# survival analysis - Kaplan-Meier estimator comparison

I am working on an assignment. We are asked to compare the probability of not having an infection in 20 days with 95% CI vs. 20 days with 75% CI. To me this question seems very strange.

I believe the estimates of the probability of not having an infection after 20 days remain the same for both cases, right? Only the confidence interval changes, namely the 75% CI should be wider.

To complete this task I have used Kaplan-Meier estimator and I have estimated the survival function. I used the log-log transformations. However, the results we not what i expected. The probabilities do remain the same, however the 95% CI is wider than the 75%. Therefore, what is going on there ? Am I not getting something ?

I have conducetd my analysis in R. Here are the results of my Kaplan-Meier estimator:

> fit_KM <- survfit(Surv(infection time,infection)~1,conf.type="log-log", data=BurnData)     # log is default
> summary(fit_KM)
Call: survfit(formula = Surv(infection time, infection) ~ 1, data = BurnData,
conf.type = "log-log")

time n.risk n.event survival std.err lower 95% CI upper 95% CI
1    154       1    0.994 0.00647        0.955        0.999
2    153       3    0.974 0.01282        0.932        0.990
3    150       4    0.948 0.01788        0.899        0.974
4    146       5    0.916 0.02240        0.859        0.950
5    141       6    0.877 0.02650        0.813        0.919
6    134       2    0.864 0.02767        0.798        0.909
7    132       3    0.844 0.02927        0.776        0.893
8    126       2    0.831 0.03030        0.761        0.881
9    121       2    0.817 0.03132        0.746        0.870
10    117       2    0.803 0.03230        0.730        0.858
11    112       3    0.781 0.03374        0.706        0.839
13    102       1    0.774 0.03426        0.698        0.833
14     96       1    0.766 0.03484        0.689        0.826
16     88       1    0.757 0.03552        0.679        0.819
17     82       2    0.738 0.03697        0.658        0.803
18     76       2    0.719 0.03847        0.636        0.787
19     70       1    0.709 0.03927        0.624        0.778
21     65       1    0.698 0.04015        0.611        0.769
23     55       1    0.685 0.04137        0.596        0.758
32     35       1    0.666 0.04458        0.570        0.745
42     19       1    0.631 0.05428        0.514        0.726
44     15       1    0.589 0.06493        0.451        0.703
47     12       1    0.539 0.07581        0.381        0.674
51      9       1    0.480 0.08795        0.302        0.637

> fit_KM_75 <- survfit(Surv(infection time,infection)~1,conf.type="log-log", data=BurnData, conf.int = 0.75)     # log is default
> summary(fit_KM_75)
Call: survfit(formula = Surv(infection time, infection) ~ 1, data = BurnData,
conf.type = "log-log", conf.int = 0.75)

time n.risk n.event survival std.err lower 75% CI upper 75% CI
1    154       1    0.994 0.00647        0.980        0.998
2    153       3    0.974 0.01282        0.954        0.985
3    150       4    0.948 0.01788        0.923        0.965
4    146       5    0.916 0.02240        0.886        0.938
5    141       6    0.877 0.02650        0.842        0.904
6    134       2    0.864 0.02767        0.828        0.892
7    132       3    0.844 0.02927        0.807        0.874
8    126       2    0.831 0.03030        0.792        0.862
9    121       2    0.817 0.03132        0.778        0.850
10    117       2    0.803 0.03230        0.763        0.837
11    112       3    0.781 0.03374        0.739        0.817
13    102       1    0.774 0.03426        0.731        0.810
14     96       1    0.766 0.03484        0.723        0.803
16     88       1    0.757 0.03552        0.713        0.795
17     82       2    0.738 0.03697        0.693        0.778
18     76       2    0.719 0.03847        0.672        0.761
19     70       1    0.709 0.03927        0.661        0.751
21     65       1    0.698 0.04015        0.649        0.741
23     55       1    0.685 0.04137        0.635        0.730
32     35       1    0.666 0.04458        0.611        0.714
42     19       1    0.631 0.05428        0.565        0.689
44     15       1    0.589 0.06493        0.510        0.659
47     12       1    0.539 0.07581        0.448        0.622
51      9       1    0.480 0.08795        0.376        0.576


and my interpretation:

The probability of not having in infection after 20 days is estimated to be to be29.1% [22.2%, 37.6%]. The same probability with 75% confidence intervals is 29.1%[24.9%, 33.9%].

Is there a procedure how to compare if the results are significantly different?