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# Incorrect confidence intervals for survival probabilities or cumulative hazard in proc phreg

Hello, I am using proc phreg to generate survival probabilities. I am following (and extending slightly) the example given here in SAS help documentation: https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_phreg_sect0...

I am choosing the confidence limit of type of, CLTYPE=LOG, which is the default, and 'The confidence limits for S(t) are obtained by back transforming the confidence limits for log(S(t))'. Source: https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_phreg_sect0...

When running my code, I generate the cumulative hazard [ cumulative hazard H(t) = -log(S(t)) ], and confidence interval around the cumulative hazard, as well as the survival probability and confidence interval around the survival probability to confirm this is how the confidence limit is being calculated.

Just as H(t) = -log(S(t)), you should expect to see the same relationship for the upper and lower confidence limits (upper limit of cumulative hazard transforms to lower limit of survival, and vice versa).

However, the relationship does not hold for the confidence limits, could anybody explain why? They should match up, which leads me to believe one is wrong.

My code to generate the survival probabilities and confidence interval is attached, and also given below. I am using SAS 9.4.

data Myeloma;
input Time VStatus LogBUN HGB Platelet Age LogWBC Frac
LogPBM Protein SCalc;
label Time='Survival Time'
datalines;
1.25 1 2.2175 9.4 1 67 3.6628 1 1.9542 12 10
1.25 1 1.9395 12.0 1 38 3.9868 1 1.9542 20 18
2.00 1 1.5185 9.8 1 81 3.8751 1 2.0000 2 15
2.00 1 1.7482 11.3 0 75 3.8062 1 1.2553 0 12
2.00 1 1.3010 5.1 0 57 3.7243 1 2.0000 3 9
3.00 1 1.5441 6.7 1 46 4.4757 0 1.9345 12 10
5.00 1 2.2355 10.1 1 50 4.9542 1 1.6628 4 9
5.00 1 1.6812 6.5 1 74 3.7324 0 1.7324 5 9
6.00 1 1.3617 9.0 1 77 3.5441 0 1.4624 1 8
6.00 1 2.1139 10.2 0 70 3.5441 1 1.3617 1 8
6.00 1 1.1139 9.7 1 60 3.5185 1 1.3979 0 10
6.00 1 1.4150 10.4 1 67 3.9294 1 1.6902 0 8
7.00 1 1.9777 9.5 1 48 3.3617 1 1.5682 5 10
7.00 1 1.0414 5.1 0 61 3.7324 1 2.0000 1 10
7.00 1 1.1761 11.4 1 53 3.7243 1 1.5185 1 13
9.00 1 1.7243 8.2 1 55 3.7993 1 1.7404 0 12
11.00 1 1.1139 14.0 1 61 3.8808 1 1.2788 0 10
11.00 1 1.2304 12.0 1 43 3.7709 1 1.1761 1 9
11.00 1 1.3010 13.2 1 65 3.7993 1 1.8195 1 10
11.00 1 1.5682 7.5 1 70 3.8865 0 1.6721 0 12
11.00 1 1.0792 9.6 1 51 3.5051 1 1.9031 0 9
13.00 1 0.7782 5.5 0 60 3.5798 1 1.3979 2 10
14.00 1 1.3979 14.6 1 66 3.7243 1 1.2553 2 10
15.00 1 1.6021 10.6 1 70 3.6902 1 1.4314 0 11
16.00 1 1.3424 9.0 1 48 3.9345 1 2.0000 0 10
16.00 1 1.3222 8.8 1 62 3.6990 1 0.6990 17 10
17.00 1 1.2304 10.0 1 53 3.8808 1 1.4472 4 9
17.00 1 1.5911 11.2 1 68 3.4314 0 1.6128 1 10
18.00 1 1.4472 7.5 1 65 3.5682 0 0.9031 7 8
19.00 1 1.0792 14.4 1 51 3.9191 1 2.0000 6 15
19.00 1 1.2553 7.5 0 60 3.7924 1 1.9294 5 9
24.00 1 1.3010 14.6 1 56 4.0899 1 0.4771 0 9
25.00 1 1.0000 12.4 1 67 3.8195 1 1.6435 0 10
26.00 1 1.2304 11.2 1 49 3.6021 1 2.0000 27 11
32.00 1 1.3222 10.6 1 46 3.6990 1 1.6335 1 9
35.00 1 1.1139 7.0 0 48 3.6532 1 1.1761 4 10
37.00 1 1.6021 11.0 1 63 3.9542 0 1.2041 7 9
41.00 1 1.0000 10.2 1 69 3.4771 1 1.4771 6 10
41.00 1 1.1461 5.0 1 70 3.5185 1 1.3424 0 9
51.00 1 1.5682 7.7 0 74 3.4150 1 1.0414 4 13
52.00 1 1.0000 10.1 1 60 3.8573 1 1.6532 4 10
54.00 1 1.2553 9.0 1 49 3.7243 1 1.6990 2 10
58.00 1 1.2041 12.1 1 42 3.6990 1 1.5798 22 10
66.00 1 1.4472 6.6 1 59 3.7853 1 1.8195 0 9
67.00 1 1.3222 12.8 1 52 3.6435 1 1.0414 1 10
88.00 1 1.1761 10.6 1 47 3.5563 0 1.7559 21 9
89.00 1 1.3222 14.0 1 63 3.6532 1 1.6232 1 9
92.00 1 1.4314 11.0 1 58 4.0755 1 1.4150 4 11
4.00 0 1.9542 10.2 1 59 4.0453 0 0.7782 12 10
4.00 0 1.9243 10.0 1 49 3.9590 0 1.6232 0 13
7.00 0 1.1139 12.4 1 48 3.7993 1 1.8573 0 10
7.00 0 1.5315 10.2 1 81 3.5911 0 1.8808 0 11
8.00 0 1.0792 9.9 1 57 3.8325 1 1.6532 0 8
12.00 0 1.1461 11.6 1 46 3.6435 0 1.1461 0 7
11.00 0 1.6128 14.0 1 60 3.7324 1 1.8451 3 9
12.00 0 1.3979 8.8 1 66 3.8388 1 1.3617 0 9
13.00 0 1.6628 4.9 0 71 3.6435 0 1.7924 0 9
16.00 0 1.1461 13.0 1 55 3.8573 0 0.9031 0 9
19.00 0 1.3222 13.0 1 59 3.7709 1 2.0000 1 10
19.00 0 1.3222 10.8 1 69 3.8808 1 1.5185 0 10
28.00 0 1.2304 7.3 1 82 3.7482 1 1.6721 0 9
41.00 0 1.7559 12.8 1 72 3.7243 1 1.4472 1 9
53.00 0 1.1139 12.0 1 66 3.6128 1 2.0000 1 11
57.00 0 1.2553 12.5 1 66 3.9685 0 1.9542 0 11
77.00 0 1.0792 14.0 1 60 3.6812 0 0.9542 0 12
;

data Inrisks;
length Id \$20;
input LogBUN HGB Id \$12-31;
datalines;
1.00 10.0 logBUN=1.0 HGB=10
1.80 12.0 logBUN=1.8 HGB=12
;

ods graphics on;
proc phreg data=Myeloma plots(overlay)=survival;
model Time*VStatus(0)=LogBUN HGB;
baseline covariates=Inrisks out=Pred_log survival=_all_ cumhaz=_all_ logsurv=LogSurvival loglogs=LogLogSurvival / cltype=log rowid=Id;
run;
ods graphics off;

proc print data=Pred_log(where=(logBUN=1 and HGB=10) obs=10);
run;

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