# P-value versus Exp(B) value in Cox regression analysis

I performed Cox regression analysis for two Biomarkers. Both were significant (p<0.05).

Biomarker 1 showed higher significance but lower Exp(B)=hazard ratio value than Biomarker 2 (see below).

Biomarker 1: HR=3.06; 95%CI=1.71-5.48; p<0.001 (sorry, SPSS doesn't show more than 3 decimals)

Biomarker 2: HR=6.05; 95%CI=1.67-21.86; p=0.006

Which of the two Biomarkers is a better (stronger) predictor? In other words: what counts more - the P-value or the Exp(B) value?

• Are we talking continuous biomarkers or dichotomoizations (e.g. biomarker #1 > threshold X)? You may have the problem that one dichotomization is a stronger predictor, but another classifies more patients as at risk (so it depends all a bit on your criterion for better). There may be considerable uncertainty around the size of the regression coefficient so simply comparing the regression coefficients is problematic for really concluding which is better. For the same reason comparing the random p-values is problematic. Perhaps you could look at the C-index (including confidence intervals). – Björn Sep 1 '15 at 11:04
• The biomarkers are dichotomic (low and high expression) – user86880 Sep 1 '15 at 12:18
• But I will look at the C-Index as well. Thanks! – user86880 Sep 1 '15 at 12:19

The two statistics that you mention (p values and hazard ratios) ask different questions, so it is not surprising that they give different answers.

If, in the population from which this sample was randomly drawn, the null hypothesis was correct (i.e. B = 1), what is the probability of getting a test statistic (B) at least as extreme as the one we got in a sample the size of the one we have.

How different are the hazards of dying for the two groups?

Opinions of p-values vary (to put it mildly!). My own view is that the question they answer is rarely the one we want to ask.

• If I understand correctly, you favor the hazard ratio over the p-value? Perhaps it helps to have the exact values in this particular case (see below) – user86880 Sep 1 '15 at 13:14
• Biomarker 1: HR=3.06; 95%CI=1.71-5.48; p<0.001 (sorry, SPSS doesn't show more than 3 decimals) Biomarker 2: HR=6.05; 95%CI=1.67-21.86; p=0.006 – user86880 Sep 1 '15 at 13:15
• 1. Yes, I favor hazard ratios over p-values. 2. The wide CI on the second biomarker is a bit worrying and is the reason that it has a higher HR with a less sig. result. 3. both p are very sig., and I believe estimates of p values that are very low are not always that accurate (but I'm not sure of that) – Peter Flom Sep 1 '15 at 13:35

Unfortunately, in terms of predictive power, you cannot tell from this output alone.

While the fact that Biomarker 2 has a larger estimated HR may lead you to believe it is better, this is incorrect reasoning. The reason for this is that this simple output alone tells you nothing about the distribution of Biomarker 2. To help think about this, consider if we measured the biomarker on a different scale, such that the new values were 10x the current values. Then the fit would be exactly the same, expect that the estimated log-hazard ratio would be 1/10 it's current estimate, despite having the exact same predictive power. So simply looking at estimated coefficients cannot tell you the predictive power of a given biomarker.

The most straight forward way to compare them is to look at an ROC curve.

In addition, is there any reason why you wouldn't use both biomarkers? Given that they were both significant in your model, it would suggest that you should get better predictions by using both biomarkers. But perhaps this is unreasonable due to costs of the two tests?

It's worth noting that @PeterFolm's answer above, I believe there is an unstated, but very important, assumption that the covariates are standardized (i.e. all have standard deviation 1). In that case, comparing regression effect sizes is more meaningful. In addition, it's important to note that just because the covariates are binary doesn't mean that they are standardized!