Modeling Criminal Careers as Departures From a Unimodal Population Age-Crime Curve: The Case of Marijuana Use
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1 Modeling Criminal Careers as Departures From a Unimodal Population Curve: The Case of Marijuana Use Donatello Telesca, Elena A. Erosheva, Derek A. Kreader, & Ross Matsueda April 15, 2014
2 extends Telesca and Inoue (2008) by applying the methods of Telesca and Inoue (2008) to a study of individual age-crime curves in the context of marijuana use over the course of young adulthood
3 Representative of criminal career trajectories Used to study: of onset of offending Timing of peak offending Duration of criminal careers Types of criminal careers Other features of criminal careers When combined with individual characteristics, can also be used to describe relationship between individual characteristics and features of criminal careers
4 Relatively new field of statistics The same/similar methods referred to as curve registration, warping regression models, structural averaging... Useful when: Data is observed as a time series for many individuals Features of the time-outcome relationship (peaks, inflection points) are of interest Substantial variation in individual curves
5 What would we need the data to look like for these questions to be easy to answer?
6 Marijuana Use
7 Marijuana Use Marijuana Use
8 Marijuana Use Marijuana Use
9 Marijuana Use Marijuana Use
10 What does the data actually look like?
11 Marijuana Use Marijuana Use
12 Marijuana Use Marijuana Use
13 Marijuana Use Marijuana Use
14 Marijuana Use Marijuana Use
15 How does the functional data analysis framework allow us to characterize data of this kind?
16 Marijuana Use Stochastic
17 Marijuana Use Stochastic
18 Marijuana Use Stochastic
19 Marijuana Use Stochastic
20 Formulation used by introduced in Ramsay and Li (1998) Introduces model where All individuals share a stochastic time-response function Variation in observed time-response curves is generated by individual, strictly monotonic functions mapping time to stochastic time, called warping functions and noise Flexibility in modeling warping functions obtained by using B-spline basis functions Smoothness of estimated warping functions obtained by penalization of relative curvature
21 Telesca and Inoue (2008) developed a framework for functional data analysis based on Ramsay and Li (1998) Allows information to be shared across curves Introduces a temporal misalignment window Incoporates smoothing penalty directly into the model Gives a formal account of amplitude and phase variability Allows for exact inference
22 Telesca etal. is an extension of Telesca and Inoue (2008) Extends the curve registration models, developed for continuous data, to count data Allows scale and time-transformations to depend on individual covariates
23 Marijuana Use Model mean trajectory as a degree two polynomial in time/age Incorporated individual variation using random effects and/or latent classes Random Effects: Raudenbush and Chan (1993) Latent Classes: Nagin and Land (1993); Roeder et al. (1999) Random Effects & Latent Classes: Muthén and Shedden (1999)
24 Marijuana Use Model mean trajectory as a degree two polynomial in time/age Incorporate individual variation using random effects and/or latent classes Random Effects: Raudenbush and Chan (1993) Latent Classes: Nagin and Land (1993); Roeder et al. (1999) Random Effects & Latent Classes: Muthén and Shedden (1999)
25 Marijuana Use λ i (t j, X i ) = a i (X i ) }{{} Amplitude Y ij λ i (t j, X i ) P (λ i (t j, X i )) S (t j, β) }{{} Shape µ i (t j, φ i ; X i ) }{{} Phase Shift = a i (X i ) S (µ i (t j, φ i ; X i ), β)
26 a i (X i ) E [a i X i ] = exp {X ib a } Gamma conditional prior distribution of a i X i, b a Conjugate prior distribution for Poisson rate parameter Naturally accommodates over dispersion
27 S (t j, β) S (t j, β) = S B (t j ) β S B (t j ) set of K B-spline basis functions of order four evaluated at time t Positivity and unimodality of S (t j, β) imposed by restrictions on β Second order shrinkage prior on a reparametrization of β shrinks S (t j, β) towards a piecewise linear trajectory
28 µ i (t j, φ i ; X i ) µ i (t j, φ i ; X i ) = S µ (t j ) φ i (X i ) S µ (t j ) set of Q B-spline basis functions of order four evaluated at time t Monotonicity and boundedness of µ i (t j, φ i ; X i ) imposed by restrictions on φ i First order random walk shrinkage prior on a reparametrization of φ i shrinks φ i towards the B-spline coefficients that yield the identity function, µ i (t j, φ i ; X i ) = t j E [φ iq X i, b φ ] = Γ q + X ib φ
29 Use Markov Chain Monte Carlo to fit model by combination of Gibbs sampling and Metropolis-Hastings sampling algorithms Use posterior predictive loss to choose number of interior knots
30 Model allows us to explicitly test: Relationship between level of offending and individual characteristics Unimodality of age-crime curve Relationship between phase shift and individual characteristics
31 Better familiarity with relevant concepts in foundational papers: Ramsay and Li (1998) Lang and Brezger (2004) Telesca and Inoue (2008) Work through derivations of: Posterior predictive loss for Poisson case Conditional posterior distributions of parameters MCMC algorithm used Begin working through code
32 Lang, S. and A. Brezger (2004, March). P-Splines. Journal of Computational and Graphical Statistics 13 (1), Muthén, B. and K. Shedden (1999, June). Finite Mixture Modeling with Mixture Outcomes Using the EM Algorithm. Biometrics 55 (June), Nagin, D. S. and K. C. Land (1993)., Criminal Careers, and Population Heterogeneity: Specification and Estimation of a Nonparametric, Mixed Poisson Model. Criminology 31 (3), Ramsay, J. O. and X. Li (1998, May). Curve Registration. Journal of the Royal Statistical Society: Series B 60 (2), Raudenbush, S. W. and W.-S. Chan (1993, December). Application of a Hierarchical Linear Model to the Study of Adolescent Deviance in an Overlapping Cohort Design. Journal of Consulting and Clinical Psychology 61 (6), Roeder, K., K. G. Lynch, and D. S. Nagin (1999). Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology. Journal of the American Statistical Association 94 (447), Telesca, D., E. A. Erosheva, D. A. Kreager, and R. L. Matsueda (2012, December). Modeling Criminal Careers as Departures From a Unimodal Population Crime Curve: The Case of Marijuana Use. Journal of the American Statistical Association 107 (500), Telesca, D. and L. Y. T. Inoue (2008, March). Hierarchical Curve Registration. Journal of the American Statistical Association 103 (481),
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