Joint modeling of a single event time response with a longitudinal covariate dates back to the 1990s. The three basic types of joint modeling formulations are selection models, pattern mixture models and shared parameter models. The shared parameter models are most widely used. One type of a shared parameter model (Joint Model I) utilizes unobserved random effects to jointly model a longitudinal sub-model and a survival sub-model to assess the impact of an internal time-dependent covariate on the time-to-event response.
Motivated by the Muscular Dystrophy Surveillance, Tracking and Research Network (MD STARnet), we constructed a new model (Joint Model II), to jointly analyze correlated bivariate time-to-event responses associated with an internal time-dependent covariate in the Frequentist paradigm. This model exhibits two distinctive features: 1) a correlation between bivariate time-to-event responses and 2) a time-dependent internal covariate in both survival models. Developing a model that sufficiently accommodates both characteristics poses a challenge. To address this challenge, in addition to the random variables that account for the association between the time-to-event responses and the internal time-dependent covariate, a Gamma frailty random variable was used to account for the correlation between the two event time outcomes. To estimate the model parameters, we adopted the Expectation-Maximization (EM) algorithm. We built a complete joint likelihood function with respect to both latent variables and observed responses. The Gauss-Hermite quadrature method was employed to approximate the two-dimensional integrals in the E-step of the EM algorithm, and the maximum profile likelihood type of estimation method was implemented in the M-step. The bootstrap method was then applied to estimate the standard errors of the estimated model parameters. Simulation studies were conducted to examine the finite sample performance of the proposed methodology. Finally, the proposed method was applied to MD STARnet data to assess the impact of shortening fractions and steroid use on the onsets of scoliosis and mental health issues.