Statistical issues in survival analysis (Comment on hazards are key)
December 3, 2025
In this short article, Per Kragh Andersen has given a rebuttal to the article by Beyersmann et al (2025) which was about hazards constituting key quantities for time-to-event data. Despite arguments by other statisticians, Beyersmann et al. (2025) have convincingly argued according to Andersen that, considering hazards as entire functions in time, contrasts between them do, indeed, allow a causal interpretation. Beyersmann et al. (2025) have first considered the situation where , the time to event, is a proper random variable, that is, the relevant multi-state model to use is the two-state model for survival analysis (e.g., Andersen and Ravn, 2023). They then moved on to more general multistate situations, such as the competing risks model, and argued that, in such models, other events (such as competing risks) may be regarded as censoring mechanisms that still allow a straightforward analysis of the hazard for “the event of interest.”
Andersen has noted that the purpose of the present note was to argue against an interpretation of a competing event as a censoring mechanism and, instead, carefully distinguish between “unavoidable” events, which are present even in a potentially completely observed population, and “avoidable” censoring events (e.g., drop-out, emigration, or being alive at end of study). If then one then wishes to make valid inference for parameters in the completely observed population in the presence of the avoidable censoring event, then he considers this to be a situation that requires an assumption of “independent censoring.”
Andersen then goes on to say that in that respect, Beyersmann et al. (2025) are entirely correct when stating that “censoring by competing risks can be regarded as independent censoring,” however, the two event types, that is, the (unavoidable) competing risks and the (avoidable) censoring at play quite different roles in the model. He explains that this is because the former are associated with intensity parameters in the complete population—parameters that are needed when evaluating marginal parameters such as and , whereas the latter is “a nuisance” for which the (statistically unverifiable) question of “independence” is crucial for valid inference. Andersen also suggested that when making direct inference for a marginal parameter via inverse probability of censoring weighted estimating equations (e.g., Andersen and Ravn, 2023, section
5.5), the censoring weights only reflect the probability of having avoided the avoidable censoring events.
Andersen then quoted from Beyersmann et al. (2025): “hazards are and will continue to be key quantities for analysing, interpreting and understanding time-to-event data.” Andersen fully agrees with this statement and argued above that transition intensities are, indeed, the basic parameters in multistate models in the sense that they specify the probabilistic properties of the process and, as such, they allow estimation of marginal parameters including state occupation probabilities and expected lengths of stay in states.
However, even though in Beyersmann et al. (2025) they argued that hazard contrasts may be given causal interpretation in a functional sense, hazards may not provide the final answer to a given scientific question. Such questions are typically more conveniently expressed in terms of marginal parameters, for example, survival probabilities or other absolute risks (the “estimands”). If one was going from the intensities to the marginal parameters, then this typically requires estimation of all intensities in the multistate model for the complete population.
In conclusion, Andersen suggests when estimating hazards from event history data, that censoring needs to be addressed, and valid inference for the hazards (and subsequent marginal parameters) builds on an assumption of independent censoring. His main point of this note was to distinguish between censoring and non-avoidable events in the multistate model that are present also in the complete population. This is since as he points out they are conceptually different, even though they technically operate in the same way in the process of estimating intensities.
Written by, Usha Govindarajulu
Keywords: survival, hazards, competing risks, multi-state models
References:
Andersen, PK (2025). “Censoring and Competing Risks: Avoidable and Non-Avoidable Events. Comment to the Article “Hazards constitute key quantities for analysing, interpreting and understanding time-to-event data” by Beyersmann, Schmoor, and Schumacher” Biometrical Journal. https://doi.org/10.1002/bimj.70099
Andersen, P. K., and H. Ravn. 2023. Models for Multi-State Survival Data. Rates, Risks, and Pseudo-Values, Chapman and Hall/CRC.
Beyersmann, J., C. Schmoor, and M. Schumacher. 2025. “Hazards Constitute Key Quantities for Analysing, Interpreting and Understanding Time-to-Event Data.” Biometrical Journal 67: e70057.
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