Estimation of higher-order two-phase regression models

Abstract

Two-phase regression models are a class of nonlinear regression models that are known for their flexibility and interpretability. An important feature of two-phase regression models is the existence of a threshold at which the relationship between an outcome and a covariate of interest changes. A standard estimation method, such as that used for generalized linear models, cannot be applied to two-phase regression models since the likelihood function is not differentiable with respect to the threshold parameter. We resolve this difficulty by using a grid search method which reduces the problem to a set of well-behaved likelihood functions for given candidate threshold values. Previously, a fast grid search algorithm that dramatically improved computational efficiency over a brute-force grid search was developed for two-phase regression models with linear trends. Here we generalize this algorithm to higher-order two-phase regression models where two separate polynomial regressions, not limited to linear, are used to model each phase (i.e., before and after the threshold). Based on the proposed fast grid search algorithm, we perform Monte Carlo simulations to examine the behavior of the parameter estimates. A real data example is also presented to illustrate the practical use of two-phase regression models.