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I am getting into survival analysis and am a bit confused about the definition of hazard ratio and odds ratios. This question implies that there is a clear difference between hazard ratios and odds ratios and that it is not straightforward to compute odds ratios from hazard ratios.

Now, I found the paper of Heimer (2016) who report the following results of a coc hazard model:enter image description here

They compute the odds ratio as the exponential of the coefficient retreived from the hazard model. For example, in column (1), the odds-ratio is retreived by exp(0.295). I though the exponent of the coefficient of a hazard model is the hazard ratio and not the odds ratio. They interpret the odds ratio as "the rate at which trades are closed increases by 34% if the position is a gain".

Are they correct in defining the odds ratio as the exponential of the coefficient? Is their interpretation of the odds ratio correct?

References: Heimer, R. Z. (2016). Peer pressure: Social interaction and the disposition effect. The Review of Financial Studies, 29(11), 3177-3209.

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2 Answers 2

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You are correct that these authors should have called the exponentiation of the Cox regression coefficient the "hazard ratio," not the "odds ratio." In practice, however, the values might not be so different from each other, as @Glen_b explains:

They're almost the same thing when doubling the odds of the event is almost the same as doubling the hazard of the event. They're not automatically similar, but under some (fairly common) circumstances they may correspond very closely.

Despite this frequent similarity in values, however, there is no reason to use terminology that isn't correct. As the report was of a Cox proportional hazards regression, "hazard ratio" is the correct terminology. For more detail on these frequently confused terms, see for example: George A, Stead T S, Ganti L (August 26, 2020) What’s the Risk: Differentiating Risk Ratios, Odds Ratios, and Hazard Ratios?. Cureus 12(8): e10047. doi:10.7759/cureus.10047

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The context of the study is insufficiently clear to say whether it is an odds ratio or a hazard ratio. In the context of a very particular setting, a Cox regression actually ends up estimating an odds ratio. This happens when events can only happen a small number of event times (e.g. time [in number of dice rolls] to rolling a 6 on the first, second or third try out of 3 rolls of a dice) and if we handle ties as actually tied event times (i.e. there's no underlying ordering in time of the events, which we just don't know because we recorded the times only a specific unit - e.g. days to death from randomization in a clinical trial is not a good example of this, because there is in truth an ordering, we just didn't record the time at a more fine-grained level than days).

From that perspective (without knowing more), it's possible the authors used the term correctly. But, it's also entirely possible that it's a wrong usage (that may happen to be a case where there's not too much of a difference, as others have pointed out).

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