Curvilinear forms of change were examined. Quadratic forms of change were significantly better fitting than a linear model (χ2[4] = 208.85, p < .001). Despite the modest variance in the quadratic curvature (σ22 = 0.002, SD = 0.05), the model that allowed the curvature to vary across individuals fit better than the model with a fixed quadratic curvature (χ2[3] = 161.88, p < .001). There was not adequate variability across individuals in the cubic curves for cubic models to converge, so we examined subsequent polynomials with fixed effects. Models with a fixed cubic term fit better than models without the cubic term (χ2[1] = 21.26, p < .001). Moreover, models with a fixed quartic term fit better than models without the quartic term (χ2[1] = 120.22, p < .001). Models with a quintic term did not fit significantly better than models without a quintic term (χ2[1] = 0.64, p = .423), so we chose the simpler quartic model for parsimony. To prevent over-fitting, we split the sample into two random subsets of cases and examined the quartic model with each
