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)

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

Our dataset: accelerated cohort design

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) 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

Theoretical background Hirschi-Gottfredson Self-control theory Self control is the essential brake on delinquent behavior Self control is the result of early childhood socialization Gender differences are invariant Open questions Delinquency and hence self control changes greatly during adolescence. How come? Kind of gender differences Differences in maturation Possible differences in the rate of change

The Age-Crime Curve 1

Age-Crime Curve 2

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 xt2 b0 c*t1 xt0 xt1 c*t0 xt3 c*t2

Modeling Percentage of delinquents Males versus Females

Testing differences Males: dxt/dt = a*xt + (b + c*t) + p*dW(t)/dt Females: dxt/dt = (a+d)*xt + ((b+e) + (c+f)*t) + (p+q)*dW(t)/dt

Testing parameters dxt/dt = (a+d). xt + ((b+e) + (c+f). t) + (p+q) Testing parameters dxt/dt = (a+d)*xt + ((b+e) + (c+f)*t) + (p+q)*dW(t)/dt Model 0a Model 0b Model 1 Model 2 Model 3 Model 4 Initial values  Equal Differ   d, e, f, q fixed (0) f free f, e free d, f and e free d, f, e, q free Chi-squared fit of model 807,22 740,265 639,638 568,319 566,236 563,720 Degrees of freedom 505 503 502 501 500 499 Probability 0,00 0,000 0,020 0,021 0,023 Akaike's Information Criterion -202,79 -265,735 -364,362 -433,681 -433,764 -434,280 RMSEA Test of X2 improvement p < .001 p < .20 Nested models Model 0-A: same model for males and females, i.e. same start value for mean and variance, d, e, f and q are zero. Model 0-B: two models for males and females, different values for start (means and variances), but d, e, f and q are zero. Model 1: two models for males and females, different value for initial mean and variance; f may differ, but d, e and q are zero. Model 2: two models for males and females, different value for initial mean and variance; f and d may differ, but e and q are zero. Model 3: two models for males and females, different value for initial mean and variance; f, d and e may differ, but q is zero. Model 4: two models for males and females, different value for initial mean and variance; f, d, e and q may differ.

Future directions and wish list Testing differences in the peak value Change from growth to decline Modeling the frequency of delinquent behavior Studying changes which are predictive of changes in delinquency Background variables / constants Feedback relations, especially control measures

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