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I want to perform a Cox Regression Analysis for the dependent variable of overall survival in years.

I want to use a categorical explanatory variable that has 3 levels. Let's say for the sake of simplicity it's 3 types of mutually exclusive radiological findings like:

  1. Good CT-Scan
  2. Neutral CT-Scan
  3. Bad CT-Scan

If my understanding is correct then for a given predictor the hazard in one group would be expected to be a constant proportion of the hazard in another group. So let's just say we pick Level 1 = Good CT-Scan to be the constant and then I get Hazard Ratios for "Neutral" and "Bad CT-Scans" when I run the Cox Regression Analysis in R.

My question is then, does it matter which level we choose to be the constant? How would I choose the level which needs to be constant?

R does this automatically, but I wonder if it makes any difference to switch.

Any help is appreciated.

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In some sense it makes no difference in a frequentist partial maximum likelihood analysis. I.e. any comparison between the categories you make afterwards will not change. On the other hand, it makes sense to parameterize things so that you directly get the comparisons you are the most interested.

If you do a Bayesian version of the analysis, it might be easier to specify prior distributions (that reflect the assumptions you want to make) in some parameterization.

Note also that specifying a reference category is just one of several parameterizations you could choose (you could e.g. also do average of the categories and difference from category 1 to 2, as well as difference from 2 to 3, or many other options). It just does not matter too much in a frequentist setting for the reasons indicated above.

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  • $\begingroup$ Thank you very much for taking the time. I'm not that versed in mathematics / statistics, so please bear with me. Am I understanding you correctly that it doesn't change the underlying hazards or their significance when I choose the reference level, but it would make sense to choose for example "Neutral" as a constant when the research question is if the other CT Scan Types predict Overall Survival better/worse/not at all compared to the neutral finding because that makes it easy to report the hazards in relation to that question? $\endgroup$
    – spx31
    Mar 28 at 12:17
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    $\begingroup$ Yes, that's it. If you set your reference category in a sensible way, you get a coefficients that are the log-hazard ratio for your comparison of interest as a direct output. You can still get other comparisons by doing things with the fitted model. E.g. for the point estimate, if you have loghr31 = 3 vs. 1 and loghr21 = 2 vs. 1, and you want 3 vs. 2, then that's just loghr32 = loghr31 - loghr21 (standard errors and confidence intervals are a little bit more complex), but you should get the exact same result, if you had used 2 as the reference category from the start. $\endgroup$
    – Björn
    Mar 28 at 12:38
  • $\begingroup$ Thank you again. This was very helpful. $\endgroup$
    – spx31
    Mar 28 at 12:52

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