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

3. Our dataset: accelerated cohort design

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)
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

5. 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

6. The Age-Crime Curve 1

7. Age-Crime Curve 2

8. 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

9. Modeling Percentage of delinquents Males versus Females

10. 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

11. 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.

12. 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

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