendix

Zero-inflated Poisson Model

Note that the ZIP model is already a special case of a finite mixture model with two classes. Treating the count outcome variable as zero-inflated at each time point means that a probability is estimated for the observation to be either in the “zero-class” or not. For the zero class a zero count occurs with probability one. For the non-zero class, the probability of a conviction is expressed with a Poisson process.

The interesting feature for the ZIP, or its expression as a two-class model, is that the probability of being in the zero class can be modeled by covariates that are different from those that predict the counts for the Poisson class. The same is true when allowing for a zero class in the growth trajectory modeling.

More formally, for the present application this model can be represented as follows: At each individual time point a count outcome variable U ti (the number of conviction at each time point t for individual i) is distributed as ZIP (Roeder et al. 1999).

$$ U_{{ti}} \sim \left\{ \begin{aligned}{} & 0{\text{ with probability }}\rho_{{ij}} \\ & {\text{Poisson}}(\lambda _{{ti}} ){\text{ with probability }}1 - \rho_{{ij}} \\ \end{aligned} \right.$$

The parameters ρ it and λ it can be represented with \( {\text{logit(}}\rho _{{ti}} {\text{) = log[}}\rho _{ti} {\text{/1}} - \rho _{ti} {\text{] = X}}_{ti} \gamma _{{t}} {\text{ }} \) and \( \log (\lambda _{{ti}} ) = {\rm X}_{{ti}} \beta _{i} \).

Notice that the mixture model within the zero-inflated Poisson is a mixture at each time point. The mixture models we discuss in the different growth models are mixtures of different growth trajectories (across all time points).

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Kreuter, F., Muthén, B. Analyzing Criminal Trajectory Profiles: Bridging Multilevel and Group-based Approaches Using Growth Mixture Modeling. J Quant Criminol 24, 1–31 (2008). https://doi.org/10.1007/s10940-007-9036-0

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