In all cases, we can think in the permutation test framework, considering the scores as fixed and permuting the treatment labels.
(Weighted) log-rank test
Following Leton & Zuluaga (2000), letting \(l_{1,j}\) and \(l_{0,j}\) denote the number of patients censored on the test treatment and control treatment, respectively, during \(\left[\left.t_j, t_{j+1}\right)\right.\), we can express the (weighted) log-rank statistic as
\[\begin{align*}
U_W &:=\sum_{j = 1}^{k} w_j\left( O_{1,j} - O_j\frac{n_{1,j}}{n_j} \right)\\
&=\sum_{j = 1}^{k} w_jO_{1,j} - \sum_{j = 1}^{k}w_j\frac{O_{j}}{n_j} \times n_{1,j}\\
&=\sum_{j = 1}^{k} w_jO_{1,j} - \sum_{j = 1}^{k}w_j\frac{O_{j}}{n_j} \times \sum_{i = j}^{k}(O_{1,i} + l_{1,i})\\
&=\sum_{j = 1}^{k} w_jO_{1,j} - \sum_{j = 1}^{k}(O_{1,j} + l_{1,j}) \times \sum_{i = 1}^{j}w_i\frac{O_{i}}{n_i}\\
&=\sum_{j = 1}^{k} O_{1,j}\left( w_j - \sum_{i = 1}^{j}w_i\frac{O_{i}}{n_i} \right) + \sum_{j = 1}^{k}l_{1,j} \left( - \sum_{i = 1}^{j}w_i\frac{O_{i}}{n_i}\right).
\end{align*}\]
This means that an observed event at time \(t_j\) is given a score of \(a_i=w_j - \sum_{i = 1}^{j}w_i\frac{O_{i}}{n_i}\), and an observation censored during \(\left[\left.t_j, t_{j+1}\right)\right.\) is given a score of \(a_i=- \sum_{i = 1}^{j}w_i\frac{O_{i}}{n_i}\).
Note that instead of using \(U_W\) one could also use \[\begin{equation*}
\tilde{U}_W = \frac{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 1 \right\rbrace a_{i}}{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 1 \right\rbrace} - \frac{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 0 \right\rbrace a_{i}}{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{(i)} = 0 \right\rbrace},
\end{equation*}\] as the test statistic in a permutation test, as \(U\) and \(\tilde{U}\) are equivalent up to a (positive) scale and shift transformation, i.e., \[\begin{equation*}
\tilde{U}_W = \frac{n}{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 1 \right\rbrace\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 0 \right\rbrace} \times U_W - \frac{\sum_{i= 1}^n a_{i} \sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 1 \right\rbrace}{\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 1 \right\rbrace\sum_{i= 1}^n \mathbb{I}\left\lbrace z_{i} = 0 \right\rbrace}.
\end{equation*}\]
Furthermore, re-scaling the scores to
\[b_i = \frac{2a_i - \max{a} - \min{a}}{\max{a} - \min{a}}\] so that \(b \in (-1,1)\) would also leave the p-value of the permutation test unchanged.
RMST / Milestone
We use the concept of pseudo-values following Andersen et al. (2017).
For the RMST, without any adjustment for covariates, the \(i\)-th pseudo-value (at time \(\tau\)) is defined as \[\widehat{\theta}_{i}^{\tiny \mbox{RMST}} = n \int_0^\tau \widehat{S}(t) dt - (n-1) \int_0^\tau \widehat{S}^{(-i)}(t) dt,\] where \(\widehat{S}^{(-i)}(t)\) is the Kaplan-Meier estimator excluding observation (or subject) \(i\). For the Milestone survival analysis rate, also without any adjustment for covariates, the \(i\)-th pseudo-value is defined as \[\widehat{\theta}_{i}^{\tiny \mbox{MLST}} = n \widehat{S}(t) - (n-1) \widehat{S}^{(-i)}(t).\] Having found pseudo-values for each patient, they can be used just like any other (continuous) outcomes. For example, they could be fed into a linear model with treatment term only. In this case, the resultant test statistic for testing the null hypothesis of zero difference between treatments would be a difference in mean pseudo-values, i.e., the pseudo-values are performing the same role as the “scores” in the weighted log-rank tests. Again these scores could be standardized to lie between -1 and 1 without affecting the p-value.
