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Weighted logrank score expansion under a common-hazard model

Statement

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 $N_{g i} (t) = 1 {U_{g i} \leq t, δ_{g i} = 1}, Y_{g i} (t) = 1 {U_{g i} \geq t},$ and satisfies the separate counting-process contract $A_{g i} (t) = \int_{0}^{t} Y_{g i} (s) λ (s) d s, M_{g i} = N_{g i} - A_{g i},$ where $λ$ is a common deterministic nonnegative hazard density with $Λ (τ) : = \int_{0}^{τ} λ (s) d s < \infty$ . Assume each $M_{g i}$ is a square-integrable martingale under this contract, $Y_{g i}$ 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 $p_{n} (t) = Y_{1, n} (t) / (Y_{0, n} (t) + Y_{1, n} (t))$ , 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, $n \to \infty$ , $n_{g} / n \to ρ_{g} \in (0, 1)$ , and, with $y_{g} (t) = ρ_{g} P (U_{g 1} \geq t)$ and $y = y_{0} + y_{1}$ , $\inf_{0 \leq t \leq τ} y (t) > 0$ . Set $\overset{ˉ}{z} (t) = y_{1} (t) / y (t)$ . Let $w_{n}$ be a common predictable weight process with $∣ w_{n} (t) ∣ \leq C$ and a measurable supremum satisfying $\sup_{t \leq τ} ∣ w_{n} (t) - w (t) ∣ \to_{P} 0$ for a deterministic bounded predictable weight $w$ . Then the weighted logrank score satisfies $\frac{U_{n, w_{n}}}{\sqrt{n}} = \frac{1}{\sqrt{n}} \sum_{g = 0}^{1} \sum_{i = 1}^{n_{g}} \int_{0}^{τ} w (t) (g - \overset{ˉ}{z} (t)) d M_{g i} (t) + o_{P} (1) .$ In particular, fixed deterministic bounded predictable weights are included by taking $w_{n} = w$ .

Facts & Assumptions

Given: A finite horizon, two fixed groups, the common-hazard model contract for each subject, no simultaneous event jumps, the uniform risk-fraction limit, and the bounded predictable weight sequence in the Statement.

[A1]

Assume the Axiom of Choice, inherited from the continuous-time martingale, bracket, and $L^{2}$ -integral interfaces. (The Axiom of Choice)

[F1]

Simple counting-process intensity and compensator contract: for a simple counting process, $A = \int Y λ$ is continuous and has finite expectation, and the model separately assumes $M = N - A$ is a square-integrable martingale.

[F2]

L² counting-process integral when the limit exists: a predictable $H$ with $E \int H^{2} d A < \infty$ has an integral defined by $L^{2}$ limits of bounded predictable step integrals when the limit exists.

[F3]

Brackets of a compensated simple counting process: under the counting-process contract, $M^{2} - A$ is a martingale and $⟨ M ⟩ = A$ .

[F4]

L² extension of the predictable counting-process integral: every predictable finite-energy integrand has a square-integrable martingale integral with $E (\int H d M)^{2} = E \int H^{2} d A$ .

[F5]

Uniform convergence of the risk-set treatment fraction: $p_{n}$ converges uniformly in probability to $\overset{ˉ}{z}$ on $[0, τ]$ .

[F6]

Two-group logrank and weighted logrank scores: $U_{n, w_{n}} = \sum_{j} w_{n} (t_{j}) (O_{j} - E_{j})$ with $E_{j} = d_{j} Y_{1, n} (t_{j}) / Y_{n} (t_{j})$ .

Proof

1.1A1F1F2F6givenalgebra

Each $Y_{g i}$ is predictable by assumption. Sums of these processes are predictable, and the ratio map $(a, b) \mapsto a / b$ for $b > 0$ , extended by zero when $b = 0$ , is Borel; hence $p_{n}$ is predictable. At an event time $t_{j}$ , the Stieltjes increment of $N_{1, n}$ is $O_{j}$ and that of $N_{n}$ is $d_{j}$ . Therefore [F6] gives $U_{n, w_{n}} = \int_{0}^{τ} w_{n} (t) {d N_{1, n} (t) - p_{n} (t) d N_{n} (t)} .$ Using $d N_{g i} = d M_{g i} + Y_{g i} λ (t) d t$ , the compensator part equals $w_{n} (t) {Y_{1, n} (t) - p_{n} (t) (Y_{0, n} (t) + Y_{1, n} (t))} λ (t) d t = 0$ . Thus, with $M_{n} = \sum_{g, i} M_{g i}$ , $U_{n, w_{n}} = \sum_{g, i} \int_{0}^{τ} w_{n} (t) (g - p_{n} (t)) d M_{g i} (t) .$

2.1A1F2F3F4F5step 1.1

No simultaneous event jumps make $N_{n} = \sum_{g, i} N_{g i}$ a simple unit-jump counting process. Its compensator is the continuous process $A_{n} = \sum_{g, i} A_{g i}$ , and $M_{n} = N_{n} - A_{n} = \sum_{g, i} M_{g i}$ is a square-integrable martingale by finite summation. Also $A_{n} (τ) \leq n Λ (τ)$ pathwise. By [F3], $M_{n}^{2} - A_{n}$ is a martingale; [F2]-[F4] apply to the common predictable integrand $H_{n} = w_{n} (p_{n} - \overset{ˉ}{z})$ , whose compensator energy is finite. The isometry gives $E [{\frac{1}{\sqrt{n}} \int_{0}^{τ} w_{n} (p_{n} - \overset{ˉ}{z}) d M_{n}}^{2}] \leq C^{2} Λ (τ) E ⁣ [\sup_{t \leq τ} ∣ p_{n} (t) - \overset{ˉ}{z} (t) ∣^{2}] ⟶ 0.$ The last expectation tends to zero because the supremum is bounded by one and converges to zero in probability by [F5]. Replacing $p_{n}$ by $\overset{ˉ}{z}$ therefore changes $U_{n, w_{n}} / \sqrt{n}$ by $o_{P} (1)$ .

3.1A1F1F2F3F4givenstep 2.1algebra∎

For the remaining weight replacement, put $D_{n} = \sum_{g, i} \int_{0}^{τ} (w_{n} - w) (g - \overset{ˉ}{z}) d M_{g i}$ . Since $g$ is constant within each group, $D_{n} = \int_{0}^{τ} (w_{n} - w) (1 - \overset{ˉ}{z}) d M_{1, n} - \int_{0}^{τ} (w_{n} - w) \overset{ˉ}{z} d M_{0, n} .$ Each group aggregate has a simple counting process, continuous compensator, and square-integrable martingale because there are no simultaneous event jumps within that group. Applying [F2]-[F4] to each aggregate and using $(a - b)^{2} \leq 2 a^{2} + 2 b^{2}$ yields $E [(D_{n} / \sqrt{n})^{2}] \leq 2 Λ (τ) E ⁣ [\sup_{t \leq τ} ∣ w_{n} (t) - w (t) ∣^{2}] ⟶ 0.$ Boundedness of $w_{n}$ and $w$ makes the squared supremum bounded, so its convergence in probability implies convergence of its expectation. Combining the risk-fraction replacement in step 2.1 with this weight replacement proves the stated expansion. No martingale process central limit theorem is used.

Depends on

Used by

Terminal normal limit for weighted logrank scores Theorem

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Sources

Lu Tian, STAT331 Unit 13: Large Sample Theory for 2-Sample Tests, PDF pp. 2-7 (standard reference, not scraped)
Michal Kulich, NMST531 Censored Data Analysis: Extended Course Notes, §5.3, PDF pp. 59-60 (standard reference, not scraped)