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.