Hazard ratio and confidence intervals from a cox PH model

Question

I have a general question: if a confidence interval for the point estimate of HR crosses 1.0, does that mean automatically that the p-value is not significant? In other words, do these two go hand in hand? If the confidence interval crosses 1, it could mean the results are not significant in one direction or another but can this be statistically significant? I got a hazard ratio of 0.67 (0.24-1.89) which crosses 1.0, but the p-value is 0.045. Thanks!

Answer 1

The answer depends on which test you use. Here’s a simple R example:

> library(survival)
> times = c(1:10)
> status = rep(1,10)
> ko = c(1,rep(0,5), rep(1,4))
> mod = coxph(Surv(times,status) ~ ko)
> summary(mod)
      coef exp(coef) se(coef)      z Pr(>|z|)  
ko -2.0659    0.1267   1.1241 -1.838   0.0661 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

   exp(coef) exp(-coef) lower .95 upper .95
ko    0.1267      7.892   0.01399     1.147

Concordance= 0.667  (se = 0.102 )
Rsquare= 0.373   (max possible= 0.951 )
Likelihood ratio test= 4.67  on 1 df,   p=0.03064
Wald test            = 3.38  on 1 df,   p=0.0661
Score (logrank) test = 4.49  on 1 df,   p=0.03419

The p-value is either 0.031 (likelihood ratio test), 0.066 (Wald test) or 0.034 (logrank/score test). The confidence interval (which includes 1) is consistent with the Wald test, but not with the other two tests.

The confidence interval is based on inverting the Wald test, and will always be consistent with its p-value. It is calculated from the point estimate and its standard error (both on the log scale), using the normal distribution:

> exp(-2.0659 + c(-1,1) * 1.96 * 1.1241)
[1] 0.01399387 1.14721355

That said, I think your example is quite strange, as the confidence limits are both very far away from 1, while the p-value is barely significant. Are you sure you are looking at the correct p-value?