Fitting bent lines to data, with applications to allometry

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.