On the win-ratio statistic in clinical trials with multiple types of event

Pocock et al. (2012), following Finkelstein & Schoenfeld (1999), has popularized the win ratio for analysis of controlled clinical trials with multiple types of outcome event. The approach uses pairwise comparisons between patients in the treatment and control groups using a primary outcome, say the time to death, with ties broken using a secondary outcome, say the time to hospitalization. In general the observed pairwise preferences and the weight they attach to the component rankings will depend on the distribution of potential follow-up time. We present expressions for the win and loss probabilities for general bivariate survival models when follow-up of all patients is limited to a specified time horizon. In the special case of a bivariate Lehmann model we show that the win ratio does not depend on this horizon. We show how the win ratio may be estimated nonparametrically or from a parametric model. Extensions to events of three or more types are described. Application of the method of marginal estimation due to Wei et al. (1989) to this problem is described.