Optimal designs for discrete-time survival models with random effects

This paper considers the optimal design for the frailty model with discrete-time survival endpoints in longitudinal studies. We introduce the random effects into the discrete hazard models to account for the heterogeneity between experimental subjects, which causes the observations of the same subject at the sequential time points being correlated. We propose a general design method to collect the survival endpoints as inexpensively and efficiently as possible. A cost-based generalized D (Ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_s$$\end{document})-optimal design criterion is proposed to derive the optimal designs for estimating the fixed effects with cost constraint. Different computation strategies based on grid search or particle swarm optimization (PSO) algorithm are provided to obtain generalized D (Ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_s$$\end{document})-optimal designs. The equivalence theorem for the cost-based D (Ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_s$$\end{document})-optimal design criterion is given to verify the optimality of the designs. Our numerical results indicate that the presence of the random effects has a great influence on the optimal designs. Some useful suggestions are also put forward for future designing longitudinal studies.