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Case influence diagnostics in the Kaplan-Meier estimator and the log-rank test

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Abstract

One or few observations can be highly influential on the Kaplan-Meier estimator, and consequently on the log-rank test statistic in comparing two survival functions. In this paper we derive case influence diagnostics for the Kaplan-Meier estimator and the log-rank test. We note that diagnostics in this context is quite different from the regression context where observations are usually assumed to be independent. Simulation studies are done to present some guidelines to determine influential observations deserving special attention. Illustrative examples are also given.

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Authors and Affiliations

  1. Department of Statistics, Pusan National University, 609-735, Pusan, Korea

    Choongrak Kim

  2. Department of Data Science, Inje University, 621-749, Kyungnam, Korea

    Whasoo Bae

Authors
  1. Choongrak Kim
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  2. Whasoo Bae
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Additional information

The first author was supported by the Korea Research Foundation Grant(KRF-2002-041-C00049), and he is a member of the Research Institute of Computer, Information and Communication at Pusan National University.

Appendix : Proof of Eq. (2.5)

Appendix : Proof of Eq. (2.5)

For Xi ≤ t,

$$\log \hat{S}(t)=\sum_{j=1}^{i-1} \delta_{j} \log (\frac{m-j}{m-j+1})+\sum_{j=i}^{k} \delta_{j} \log (\frac{m-j}{m-j+1})$$

and

$$\begin{aligned} \log \hat{S}_{-i}(t) &=\sum_{j=1}^{k-1} \delta_{j(i)}^{*} \log (\frac{m-j-1}{m-j}) \\ &=\sum_{j=1}^{i-1} \delta_{j} \log (\frac{m-j-1}{m-j})+\sum_{j=i}^{k-1} \delta_{j(i)}^{*} \log (\frac{m-j-1}{m-j}) \\ &=\sum_{j=1}^{i-1} \delta_{j} \log (\frac{m-j-1}{m-j})+\sum_{j=i+1}^{k} \delta_{j} \log (\frac{m-j}{m-j+1}) \end{aligned}$$

Also, for Xi > t,

$$\log \hat{S}(t)-\log \hat{S}_{-i}(t)=\sum_{j=1}^{k} \delta_{j} \log (\frac{m-j}{m-j+1})-\sum_{j=1}^{k} \delta_{j} \log (\frac{m-j-1}{m-j})$$

which completes the proof.

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Kim, C., Bae, W. Case influence diagnostics in the Kaplan-Meier estimator and the log-rank test. Computational Statistics 20, 521–534 (2005). https://doi.org/10.1007/BF02741312

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