Department of Mathematics - Mathematics Colloquium - Change-point regression models: new results, diverse applications, and some history

A vast number of methods have been proposed to accommodate nonlinearity in linear models. These include transforming the dependent variable, adding higher-order terms, spline models, and kernel smoothing. All of the techniques just mentioned, along with most others, have in common the assumption of smoothness. However, in many cases there is reason to believe the underlying pattern to be discontinuous, either on the raw scale ("broken line") or its derivative ("bent line"). These change-point models have a long history. They were first posed, to my knowledge, in the 1950s in order to estimate changes in stock prices. In 1989 I became interested in applying them to model the mean in a biological context, which I will use in this talk as a motivating example. It then became clear that many questions are better addressed using quantiles rather than means and that accounting for repeated measures would be useful. Both of these extensions are used to model cognitive changes in Alzheimer's disease. The next step for that example is to allow the change-points to vary via a random effect. All of these situations involve interesting theoretical issues due to the likelihood's irregularity as well as computational challenges of varying complexity. Time permitting, I will discuss a related question in the area of non-inferiority clinical trials.