On profile MM algorithms for gamma frailty survival models

Gamma frailty survival models have been extensively used for the analysis of such multivariate failure time data as clustered failure times and recurrent events. Estimation and inference procedures in these models often center on the nonparametric maximum likelihood method and its numerical implementation via the EM algorithm. Despite its success in dealing with incomplete data problems, the algorithm may not fare well in high-dimensional situations. To address this problem, we propose a class of profile MM algorithms with good convergence properties. As a key step in constructing minorizing functions, the high-dimensional objective function is decomposed as a sum of separable low-dimensional functions. This allows the algorithm to bypass the difficulty of inverting large matrix and facilitates its pertinent use in high-dimensional problems. Simulation studies show that the proposed algorithms perform well in various situations and converge reliably with practical sample sizes. The method is illustrated using data from a colorectal cancer study.