Extension of the association structure in joint models to include weighted cumulative effects.

Motivated by a study measuring diabetes-related risk factors and complications, we postulate an extension to the standard formulation of joint models for longitudinal and survival outcomes, wherein the longitudinal outcome has a cumulative effect on the hazard of the event, weighted by recency. We focus on the relationship between the biomarker HbA1c and the development of sight threatening retinopathy, since the impact of the HbA1c marker on the risk of sight threatening retinopathy is expected to be cumulative, with the evolution of the HbA1c marker over time contributing to progressively greater damage to the vascular structure of the retina. Opting for a parametric approach, we propose the use of the normal and skewed normal probability density functions as weight functions, estimating the relevant parameters directly from the data. The use of the recency-weighted cumulative effect specification allows us to incorporate differences in the development of the longitudinal profile over time in the calculation of hazard ratios between subjects. The proposed functions provide us with parameters with clinically relevant interpretations while retaining a degree of flexibility. In addition, they also allow answering of important clinical questions regarding the relative importance of various segments of the biomarkers history in the estimation of the risk of the event.