Analysis of ordered composite endpoints

Composite endpoints are frequently used in clinical trials, but simple approaches, such as the time to first event, do not reflect any ordering among the endpoints. However, some endpoints, such as mortality, are worse than others. A variety of procedures have been proposed to reflect the severity of the individual endpoints such as pairwise ranking approaches, the win ratio, and the desirability of outcome ranking. When patients have different lengths of follow-up, however, ranking can be difficult and proposed methods do not naturally lead to regression approaches and require specialized software. This paper defines an ordering score O to operationalize the patient ranking implied by hierarchical endpoints. We show how differential right censoring of follow-up corresponds to multiple interval censoring of the ordering score allowing standard software for survival models to be used to calculate the nonparametric maximum likelihood estimators (NPMLEs) of different measures. Additionally, if one assumes that the ordering score is transformable to an exponential random variable, a semiparametric regression is obtained, which is equivalent to the proportional hazards model subject to multiple interval censoring. Standard software can be used for estimation. We show that the NPMLE can be poorly behaved compared to the simple estimators in staggered entry trials. We also show that the semiparametric estimator can be more efficient than simple estimators and explore how standard Cox regression maneuvers can be used to assess model fit, allow for flexible generalizations, and assess interactions of covariates with treatment. We analyze a trial of short versus long-term antiplatelet therapy using our methods.