Kaplan-Meier estimate and Practice Question

The Kaplan-Meier estimate is a non-parametric method used to estimate the survival function (or the probability of survival) from observed time-to-event data. It is commonly used in survival analysis when dealing with censored data, where not all events have occurred by the end of the study.

Here are the main steps in calculating the Kaplan-Meier estimate:

Sort Data:

• Arrange the observed time-to-event data in ascending order.

• Set the initial number of individuals at risk  to the total sample size.

• For each event time, calculate the Kaplan-Meier estimate (S^(t)) using the formula:

Where:

• ti​ is the time of the i-th event.

• di​ is the number of events at time ti​.

• ni​ is the number of individuals at risk just before time ti​.

• Update the number of individuals at risk (ni​) for each event time.

• Plot the Kaplan-Meier survival curve using the calculated survival probabilities.

Interpretation:

• The Kaplan-Meier curve visually represents the estimated survival function over time.

• Each step in the curve corresponds to an observed event time.

• The curve may drop when an event occurs and remains constant between event times.

Censoring:

• The Kaplan-Meier estimate accommodates censored data, where individuals have not experienced the event by the end of the study.

• For censored observations, the survival probability is not affected at that specific time point.

Confidence Intervals:

• Confidence intervals for the Kaplan-Meier estimate can be calculated using Greenwood's formula.

Log-Rank Test:

• The log-rank test is commonly used with Kaplan-Meier curves to compare survival distributions between different groups.

Python Notes for Kaplan-Meier Estimates

Example Questions

Q2. An electronics company developed a revolutionary new battery which it believed would make it enormous profits. It commissioned a sub-contractor to estimate the survival function of battery life for the first 12 prototypes. The sub-contractor inserted each prototype battery into an identical electrical device at the same time and measured the duration elapsing between the time each device was switched on and the time its battery ran out. The subcontractor was instructed to terminate the test immediately after the failure of the 8th battery, and to return all 12 batteries to the company.

When the test was complete, the sub-contractor reported that he had terminated the test after 150 days. He further reported that:

·        two batteries had failed after 97 days

·        three further batteries had failed after 120 days

·        two further batteries had failed after 141 days

·        one further battery had failed after 150 days.

However, he reported that he was only able to return 11 batteries, as one had exploded after 110 days, and he had treated this battery as censored at that duration when working out the Kaplan-Meier estimate of the survival function.

(i) State, with reasons, the forms of censoring present in this study. [2]

(ii) Calculate the Kaplan-Meier estimate of the survival function based on the information supplied by the sub-contractor. [5]

In his report, the sub-contractor claimed that the Kaplan-Meier estimate of the survival function at the duration when the investigation was terminated was 0.2727.

(iii) Explain why the sub-contractors Kaplan-Meier estimate would be consistent with him having stolen the battery he claimed had exploded. [4]                    [Total 11]

CS2 Practice Question Q2.pdf

Download PDF • 216KB