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My understanding is that a hazard ratio from a Cox proportional hazards model compares the effect on the hazard rate of a given factor to a reference group. How would you report that to an audience that doesn't know statistics?

Let's try to phrase an example. Say we enroll people in a study of how long before they buy a couch. We right-censor at 3 years. For this example we have two factors: age < 30 or >= 30, whether they own a cat. It turns out the hazard ratio of "owns cat" to the reference group (age < 30, "doesn't own cat") is 1.2, and significant (say p<0.05).

Am I correct to say that means all of these: cat owners have more events (couch purchase) within 3 years, OR that time-to-event (couch purchase) is faster for cat owners, OR some combination of those two things?

Edit: Suppose the event is their first purchase of the couch within the period (if one occurs). This model does not help us analyze multiple purchases within the time period.

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A hazard ratio is a rate ratio. A rate is "events per unit time". Given that the Cox model specifies proportional hazards at all time points, a hazard ratio of 1.2 means that the rate of couch-buying in the "owns cat" group is 20% higher at any given time point studied than the rate in the "doesn't own cat" group.

So I would say your first assertion (cat owners have more events [couch purchase] within 3 years) is correct, except that in addition to having more events within 3 years, they also have more events at any given time within that years (instantaneous hazard). A subtle difference, perhaps.

I guess the conclusion is that damage caused by cats might lead to more couch purchases? :)

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    $\begingroup$ So if the two groups have the same number of events, but one has them happen all right away, and the other right at the end, the hazard ratio will be 1? That is, the time-to-event doesn't affect the hazard ratios? $\endgroup$ – dfrankow Aug 17 '11 at 15:53
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    $\begingroup$ That type of data would not meet the proportional hazards assumption of the Cox model and would be better off modelled using a different assumed distribution. $\endgroup$ – pmgjones Aug 17 '11 at 15:56
  • $\begingroup$ aha, good point. So, it is true that time-to-event doesn't affect hazard ratios (except indirectly through a difference in # of events)? $\endgroup$ – dfrankow Aug 17 '11 at 16:08
  • $\begingroup$ .. because that is the assumption of proportional hazards (the Cox model assumption)? $\endgroup$ – dfrankow Aug 17 '11 at 16:14
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    $\begingroup$ The statement that 'cat owners have more events within 3 years' might be misconstrued as some individual cat owners buying more than one couch (as a cat owner, I do not recommend this!). The Cox model is usually applied to mortality (you evidently only die once) where there should be no such ambiguity, however. $\endgroup$ – shabbychef Aug 17 '11 at 16:30
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To a pure lay-audience, I'd go with "Cat owners are 1.2 times as likely to purchase a couch than non-cat owners."

Things like "at any point t during the study period", or trying to define the idea of a hazard, is getting a bit close to sausage making for most people, and will not advance them understanding the core of your results any more - which is the actual point of a summary like this.

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    $\begingroup$ To whomever downvoted this, it seems like a perfectly fine response to me, and deserves justification for why the downvote was given. @EpiGrad, I've never heard the colloquialism "close to sausage making", do you know where this saying comes from? $\endgroup$ – Andy W Aug 17 '11 at 18:31
  • $\begingroup$ Thanks for the vote of confidence :) Its an adaptation of the saying "There are two things that you do not want to know how they were made, law and sausages" - I've heard several people adapt it to statistics. $\endgroup$ – Fomite Aug 17 '11 at 20:30
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Let $X$ be an indicator variable, being $1$ if the guy has a cat, and $0$ otherwise. Your result is

$\frac{h(t | X = 1)}{h(t | X = 0)} = 1.2 \hspace{2cm}$ (1)

where $h(t | X=x)$ is the hazard function evaluated at time $t$ for those with $X = x$.

Here, the hazard at time $t$, h(t), is the conditional instantanenous probability of buying a couch at time $t$, given that you still did not buy it just prior to time $t$.

In words, (1) is the ratio of the hazards of buying a couch at any time for an individual who has a cat relative to an individual who has not a cat.

Alternatively, this states that the hazard of buying a couch at any time $t$ for a person who has a cat is superior than that of a person who does not have a cat.

Now, it might be interesting to investigate whether this hazard ratio is significantly different from $1$. If not, having a cat does not impact the hazard of buying a couch. This could be done by constructing the confidence interval for that hazard ratio.

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    $\begingroup$ I don't disagree with your facts, but your English summaries do not seem to be easy to read for a non-statistician: 1) "the ratio of the hazards of buying a couch at any time for an individual who has a cat relative to an individual who has not a cat" ??; 2) "the hazard of buying a couch at any time t for a person who has a cat is superior than that of a person who does not have a cat" ?? Remember, this question is about how to phrase in English for a non-technical audience. $\endgroup$ – dfrankow Aug 17 '11 at 16:07
  • $\begingroup$ @dfrankow: I disagree: this is not technical, but "rigourous". If you do not want to speak about hazard, then you should not use the Cox model... $\endgroup$ – ocram Aug 17 '11 at 16:26
  • $\begingroup$ I'm in agreement with dfrankow - there is a vast difference between choosing the appropriate statistical test, and communicating that result to a lay audience. And in this case, "rigorous" is technical - and counter-productive for many audiences. $\endgroup$ – Fomite Aug 17 '11 at 23:57
  • $\begingroup$ @EpiGard: I agree it is difficult to communicate statistics to a lay audience. But still, it is the duty of the statistician to interpret results rigourously. Otherwise, software would replace them! "Cat owners are 1.2 times as likely to purchase a couch than non-cat owners." would be translated by "Pr(purchase a couch | cat) = 1.2 Pr(puchase a couch | not cat)". This is not what dfrankow wants to communicate... $\endgroup$ – ocram Aug 18 '11 at 5:39

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