Data-driven choice of a model selection method in joinpoint regression

Abstract

Selecting the number of change points in segmented line regression is an important problem in trend analysis, and there have been various approaches proposed in the literature. We first study the empirical properties of several model selection procedures and propose a new method based on two Schwarz type criteria, a classical Bayes Information Criterion (BIC) and the one with a harsher penalty than BIC (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mstyle><mml:mtext>BIC</mml:mtext></mml:mstyle><mml:mn>3</mml:mn></mml:msub></mml:math>). The proposed rule is designed to use the former when effect sizes are small and the latter when the effect sizes are large and employs the partial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> to determine the weight between BIC and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mstyle><mml:mtext>BIC</mml:mtext></mml:mstyle><mml:mn>3</mml:mn></mml:msub></mml:math>. The proposed method is computationally much more efficient than the permutation test procedure that has been the default method of Joinpoint software developed for cancer trend analysis, and its satisfactory performance is observed in our simulation study. Simulations indicate that the proposed method performs well in keeping the probability of correct selection at least as large as that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mstyle><mml:mtext>BIC</mml:mtext></mml:mstyle><mml:mn>3</mml:mn></mml:msub></mml:math>, whose performance is comparable to that of the permutation test procedure, and improves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mstyle><mml:mtext>BIC</mml:mtext></mml:mstyle><mml:mn>3</mml:mn></mml:msub></mml:math> when it performs worse than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle><mml:mtext>BIC</mml:mtext></mml:mstyle><mml:mo>.</mml:mo></mml:math> The proposed method is applied to the U.S. prostate cancer incidence and mortality rates.