Using cubic or quartic polynomials to produce trajectories is popular due to being able to produce smooth curves and interesting shapes. However, the use of
0 1 2 3 4 5 6 13 15 17 19 21 23 25 27 29 31 33 35 M e an num be r of convi ct ions pe r 2 ye ar Age
Class1 - Low Rate Persistent Class2 - Adolescent Limited Class3 - Late Starters Class4 - High Rate Persistent 0 1 2 3 4 5 6 13 15 17 19 21 23 25 27 29 31 33 35 M e an num be r of convi ct ions pe r 2 ye ar Age
Class1 - Low Rate Persistent Class2 - Low Rate Adolescent Limited Class3 - High Rate Adolescent Limited Class4 - High Rate Persistent
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polynomial curves can lead to modelling difficulties. Unusual and unexpected changes in the direction of the plotted estimated trajectories can sometimes occur when using polynomial curves, which are not supported by the data. When using polynomials to estimate a curve through a given set of data points, it would seem reasonable to assume increasing the degree of the polynomial (or number of
interpolating points) would reduce the error in the polynomial interpolation. However this is not always the case and actually increasing the degree of the polynomials does not always improve the accuracy of the interpolating polynomial (Epperson, 1987). Several researchers have illustrated that when using polynomial curves, the estimated trajectories show a pattern of increase, followed by a decrease, followed by an increase or uptick towards the end of the observed period (Blokland et al., 2005, Blokland and Nieuwbeerta, 2005, Nieuwbeerta et al., 2011). This problem can be presented in Figure 5.7 and Figure 5.8. These are examples from the literature, by Marshall (2006) estimated juvenile trajectories for indigenous (native Australian population) and non-indigenous juveniles, and found there was an upward turn in their high rate trajectory at age 19 after an earlier increase and decrease before age 19. Bushway et al. (2003) also model trajectories of offending behaviour from age 13 to age 22, and estimate an uplift for three of their trajectories. The two papers take different approaches to these trajectory shapes; Marshall comments in the text on the change of shape without suggesting a reason, whereas Bushway et al. (2003) also comment, but suggest such behaviour to be evidence of intermittency. It is clear that when trajectories are estimated which show a number of changes of direction, then authors are sometimes uncertain how to interpret these shapes and whether such changes in direction are real.
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Figure 5.7 Example of cubic polynomial trajectories from the literature showing uplifts on some trajectories at the end of the age scale. This example is from Marshall (2006) with an uplift for the 'high'
group.
Figure 5.8 Example of cubic polynomial trajectories from the literature showing uplifts on some trajectories at the end of the age scale. This example is from Bushway et al (2003) using the Rochester
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Fitting polynomial curves is “non-local”, meaning that a data point in one part of the time axis can influence the shape of the curve in a distant part of time axis. This is generally not a desirable characteristic and something researchers would want to avoid. Cubic polynomials are not able to generate curves that are randomly shaped. It is not possible for a trajectory curve to be produced that rises steeply in the first part of the time axis, and then exhibits a constant rate thereafter. To gain more variation in the shape of the curves would mean increasing the degree of the polynomial, which does not always improve the accuracy, and the function behaves abnormally at the extremes of the curves (Liu, 2015).
There are a few approaches that can be used to deal with the issues caused by using cubic polynomials. Firstly age can be fitted as a categorical factor. This would mean age is fitted as a stepwise function where the levels of the steps are constant within each time period, and jump between the time periods. Even though this approach offers local fitting it unfortunately does not use information provided by the previous time period, so in effect the fitting is too local. This approach unfortunately can be problematic as it normally requires a large number of parameters to be estimated for each trajectory. Secondly, higher order polynomials could be used. Sweeten (2014) reports that PROC TRAJ, the SAS software add on for group based trajectory modelling (Jones and Nagin, 2007), allows polynomials up to order five to be fitted. While using high order polynomials may allow more flexibility in the shape of the trajectory, the method still fails to solve the problem of the non-locality of polynomial curves, where a data point at a low age can have a large effect on the fitted curve at a high age. A third option is to use cubic B-splines, which have not previously been used in GBTMs. Employing cubic B-splines provides a flexible approach to estimating curves (Silverman, 1985). Fitting of cubic B-splines is relatively straightforward and together with the flexibility of shape, makes it suitable for group-based trajectory models. (See Francis, Elliott and Weldon, 2016 for a
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discussion of smoothing GBTM through B-splines and which is a publication arising from this thesis).