Computes the average hazard with survival weight (AHSW) of Uno and Horiguchi
for two groups and the between-group contrasts. The average hazard on the
window from 0 to tau is the ratio of the cumulative event probability
at tau to the restricted mean survival time at tau, both based
on the Kaplan-Meier estimate. The function returns the per-group average
hazard, the ratio of average hazards (RAH, treatment over control) on the log
scale, and the difference of average hazards (DAH, treatment minus control)
on the identity scale, each with a confidence interval and a two-sided test.
The C++ backend walks the pooled sorted data once per group, so the function
is suitable for simulation loops with presorted = TRUE.
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
grouprepresents the control group.- side
1 for a one-sided test in the direction of treatment benefit (lower average hazard in the treatment group, i.e. a ratio below 1 and a negative difference) or 2 for a two-sided test (default 2). The reported p-values follow this choice; the confidence intervals are always two-sided at
conf.level.- conf.level
A single numeric value in (0, 1) specifying the confidence level. Defaults to 0.95.
- tau
A single positive numeric value, the truncation time point for the average hazard. Both groups must have positive Kaplan-Meier survival at
tau.- presorted
A logical value. If
TRUE,time,event, andgroupare assumed to be sorted in ascending order oftime, and the internalorder()call is skipped. IfFALSE(default), sorting is handled internally.
Value
An object of class "ahsw_fast", a named numeric vector
containing the per-group average hazards (ah.ctrl, ah.trt),
the ratio contrast (rah, rah.lower, rah.upper,
p.rah), and the difference contrast (dah, dah.lower,
dah.upper, p.dah). The truncation time and confidence level
are stored as attributes tau and conf.level, and the
control label is also stored. Returns NA values (still with
class "ahsw_fast") when either group has zero survival at tau
or a non-finite variance.
Details
The average hazard with survival weight is
AH(tau) = (1 - S(tau)) / integral over [0, tau] of S(u) du,
the ratio of the cumulative event probability at tau to the restricted
mean survival time at tau. It can be read as a general censoring-free
incidence rate on the window from 0 to tau and stays interpretable
under non-proportional hazards. This is a different quantity from the average
hazard ratio of Kalbfleisch, which averages the time-varying ratio of hazards
rather than forming a single average hazard per group and then contrasting.
Writing the treatment and control average hazards as a1 and a0, the ratio contrast is RAH = a1 / a0, formed on the log scale with variance v_Q1 / n1 + v_Q0 / n0, and the difference contrast is DAH = a1 - a0, with variance v_U1 / n1 + v_U0 / n0, using the independence of the two groups. The per-group variance terms v_Q (log scale) and v_U (identity scale) follow the asymptotic variance of Uno and Horiguchi, computed from the Nelson-Aalen increments, the running restricted mean survival time and the at-risk fraction. The confidence interval for RAH is exponentiated from the log scale, and the two-sided p-values are based on the normal approximation.
When presorted = TRUE, the inputs are assumed to be sorted in
ascending order of time, so the internal order() call is
skipped. Splitting into groups preserves the ascending order within each
group.
References
Uno, H., & Horiguchi, M. (2023). Ratio and difference of average hazard with survival weight: new measures to quantify survival benefit of new therapy. Statistics in Medicine, 42(7), 936-952.
See also
rmst_fast for the restricted mean survival time, and
survfit_fast for the Kaplan-Meier estimate at a time point.
Examples
library(survival)
# Average hazard contrasts on the ovarian data
ahsw_fast(ovarian$futime, ovarian$fustat, ovarian$rx, control = 1, tau = 600)
#> Average hazard with survival weight (two-group)
#>
#> tau = 600, control = 1
#> alternative = two.sided
#>
#> AH
#> control 0.0011
#> treatment 0.0008
#>
#> Est. lower 95% upper 95% p.value
#> ratio (treatment / control) 0.7548735 0.2622750 2.1726579 0.602
#> difference (treatment - control) -0.0002646 -0.0012914 0.0007622 0.613
# presorted = TRUE: sort once outside, reuse inside a loop
ord <- order(ovarian$futime)
ahsw_fast(ovarian$futime[ord], ovarian$fustat[ord], ovarian$rx[ord],
control = 1, tau = 600, presorted = TRUE)
#> Average hazard with survival weight (two-group)
#>
#> tau = 600, control = 1
#> alternative = two.sided
#>
#> AH
#> control 0.0011
#> treatment 0.0008
#>
#> Est. lower 95% upper 95% p.value
#> ratio (treatment / control) 0.7548735 0.2622750 2.1726579 0.602
#> difference (treatment - control) -0.0002646 -0.0012914 0.0007622 0.613
# \donttest{
# Validation against survAH
if (requireNamespace("survAH", quietly = TRUE)) {
arm <- as.numeric(ovarian$rx == 2)
survAH::ah2(time = ovarian$futime, status = ovarian$fustat,
arm = arm, tau = 600)
}
#>
#> The time window: [eta, tau] = [0, 600] was specified. Warning: The normal approximation may be questionable with the specified tau. A smaller value of tau would be recommended for this data.
#>
#> Number of observations:
#> Total N Event by tau Censor by tau At risk at tau
#> arm0 13 6 2 5
#> arm1 13 5 2 6
#>
#>
#> Average Hazard (AH) by arm:
#> Est. Lower 0.95 Upper 0.95
#> AH (arm0) 0.001 0 0.002
#> AH (arm1) 0.001 0 0.002
#>
#>
#> Between-group contrast:
#> Est. Lower 0.95 Upper 0.95 P-value
#> Ratio of AH (arm1/arm0) 0.755 0.262 2.173 0.602
#> Difference of AH (arm1-arm0) 0.000 -0.001 0.001 0.613
# }