Chappell R
Department of Statistics, University of Chicago, Illinois 60637.
J Theor Biol. 1989 May 22;138(2):235-56. doi: 10.1016/s0022-5193(89)80141-9.
Change-point models, in which a linear or non-linear relation is generalized by allowing it to change at a point not fixed in advance, are of growing importance in allometric and other types of modeling. Frequently, the change-point is picked "by eye" and separate regressions are run for each resultant subdomain. This procedure is deficient, however, for the following reasons: first, a repeatable and objective procedure for estimating the change-point has not been used; second, the subsequent analysis usually does not take into account the fact that the change-point is estimated from the data; and last, the usually desirable requirement of continuity at the change-point is ignored. This paper describes various methods for jointly estimating linear relations and the intervening change-point from the data. In the simplest case, with normal errors and a linear relation of one variable upon another, this amounts to fitting a "bent line" via least squares techniques. In addition, tests and graphical diagnostics for the presence of change-points are presented. An example is given where a change-point and slopes are estimated for the relation of running speed with size among land mammals. In the past, these data have been fit with a straight line or a parabola. It is shown here that superior fit and interpretability are achieved using a change-point model.
变点模型在异速生长及其他类型的建模中愈发重要，在这类模型中，线性或非线性关系通过允许其在一个事先未确定的点发生变化来进行推广。通常，变点是“凭肉眼”选取的，然后对每个所得子域进行单独回归。然而，该过程存在以下缺陷：首先，未使用用于估计变点的可重复且客观的程序；其次，后续分析通常未考虑变点是根据数据估计得出这一事实；最后，变点处通常期望的连续性要求被忽略。本文描述了从数据中联合估计线性关系及中间变点的各种方法。在最简单的情况下，对于具有正态误差且一个变量对另一个变量呈线性关系的情况，这相当于通过最小二乘法拟合一条“折线”。此外，还给出了用于检验变点是否存在的测试和图形诊断方法。给出了一个示例，其中估计了陆地哺乳动物奔跑速度与体型关系的变点和斜率。过去，这些数据是用直线或抛物线拟合的。这里表明，使用变点模型可实现更好的拟合和可解释性。
