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On Weighted Log-Rank Combination Tests and Companion Cox Model Estimators

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Abstract

In randomized clinical trials, the log-rank test and Cox proportional hazards model are the gold standard in survival data analyses. While the log-rank test is generally valid, in the presence of non-proportional hazards, the power can be substantially decreased relative to the proportional hazards assumptions under which studies are usually designed. In contrast, weighted log-rank tests can be more powerful for specific treatment differences under non-proportional hazards scenarios. However, a poor choice of the weighting form can be detrimental. Recent work on combining various weighted log-rank tests allows for tests that are capable of detecting treatment effects across a broad range of non-proportional hazards scenarios. In this paper, we expand on these ideas with a framework based on a flexible resampling approach [5] which allows for the combination of various testing procedures in addition to weighted log-rank tests. In particular, we describe how tests based on restricted mean survival time (RMST) comparisons can be included within combinations of weighted log-rank tests as well as other test statistics such as Tarone-Ware and Renyi-type supremum families. For estimation, we propose companion weighted Cox model estimators [14, 21, 22] which utilize the weighting form that is “selected” through the combination test and provide simultaneous confidence intervals. The performance of various combinations and their companion Cox estimators as well as RMST are evaluated in simulation studies under null, proportional hazards, late-separation, and early-separation scenarios. We find the combination tests perform quite well in controlling type-1 error rates and in achieving higher power than individual tests across the scenarios considered here. We suggest the companion Cox estimators are a natural link to the testing procedures and can be a useful complementary summary of treatment effects with careful interpretation. For illustration, we apply the proposals to a randomized clinical trial study of the PD-L1-targeted therapy atezolizumab in comparison with docetaxel in previously treated non-small-cell lung cancer patients. R code can be found at https://github.com/larryleon/combination-tests-and-estimators.

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Acknowledgements

The authors are grateful to the Atezolizumab Lung Team as well as Zhengrong Li, Kaspar Rufibach, Marcel Wolbers, and Hans Ulrich Burger for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Biostatistics, Celgene, San Francisco, CA, USA

    Larry F. León

  2. Department of Biostatistics, Genentech, San Francisco, CA, USA

    Ray Lin

  3. Late Development Statistics, Merck Research Laboratories, North Wales, PA, USA

    Keaven M. Anderson

Authors
  1. Larry F. León
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  2. Ray Lin
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  3. Keaven M. Anderson
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Correspondence to Larry F. León.

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Appendix

Appendix

1.1 Weighted Cox Model Variance Estimation and Simultaneous Confidence Interval Calculations

We briefly outline some details on the weighted Cox estimator and simultaneous confidence intervals based on synthetic-martingale resampling. By Taylor series expansion

$$\begin{aligned} \sqrt{n_{0}+n_{1}}({\hat{\beta }}_{w}-\beta _{w}^{*}) \approx \left\{ i_{w}(\beta _{w}^{**})/(n_{0}+n_{1})\right\} ^{-1}(n_{0}+n_{1})^{-1/2}U_{w}(\beta _{w}^{*}), \end{aligned}$$

where \(\beta _{w}^{**}\) lies between \({\hat{\beta }}_{w}\) and \(\beta _{w}^{*}\), and \(i_{w}(\beta _{w}^{**})=(-1){\partial U_{w}(\beta ) \over \partial \beta }|_{\beta =\beta _{w}^{**}}\) with \(i_{w}(\beta )\) given by

$$\begin{aligned} i_{w}(\beta )= \int _{0}^{\infty }w(t)\left\{ {{\bar{Y}}_{0}(t)\exp (\beta ) {\bar{Y}}_{1}(t)\over ({\bar{Y}}_{0}(t)+\exp (\beta ){\bar{Y}}_{1}(t))^{2}}\right\} d {\bar{N}}(t). \end{aligned}$$
(10)

Define \(a_{j}=n_{j}/(n_{0}+n_{1})\) for \(j=0,1\) and

$$\begin{aligned} K(t,\beta )=\sqrt{{n_{0}+n_{1} \over n_{0}n_{1}}} w(t){{\bar{Y}}_{0}(t)\exp (\beta ) {\bar{Y}}_{1}(t)\over {\bar{Y}}_{0}(t)+ \exp (\beta ){\bar{Y}}_{1}(t)}, \end{aligned}$$

where \(K(t,\beta )\) is equivalent to K(t) in the weighted log-rank statistic defined in (1) but with \({\bar{Y}}_{1}(t)\) substituted with \(\exp (\beta ){\bar{Y}}_{1}(t)\).

Recall that \({\bar{M}}_{j}(t)=\sum _{i=1}^{n}M_{i,j}(t)\) are martingale processes for the control (\(j=0\)) and experimental groups (\(j=1\)) with \(M_{i,j}(t)= N_{i,j}(t)-\int _{0}^{t}Y_{i,j}(s)\lambda _{j}(s)ds\) where \(\lambda _{j}\) denotes the hazard function for group \(j=0,1\). We can then write

$$\begin{aligned}&(n_{0}+n_{1})^{-1/2}U_{w}(\beta )\\&\quad = \sqrt{a_0 a_1} \int _{0}^{\infty }{K(t,\beta ) \over \exp (\beta ){\bar{Y}}_{1}(t)}d{\bar{M}}_{1}(t) - \sqrt{a_0 a_1} \int _{0}^{\infty }{K(t,\beta ) \over {\bar{Y}}_{0}(t)}d{\bar{M}}_{0}(t) \\&\qquad + \sqrt{a_0 a_1}\int _{0}^{\infty }K(t,\beta ){(\lambda _{1}(t)-\exp (\beta )\lambda _{0}(t)) \over \exp (\beta )}dt. \end{aligned}$$

We assume uniform convergence in probability for \(K(\cdot ,\cdot )\) and

$$\begin{aligned} \sqrt{a_0 a_1}\int _{0}^{\infty }K(t,\beta ){(\lambda _{1}(t)-\exp (\beta )\lambda _{0}(t)) \over \exp (\beta )}dt \rightarrow \zeta (\beta ) \quad \hbox {(say)}. \end{aligned}$$

In addition, let \(\pi _{j}\) denote the limits of \(a_{j}\) for \(j=0,1\). Then,

$$\begin{aligned}&(n_{0}+n_{1})^{-1/2}U_{w}(\beta _{w}^{*}) \nonumber \\&\quad \approx \sqrt{\pi _0 \pi _1} \int _{0}^{\infty }{K(t,\beta _{w}^{*}) \over \exp (\beta _{w}^{*}){\bar{Y}}_{1}(t)}d{\bar{M}}_{1}(t) \nonumber \\&\qquad - \sqrt{\pi _0 \pi _1} \int _{0}^{\infty }{K(t,\beta _{w}^{*}) \over {\bar{Y}}_{0}(t)}d{\bar{M}}_{0}(t) + \zeta (\beta _{w}^{*}). \end{aligned}$$
(11)

Now, the first two terms are of the form \(\int _{0}^{\infty }H_{1}d{\bar{M}}_{1}(t) - \int _{0}^{\infty }H_{0}d{\bar{M}}_{0}(t)\) where \(H_{1}=\sqrt{\pi _0 \pi _1}{K(t,\beta _{w}^{*}) \over \exp (\beta _{w}^{*}){\bar{Y}}_{1}(t)}\) and \(H_{0}=\sqrt{\pi _0 \pi _1}{K(t,\beta _{w}^{*}) \over {\bar{Y}}_{0}(t)}\). Let \(h_{1}\) denote the limit of \(H_{1}^{2}\exp (\beta _{w}^{*}){{\bar{Y}}}_{1}\) and \(h_{0}\) denote the limit of \(H_{0}^{2}{\bar{Y}}_{0}\), then following arguments as in Fleming and Harrington [6] (See page 268) the variance of \((n_{0}+n_{1})^{-1/2}U_{w}(\beta _{w}^{*})\) can be estimated by (upon plugging in empirical counterparts for limits)

$$\begin{aligned}&\hat{\sigma }^{2}(U_{w}(\beta _{w}^{*}))\nonumber \\&\quad = (\pi _0 \pi _1) \int _{0}^{\infty } \left\{ {K(t,\beta _{w}^{*})^2 \over {\bar{Y}}_{0}(t)}+{K(t,\beta _{w}^{*})^2 \over \exp (\beta _{w}^{*}){\bar{Y}}_{1}(t)} \right\} \left\{ 1-{\varDelta {\bar{N}}(t) -1 \over {\bar{Y}}_{0}(t)+\exp (\beta _{w}^{*}){\bar{Y}}_{1}(t)-1} \right\} \nonumber \\&\qquad \times {d {\bar{N}}(t) \over {\bar{Y}}_{0}(t)+ \exp (\beta _{w}^{*}){\bar{Y}}_{1}(t)}. \end{aligned}$$
(12)

Note that for \(\beta _{w}^{*}=0\) the score \(U_{w}(0)\) and \(\hat{\sigma }^{2}(U_{w}(0))\) are equivalent to the score statistic (1) and its variance estimate (2). The variance of \({\hat{\beta }}_{w}\) is estimated by plugging-in \({\hat{\beta }}_{w}\) in Eqs. (10) and (12).

Now, for calculating the constant \(c_{max}\) in the simultaneous confidence interval described in (6), we use

$$\begin{aligned} \sqrt{n_{0}+n_{1}}({\hat{\beta }}_{w}-\beta _{w}^{*}) \approx \left\{ i_{w}({\hat{\beta }}_{w})/(n_{0}+n_{1})\right\} ^{-1}(n_{0}+n_{1})^{-1/2}U_{w}^{\dagger }({\hat{\beta }}_{w}), \end{aligned}$$

where \(U_{w}^{\dagger }({\hat{\beta }}_{w}):= \sum _{i=1}^n\int _{0}^{\infty }w(t)\left\{ Z_{i}-{S^{(1)}(t,{\hat{\beta }}_{w}) \over S^{(0)}(t,{\hat{\beta }}_{w})} \right\} dG_{i}N_{i}(t)\), and the same \(G_{i}\)’s (in analogy to (3)) would be used for each weighted estimator within (6).

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León, L.F., Lin, R. & Anderson, K.M. On Weighted Log-Rank Combination Tests and Companion Cox Model Estimators. Stat Biosci 12, 225–245 (2020). https://doi.org/10.1007/s12561-020-09276-1

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