Two-group logrank and weighted logrank scores

Fix a follow-up horizon $\tau >0$ and a finite set of subjects $\mathcal{I}=\{1,\dots ,n\}$. A right-censored record for subject $i$ is a pair $(U_i,{\delta }_i)\in [0,\tau ]\times \{0,1\}$, where $U_i=\min (T_i,C_i)$ is the observed time and ${\delta }_i=1\{T_i\le C_i\}$ records an event, including an event-censoring tie; ${\delta }_i=0$ records right censoring. A two-group assignment is a map $z:\mathcal{I}\to \{0,1\}$. These conventions are the right-censoring observation scheme; the score below uses the observed record and does not infer event or censoring laws from it.

Let $t_1<\cdots <t_K$ be the distinct observed failure times in $[0,\tau ]$. For $g\in \{0,1\}$ define the risk count and event count at $t_j$ by $Y_{gj}(z):=\mathrm{\#}\{i:z_i=g,U_i\ge t_j\},d_{gj}(z):=\mathrm{\#}\{i:z_i=g,U_i=t_j,{\delta }_i=1\}\mathrm{.}$ Write $Y_j=Y_{0j}+Y_{1j}$, $d_j=d_{0j}+d_{1j}>0$, and $O_j=d_{1j}$. Since $t_j$ is an observed failure time, $1\le d_j\le Y_j$. The expected group-1 count under the conditional hypergeometric model is $E_j:=d_j\frac{Y_{1j}}{Y_j}\mathrm{.}$ Under that model the $d_j$ failing subjects are a uniform $d_j$-subset of the $Y_j$ subjects at risk, conditional on the risk-set and event-count margins. The corresponding conditional variance is $V_j:=\{\begin{array}{ll}{\displaystyle \frac{Y_{1j}{Y}_{0j}{d}_j(Y_j-d_j)}{Y_j^2(Y_j-1)},}& Y_j>1,\\ 0,& Y_j=1.\end{array}$ If the study has no observed failures, all sums below are empty and equal zero. The conditional hypergeometric model is an explicit finite risk-set assumption; equality of survival laws with arbitrary censoring or subject heterogeneity does not by itself assert this conditional allocation law. The uniform law on the finite set of $d_j$-subsets is the uniform finite probability space of The uniform probability space on a nonempty finite set.

A weight rule assigns a real value $w_j$ using only the predeclared time and information available strictly before the failures at $t_j$; deterministic bounded time weights are a special case. If ${\sup }_j\mathrm{\mid }{w}_j\mathrm{\mid }<\mathrm{\infty }$, define the weighted logrank score and its conditional variance estimate by $U_w(z;U,\delta ):={\sum }_{j=1}^K{w}_j(O_j-E_j),{\widehat{V}}_w(z;U,\delta ):={\sum }_{j=1}^K{w}_j^2{V}_j\mathrm{.}$ When ${\widehat{V}}_w>0$, set $Z_w:=U_w\mathrm{/}\sqrt{{\widehat{V}}_w}$. If ${\widehat{V}}_w=0$, this standardized statistic is left undefined; the unstandardized score remains defined. The normal calibration is only an approximation until the separate terminal limit hypotheses are verified. It is not the finite-sample Fisher randomization p-value.

For a fixed observed record, the displayed scores are statistics in the measurable-map sense of Statistics and estimators: each risk count is a finite sum of Borel indicator functions, and every denominator is positive at an observed event time. A weight rule is fixed before assignments or outcomes are inspected and is recomputed by the same rule for any candidate assignment.

The risk-set table concerns one predeclared event type. If simultaneous records contain distinct event causes that are not exchangeable, form separate cause-specific counts $O_{jc}$ and $d_{jc}$ using the same pre-event risk set; use the displayed hypergeometric variance only for a cause whose failures are conditionally exchangeable. This definition does not pool different causes or add their variances without a specified cross-cause covariance model.