A prospective study with a 2-year (24-month) follow-up was conducted. The table shows the results for individuals who either died or were censored before the end of the follow-up period.

Survival data for 20 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

1. Using the data from the table, for all deaths, calculate (a) the probability of death at the exact time when each death occurred, (b) the probability of survival beyond the time when each death occurred, and (3) the cumulative probabilities of survival.
2. What is the cumulative survival probability at the end of the follow-up period?
3. What is the simple proportion of individuals apparently surviving (i.e., not observed to die) through the end of the study's observation period?
4. Why are the simple proportion surviving and the cumulative probability of survival different?
5. Using the same data, calculate the overall death rate per 100 person-years. (To facilitate your calculations, you may wish to calculate the number of person-months and then convert that to the number of person-years.)
6. 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.)
7. Assuming no random variability, was it appropriate to calculate the rate per person-year for the total 2-year duration of follow-up?
8. What is the most important assumption underlying the use of both survival analysis and the person-time approach?
9. Now, assume that the length of follow-up was the same for all individuals (except those who died). Calculate the proportion of dying and the odds of death in this cohort.
10. Why are these figures so different in this study?