Robust Modestly-Weighted Log-Rank Test for Two-Group Survival Data

Computes the robust modestly-weighted (rMW) log-rank test of Magirr and Ohrn, which combines the standard log-rank test with a single modestly-weighted log-rank test. The test statistic is the maximum of the two standardized components, evaluated against their joint null distribution. Because the standard log-rank statistic is included as one of the two components and the components are strongly correlated under the null, the multiplicity adjustment is small, so the test loses little power relative to the log-rank test in worst-case scenarios while gaining substantial power under delayed effects. The two component numerators, their variances, and their null covariance are computed in a single pass by the C++ core rmw_core.

Usage

rmw_fast(
  time,
  event,
  group,
  control,
  side = 2,
  s_star = 0.5,
  presorted = FALSE
)

Arguments

time: A numeric vector of follow-up times for all subjects.
event: An integer or numeric vector of event indicators (1 = event, 0 = censored), aligned with time.
group: A vector of group labels aligned with time.
control: A scalar value indicating which level of group represents the control group.
side: An integer, either 1 or 2 (default 2). If side = 1, the statistic is the minimum of the two standardized components and a one-sided p-value is returned. If side = 2, the statistic is the maximum of their absolute values and a two-sided p-value is returned.
s_star: A single numeric value in (0, 1], the survival-probability threshold of the modestly-weighted component. The weight is capped at 1 / s_star. Defaults to 0.5. A value of 1 caps the weight at one, so the modestly-weighted component equals the log-rank component and the test reduces to the ordinary log-rank test.
presorted: A logical value. If TRUE, time, event, and group are assumed to be already sorted by ascending time, and the internal order() call is skipped. If FALSE (default), sorting is handled internally.

Value

An object of class "rmw_fast", a length-two numeric vector c(statistic, p.value) with attributes z (the named component Z-scores c(logrank, mwlrt)), corr (the 2 by 2 null correlation matrix of the two components), s_star, O1 (the observed number of events in the treatment group), side, and n (the total sample size). Returns NA statistic and p-value (still with class "rmw_fast") when either component variance is zero (e.g., all events in one group).

Details

The first component is the ordinary log-rank statistic, with weight one at every event time. The second component is a modestly-weighted log-rank statistic with weight min(1 / S(t-), 1 / s_star), where S(t-) is the left-continuous pooled Kaplan-Meier estimate just prior to each event time and s_star is a survival-probability threshold. This is the survival-threshold parameterization of the modestly-weighted test of Magirr and Burman, in which the weight is capped at 1 / s_star. It differs from the timepoint parameterization exposed by survdiff_fast() with weight = "mwlrt", where the cap is derived from a timepoint t_star. The choice s_star = 0.5 caps the weight at 2 and is the value used in the original rMW article.

Writing the two standardized components as Z_lr and Z_mw, under the null hypothesis of equal survival the pair is asymptotically bivariate normal with zero means, unit variances, and correlation rho = C / sqrt(V_lr V_mw), where C is the covariance of the two numerators returned by the C++ core. When side = 1 the statistic is min(Z_lr, Z_mw), so a protective treatment effect (fewer events than expected) yields a small, negative value, and the one-sided p-value is P(min(Z_lr, Z_mw) <= observed) under the joint null. When side = 2 the statistic is max(abs(Z_lr), abs(Z_mw)) and the two-sided p-value is P(max(abs(Z_lr), abs(Z_mw)) >= observed). The joint normal probability is evaluated with mvtnorm::pmvnorm, using the exact TVPACK algorithm for the one-sided half-space and the deterministic Miwa algorithm for the two-sided rectangle.

When presorted = TRUE, the input vectors are assumed to be sorted by ascending time and the internal order() call is skipped, which avoids one O(n log n) pass in simulation loops where the data are already generated in time order.

References

Magirr, D., & Öhrn, F. (2026). Robust modestly weighted log-rank tests. Pharmaceutical Statistics, 25(1), e70066.

Magirr, D., & Burman, C.-F. (2019). Modestly weighted logrank tests. Statistics in Medicine, 38(20), 3782-3790.

Examples

library(survival)

# One-sided robust modestly-weighted test with s_star = 0.5
fit <- rmw_fast(ovarian$futime, ovarian$fustat, ovarian$rx,
                control = 1, side = 1, s_star = 0.5)
fit
#> Robust modestly-weighted log-rank test (two-group)
#> 
#>   N = 26,  control = 1,  s_star = 0.5
#> 
#>                         Z
#> log-rank          -1.0309
#> modestly-weighted -0.7583
#> 
#>   Null correlation = 0.9821
#>  min Z = -1.031,  one-sided p-value = 0.169  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# The log-rank component matches survdiff_fast with weight = "logrank"
z_lr <- as.numeric(survdiff_fast(ovarian$futime, ovarian$fustat, ovarian$rx,
                                 control = 1, side = 1))
cat("rmw_fast log-rank component:", attr(fit, "z")[["logrank"]], "\n")
#> rmw_fast log-rank component: -1.030893 
cat("survdiff_fast log-rank Z   :", z_lr, "\n")
#> survdiff_fast log-rank Z   : -1.030893 

# Two-sided test
rmw_fast(ovarian$futime, ovarian$fustat, ovarian$rx,
         control = 1, side = 2, s_star = 0.5)
#> Robust modestly-weighted log-rank test (two-group)
#> 
#>   N = 26,  control = 1,  s_star = 0.5
#> 
#>                         Z
#> log-rank          -1.0309
#> modestly-weighted -0.7583
#> 
#>   Null correlation = 0.9821
#>  Max |Z| = 1.031,  two-sided p-value = 0.338  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# presorted = TRUE: sort once outside, reuse inside a loop
ord <- order(ovarian$futime)
rmw_fast(ovarian$futime[ord], ovarian$fustat[ord], ovarian$rx[ord],
         control = 1, side = 1, s_star = 0.5, presorted = TRUE)
#> Robust modestly-weighted log-rank test (two-group)
#> 
#>   N = 26,  control = 1,  s_star = 0.5
#> 
#>                         Z
#> log-rank          -1.0309
#> modestly-weighted -0.7583
#> 
#>   Null correlation = 0.9821
#>  min Z = -1.031,  one-sided p-value = 0.169  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1