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I have a cox regression with two predictors:

Predictor A: HR = 2.00, Z = 15.24, range is 0-1 (continuous)

Predictor B: HR = 0.97, Z = -6.68, range is 1-17,

Based on this information, can I say that the first variable has a stronger effect on the outcome? If not, how can I determine which predictor carries a stronger effect? Essentially, I'm looking for something like Beta in OLS.

Thanks!

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3 Answers 3

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To move away from assumptions about ranges and units, I suggest using relative log-likelihood ratio $\chi^2$ explained or relative explained variation as detailed here. These measure relative predictor importance on a 0-1 scale. See also this and partial $R^2$ measures.

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  • $\begingroup$ Thank you Professor Harrell. If I understand correctly, what I need to do is to construct two reduced models, one without predictor A and one without predictor B, and then divide each model's Wald with the full model's Wald. The predictor with a bigger "loss" (i.e., difference between the reduced and full Walds) is more important in explaining the outcome. Am I correct? $\endgroup$
    – Eran
    Commented Jun 25 at 5:46
  • $\begingroup$ Correct, except use likelihood ratio $\chi^2$ instead of Wald. $\endgroup$ Commented Jun 25 at 12:00
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I don't know of any equivalent of $\beta$ for Cox regression.

The results you have shown do not allow you to say one is stronger than the other, because the HR is per unit of the independent variable and the second has much bigger range.

One thing you could do is divide the second measure by 17, so the ranges are equal, and then redo the analysis. This at least gives you comparable measures.

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    $\begingroup$ Our answers crossed ! (+1) $\endgroup$ Commented Jun 24 at 11:34
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I was about the post essentially the same answer as Peter. I will go into a bit more detail.

It may be worth noting that the associated p-values are very small (around 1e-16 and 2e-11 respectively), meaning that if there were actually no association between the predictors and the outcome then the probability of observing these, or more extreme, test statistics, is vanishingly small.

can I say that the first variable has a stronger effect on the outcome?

While it is tempting to notice that the coefficient estimates are different. for A, where HR = 2.00 implies that a one-unit increase in Predictor A is associated with a doubling of the hazard rate, and for B, HR = 0.97 implies that a one-unit increase in Predictor B is associated with a decrease in the hazard rate by 3%. The temptation is therefore to conclude that A has the stronger effect.

However that would be wrong because the variables are on very different scales: [0,1] for A and [1,17] for B

If the variables were standardised, so that they are on the same scale, the comparison above would make sense.

When the scales are markedly different, as they are here, the comparison obviously doesn't make sense. Without access to the data, and to run the analysis with standardised variables, one thing we can do is to look at the full range of each predictor and examine the association of each predictor with the outcome over it's entire range. This is basically the same suggestion that Peter made. For A, it's range is 0 to 1 so over this range we have the same association - the hazard rate doubles. For B on the other hand, over it's entire range of $17-1=16$ units, the hazard rate changes by $0.97^{16} \approx 0.61$.

So we are comparing a doubling of the hazard rate for A (100% change), and a reduction of 61% for B. Hence A has the stronger "effect", since $100 > 61$, when considering the association between a change of the entire range of each variable.

A better approach would be to standardise both predictors to be on the same scale

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