Sept 2021 Question 5(iii) Kaplan-Meier: Nelson-Aalen estimate

[Bill SD]

Hi,
The Question provides the Nelson-Aalen estimate of the Survival function and asks to determine the Kaplan-Meier estimate.

The Examiners Report answers this question by calculating lambda(hat) and then 1 - d_j / n_j to produce the exact Kaplan-Meier estimate.

However, Acted Notes on Chapter 7 (pg 26 in 2019 version)include that the KM and NA estimates can both approximate each other.
Would this also be an acceptable answer if one explained the approximation [exp^x~1+x; so 1-dj/nj ~ exp^-(dj/nj)] and demonstrated how the formulae are equivalent?

If it wouldn't gain marks for this question, when would this approximation be acceptable?

 

[Dave Johnson]

Hi

The approximation you mention is central to this question. It can be written as:

\( \hat S_{KM} = \prod_{t_j \le t} 1 - \hat \lambda_j \approx \prod_{t_j \le t} e^{- \hat \lambda_j} = \hat S_{NA} \)

So what we must do is recover \( \hat \lambda_j \) from the Nelson-Aalen estimate, and then use it to construct the Kaplan-Meier estimate. As you will see, the results are very close to each other because of the approximation rule you mentioned.

I don't think noting the approximation and explaining it would have been worth any marks based on the marking schedule for the exam, though of course a similar question might attract marks based on the wording.

For your second question - I can't think of a time when you would want to use this approximation without being asked to. Both estimates are very easy to calculate, so I would recommend you make sure you are comfortable with both of them.

Hope that helps.

Dave

 

[Bill SD]

Hi

The approximation you mention is central to this question. It can be written as:

\( \hat S_{KM} = \prod_{t_j \le t} 1 - \hat \lambda_j \approx \prod_{t_j \le t} e^{- \hat \lambda_j} = \hat S_{NA} \)

So what we must do is recover \( \hat \lambda_j \) from the Nelson-Aalen estimate, and then use it to construct the Kaplan-Meier estimate. As you will see, the results are very close to each other because of the approximation rule you mentioned.

I don't think noting the approximation and explaining it would have been worth any marks based on the marking schedule for the exam, though of course a similar question might attract marks based on the wording.

For your second question - I can't think of a time when you would want to use this approximation without being asked to. Both estimates are very easy to calculate, so I would recommend you make sure you are comfortable with both of them.

Hope that helps.

Dave

Thanks very much Dave - i had interpreted the approximation to mean that for the Sept 21 question, i could simply take the Kaplan-Meier survival estimates to be (approximately) the same as the Nelson-Aalen estimates (without any further calcs). Now understand from your reply and the Examiners Report that would actually need to recover \( \hat \lambda_j \) etc. and construct the Kaplan-Meier estimate. Yes, appreciate if exam question doesn't mention the approximation then need to calculate KM and NA in the usual way. Thanks