I am puzzled about the interpretation of P value in the Cox hazard ratio analysis. I read from literature that the P value is to "reject the null hypothesis that HR=1". However, in many cases, we have tested multiple variables in the cox analysis, and therefore have multiple P values. For example

X = cbind(lcx,lvef)
coxph(formula = Surv(time_to_therapy, therapy) ~ X)

  n= 174, number of events= 54 

           coef exp(coef)  se(coef)      z Pr(>|z|)    
Xlcx   1.218259  3.381297  0.324619  3.753 0.000175 ***
Xlvef -0.004575  0.995436  0.016626 -0.275 0.783187    
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

      exp(coef) exp(-coef) lower .95 upper .95
Xlcx     3.3813     0.2957    1.7896     6.389
Xlvef    0.9954     1.0046    0.9635     1.028

Here X is a 2-column variable with first column a binary vector named lcx, and the second column a continuous vector named lvef.

How can I interpret the two p values (0.000175 and 0.783187)? In my idea, it should be related to the predictive value of two variables (lcx and lvef)?

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    $\begingroup$ What is Xlvef? I don't see it in the call for the model fit. FWIW, p-values here mean the same as p-values anywhere. $\endgroup$ – gung - Reinstate Monica Mar 21 '16 at 15:48
  • $\begingroup$ So here I have two p values, I assume they should link to the two variables in the Cox model? $\endgroup$ – alize Mar 21 '16 at 16:04
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    $\begingroup$ This is as for any multiple regression, one p-value per predictor variable in the part of the coxph output that you show. The associated hazard ratios (exp(coef)) are probably more informative. There should also be several p-values for the overall model fit (based on likelihood-ratio, Wald, and logrank tests) somewhere in the output; those should always be examined before interpreting p-values for individual predictors. $\endgroup$ – EdM Mar 21 '16 at 16:20
  • $\begingroup$ @gung, I'm not sure that's the best duplicate. The p-values qua p-values are the same as always, but coxph returns $p+3$ of them and it's potentially worth explaining the differences between the hypotheses that each of them purports to test. $\endgroup$ – Matt Krause Mar 21 '16 at 17:22
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    $\begingroup$ @gung I think the OP may be familiar with testing, but the question concerns interpretation of the hazard ratio, therefore may be a duplicate of a different question. $\endgroup$ – AdamO Mar 21 '16 at 17:57

Just as in a linear regression model, where a model coefficient represents a slope parameter, a model coefficient in a Cox model represents a "hazard ratio". Recall the Cox proportional hazards model accounts for an arbitrary hazard function, which represents an undulating, inestimable instantaneous risk for a failure/death at any point in time. The assumption we make is that the risk for the event of interest is continually proportional to this function according to groups defined by your exposures (such as a one-unit higher (1 mL/s) left ventricular ejection fraction... ).

The p-value comes from testing the null hypothesis that this hazard ratio is 1, or that there is no difference in the relative risk of the event comparing individuals with varying levels of LVEF.

When you control for multiple covariates at the same time, the interpretation of the hazard ratio changes somewhat. The p-value for left circumflex (which is significant) comes from testing the hazard ratio for LCX controlling additionally for LVEF. Here, the hazard ratio interpretation is the relative risk comparing individuals differing by 1 unit in left circumflex who have the same LVEF. So think of it as conditioning upon the effect of LVEF by comparing groups with varying LCX and similar LVEF.

Just the same, the $p$-value from LVEF comes from the test of relative risk for groups differing by one unit in LVEF having the same LCX.

As a point, I would consider adjusting for at least sex and age.

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