Purpose
This paper investigates the use of B-spline smoothers as an alternative to polynomials in time 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 end.

Methods
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.

Results
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. However ,the shapes of the two chronic trajectory curves 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.

Conclusions
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.