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In a cox regression, modelled by the coxph function in the survival package, I got the following output for group only:

               coef exp(coef) se(coef)     z Pr(>|z|)  
Group2         0.2972    1.3461   0.1714 1.734    0.083 .

However, when I add a covariate, the effect of group gets significant:

                        coef exp(coef) se(coef)     z   Pr(>|z|)    
Group2              0.35135   1.42098  0.17193 1.986     0.0471 *  
covariate1          0.08305   1.08659  0.01576 4.639 0.00000351 ***  

This seems odd to me. In my understanding, in the second model, the effect of group is held constant for values of my covariate1. However, the effect of covariate1 points in the same direction as that of group (i.e., an increase in covariate1 increases the hazards; a change from reference group to Group2 increases the hazards, too) - in my understanding, it should have pointed in the other direction (i.e., HR<1), so that after controlling for it the effect of Group gets significant.

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1 Answer 1

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First, don't overemphasize the "p < 0.05" cutoff. The hazard ratios associated with Group2 in both models are almost identical (1.35 versus 1.42). It's just that in the second model the coefficient estimate happened to pass (barely) that cutoff. Don't confuse statistical significance with practical significance.

Second, what you see can happen with Cox models. If you omit any outcome-associated predictor from a Cox model, the coefficients for the included predictors can be biased in magnitude toward zero even if the omitted predictor isn't correlated with the included predictors. That's similar to what happens in binomial regression. In this case, adding the outcome-associated Covariate1 did move the coefficient for Group2 slightly farther from 0, sufficient in this case to pass (barely) the "p < 0.05" cutoff.

It's thus good practice to include as many outcome-associated predictors in a Cox model as you can, provided that you don't overfit the data.

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  • $\begingroup$ Dear EdM, thank you very much for this clear and detailed answer! I read the article you linked and have another two questions. 1) would you agree with my understanding of the link you posted that this is an attenuation bias, following the missspecification of the residual term which comprises the errors of my covariate1 when I only include the group term? 2) if I am only interested in group effects and the result is deemed to decide whether my grouping variable is signficant or not, the differnce in stat. significance can change a lot. Which model would you stick to (if not prespecified) $\endgroup$
    – Sebastian
    Feb 21, 2023 at 7:49
  • $\begingroup$ @Sebastian for (1) you can think of this as a type attenuation bias, but with Cox models it doesn't come from misspecified residuals in the way it does in ordinary least squares. The problem is worse in Cox and binomial regression models, where noncollapsibility plays an important role. For (2), it's best to use a model as complex as reasonable without overfitting. The "difference in stat. significance" isn't big between your models, it's just that one p-value is slightly below and one slightly above 0.05. The HRs are almost identical. $\endgroup$
    – EdM
    Feb 21, 2023 at 13:24
  • $\begingroup$ thanks alot for your valuable answers. One last question would be if you have any reference (paper or book) which comprises this problem and leads to the conclusion of a model as complex as possible without overfitting? $\endgroup$
    – Sebastian
    Feb 23, 2023 at 16:49
  • $\begingroup$ @Sebastian I'd recommend Frank Harrell's notes on Regression Modeling Strategies and the associated book. Chapter 4 is particularly on point. That's true for all type of regression, not just Cox models. $\endgroup$
    – EdM
    Feb 23, 2023 at 19:45

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