Accelerated fitting of joint models of survival and longitudinal data with cumulative variations

It has been well recognized that not only biomarkers but also their variability are important for predicting biomarker-related diseases. Understanding and adequately modeling the variability of biomarkers is crucial for detecting and predicting health risks, leading to improved health outcomes and patient care. However, biomarker variability modeling comes with a high computational cost, as statistical models incorporating biomarkers’ variability rely on double integrals with two nested integrations, which must be repeatedly calculated during modeling. To reduce the computational burden, we propose a novel approach aligned with arc length in mathematics to approximate and model biomarker fluctuations. Furthermore, we propose an algorithm that aligns with fast arc length evaluations for the joint modeling of survival and longitudinal data. We synthesize multiple efficient computing methods into a unified framework to accelerate the entire computational process. The core component of the acceleration is the computational efficiency of the double integrals, even when the iterated integral representation of the double integral is not possible. Finally, we illustrate the usage and benefit of our algorithm in joint models in numerical examples and the primary biliary cholangitis clinical study.