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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?

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The 75% confidence interval should be narrower. To have more confidence you must incorporate more uncertainty. Think of it this way: you can always be 100% confident that a survival is between 0 and 1.

You are overthinking what it means to "compare" them. Just remark on the actual values and try to interpret them for a non-statistical audience. Note as you have done here that the less confident interval is correspondingly narrower.

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