Question

A pilot study was designed to evaluate the potential efficacy of a program designed to reduce prison recidivism amongst inmates who have a documented long-term history of drug and/or alcohol problems. A sample of 11 prisoners was followed for up to 24 months after their most recent release from prison. Six of the inmates returned to prison at 3, 7 9, 11, 14 and 21 months respectively. Five of the inmates had not returned to prison as of the last time they were last contacted which was at 4, 8, 16, 24, and 24 months respectively. Use the Kaplan Meier approach to estimate the survival curve for this set of inmates (which tracks the proportion who have not yet returned to prison over time). Please show your work.

What is the estimated proportion of the total sample who had not returned to prison by 7 months after enrolling in the study?

Answer 1

The Kaplan-Meier estimator is a non parametric statistic used to estimate the survival function from the lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment. In other fields, Kaplan–Meier estimators may be used to measure the length of time people remain unemployed after a job loss, the time-to-failure of machine parts, or how long fleshy fruits remain on plants before they are removed by frugivores.

The estimator of the survival function (the probability that life is longer than ) is given by:

with a time when at least one event happened, d_ithe number of events (i.e., deaths) that happened at time , and the individuals known to have survived (have not yet had an event or been censored) up to time .

A.T.Q.

For each time interval, survival probability is calculated as the number of subjects surviving divided by the number of patients at risk. Subjects who have died, dropped out, or move out are not counted as “at risk” i.e., subjects who are lost are considered “censored” and are not counted in the denominator. Total probability of survival till that time interval is calculated by multiplying all the probabilities of survival at all time intervals preceding that time (by applying law of multiplication of probability to calculate cumulative probability).

Hence, the result of the question.