Question

Text book: Essentials of Biostatistics in Public Health 3 ed. by Lisa Sullivan.

Instructor provided problem: An observational study is conducted to compare experiences of men and women between the ages of 50-59 years following coronary artery bypass surgery. Participants undergo the surgery and are followed until the time of death, until they are lost to follow-up or up to 30 years, whichever comes first. The following table details the experiences of participating men and women. The data below are years of death or years of last contact for men and women. Estimate the Estimate the survival functions for each treatment group using the Kaplan-Meier approach. Test if there is a significant difference in survival between treatment groups using the log rank test and a 5% level of significance.

Men			Women
Year of Death	Year of Last Contact		Year of Death	Year of Last Contact
5	8		19	4
12	17		20	9
14	24		21	14
23	26		24	15
29	26			17
	27			19
	29			21
	30			22
	30			24
	30			25
				30

Answer 1

a)We assume that censoring of data is done after the death occurs.

Terminology:
t_j = Time of j^th death
d_j = No. of death at �t_j�
n_j = No. of people alive just before j^th death.
?_j = Probability of death at t_j.

MEN

j	tj	dj	nj	?j = dj / nj	1 - ?j	S(t)
1	5	1	15	0.0667	0.9333	0.9333
2	12	1	13	0.0769	0.9231	0.8615
3	14	1	12	0.0833	0.9167	0.7897
4	23	1	10	0.1000	0.9000	0.7108
5	29	1	5	0.2000	0.8000	0.5686

Survival Function = S(t)	1	0 < t < 5
	0.9333	5 < t < 12
	0.8615	12 < t < 14
	0.7897	14 < t < 23
	0.7108	23 < t < 29
	0.5686	t > 29

WOMEN

j	tj	dj	nj	?j = dj / nj	1 - ?j	S(t)
1	19	1	10	0.1000	0.9000	0.9000
2	20	1	8	0.1250	0.8750	0.7875
3	21	1	7	0.1429	0.8571	0.6750
4	24	1	4	0.2500	0.7500	0.5063

Survival Function = S(t)	1	0 < t < 19
	0.9	19 < t < 20
	0.7875	20 < t < 21
	0.675	21 < t < 24
	0.5063	t > 24