A prospectus study with a 2-year (24 month) follow up was conducted. Results are shown in the table for individuals who either died or were censored before the end of the follow-up period.

Survival data for 24 participants of a hypothetical prospective study.
Follow- up time (months)	Event
2	Death
4	Censored
7	Censored
8	Death
12	Censored
15	Death
17	Death
19	Death
20	Censored
23	Death

1a. Using the data from the table, for all deaths calculate (1) the probability of death at the exact time when each death occurred.

Follow-up time (months) (t)	Event (e)	Death (d)	Number of participants (n)	Probability of death (d/n)
2	Death	1	20	0.05
8	Death	1	19	0.05
15	Death	1	18	0.06
17	Death	1	17	0.06
19	Death	1	16	0.06
23	Death	1	15	0.07

(2) The probability of survival beyond the time when each death occurred.

Follow-up time in (months) (t)	Event (e)	Death (d)	Number of participants (n)	Probability of death (d/n)	Probability of survival (1-d/n)
2	Death	1	20	0.05	0.95
8	Death	1	19	0.05	0.95
15	Death	1	18	0.06	0.94
17	Death	1	17	0.06	0.94
19	Death	1	16	0.06	0.94
23	Death	1	15	0.07	0.93

(3) The cumulative probabilities of survival

Follow-up time in (months) (t)	Event (e)	Death (d)	Number of participants (n)	Probability of death (d/n)	Probability of survival (1-d/n)	Cumulative Probability of survival
2	Death	1	20	0.05	0.95	0.95
8	Death	1	19	0.05	0.95	0.90
15	Death	1	18	0.06	0.94	0.85
17	Death	1	17	0.06	0.94	0.80
19	Death	1	16	0.06	0.94	0.75
23	Death	1	15	0.07	0.93	0.70

Reference:

• Szklo, M., & Nieto, F. J. (2019). Measuring disease occurrence. In Epidemiology: Beyond the basics (4th ed., pp. 51-86). Jones & Bartlett Learning.

f. Using the same data, calculate the overall death rate per 100 persons-years. (To facilitate

you calculations, you may wish to calculate the number of persons-months and then

convert that to the number of person-years.)

g. Calculate the rates separately for the first and second years of follow-up. (For this

calculation, assume that the individual who withdrew at month 12 withdrew just after

midnight on the last day of the month.

h. Assuming that there was no random variability, was it appropriate to calculate the rate

per person- year for the total 2-yeat duration of the follow-up?

i.What is the most important assumption underlying the use of both survival analysis and

the person time approach?

j. Now, assume that the length of the follow-up was the same for all individuals (except for

those who died). Calculate the proportion dying, and the odds of death in this cohort.

k. Why are these figures so differently in this study?