A new MM algorithm for constrained estimation in the proportional hazards model

• Published in: Computational statistics & data analysis Vol. 84; pp. 135 - 151
• Main Authors: Ding, Jieli, Tian, Guo-Liang, Yuen, Kam Chuen
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2015
• Subjects: algorithms; Asymptotic properties; Bootstrap approach; Constrained estimation; data collection; Karush–Kuhn–Tucker conditions; MM algorithm; Proportional hazards model; regression analysis; MM algorithm; Constrained estimation; Asymptotic properties; Bootstrap approach; Karush–Kuhn–Tucker conditions; Proportional hazards model
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2014.11.005

The constrained estimation in Cox’s model for the right-censored survival data is studied and the asymptotic properties of the constrained estimators are derived by using the Lagrangian method based on Karush–Kuhn–Tucker conditions. A novel minorization–maximization (MM) algorithm is developed for calculating the maximum likelihood estimates of the regression coefficients subject to box or linear inequality restrictions in the proportional hazards model. The first M-step of the proposed MM algorithm is to construct a surrogate function with a diagonal Hessian matrix, which can be reached by utilizing the convexity of the exponential function and the negative logarithm function. The second M-step is to maximize the surrogate function with a diagonal Hessian matrix subject to box constraints, which is equivalent to separately maximizing several one-dimensional concave functions with a lower bound and an upper bound constraint, resulting in an explicit solution via a median function. The ascent property of the proposed MM algorithm under constraints is theoretically justified. Standard error estimation is also presented via a non-parametric bootstrap approach. Simulation studies are performed to compare the estimations with and without constraints. Two real data sets are used to illustrate the proposed methods.