Two-Sample Comparison of Average Hazard

The ah2 function performs two-sample comparisons using the average hazard (AH) as a summary measure of the survival time distribution. Two kinds of between-group contrast metrics, the ratio of AH (RAH) and the difference in AH (DAH), are calculated.

Usage

ah2(time, status, arm, tau=NULL, conf.int=0.95, eta=0, strata=NULL)

Arguments

time: A numeric vector of follow-up times for right-censored data.
status: A numeric vector indicating the event status; 1 = event occurred, 0 = right censored.
arm: A binary vector indicating group assignment; elements should be either 0 or 1. Typically, 0 = control group and 1 = treatment group.
tau: A scalar specifying the end time point (tau) for calculating the average hazard. If tau = NULL, the default is the maximum time point at which the risk set size in both groups is at least 10.
conf.int: A numeric value specifying the confidence level for confidence intervals. The default is 0.95.
eta: A scalar specifying the start time point (eta) for calculating the average hazard over the time window [eta, tau]. The default is 0.
strata: An optional numeric vector specifying a stratification factor for stratified analysis. If strata = NULL (default), an unstratified analysis is performed.

Value

An object of class ah2, which contains the following components:

note: A note indicating the time window used in the analysis, specified as [eta, tau].
n.obs: A summary of the number of observations, including the total number, number of events by tau, number of censored observations by tau, and the size of the risk set at tau.
ah: Estimated average hazard in each arm.
rah: Ratio of average hazards (RAH), calculated as treatment over control.
dah: Difference of average hazards (DAH), calculated as treatment minus control.
conventional_rah: Ratio of average hazards based on the conventional stratified analysis method.
conventional_dah: Difference of average hazards based on the conventional stratified analysis method.
stratified_ah: Estimated average hazard by arm based on the proposed stratified analysis method.
stratified_rah: Ratio of average hazards based on the proposed stratified analysis method.
stratified_dah: Difference of average hazards based on the proposed stratified analysis method.

Details

The function provides the AH for each of the two groups, the absolute difference and the absolute ratio of AH (DAH and RAH) between the two groups, and the corresponding confidence intervals. It also calculates p-values for the two-sided tests based on the RAH and DAH.

References

Uno H and Horiguchi M. Ratio and difference of average hazard with survival weight: new measures to quantify survival benefit of new therapy. Statistics in Medicine. 2023;1-17. <doi:10.1002/sim.9651> Horiguchi M, Tian L, Kehl K.L., and Uno H. Assessing delayed treatment benefits of immunotherapy using long-term average hazard: a novel test/estimation approach. <arXiv:2403.10742> Qian Z, Tian L, Horiguchi M, and Uno H. A novel stratified analysis method for testing and estimating overall treatment effects on time-to-event outcomes using average hazard with survival weight. Statistics in Medicine. 2025. <arXiv:2404.00788>

Author

Hajime Uno, Miki Horiguchi, Zihan Qian

Examples

#====================================================================
# cm214_pfs: The sample reconstructed data of the CheckMate214 study.
#====================================================================
# The code below reproduces the results reported by
# Uno H and Horiguchi M (StatMed; 2023) in Table 6.
D      = cm214_pfs
time   = D$time
status = D$status
arm    = D$arm
tau    = 21

a = ah2(time=time, status=status, arm=arm, tau=tau, conf.int=0.95, eta=0)
print(a, digits=3)
#> 
#> The time window: [eta, tau] = [0, 21] was specified.
#> 
#> Number of observations: 
#>      Total N Event by tau Censor by tau At risk at tau
#> arm0     422          225           163             34
#> arm1     425          219           160             46
#> 
#> 
#> Average Hazard (AH) by arm: 
#>            Est. Lower 0.95 Upper 0.95
#> AH (arm0) 0.066      0.057      0.076
#> AH (arm1) 0.049      0.042      0.057
#> 
#> 
#> Between-group contrast: 
#>                                Est. Lower 0.95 Upper 0.95 P-value
#> Ratio of AH (arm1/arm0)       0.747      0.608      0.917   0.005
#> Difference of AH (arm1-arm0) -0.017     -0.029     -0.005   0.006


# The code below reproduces the results reported by
# Horiguchi M, Tian L, Kehl K.L, Uno H (arXiv; 2024) in Table 3.
b = ah2(time=time, status=status, arm=arm, tau=21, conf.int=0.95, eta=7)
print(b, digits=3)
#> 
#> The time window: [eta, tau] = [7, 21] was specified.
#> 
#> Number of observations: 
#>      Total N Event by tau Censor by tau At risk at tau
#> arm0     171           67            70             34
#> arm1     213           61           106             46
#> 
#> 
#> Average Hazard (AH) by arm: 
#>            Est. Lower 0.95 Upper 0.95
#> AH (arm0) 0.051      0.040      0.065
#> AH (arm1) 0.028      0.022      0.037
#> 
#> 
#> Between-group contrast: 
#>                                Est. Lower 0.95 Upper 0.95 P-value
#> Ratio of AH (arm1/arm0)       0.553      0.387      0.791   0.001
#> Difference of AH (arm1-arm0) -0.023     -0.037     -0.008   0.002