Question

1. The data below are survival times (in years) for cancer patients receiving experimental therapy (TREAT) or standard of care (SOC) in a time-to-mortality study.

ID	Therapy	Years	Event
1	TREAT	1.6	1
2	TREAT	2.7	1
3	TREAT	4.9	1
4	TREAT	5.3	0
5	TREAT	6.2	1
6	TREAT	6.7	0
7	TREAT	7.1	1
8	TREAT	7.9	1
9	SOC	0.6	1
10	SOC	1.4	1
11	SOC	1.6	1
12	SOC	3.5	1
13	SOC	4.8	1
14	SOC	5.7	1
15	SOC	6.3	1
16	SOC	7.3	0

1. By hand, calculate the Kaplan-Meier estimate of the survival function for each therapy:
2. By hand, calculate the pointwise 95% confidence intervals for the Kaplan-Meier survival estimate for each therapy.
3. By hand, calculate the point and 95% confidence interval estimates of the 25^th, 50^th, and 75^th percentiles of the survival distribution for each therapy.

Answer 1

First we have to split the data with two datasets. One is for TREAT and another is for SOC.

ID	Therapy	Years	Event(d)	At risk(n)	k_M estimates(st)	S.E	CI_upper	CI_lower
1	TREAT	1.6	1	8	0.875	0.709955	0.960816	0.640137
2	TREAT	2.7	1	7	0.75	0.608533	0.848444	0.604375
3	TREAT	4.9	1	6	0.625	0.507111	0.716311	0.515768
4	TREAT	5.3	0	5	0.75	0.608533	0.848444	0.604375
5	TREAT	6.2	1	4	0.5625	0.4564	0.64734	0.467094
6	TREAT	6.7	0	3	0.5625	0.4564	0.64734	0.467094
7	TREAT	7.1	1	2	0.28125	0.2282	0.327176	0.236871
8	TREAT	7.9	1	1	0	0
ID	Therapy	Years	Event	At risk	k_M estimates	S.E	CI_upper	CI_lower
1	SOC	0.6	1	8	0.875	1.042631	0.977541	0.456126
2	SOC	1.4	1	7	0.857143	1.021353	0.967313	0.489177
3	SOC	1.6	1	6	0.833333	0.992982	0.951399	0.513145
4	SOC	3.5	1	5	0.8	0.953262	0.925602	0.525154
5	SOC	4.8	1	4	0.75	0.893684	0.881238	0.519644
6	SOC	5.7	1	3	0.666667	0.794385	0.797511	0.483547
7	SOC	6.3	1	2	0.5	0.595789	0.610943	0.377177
8	SOC	7.3	0	1	0.5	0.595789	0.610943	0.377177

Kaplan-Meier survival estimates(Group TREAT):

We get the estimates as

k_M estimates(st)
0.875
0.75
0.625
0.75
0.5625
0.5625
0.28125
0

We get

S.E
0.709955
0.608533
0.507111
0.608533
0.4564
0.4564
0.2282
0

Same for Kaplan-Meier survival estimates(Group SOC):

The results are:

ID	Therapy	Years	Event	At risk	k_M estimates	S.E	CI_upper	CI_lower
1	SOC	0.6	1	8	0.875	1.042631	0.977541	0.456126
2	SOC	1.4	1	7	0.857143	1.021353	0.967313	0.489177
3	SOC	1.6	1	6	0.833333	0.992982	0.951399	0.513145
4	SOC	3.5	1	5	0.8	0.953262	0.925602	0.525154
5	SOC	4.8	1	4	0.75	0.893684	0.881238	0.519644
6	SOC	5.7	1	3	0.666667	0.794385	0.797511	0.483547
7	SOC	6.3	1	2	0.5	0.595789	0.610943	0.377177
8	SOC	7.3	0	1	0.5	0.595789	0.610943	0.377177