Cox regression when reference group had zero events I would appreciate some advise on an a problem I ran into.
I use SPSS for statistical analysis of a study.
The study look into how a blood test predicts mortality with patients followup of 1 year. Patients were divided into quartiles, and using the first quartile as a reference group I used Cox regression to determine the unadjusted Hazard ratios (HR) and the adjusted HR for quartile 2-4.
The problem is that the reference group (Quartile 1–the first 25%) did not have any events.
Therefore I am getting Hazard ratios of 80,000+ for the 2nd, 3rd and 4th quartiles as well as error messages under SPSS.
Can anyone advise me on what Im doing wrong?
 A: Well, what you're doing wrong is using as the reference group a group with zero events. Instead of hazard ratios, think in simpler terms (in my opinion) of incident rate ratios (IRRs), where the incident rate (IR) is $IR=\text{number of cases }/\text{ total person-time}$.
$$IRR_{\text{quartile 4 vs. quartile 1}}=\frac{IR_{\text{quartile 4}}}{IR_{\text{quartile 1}}}$$
What happens if $IR_{\text{quartile 1}}=0$?
You can change your categorisation (use tertiles or some other meaningful categorisation) or, even better, if you have a continuous predictor you can treat it as such and examine potential nonlinear relationships using polynomial terms, fractional polynomials or restricted cubic splines, for example. 
A: To supplement andrea's response by extending it a bit to hazard ratios:
The hazard of an event is the instantaneous probability of an event occurring at time t, conditional on it not having previously occurred.
Your problem should be clear instantly - with no events, the probability is zero. Borrowing from andrea's example, the incident rate is equivalent to a constant hazard - in your case, a constant hazard of zero.
Dividing by zero tends to make software angry.
You need to switch your reference category. My suggestion is to use "Quartile 4" or the other high value of the category, and step down, rather than using Quartile 1 and stepping up. If you were hoping to, for example, show an increase in the HR as you moved up a category, you're now showing the equivalent protective effect from moving down one.
I would also suggest taking a moment to consider why you have no events.
It's possible you're simply having a run of "bad luck", at which point there's nothing you can do but increase the study size or follow the population for longer in hopes of accumulating more events. But you should make sure there's no reason that the probability of having an outcome in your population isn't zero for a reason. For cardiac events I can imagine one, but it is always worth stopping to consider when you have zero events in some level of a covariate.
