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To jointly model multivariate longitudinal biomarkers and a time-to-event outcome, we propose a novel joint model based on linear QR mixed models. The longitudinal submodel accounts for associations among multivariate responses by assuming a multivariate asymmetric Laplace distribution for the errors, utilizing its location-scale mixture representation for computational efficiency. For the event-time process, a Cox proportional hazards model is employed, which incorporates shared random effects from the longitudinal submodel to link the two processes. Estimation is performed using a Monte Carlo expectation-maximization algorithm, with Metropolis-Hastings sampling to approximate the E-step. Comprehensive simulations under various quantile levels, censoring rates, error distributions, and correlation structures demonstrate that the proposed method yields accurate estimates for the regression parameters in both the longitudinal and survival submodels. 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