A novel block‐coordinate gradient descent algorithm for simultaneous grouped selection of fixed and random effects in joint modeling

Joint models for longitudinal and time‐to‐event data are receiving increasing attention owing to its capability of capturing the possible association between these two types of data. Typically, a joint model consists of a longitudinal submodel for longitudinal processes and a survival submodel for the time‐to‐event response, and links two submodels by common covariates that may carry both fixed and random effects. However, research gaps still remain on how to simultaneously select fixed and random effects from the two submodels under the joint modeling framework efficiently and effectively. In this article, we propose a novel block‐coordinate gradient descent (BCGD) algorithm to simultaneously select multiple longitudinal covariates that may carry fixed and random effects in the joint model. Specifically, for the multiple longitudinal processes, a linear mixed effect model is adopted where random intercepts and slopes serve as essential covariates of the trajectories, and for the survival submodel, the popular proportional hazard model is employed. A penalized likelihood estimation is used to control the dimensionality of covariates in the joint model and estimate the unknown parameters, especially when estimating the covariance matrix of random effects. The proposed BCGD method can successfully capture the useful covariates of both fixed and random effects with excellent selection power, and efficiently provide a relatively accurate estimate of fixed and random effects empirically. The simulation results show excellent performance of the proposed method and support its effectiveness. The proposed BCGD method is further applied on two real data sets, and we examine the risk factors for the effects of different heart valves, differing on type of tissue, implanted in the aortic position and the risk factors for the diagnosis of primary biliary cholangitis.