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15 Examples 284The independence of ɛ and u vc and dependence between δ and u vc is clearlyillustrated in Figures 15.10 and 15.11. In this example the two measures haveprovided very different rankings of the varieties. The choice of tolerance measuredepends on the aim of the experiment. In this experiment the aim was toidentify tolerance which is independent of inherent vigour so the deviations fromregression measure is preferred.15.9 Balanced longitudinal data - Random coefficients and cubicsmoothing splines - OrangesWe now illustrate the use of random coefficients and cubic smoothing splinesfor the analysis of balanced longitudinal data. The implementation of cubicsmoothing splines in ASReml was originally based on the mixed model formulationpresented by Verbyla et al. (1999). More recently the technology has beenenhanced so that the user can specify knot points; in the original approach theknot points were taken to be the ordered set of unique values of the explanatoryvariable. The specification of knot points is particularly useful if the number ofunique values in the explanatory variable is large, or if units are measured atdifferent times.The data we use was originally reported by Draper and Smith (1998, ex24N, p559)and has recently been reanalysed by Pinheiro and Bates (2000, p338). The dataare displayed in Figure 15.12 and are the trunk circumferences (in millimetres) ofeach of 5 trees taken at 7 times. All trees were measured at the same time so thatthe data are balanced. The aim of the study is unclear, though, both previousanalyses involved modelling the overall ‘growth’ curve, accounting for the obviousvariation in both level and shape between trees. Pinheiro and Bates (2000) useda nonlinear mixed effects modelling approach, in which they modelled the growthcurves by a three parameter logistic function of age, given byy =φ 11 + exp [−(x − φ 2 )/φ 3 ](15.11)where y is the trunk circumference, x is the tree age in days since December 311968, φ 1 is the asymptotic height, φ 2 is the inflection point or the time at whichthe tree reaches 0.5φ 1 , φ 3 is the time elapsed between trees reaching half andabout 3/4 of φ 1 .The datafile consists of 5 columns viz, Tree, a factor with 5 levels, age, tree agein days since 31st December 1968, circ the trunk circumference and season. Thelast column season was added after noting that tree age spans several years and if