# 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?

• I think the solution here is @andrea's suggestion to use your continuous covariate as continuous, rather than categorizing it. Categorizing continuous covariates is commonly done; it may make interpretation easier for clinicians, but statistically it is always a poor choice. Your situation is an example of that. Although the context differs substantially, I discussed some reasons to not categorize continuous covariates here: How to choose between ANOVA and ANCOVA in a designed experiment (especially below "Update"). Oct 14 '12 at 14:56

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.

• Thanks for the advice. Quartiles really capture the data well. Especially when I have 2 outcomes (Death and heart attacks) and the Kaplan Maier curves come out really well. Its frustrating that after a full year there has been no events in the reference group for heart attacks (ok not frustrating its actually great for the patients) and there is an incremental rise across the remaining quartiles. Was hoping there was a small trick to it that I'm missing... But if what you say is true then there is no way I can use Cox regression to get the HR when the reference has zero events Thanks again Oct 14 '12 at 8:24
• +1 for "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", which I think is the real solution here. Oct 14 '12 at 14:47

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.

• My population is high risk for heart attacks. The test used is a new one. Its remarkable that all the patients in Q1 didn't get a heart attack after 1 year (in a population known to have a 30% event rate in the 1st year). This is with n=400. The data is exciting but not sure the best way to present it with 0 events in Q1.Tried using Q4 as reference but still getting HR of 1000 for Q1. I think the best suggest is Andreas-tertiles & reanalyse the whole data.Or wait until something happens in Q1 :p Thanks for taking the time to explain things. Happy to hear any others suggestions or advise... Oct 14 '12 at 9:54
• @HishamC.Hassan Instead of dividing by zero by using Q1 as a reference, you're dividing zero by the reference. You'll still get unstable results for the HRs involving Q1. The difference is that's one unstable result, instead of all unstable results. Oct 14 '12 at 9:57