# How to calculate the HR and 95%CI using the log-rank test in R?

The R survival package is very useful to do survival analysis. And I know the survdiff function can be used to compare the difference of survival time in two or more groups. And the p-value number can also be calculated as below. However, how can I calculate the HR and 95% CI using the log-rank test.

And I also know I can use the coxph() function to calculate the HR and 95% CI using the Cox regression. However, as the assumption of both the Cox model and log-rank test are that the hazard ratio stay constant over time, so I think I can also calculate the HR and 95% CI using the log-rank test.

According to the book Survival Analysis: A Practical Approach, I got two formulas on Page 62 and 66 to do this (as shown below). So I wrote the R code as below, is there anybody know whether I'm right?

Input codes:

library(survival)
data.survdiff <- survdiff(Surv(time, status) ~ group)
p.val = 1 - pchisq(data.survdiff$chisq, length(data.survdiff$n) - 1)
HR = (data.survdiff$obs[2]/data.survdiff$exp[2])/(data.survdiff$obs[1]/data.survdiff$exp[1])
up95 = exp(log(HR) + qnorm(0.975)*sqrt(1/data.survdiff$exp[2]+1/data.survdiff$exp[1]))
low95 = exp(log(HR) - qnorm(0.975)*sqrt(1/data.survdiff$exp[2]+1/data.survdiff$exp[1]))


Output results:

> data.survdiff
Call:
survdiff(formula = Surv(data[, "os_whw"], data[, "status_whw"] ==
1) ~ data[, "pcascore"] >= median(data[, "pcascore"]))

N Observed Expected (O-E)^2/E (O-E)^2/V
data[, "pcascore"] >= median(data[, "pcascore"])=FALSE 4        3     4.33     0.411     0.974
data[, "pcascore"] >= median(data[, "pcascore"])=TRUE  5        5     3.67     0.486     0.974

Chisq= 1  on 1 degrees of freedom, p= 0.324
> p.val
[1] 0.3235935
> HR
[1] 1.970484
> up95
[1] 7.917248
> low95
[1] 0.4904239


If your research question is whether groups differ in survival, regardless of other characteristics, you could use the ordinary Kaplan-Meier estimation and obtain p values. That p value is derived from the log rank test. It will merely tell you if groups differ in survival time. That is appropriate if (1) groups are randomized, or (2) you only want to know if groups differ in survivalor (3) you'd like to inspect survival curves.

However, if groups are not randomized and they differ in baseline charactersitics (e.g age, sex etc) then crude survival differences are of little use (e.g differences in survival might be due to age differences); thats when Cox regression comes in. You can use Cox regression to compare survival and adjust for confounders. If your Cox regression does not contain any other predictor than 'group', it will yied same p value as the log rank test from the Kaplan-Meier estimation. But the Cox regession allows you to obtain hazard ratios with confidence intervals for each predictor, along with p values.

Use the coxph function from survival package or the cph function from rms package.

In general there is no reason to tweak this in any other way. I think you'll be fine using the functions from the Rms package, which is accompanied by an excellent book with the same name as the package (Springer; FE Harrell). The package comes with functions that let you visualize and tabulate your results easily.

There might be theoretical reasons for your approach which is beyond me, but in that case one of the experts might guide us. In other scenarios, you'll be fine with this approach.

(Btw: you can obtain Cox adjusted survival curves; google it or check out RMS package documentation).

Posting here as I do not have enough rep. to comment. Your implementation seems correct, and you are right in thinking that the log-rank can be used to find the HR and 95%-Interval, I do have a couple of questions thought:

1. Is there any particular reason why you are recalculating the p-value? (as the survdiff()-function already provides it, it just seems like extra work).
2. Is the HR and accompanying 95% interval, found using your functions, in accordance with that found using the coxph()-function? That is use the coxph()-function to validate your own implementation.

Note: Why use the log-rank test to calculate the HR and 95%CI, when the coxph()-function does it for you? In order to practise, to see if your understood the material or just for the lulz?

• 1. It's ture that the survdiff() function already provides the p-value, but you want to extract the p-value when you deal with many data. For example, you want to compare the difference between age>= 60 or not, sex, race, cohort, treatment protocol and so on. You can put them into a "for" cycle and only output the p-value every time. 2. Both the Cox and log-rank test can give HR and 95% CI. I just want to use the two methods to select more robust predictors. I used both the two methods at the same time. – zjuwhw Nov 18 '14 at 12:09
• I could be wrong - but it seems to me that log-rank test is primarily used as a hypothesis test for the null hypothesis "two groups having identical survival and hazard functions" (en.wikipedia.org/wiki/Log-rank_test), it is probably not able to give HR estimate. BTW, what do you mean by "more robust predictor"? Variable selection is in general a bad idea, see for example 4.3 in: biostat.mc.vanderbilt.edu/wiki/pub/Main/RmS/rms.pdf See Sec 4.10 for general modeling strategy and Sec 18 of the same for a tutorial in Cox Regression. Hope this helps. – Clark Chong Nov 19 '14 at 5:50
• Hi, Clark. Thank you for your kind and helpful comments. I agree with your idea that log-rank test is used to compare the survival of two groups. However, I think it can be used to give HR, the reason has been included in the question. And I'm a bioinformatic to deal with gene expression data. Your material is useful and I realize the Harrell's package "rms" is popular after I begin to use CrossValidated. To select more robust predictor from many predictors means that I want use both log-rank test and Cox to calculate the p-value of a predictor. And I'm also going to use cross validation. – zjuwhw Nov 20 '14 at 6:53