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## Why does stsum sometimes report missing values for the percentiles of survival time?

 Title Missing values reported by stsum Author Mario Cleves, StataCorp

Look at the following stsum output:

. use https://www.stata.com/support/faqs/dta/stsum.dta

. stsum, by(exp)

Failure _d: disease
Analysis time _t: time

Incidence     Number of   |       Survival time       |
exp   Time at risk       rate      subjects        25%       50%       75%

0             83   .2650602            29          1         2         5
1             93   .1935484            28          1         2         .

Total            176   .2272727            57          1         2         .


See the missing value for the 75th percentile of survival time for the group exp==1? That is the question: why is that value missing? In more extreme cases, you might see both the 50th and 75th percentile estimates, or even all three, missing.

In any case, the short answer is that the estimates are missing because the percentiles cannot be estimated. The percentiles of survival time reported are for completed survival times, and they can be obtained from the estimate of the failure function, F(t). This curve is computed as 1-(S(t)), where S(t) is the Kaplan–Meier product-limit estimate of the survival curve. Let's look at F(t) for this group:

. sts list if exp==1, failure

Failure _d: disease
Analysis time _t: time

Kaplan–Meier failure function

At                   Failure      Std.
Time     risk   Fail   Lost    function     error     [95% conf. int.]

1       28      7      0      0.2500    0.0818     0.1279    0.4539
2       21      8      0      0.5357    0.0942     0.3667    0.7244
3       13      2      0      0.6071    0.0923     0.4349    0.7833
4       11      1      4      0.6429    0.0906     0.4703    0.8114
5        6      0      2      0.6429    0.0906     0.4703    0.8114
6        4      0      2      0.6429    0.0906     0.4703    0.8114
10        2      0      1      0.6429    0.0906     0.4703    0.8114
12        1      0      1      0.6429    0.0906     0.4703    0.8114


. sts graph if exp==1, failure xlabel(1(1)15)


The failure function F(t) reports the probability of failing before or at time t. In other words, F(t) gives us the expected proportion of individuals that would fail by time t.

The failure function here becomes flat at F(t) = 0.6429. It does that because not all the subjects have died yet—this is called right-censoring.

What is the 25th percentile of completed survival times? The 25th percentile occurs where F(t) = .25 (meaning that 25% have failed and 75% have yet to fail), and that is t=1. Look back at the stsum output, and you will see that 1 is reported.

What is the 50th percentile of completed survival times? The 50th percentile occurs where F(t) = .50, and that is t=2. stsum reports that number too.

What is the 75th percentile? We do not know, because less than 75% of our individuals have already failed in our data. Thus, stsum reports the 75th percentile for survival time as missing.