Fig. 1.

Outline of the joint frailty model for recurrent exacerbations and early treatment discontinuation risks. Baseline hazard functions are denoted by r₀(t) and λ₀(t) for exacerbations and early treatment discontinuations, respectively. The frailty u_i is a random effect from a gamma distribution and α represents the strength of the association between exacerbations and discontinuations. Vectors X_{ex, ij} and X_{ed, i} denote covariates related to exacerbations and early discontinuations, respectively, β_ex and β_ed are the corresponding regression coefficients. The at-risk process for exacerbations (subjects not at risk during an ongoing exacerbation episode) is denoted $Y_i^{\mathit{ex}}\left(t\right)$