Many brain structures show a complex, non-linear pattern of maturation and age-related change. Often, quadratic models (β 0 + β 1 age + β 2 age 2 + ε) are used to describe such relationships. Here, we demonstrate that the fitting of quadratic models is substantially affected by seemingly irrelevant factors, such as the agerange sampled. Hippocampal volume was measured in 434 healthy participants between 8 and 85 years of age, and quadratic models were fit to subsets of the sample with different age-ranges. It was found that as the bottom of the age-range increased, the age at which volumes appeared to peak was moved upwards and the estimated decline in the last part of the age-span became larger. Thus, whether children were included or not affected the estimated decline between 60 and 85 years. We conclude that caution should be exerted in inferring age-trajectories from global fit models, e.g. the quadratic model. A nonparametric local smoothing technique (the smoothing spline) was found to be more robust to the effects of different starting ages. The results were replicated in an independent sample of 309 participants. © 2010 Elsevier Inc. All rights reserved.
IntroductionOver the past few years, research has demonstrated that most brain structures undergo a complex pattern of maturation and agerelated change. For instance, the hippocampus shows a marked nonlinear pattern of change throughout the lifespan (Allen et al., 2005;Jernigan and Gamst, 2005;Kennedy et al., 2008;Walhovd et al., 2005). Very often, non-linearity of age relationships is tested using quadratic or other polynomial models. A quadratic term is added to the list of predictors in a regression analysis, yielding a higher order polynomial function. If the quadratic term is significant, the brain structure in question can be said to have a non-linear age-trajectory (Allen et al., 2005;Good et al., 2001;Jernigan and Gamst, 2005;Kennedy et al., 2008;Lupien et al., 2007;Sowell et al., 2003;Sullivan et al., 1995;Terribilli et al., 2009;Walhovd et al., 2005). In addition, the trajectory of the curve may be used to describe the relationship between age and the brain structure, e.g. to determine when the hippocampus reaches its maximum volume, or how steep the subsequent decline is. This, however, may be problematic: First, to say that a relationship is nonlinear is not the same as saying that it is quadratic. This is an example of a specification effect. Second, if the same quadratic model is fit to different sets of data, one will get different results. For example, the observed peaks of quadratic functions will inherently depend on the age range sampled. This can lead to completely erroneous inferences about features of the trend, for example the location of peaks. This is a localization effect. The aim of this report is to demonstrate biases associated with quadratic model fits, and hint at possible solutions.The quadratic function is always a parabola, and the basic shape only differs in curvature and direction (whether it has a peak or a dip). Thus...