How to report hazard ratios from a Cox proportional hazards model in English? 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.
 A: 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. 
A: 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? :)
A: 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.
