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.
Best Answer
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? :)