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Fig. 1 | Respiratory Research

Fig. 1

From: Improving the evaluation of COPD exacerbation treatment effects by accounting for early treatment discontinuations: a post-hoc analysis of randomized clinical trials

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 ui is a random effect from a gamma distribution and α represents the strength of the association between exacerbations and discontinuations. Vectors Xex, ij and Xed, i denote covariates related to exacerbations and early discontinuations, respectively, βex and βed are the corresponding regression coefficients. The at-risk process for exacerbations (subjects not at risk during an ongoing exacerbation episode) is denoted \( {Y}_i^{ex}(t) \)

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