Long-Term Restricted Mean Survival Times with Adaptively Selected Lower End of Time Window

This function performs the method proposed by Horiguchi, Tian, and Uno. (2023) <doi:10.1002/sim.9662>. Specifically, it estimates the restricted mean survival time (RMST) within a prespecified time window (from tau1 to tau2) to quantify the long-term treatment benefit. Instead of choosing one specific time point as the lower end of the time window (tau1), the procedure allows users to prespecify a set of time points. The procedure picks one time point among the set of tau1 values that gives the most significant difference in the long-term RMST between the two groups. It then performs testing for equality of the long-term RMSTs between the two groups and estimates the difference in RMST within the adaptively selected time window [tau1, tau2] between the two groups. Multiplicity as a result of specifying several values for tau1 is taken into account in this procedure.

Usage

ltrmst2adapt(indata, tau1, tau2, iteration=50000, seed=NULL, test="2_side", conf.int=0.95)

Arguments

indata: A data matrix (data frame). The 1st column is the time-to-event variable, the 2nd column is the event indicator (1=event, 0=censor), and the 3rd column is the treatment indicator (1=treatment, 0=control). No missing values are allowed in this data matrix.
tau1: An integer value or a vector indicating a set of tau1 values for the lower end of the time window.
tau2: An integer value indicating the upper end of the time window. When tau2 = NULL, the default value is used. See Details for the definition of the default value for tau2.
iteration: A number of iterations for the resampling (the multivariate normal distribution-based perturbation method). It is recommended to specify at least 50000 (default) or larger.
seed: An integer value used for random number generation in the resampling procedure. Default is NULL.
test: Specify "1_side" for the one-sided test where the alternative hypothesis is that the treatment group is superior to the control group with respect to survival time. Specify "2_side" for the two-sided test where the alternative hypothesis is that the treatment group is not equal to the control group with respect to survival time. Default is "2_side".
conf.int: Specify a confidence coefficient for calculating confidence bands for the differences in long-term RMST. Default is 0.95.

Value

an object of class ltrmst2adapt.

iteration: The number of iterations for resampling from a multivariate normal distribution with a mean 0 and a variance-covariance matrix.
test: The type of test used in the analyses
arm1: The RMST [tau1, tau2] estimation for arm1
arm0: The RMST [tau1, tau2] estimation for arm0
diff10: The difference in RMST [tau1, tau2] between the two groups (arm1 minus arm0)
diff10_selected: The difference in RMST within the selected time window [tau1, tau2] between the two groups (arm1 minus arm0) with the normal confidence interval considering the randomness of selecting one time window. The p-value for the RMST difference test within the selected time window [tau1, tau2] is also provided.

Details

The definition of the default value for tau2. Let x1 and x0 be the maximum observed time in Group 1 and Group 0, respectively. Case 1: If the last observations in Group 1 and Group 0 are "event," then tau = max(x1, x0). Case 2-1: If the last observation in Group 1 is "event," the last observation in Group 0 is "censor," and x1 <= x0, tau2 = max(x1, x0) = x0. Case 2-2: If the last observation in Group 0 is "event," the last observation in Group 1 is "censor," and x1 > x0, tau2 = max(x1, x0) = x1. Case 3-1: If the last observation in Group 1 is "event," the last observation in Group 0 is "censor," and x1 > x0, tau2 = min(x1, x0) = x0. Case 3-2: If the last observation in Group 0 is "event," the last observation in Group 1 is "censor," and x1 <= x0, tau2 = min(x1, x0) = x1. Case 4: If the last observations in Group 1 and Group 0 are "censor," then tau = min(x1, x0).

References

Horiguchi M, Tian L, Uno H. On assessing survival benefit of immunotherapy using long-term restricted mean survival time. Statistics in Medicine 2023. DOI:10.1002/sim.9662.

Author

Miki Horiguchi, Lu Tian, Hajime Uno

Examples

#--- sample data ---#
data = cm214_pfs
b    = ltrmst2adapt(indata=data, tau1=c(0,1,2,3), tau2=10, 
               test="2_side", seed=123)
print(b)
#> 
#> RMST over the time window [3,10] 
#> 
#> RMST(arm1) RMST(arm0) 
#>      4.415      4.191 
#> 
#> RMST(arm1-arm0)      lower 0.95      upper 0.95         p_value 
#>           0.224          -0.218           0.666           0.332 
#> 
#>