Abstract
This paper investigates the use of B-spline smoothers as an alternative to polynomials when estimating trajectory shape in group-based trajectory models. The use of polynomials in these models can cause undesirable curve shapes, such as uplifts at the end of the trajectory, which may not be present in the data. Moreover, polynomial curves are global, meaning that a data point at one end of the trajectory can affect the shape of the curve at the other end. We use the UK Offenders Index 1963 birth cohort to investigate the use of B-splines. The models are fitted using Latent Gold, and two information criteria (BIC and ICL-BIC) are used to estimate the number of knots of the B-spline, as well as the number of groups. A small simulation study is also presented. A three-group solution was chosen. It is shown that B-splines can provide a better fit to the observed data than cubic polynomials. The offending trajectory groups correspond to the classic groups of adolescent-limited, low-rate chronic and high-rate chronic which were proposed by Moffitt. The shapes of the two chronic trajectory curves from the B-spline fitting are more consistent with the life-course persistent nature of chronic offending than the traditional cubic polynomial curves. The simulation shows improved performance of the B-spline over cubic polynomials. The use of B-splines is recommended when fitting group-based trajectory models. Some software products need further development to include such facilities, and we encourage this development.
Highlights
In the context of criminology, group-based trajectory models (GBTM) provide a method of examining and understanding the evolving nature of offending over the life course
Trajectory models most commonly examine frequency measures of offending over the life course through arrest or conviction data, the method has been used for examining other processes, such as changing offending seriousness over the criminal career [16]
In contrast to other methods, such as the growth curve model [37], such models assume that the underlying population consists of a fixed but unknown number of groups, with distinct trajectories which give the changing estimated mean values for each group over time
Summary
We use the UK Offenders Index 1963 birth cohort to investigate the use of B-splines. The models are fitted using Latent Gold, and two information criteria (BIC and ICL-BIC) are used to estimate the number of knots of the B-spline, as well as the number of groups. It is shown that B-splines can provide a better fit to the observed data than cubic polynomials. The offending trajectory groups correspond to the classic groups of adolescent-limited, low-rate chronic and high-rate chronic which were proposed by Moffitt. The shapes of the two chronic trajectory curves from the B-spline fitting are more consistent with the life-course persistent nature of chronic offending than the traditional cubic polynomial curves. The simulation shows improved performance of the B-spline over cubic polynomials
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