An illness–death multistate model to implement delta adjustment and reference-based imputation with time-to-event endpoints

Abstract

With a treatment policy strategy, therapies are evaluated regardless of the disturbance caused by intercurrent events (ICEs). Implementing this estimand is challenging if subjects are not followed up after the ICE. This circumstance can be dealt with using delta adjustment (DA) or reference-based (RB) imputation. In the survival field, DA and RB imputation have been researched so far using multiple imputation (MI). Here, we present a fully analytical solution. We use the illness–death multistate model with the following transitions: (a) from the initial state to the event of interest, (b) from the initial state to the ICE, and (c) from the ICE to the event. We estimate the intensity function of transitions (a) and (b) using flexible parametric survival models. Transition (c) is assumed unobserved but identifiable using DA or RB imputation assumptions. Various rules have been considered: no ICE effect, DA under proportional hazards (PH) or additive hazards (AH), jump to reference (J2R), and (either PH or AH) copy increment from reference. We obtain the marginal survival curve of interest by calculating, via numerical integration, the probability of transitioning from the initial state to the event of interest regardless of having passed or not by the ICE state. We use the delta method to obtain standard errors (SEs). Finally, we quantify the performance of the proposed estimator through simulations and compare it against MI. Our analytical solution is more efficient than MI and avoids SE misestimation—a known phenomenon associated with Rubin's variance equation.