1. 1 Modeling age-related development of delinquency during adolescence and early adulthood with an autoregressive growth curve Johannes A. Landsheer, Utrecht University Johan H. L. Oud, University of Nijmegen Cor van Dijkum, Utrecht University (A work in progress)

2. 2 Modeling age-related development Longitudinal survey Complex dataset Inherently incomplete Necessity to generalize over age

3. 3 Our dataset: accelerated cohort design

4. 4 Features of the dataset Each wave a large cross-section with multiple age-cohorts Start: 12 to 25 (N = 3393) Resampling of lost age groups in wave 2 and 3 Wave 1: 12 to 25 Wave 2: 12 to 28 (resample 12 – 15) Wave 3: 12 to 31 (resample 12 – 15) Each wave consists of a relatively large number of age-groups Large time lag of three years Limited longitudinal data Not all subjects, but a large part followed longitudinally Attrition of about 18% Only three points of measurement: 1991, 1994 and 1997. That is: for each individual are at best 3 out of 20 measurements available. The rest is missing by design.

5. 5 Dataset on a single time axe (means)

6. 6 Advantages of the data collection Limited attrition problem Each wave consists of a representative cross- section Acceleration: from three measurements to developmental curves from ages 12 through 31 Combination of different cohorts with overlapping developmental periods

7. 7 Theoretical background Hirschi-Gottfredson Self-control theory Age differences in delinquency are invariant over time and place Male-Female differences in delinquency are invariant over time and place Research question Are differences in delinquency between males and females invariant over age?

8. 8 The Age-Crime Curve 1

9. 9 Age-Crime Curve 2

10. 10 Model dxt/dt = a*xt + (b + c*t) + v(t) a = drift (continuous model) or auto-regression (discrete model) b = constant intercept c = linear increase in time of intercept v(t) = error term (expected value 0) a x t2 b0b0 c*t 1 a x t0 x t1 b0b0 c*t 0 x t3 b0b0 c*t 2 a

11. 11 Modeling Percentage of delinquents Males versus Females

12. 12 Testing differences between males and females Males: dxt/dt = a*xt + (b + c*t) + v(t) Females: dxt/dt = a*xt + ((b+d) + (c+e)*t) + v(t) Note: differences in slope (a) have not yet been tested

13. 13 Future directions and wish list Modeling the frequency of delinquent behavior Studying changes which are predictive of changes in delinquency Background variables / constants Feedback relations, especially control measures

14. 14 General autoregressive growth model dx(t)/dt = A(t)x(t) + γ + B(t)u(t) + G(t)(dW(t)/dt)) A(t) specifies how the change in the state x(t) depends on itself. γ specifies random subject effects B(t) specifies how the fixed input variables in u(t) accommodate for nonzero and non-constant mean trajectories W(t) Wiener process or limiting form of the discrete time random walk process. G(t) is the transformation matrix