Hazards Constitute Key Quantities for Analyzing, Interpreting and Understanding Time-to-Event Data

Abstract

Censoring makes time-to-event data special and requires customized statistical techniques. Survival and event history analysis therefore builds on hazards as the identifiable quantities in the presence of rather general censoring schemes. The reason is that hazards are conditional quantities, given previous survival, which enables estimation based on the current risk set—those still alive and under observation. But it is precisely their conditional nature that has made hazards subject of critique from a causal perspective: A beneficial treatment will help patients survive longer than had they remained untreated. Hence, in a randomized trial, randomization is broken in later risk sets, which, however, are the basis for statistical inference. We survey this dilemma—after all, mapping analyses of hazards onto probabilities in randomized trials is viewed as still having a causal interpretation—and argue that a causal interpretation is possible taking a functional point of view. We illustrate matters with examples from benefit–risk assessment: Prolonged survival may lead to more adverse events, but this need not imply a worse safety profile of the novel treatment. These examples illustrate that the situation at hand is conveniently parameterized using hazards, that the need to use survival techniques is not always fully appreciated and that censoring not necessarily leads to the question of “what, if no censoring?” The discussion should concentrate on how to correctly interpret causal hazard contrasts and analyses of hazards should routinely be translated onto probabilities.