Log-Rank Test

Log-rank test (also known as the Mantel–Cox test or Peto–Peto log-rank statistic) is the foundational non-parametric hypothesis test in biostatistics, randomized clinical trials, and longitudinal epidemiology for testing whether the time-to-event distributions of two or more independent groups are identical in the presence of right-censored observations. First derived by Nathan Mantel in 1966 as a time-stratified extension of the Cochran–Mantel–Haenszel test and subsequently named and proved to be locally most powerful under proportional hazards alternatives by Richard Peto and Julian Peto in 1972, the log-rank test serves as the inferential companion to the Kaplan–Meier estimator. By constructing a conditional 2 × 2 contingency table at every distinct failure time and comparing the observed event count in each arm against its hypergeometric expectation under the null hypothesis of equal hazard rates, the log-rank test evaluates whole-curve survival differences without assuming any parametric form for the underlying baseline hazard function.

Risk-Set Stratification at Ordered Event Times

Suppose two independent treatment or exposure groups—Group 1 (experimental intervention) and Group 2 (control or standard of care)—are followed prospectively over time t ≥ 0 in a clinical trial or cohort study. Individual failure times are subject to independent, non-informative right-censoring (administrative study termination, withdrawal, or loss to follow-up). Let the pooled sample across both arms exhibit J distinct, ordered failure times t₁ < t₂ < ··· < t_J.

At each distinct event time t_j (j = 1, …, J), the investigator constructs a 2 × 2 contingency table conditioned on the marginal risk set just prior to t_j (t_j⁻) and the total number of failures occurring at t_j:

• Let n_1j and n_2j denote the number of individuals still at risk (alive and uncensored) just prior to time t_j in Group 1 and Group 2, respectively, with total risk set n_j = n_1j + n_2j.
• Let d_1j and d_2j denote the observed number of failure events occurring at time t_j in Group 1 and Group 2, respectively, with total events d_j = d_1j + d_2j.

Under the null hypothesis H₀ : S₁(t) = S₂(t) for all t ≥ 0 (equivalently, equal instantaneous hazard rates h₁(t) = h₂(t)), every individual in the pooled risk set n_j shares the same conditional probability of failing at t_j. Conditioned on the fixed row margins (d_j total events and n_j − d_j survivors) and column margins (n_1j and n_2j), the random variable O_1j = d_1j follows a central hypergeometric distribution with conditional expectation E_1j and conditional hypergeometric variance V_1j:

When survival times are continuous and untied (d_j = 1 at every event time t_j), the finite-population tie-correction factor (n_j − d_j) / (n_j − 1) simplifies to unity, reducing V_1j to the Bernoulli variance (n_1j n_2j) / n_j² = p_1j (1 − p_1j), where p_1j = n_1j / n_j is the proportion of the active risk set belonging to Group 1.

Summing Observed-Minus-Expected Increments and Martingale Interpretation

Summing the discrepancy O_1j − E_1j across all J ordered failure times yields the unstandardized log-rank score statistic U_L = ∑_j=1^J (d_1j − E_1j). In the counting-process and martingale framework formalized by Per Kragh Andersen and Richard D. Gill (1982), U_L is a stochastic integral of predictable risk-set processes with respect to a zero-mean counting-process martingale under H₀. Consequently, the increments (d_1j − E_1j) at distinct failure times t₁ < ··· < t_J are uncorrelated, and the variance of their sum equals the sum of their conditional hypergeometric variances V_L = ∑_j=1^J V_1j.

Standardizing U_L yields the two-sided Mantel–Cox log-rank test statistic:

A simpler conservative approximation often used in descriptive epidemiology replaces the hypergeometric denominator with the Poisson expected counts E₁ = ∑ E_1j and E₂ = ∑ E_2j, giving χ²_approx = (O₁ − E₁)² / E₁ + (O₂ − E₂)² / E₂, though the hypergeometric denominator V_L is always preferred because it remains unbiased when tied events occur or when group risk sets become unbalanced late in follow-up.

Local Optimality Under Lehmann Alternatives

Peto and Peto (1972) derived the name log-rank test by connecting the rank order of censored observations to the score function of the exponential and Weibull family under Lehmann alternatives, where the survival functions of two groups satisfy S₁(t) = [S₀(t)]^ψ for a constant hazard ratio ψ = h₁(t) / h₀(t) = exp(β). In the proportional hazards model introduced by David R. Cox in 1972, h(t | X) = h₀(t) exp(β X) with a single binary group indicator X ∈ {0, 1}, the partial log-likelihood score function U(β) evaluated at the null value β = 0 is identical to the Mantel–Cox numerator:

and the observed Fisher information I(β = 0) under the discrete-time Cox partial likelihood equals ∑_j=1^J V_1j. Thus, the log-rank test is the Rao score test of β = 0 in a univariate Cox regression model. By the Neyman–Pearson theory of rank-invariant tests, the log-rank test is asymptotically Pitman-efficient (locally most powerful) whenever the true data-generating mechanism obeys proportional hazards.

K-Sample, Stratified, and Trend Log-Rank Tests

When comparing K > 2 treatment arms, the scalar difference O_1j − E_1j generalizes to a (K − 1)-dimensional vector u_j with components O_kj − E_kj (k = 1, …, K − 1) and a (K − 1) × (K − 1) multivariate hypergeometric covariance matrix V_j whose diagonal entries are V_kk,j and off-diagonal entries are:

The quadratic form U^⊤ V⁻¹ U follows an asymptotic χ² distribution with K − 1 degrees of freedom. In multicenter trials or observational studies confounded by baseline prognostic strata s = 1, …, S (such as clinical tumor stage or age decile), the stratified log-rank test computes U_s and V_s separately within each stratum s and sums them prior to squaring: χ²_strat = (∑_s=1^S U_s)² / (∑_s=1^S V_s). If the K groups represent ordered dose levels z₁ < z₂ < ··· < z_K, the 1-degree-of-freedom log-rank test for trend evaluates whether hazard rates increase or decrease monotonically across doses.

Because the standard log-rank test assigns weight w_j = 1 to every failure time t_j regardless of how many subjects remain at risk, its statistical power degrades markedly when the proportional hazards assumption fails—such as when two Kaplan–Meier survival curves cross, when a surgical procedure carries early perioperative mortality followed by long-term cure, or when an immunotherapy agent exhibits a delayed onset of clinical benefit. In those settings, investigators choose members of the Fleming–Harrington test family or summary metrics such as restricted mean survival time (RMST).

Schoenfeld's Event-Driven Power Formula

In randomized controlled trials with a time-to-event primary endpoint, statistical power is governed by the total number of observed events (D) rather than the total number of enrolled patients (N) alone. Using the asymptotic distribution of the log-rank statistic under local proportional hazards alternatives with true log hazard ratio θ = ln(HR), David A. Schoenfeld (1981) derived the required total event count D for a two-sided significance level α and statistical power 1 − β:

where p is the allocation proportion randomized to Group 1 (for 1:1 randomization, p (1 − p) = 0.25, so the numerator is multiplied by 4). For example, detecting a hazard ratio of HR = 0.75 (ln 0.75 = −0.2877) with 80% power (z_0.80 = 0.8416) at a two-sided α = 0.05 (z_0.975 = 1.9600) under 1:1 randomization requires D = 4 × (1.9600 + 0.8416)² / (−0.2877)² ≈ 380 total observed events across both arms. Once D is fixed, trialists choose total enrollment N and accrual/follow-up duration to achieve D expected failures given the anticipated baseline hazard and dropout rate.

Validity Conditions and Competing Risks

Like the Nelson–Aalen estimator and the Kaplan–Meier curve, the log-rank test requires independent (non-informative) censoring within each group: patients censored at time t must carry the same conditional hazard of subsequent failure as uncensored patients remaining in the risk set. When a secondary event precludes observation of the primary endpoint (such as non-cancer mortality in an oncology relapse trial), applying the standard log-rank test to cause-specific censoring tests only the cause-specific hazard functions rather than the cumulative probability of the event; comparing cumulative incidence across groups in the presence of competing risks requires Gray's test (1988) on subdistribution hazards.

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