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

Discussion in 'CS2' started by Bill SD, Apr 3, 2022.

  1. Bill SD

    Bill SD Very Active Member

    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?
     
  2. Dave Johnson

    Dave Johnson ActEd Tutor Staff Member

    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
     
  3. Bill SD

    Bill SD Very Active Member

    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
     

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