Efficient estimation of the marginal mean of recurrent events in randomized clinical trials

Recurrent events data is often encountered in bio-medical settings, for example, as in the LEADER clinical trial that considered if the use of Liraglutide to the reduce the number of myocardial infarctions. To address such a question a useful estimand and analysis for recurrent event data is to estimate the marginal mean number of events of a specific type, also in the presence of a terminal event. There has recently been renewed interest in the pharmaceutical industry to use and study recurrent events data more efficiently and correctly Fritsch et al. (2021); Akacha et al. (2018); Schmidli et al. (2021). In this work we will focus on randomized clinical trials (RCT), where it is often of interest to estimate the treatment effect of a new drug. In our contest this is the difference of the marginal means of recurrent events between the treated and untreated subjects. In this work we will show how to improve the efficiency of such estimand developing a double augmented estimator, where the first augmentation due to right-censorings is utilizing information about each subject dynamically collected over time and the auxiliary covariates, whereas the second augmentation exploits the randomisation of the treatment. We demonstrate the important result that the the two different augmentation terms are orthogonal and thus contribute different sources to reducing the variance of our estimators. As a consequence, we can study them separately. The two augmentation terms are based on working models and we show that no matter how they are chosen we will reduce the variance, at least asymptotically, compared to the simple RCT estimator. The first augmentation that improves the efficiency with respect to the right-censoring is based on a regression augmentation that is easy to compute, but as pointed out in Cortese and Scheike (2022) to get an improvement it’s crucial that this is done dynamically over time. We provide specific formula for the optimal gain that can achieved from this augmentation and demonstrate that the first regression augmentation will always improve the asymptotic variance of the estimator. The second augmentation is achieved by utilizing the randomization and has a structure that is well understood, see for example (Tsiatis, 2006; Van der Laan and Robins, 2003; Robins and Rotnitzky, 1992). We provide a specific formula for the optimal gain that can be obtained due to randomization. In practice the optimal gain can be obtained if the conditional model of the response given covariates and treatment is known. In practice we must use a working model and if the working model is chosen appropriately we will still obtain consistent estimators that will asymptotically improve on the simple RCT estimator. We show directly that when the working model is chosen appropriately it will still lead to reduction in the asymptotic variance compared to the simple RCT estimator. We will apply our methods to the LEADER clinical trial, a study that experiments the use of Liraglutide drug (the treatment) to reduce cardiovascular disease in type-two diabetes patients. The study considers two different outcomes of interest: the MI outcome that is the number of non-fatal myocardial infarction and is a classical recurrent events outcome considering death as a terminal event; and the 3-p MACE outcome that counts the number of non-fatal myocardial infraction and cardiovascular death and still has death a terminal event. The 3-p MACE outcome is a so-called composite outcome, see Mao and Lin (2016). Our procedure for improving the efficiency can be applied to both types of outcomes. Extensive finite sample simulations demonstrate that there are indeed important gains in efficiency in settings that mimics those of the LEADER data for both the MI outcome and the composite 3-p MACE outcome that are considered in different simulations.