Background and objectives

We develop a flexible modelling framework and associated inferential tools for general continuous-time state-space models (SSMs), which is motivated from a longitudinal data set on delinquent behaviour of adolescents in Germany. The study aim is to investigate the persistence of an individual’s delinquency level over time. We assume the latter to be a latent trait underlying the trajectories of adolescents' delinquency, which is observed at irregular time intervals. For analysing such irregularly sampled sequential observations that are driven by an underlying state process, continuous-time SSMs constitute a flexible tool. However, corresponding applications typically involve restrictive assumptions concerning linearity and Gaussianity to facilitate inference on the model parameters via the Kalman filter. In this contribution, we provide a general continuous-time SSM framework, allowing both the observation and the state process to be non-linear and non-Gaussian. The modelling approach is illustrated via the analysis of delinquent behaviour in adolescence and young adulthood.

Methodology

We measure the adolescents’ delinquency as the total number of offences committed in the twelve months prior to the survey occasion. To allow for possible overdispersion, we assume the number of offences to follow a negative binomial distribution (conditional on the states). As the study participants' age and gender are known to affect their delinquent behaviour (e.g., Reinecke & Weins, 2013), we additionally include these covariates in the observation process. We further specify the state process to be an Ornstein-Uhlenbeck process with mean zero, such that an individual's delinquency level – or, more precisely, the deviation of the individual's delinquency level from the population mean – is persistent over time and changes gradually.

Statistical inference is carried out by maximum approximate likelihood estimation, where multiple numerical integration within the likelihood evaluation is performed via a fine discretisation of the state process (Kitagawa, 1987). The corresponding reframing of the SSM as a continuous-time hidden Markov model enables us to apply the associated efficient algorithms for parameter estimation and state decoding (Langrock, 2011).

Results

The results reveal temporal persistence in the deviation of an individual's delinquency level from the population mean. Moreover, trajectories of the expected number of offences, based on the decoded delinquency levels at each observation time, reveal different types of trajectories: while some adolescents have a permanently increased or reduced level of delinquency, others show early or late periods of increased delinquency levels.

Conclusions and implications

While the presented continuous-time SSM is tailored to the analysis of adolescents’ delinquent behaviour, the modelling framework generally allows for any continuous or discrete (and even categorical) distribution within the observation process. As neither the observation process nor the state process need not to be Gaussian or linear, our approach enables various possible model specifications, requiring only that the transition density of the state process has an explicit analytic form. In conclusion, our approach constitutes an accessible and very flexible framework for modelling irregularly spaced sequential data driven by a one-dimensional underlying state process.

References

Kitagawa, G. (1987). Non-Gaussian state-space modeling of nonstationary time series. Journal of the American Statistical Association, 82(400), 1032–1041.

Langrock, R. (2011). Some applications of nonlinear and non-Gaussian state-space modelling by means of hidden Markov models. Journal of Applied Statistics, 38(12), 2955–2970.

Reinecke, J. and Weins, C. (2013). The development of delinquency during adolescence: a comparison of missing data techniques. Quality & Quantity, 47(6), 3319–3334.