# p-value zero in hypothesis testing for survival curves

I have done survival analysis. I used Kaplan-Meir to do the survival analysis.

Description of data: My data set is large and data table has close 120,000 records of survival information belong to 6 groups.

Sample:

   user_id   time_in_days   event total_likes total_para_length group
1:       2          4657     1       38867        431117212   AA
2:       2          3056     1       31392        948984460   BB
3:       2            49     1          15            67770   CC
4:       3          4181     1       15778        379211806   BB
5:       3            17     1           3            19032   CC
6:       3          2885     1       12001        106259666   EE


After fitting the survival curves and plotting it, I see they are similar but yet at any given point in time their survival proportions don't seem to look like identical.

Here is the plot:

I ran a hypothesis test where my H0: There is not difference between the survival curves and here is the results that I got.

> survdiff(formula= Surv(time, event) ~ group, rh=0)
Call:
survdiff(formula = Surv(time, event) ~ group, rho = 0)

N Observed Expected (O-E)^2/E (O-E)^2/V
group=FF 28310    27993    28632      14.3      19.0
group=AA 64732    63984    67853     220.6     460.1
group=BB 19017    18690    16839     203.4     245.6
group=CC  9687     9536     8699      80.6      91.0
group=DD 13438    13187    11891     141.3     164.2
group=EE  3910     3847     3324      82.4      89.7

Chisq= 788  on 5 degrees of freedom, p= 0


I am little confuse by trying to figure out what it means, specially since I got p-value=0.

I am fairly new to survival analysis so after reading and digging through I realized that this is a non-parametric as I understand which means that it doesn't make any assumptions of the underline distributions of the time.

After reading about cox-proportional hazard function and going over c-cran pdf I performed a cox regression test and here is what I got from that:

> cox_model <- coxph(Surv(time, event) ~ X)
> summary(cox_model)
Call:
coxph(formula = Surv(time, event) ~ X)

n= 139094, number of events= 137237

coef  exp(coef)   se(coef)       z Pr(>|z|)
X1 -7.655e-05  9.999e-01  1.504e-06 -50.897   <2e-16 ***
X2 -1.649e-10  1.000e+00  5.715e-11  -2.886   0.0039 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

exp(coef) exp(-coef) lower .95 upper .95
X1    0.9999          1    0.9999    0.9999
X2    1.0000          1    1.0000    1.0000

Concordance= 0.847  (se = 0.001 )
Rsquare= 0.111   (max possible= 1 )
Likelihood ratio test= 16307  on 2 df,   p=0
Wald test            = 7379  on 2 df,   p=0
Score (logrank) test = 4628  on 2 df,   p=0


My big X is generated by doing rbind on total_like and total_para_length. Looking at Rsquare and P-Values I am not sure what really is going on here. If I can't throw away the Null-Hypothesis I should give a larger p-value.

-

Your $p$-value is not actually zero, it's just very close to it. If you look at your test statistics ($\chi^{2}=788$ in the Kaplan-Meier model, and the Wald $\chi^{2}=7379$ in the CPH model) they are ginormous! Just really, really big. So the associated $p$-values are tiny.
But perhaps you wonder why, if your survival curves were so similar visually, you get such a significant difference with these tests? If so, consider: even tiny differences can obtain miniscule $p$-values if the sample size is large enough. And how big is your sample? It's about 120,000 observations: big! So you might expect that even very small differences will be found "significant."