Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).

Mats Julius Stensrud

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause specific or subdistribution hazards, are fundamentally hard to be interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause specific hazard functions. We describe the asymptotic properties of these estimators, and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we re-analyze the prostate cancer trial from Stensrud et al (2020).

2020

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

This work is concerned with the estimation of the spreading potential of the disease in the initial stages of an epidemic. A speedy and accurate estimation is important for determining whether or not interventions are necessary to prevent a major outbreak. At the same time, the information available in the early stages is scarce and data collection imperfect. We consider an epidemic in a large susceptible population, and address the estimation based on temporally aggregated counts of new cases that are subject to unknown random under-reporting. We allow for an influence of the detection process on the evolution of the epidemic. While the proportion of infectious individuals in the population is small, the role of chance in the spread of the disease may be substantial. Therefore, stochastic epidemic models are applied. As these are difficult to analyse, the time evolution of the number of infectious individuals is approximated by branching processes. We study the estimation in a partially observed Galton–Watson process and in a partially observed linear birth and death process; and in each case focus on the parameter characterising the growth of the process. We aim at estimators that perform well in the asymptotic sense where a single trajectory is observed over a long period of time, and study the asymptotics conditionally on the eventual explosion of the process. The partially observed Galton–Watson process has been recently proposed in the literature as a model for the initial stages of an epidemic. Its probabilistic structure has been explored and estimation has been partially addressed, in that consistent estimators have been constructed. However, the estimation-related uncertainty has not been evaluated. We address this issue here by constructing estimators that are motivated from the asymptotic dependence structure of the process. We show that they are consistent and asymptotically normal, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. In addition, we evaluate their finite-sample performance in a simulation study and their practical performance on real data. The observation mechanism in the partially observed Galton–Watson process is inherently discrete. To allow for continuous-time dynamics, we incorporate partial observation in the linear birth and death process. In particular, we propose a model where the birth process is completely unobserved, while a random proportion of the death process is observed at discrete time points. We study the estimation in this model. Motivated by its counting process structure, we arrive at consistent and asymptotically normal estimators, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. We also evaluate the finite-sample and practical performance of the estimators in a simulation study and on real data.

EPFL2013

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

EPFL2013