Logrank and weighted logrank

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.

For $g\in \{0,1\}$, let $X_{g1},X_{g2},\dots$ be IID measurable $[0,\mathrm{\infty })$-valued observed times with law $P_g$. For deterministic $n_g\to \mathrm{\infty }$, define $Y_{g,n}(t):={\sum }_{i=1}^{n_g}1\{X_{gi}\ge t\},s_g(t):=P_g(X_{g1}\ge t),0\le t\le \tau \mathrm{.}$ Then, for each group, ${\sup }_{0\le t\le \tau }\mid \frac{Y_{g,n}(t)}{n_g}-s_g(t)\mid {⟶}_{P}0.$

Fix a risk set with $Y_1$ group-1 subjects, $Y_0$ group-0 subjects, and $Y=Y_1+Y_0$. Conditional on the margins, suppose the $d$ failing subjects are a uniform $d$-subset of these $Y$ subjects, where $0\le d\le Y$. Let $O$ be the number of failures in group 1. If $Y>1$, then $\mathbb{E}[O]=d\frac{Y_1}{Y},\mathrm{Var}(O)=\frac{Y_1{Y}_{0}d(Y-d)}{Y^2(Y-1)}\mathrm{.}$ If $Y=0$ or $Y=1$, set the contribution to the variance to zero; the expectation remains $d{Y}_1\mathrm{/}Y$ when $Y>0$, and is zero when $Y=0$.

For the sequential assertion, take a finite sequence $j=1,\dots ,J$ on a probability space with a filtration ${\mathcal{H}}_0\subseteq {\mathcal{H}}_1\subseteq \cdots \subseteq {\mathcal{H}}_J$. Before the failures at step $j$ are revealed, the current risk-set subjects, their group counts $Y_{1j},Y_{0j}$, and the total $d_j$ are ${\mathcal{H}}_{j-1}$-measurable, with $0\le d_j\le Y_j:=Y_{1j}+Y_{0j}$. Conditionally on ${\mathcal{H}}_{j-1}$, the failing subjects are a uniform $d_j$-subset of the current $Y_j$ subjects; their identities are revealed in ${\mathcal{H}}_j$. If $Y_j=0$, necessarily $d_j=0$ and the failing subset is empty. Put $E_j=d_j{Y}_{1j}\mathrm{/}{Y}_j$ when $Y_j>0$ and $E_j=0$ when $Y_j=0$; let $V_j$ be the displayed risk-set variance when $Y_j>1$ and zero when $Y_j\le 1$. Suppose each real weight $w_j$ is ${\mathcal{H}}_{j-1}$-measurable and $\mathbb{E}\text{\hspace{0.17em}⁣}\left[{\sum }_{j=1}^J{w}_j^2{V}_j\right]<\mathrm{\infty }\mathrm{.}$ Then $D_j=w_j(O_j-E_j)$ are square-integrable martingale differences and $\mathbb{E}\text{\hspace{0.17em}⁣}\left[{\sum }_j{D}_j\right]=0,\mathrm{Var}\text{\hspace{0.17em}⁣}\left({\sum }_j{D}_j\right)=\mathbb{E}\text{\hspace{0.17em}⁣}\left[{\sum }_j{w}_j^2{V}_j\right],$ where the nonempty-risk-set variance contribution is the one in Two-group logrank and weighted logrank scores.

For each $g\in \{0,1\}$, let $(U_{gi}{)}_{i\ge 1}$ be IID measurable $[0,\mathrm{\infty })$-valued observed times with law $P_g$; independence between the two groups is not required. Along a sequence of deterministic positive sample sizes, let $n=n_0+n_1\to \mathrm{\infty }$ and $n_g\mathrm{/}n\to {\rho }_g\in (0,1)$, where ${\rho }_0+{\rho }_1=1$. Set $s_g(t):=P_g(U_{g1}\ge t)$, $y_g(t)={\rho }_g{s}_g(t)$, and $y(t)=y_0(t)+y_1(t)$. Assume ${\inf }_{0\le t\le \tau }y(t)>0$. Write $Y_{g,n}(t)={\sum }_{i=1}^{n_g}1\{U_{gi}\ge t\},p_n(t)=\{\begin{array}{ll}{Y}_{1,n}(t)\mathrm{/}(Y_{0,n}(t)+Y_{1,n}(t)),& Y_{0,n}(t)+Y_{1,n}(t)>0,\\ 0,& Y_{0,n}(t)+Y_{1,n}(t)=0.\end{array}$ Then, for $g=0,1$, ${\sup }_{t\le \tau }\mid \frac{Y_{g,n}(t)}{n}-y_g(t)\mid {⟶}_{P}0,$ and ${\sup }_{0\le t\le \tau }\mid p_n(t)-\bar{z}(t)\mid {⟶}_{P}0,\bar{z}(t):=\frac{y_1(t)}{y(t)}\mathrm{.}$

Fix a finite randomized treatment-assignment model with allowed assignments $\mathcal{Z}$ and known law $p$, with unit set $\mathcal{I}$ as in Finite randomized treatment-assignment models. Fix a map $\gamma :\mathcal{Z}\to \{0,1{\}}^{\mathcal{I}}$ assigning each allowed assignment a binary logrank group map. For each subject $i$, let $(U_i(z),{\delta }_i(z))\in [0,\tau ]\times \{0,1\}$ be the full right-censored record under assignment $z$. Assume the record sharp null: for every $i$, the pair $(U_i(z),{\delta }_i(z))$ is constant over all $z\in \mathcal{Z}$. Fix before observing assignment or records a bounded logrank weight rule and the two-sided ordering $T(z)=\mathrm{\mid }{U}_w(\gamma (z);U,\delta )\mathrm{\mid }$.

The inclusive assignment-tail p-value $p_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}(z):={\sum }_{z^{\mathrm{\prime }}\in \mathcal{Z}}p(z^{\mathrm{\prime }})1\{\mathrm{\mid }{U}_w(\gamma (z^{\mathrm{\prime }});U,\delta )\mathrm{\mid }\ge \mathrm{\mid }{U}_w(\gamma (z);U,\delta )\mathrm{\mid }\}$ is superuniform under the assignment law, so it is a valid finite-sample p-value for the record sharp null. If an auxiliary $V$ is independent of the assignment and uniform on $[0,1]$, replacing the inclusive tail by the strict-tail mass plus $V$ times the tied-score mass gives an exactly uniform p-value. The result uses the actual assignment probabilities and asserts no validity under a weak average-effect null or under an equal-hazard null alone.

Assume the Axiom of Choice. Let $n_0,n_1$ be deterministic, $n=n_0+n_1$, and let $Z_i=g$ for subject $i$ in group $g\in \{0,1\}$. On a common finite-horizon filtration, each subject has the one-event process $N_{gi}(t)=1\{U_{gi}\le t,{\delta }_{gi}=1\},Y_{gi}(t)=1\{U_{gi}\ge t\},$ and satisfies the separate counting-process contract $A_{gi}(t)={\int }_0^t{Y}_{gi}(s)\lambda (s)ds,M_{gi}=N_{gi}-A_{gi},$ where $\lambda$ is a common deterministic nonnegative hazard density with $\mathrm{\Lambda }(\tau ):={\int }_0^{\tau }\lambda (s)ds<\mathrm{\infty }$. Assume each $M_{gi}$ is a square-integrable martingale under this contract, $Y_{gi}$ is predictable, and no two subjects have simultaneous event jumps. Write $N_{g,n}$ and $Y_{g,n}$ for the group sums, $N_n=N_{0,n}+N_{1,n}$, and $p_n(t)=Y_{1,n}(t)\mathrm{/}(Y_{0,n}(t)+Y_{1,n}(t))$, with $p_n(t)=0$ on an empty pooled risk set.

Assume the observed-time and allocation hypotheses of Uniform convergence of the risk-set treatment fraction: within each group the observed times are IID, $n\to \mathrm{\infty }$, $n_g\mathrm{/}n\to {\rho }_g\in (0,1)$, and, with $y_g(t)={\rho }_{g}P(U_{g1}\ge t)$ and $y=y_0+y_1$, ${\inf }_{0\le t\le \tau }y(t)>0$. Set $\bar{z}(t)=y_1(t)\mathrm{/}y(t)$. Let $w_n$ be a common predictable weight process with $\mathrm{\mid }{w}_n(t)\mathrm{\mid }\le C$ and a measurable supremum satisfying ${\sup }_{t\le \tau }\mathrm{\mid }{w}_n(t)-w(t)\mathrm{\mid }{\to }_{P}0$ for a deterministic bounded predictable weight $w$. Then the weighted logrank score satisfies $\frac{U_{n,w_n}}{\sqrt{n}}=\frac{1}{\sqrt{n}}{\sum }_{g=0}^1{\sum }_{i=1}^{n_g}{\int }_0^{\tau }w(t)(g-\bar{z}(t))d{M}_{gi}(t)+o_P(1)\mathrm{.}$ In particular, fixed deterministic bounded predictable weights are included by taking $w_n=w$.

Assume the two-group common-hazard model with finite-horizon predictable risk processes and the explicit square-integrable martingale contracts $M_{gi}=N_{gi}-A_{gi}$, $A_{gi}(t)={\int }_0^t{Y}_{gi}(s)\lambda (s)ds$; assume no simultaneous event jumps. Let $Y_{g,n}={\sum }_i{Y}_{gi}$, $Y_n=Y_{0,n}+Y_{1,n}$, and assume the uniform risk-fraction convergence $Y_n\mathrm{/}n\to y$ and $p_n=Y_{1,n}\mathrm{/}{Y}_n\to \bar{z}$ in probability on $[0,\tau ]$, with $p_n=0$ on an empty risk set (as proved in Uniform convergence of the risk-set treatment fraction). Let $w_n\to w$ uniformly in probability, where the predictable weights are uniformly bounded and $w$ is deterministic. Suppose event times are continuous, so every observed failure time has $d_j=1$ almost surely. Here ${\widehat{V}}_{n,w_n}$ is the plug-in statistic defined in Two-group logrank and weighted logrank scores; the claim is its asymptotic consistency and does not assert an exact finite-sample conditional variance law. Here $y_0,y_1$ are deterministic nonnegative Borel functions, $y=y_0+y_1$, ${\inf }_{0\le t\le \tau }y(t)>0$, and $\bar{z}=y_1\mathrm{/}y$. Define ${\sigma }_w^2:={\int }_0^{\tau }w(t{)}^2\frac{y_0(t)y_1(t)}{y(t)}\lambda (t)dt\mathrm{.}$ Then the weighted plug-in variance sum from Two-group logrank and weighted logrank scores satisfies $\frac{{\widehat{V}}_{n,w_n}}{n}{⟶}_P{\sigma }_w^2\mathrm{.}$

Assume the Axiom of Choice. For $g\in \{0,1\}$, let $(T_{gi},C_{gi}{)}_{i\ge 1}$ be IID within group, with $T_{gi},C_{gi}\in [0,\mathrm{\infty })$ almost surely and the two group families independent. Assume conditional independent censoring $T_{gi}\perp C_{gi}\mid g$, and a common absolutely continuous event-time law with hazard density $\lambda$ on $[0,\tau ]$, where $\mathrm{\Lambda }(\tau )={\int }_0^{\tau }\lambda (t)dt<\mathrm{\infty }$. Censoring distributions may differ between groups. Use the horizon-truncated record $U_{gi}:=\min (T_{gi},C_{gi},\tau )$ and ${\delta }_{gi}:=1\{T_{gi}\le C_{gi},T_{gi}\le \tau \}$, which lies in $[0,\tau ]\times \{0,1\}$ as required by Two-group logrank and weighted logrank scores.

For deterministic positive sample sizes $n_g$ along a sequence with $n=n_0+n_1\to \mathrm{\infty }$ and $n_1\mathrm{/}n\to \rho \in (0,1)$, set ${\rho }_1=\rho$, ${\rho }_0=1-\rho$, $y_g(t)={\rho }_{g}P(U_{g1}\ge t)$, $y(t)=y_0(t)+y_1(t)$, and $\bar{z}(t)=y_1(t)\mathrm{/}y(t)$. Assume ${\inf }_{t\le \tau }y(t)>0$. For each subject define $N_{gi}(t)=1\{U_{gi}\le t,{\delta }_{gi}=1\}$ and $Y_{gi}(t)=1\{U_{gi}\ge t\}$. Under one common filtration, assume $Y_{gi}$ is predictable and, separately, that $A_{gi}(t)={\int }_0^t{Y}_{gi}(s)\lambda (s)ds,M_{gi}=N_{gi}-A_{gi}$ satisfies the finite-horizon square-integrable martingale contract of Simple counting-process intensity and compensator contract. Assume no two subjects have simultaneous event jumps. Conditional independent censoring and the displayed hazard are scientific model premises; they are not used as substitutes for the explicit martingale contract.