Joint modeling of two longitudinal outcomes and competing risk data.

Aortic gradient and aortic regurgitation are echocardiographic markers of aortic valve function. Both are biomarkers repeatedly measured in patients with valve abnormalities, and thus, it is expected that they are biologically interrelated. Loss of follow-up could be caused by multiple reasons, including valve progression related, such as an intervention or even the death of the patient. In that case, it would be of interest and appropriate to analyze these outcomes jointly. Joint models have recently received much attention because they cover a wide range of clinical applications and have promising results. We propose a joint model consisting of two longitudinal outcomes, one continuous (aortic gradient) and one ordinal (aortic regurgitation), and two time-to-events (death and reoperation). Moreover, we allow for more flexibility for the average evolution and the subject-specific profiles of the continuous repeated outcome by using B-splines. A disadvantage, however, is that when adopting a non-linear structure for the model, we may have difficulties when interpreting the results. To overcome this problem, we propose a graphical approach. In this paper, we apply the proposed joint models under the Bayesian framework, using a data set including serial echocardiographic measurements of aortic gradient and aortic regurgitation and measurements of the occurrence of death and reoperation in patients who received a human tissue valve in the aortic position. The interpretation of the results will be discussed.