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I did a multivariable cox proportional hazards analysis including variables with univariable p<0.2 , and 3 variables gave statistical significance. Their HRs are around 5 to 8, which is pretty high, but their 95% confidence levels, especially for the HR=8, goes up to almost 50. How do I explain this? Can this be attributed to small sample size (n=46), and/or what others reasons might there be? Thanks!

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    $\begingroup$ Do a search for complete separation ... $\endgroup$
    – Ben Bolker
    May 24, 2021 at 18:59

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Note that a hazard ratio (HR) estimate of 8 with a top 95% confidence limit of 50* isn't quite as bad as it might seem. Cox regression coefficients are estimated in the log-hazard scale. A quick calculation suggests that corresponds to a regression coefficient of 2.08 with a standard error of 0.93. On that scale, things don't seem so extreme.

Your model, however, is probably overfit, with a small number of events in a combination of cases that is helping to determine that coefficient value. Even if the "3 variables [that] gave statistical significance" were the only ones in your model and almost all of your 46 cases had events, you would be at the usual overfitting limit of about 15 events per predictor variable. (It's the number of events, not the total number of cases, that provides power in Cox regressions.) It sounds like you started out considering a lot more than 3 predictors and that there were more than 3 in your final multiple-regression model, so you have even a greater risk of having overfit your data.

With your predictor-selection approach and an overfit model, you will tend to find predictors that appear "significant" in your data set but that don't necessarily work so well on new data samples. I'd worry more about that. More data and a more principled approach to regression, as explained in Harrell's course notes, will give you more generally reliable results.


*When you go to the other end of the confidence interval, the lower 95% limit on the HR scale consistent with that point estimate and upper limit is about 1.28. Yes, you have a very broad confidence interval on the (exponentiated) HR scale, as expected with a limited number of events.

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