Estimating joinpoints in continuous time scale for multiple change-point models

Joinpoint models have been applied to the cancer incidence and mortality data with continuous change points. The current estimation method [Lerman, P.M., 1980. Fitting segmented regression models by grid search. Appl. Statist. 29, 77-84] assumes that the joinpoints only occur at discrete grid points. However, it is more realistic that the joinpoints take any value within the observed data range. Hudson [1966. Fitting segmented curves whose join points have to be estimated. J. Amer. Statist. Soc. 61, 1097-1129] provides an algorithm to find the weighted least square estimates of the joinpoint on the continuous scale. Hudson described the estimation procedure in detail for a model with only one joinpoint, but its extension to a multiple joinpoint model is not straightforward. In this article, we describe in detail Hudson's method for the multiple joinpoint model and discuss issues in the implementation. We compare the computational efficiencies of the LGS method and Hudson's method. The comparisons between the proposed estimation method and several alternative approaches, especially the Bayesian joinpoint models, are discussed. Hudson's method is implemented by C ++ and applied to the colorectal cancer incidence data for men under age 65 from SEER nine registries.