Elena A. Erosheva, Donatello Telesca, Ross L. Matsueda, and Derek Kreager

"Hierarchical Bayesian modeling of marijuana use trajectories in young adults and adolescents"

Presented at the Case Studies in Bayesian Statistics, Workshop 9

October 19, 2007, Pittsburgh, PA

Understanding within- and between-individual variability in longitudinal data on crime and substance use remains an important topic in criminology. A popular method for analyzing life-course crime data accounts for between-individual variability by assuming existence of several distinct groups of offenders, and group-specific polynomial relationships between age and behavior (Roeder, Lynch and Nagin 1999). An extension of this method adds individual-specific random effects such as age, age-squared and age-cubed (Muthen and Shedden 1999). We begin this study with an observation that the usual mixture models with polynomial age-dependence explain little variation in the individual trajectories. We develop an alternative model by assuming the existence of a single natural age-crime curve, and then modeling individual departures from that curve. Following Sampson and Laub (2003), who observed that between-individual heterogeneity in the age-crime relationship ``seems to be the age at desistance and level of offending,'' we introduce individual-specific parameters of phase (to capture desistance) and amplitude (to capture level). We model the natural age-crime curve nonparametrically using B-splines, and factor out individual-specific temporal misalignment and amplitude via Bayesian curve registration methods. We apply this method for analyzing longitudinal data of marijuana use from the Denver Youth Survey. As expected, we find the estimated natural age-crime curve of marijuana use to have one major hump. The individual-specific predictions of marijuana use trajectories fit the observed data reasonably well.