Compare Milestone Survival Probabilities Between Two Groups

Compares the Kaplan-Meier survival probabilities of two groups at a prespecified milestone timepoint. The point estimate of interest is the difference in milestone survival, treatment minus control. Three inference methods are provided. The "wald" method uses the unpooled Greenwood variance directly. The "loglog" and "mover" methods build the confidence interval for the difference with the method of variance estimates recovery (MOVER), recovering the variance from the one-sample complementary log-log and log transformed confidence intervals respectively. When the estimate of a group is 0 or 1 at tau, its one-sample interval degenerates to the estimate, so the interval of the difference is still given; the "loglog" test statistic is then NA. See Tang (2021) for the MOVER difference interval and Tang (2022) for the use of milestone survival in trial design.

Usage

milestone_fast(
  time,
  event,
  group,
  control,
  side = 2,
  conf.level = 0.95,
  tau,
  method = c("wald", "loglog", "mover"),
  presorted = FALSE,
  data = NULL
)

Arguments

time: A numeric vector of follow-up times. Alternatively, a formula Surv(time, event) ~ group, with the variables in data; see formula_interface.
event: An integer vector of event indicators, 1 for an event and 0 for a censored observation.
group: A vector with exactly two distinct values identifying the group.
control: The value of group that denotes the control group. The other value is the treatment group and the difference is reported as treatment minus control. With the formula interface it defaults to the first level of the group variable.
side: 1 for a one-sided test in the direction of treatment benefit (treatment milestone survival larger than control) or 2 for a two-sided test (default 2). The confidence interval is always reported as a two-sided interval at conf.level.
conf.level: The confidence level for the reported intervals.
tau: The milestone timepoint at which the survival probabilities are compared. A single positive number. A warning is given when it exceeds the largest observed time of either group, because the Kaplan-Meier curve is not estimated beyond that time.
method: The inference method for the difference in milestone survival, one of "wald", "loglog", or "mover".
presorted: Logical. If TRUE the input is assumed to be sorted by time in ascending order and the internal sort is skipped. This is intended for repeated calls inside simulation loops. The order is checked, and an error is given when it does not hold.
data: A data frame in which the variables of a formula given as time are evaluated. Used only with the formula interface; see formula_interface.

Value

An object of class "milestone_fast", a list with the per-group milestone survival estimates and standard errors, the difference estimate with its confidence interval, the test statistic, and the p-value. The object also has the class "fastsurv", whose as.data.frame(), coef(), and confint() methods are described in as.data.frame.fastsurv.

Details

The "wald" statistic is the difference divided by its Greenwood standard error. The "loglog" statistic is the difference of the complementary log-log transforms of the two estimates divided by its standard error, so it is negative when the treatment group has the higher milestone survival, and its one-sided p-value is the lower tail. The "mover" statistic is the upper (1 - conf.level) / 2 quantile of the standard normal distribution (1.96 for conf.level = 0.95) times the difference divided by the half-width of the MOVER interval on the side of zero. It is obtained by inverting the interval at conf.level, so the statistic and its p-value depend on conf.level, and the two-sided p-value is below 1 - conf.level exactly when the interval excludes zero. For "wald" and "mover" a positive statistic favors treatment and the one-sided p-value is the upper tail.

References

Tang, Y. (2021). Some new confidence intervals for Kaplan-Meier based estimators from one and two sample survival data. Statistics in Medicine, 40(23), 4961-4976.

Tang, Y. (2022). Complex survival trial design by the product integration method. Statistics in Medicine, 41(4), 798-814.

Examples

set.seed(1)
time <- c(rexp(50, 0.1), rexp(50, 0.07))
event <- rep(1, 100)
group <- rep(c(0, 1), each = 50)
milestone_fast(time, event, group, control = 0, tau = 10, method = "loglog")
#> Milestone survival (two-group)
#> 
#>   tau = 10,  control = 0
#>   method = loglog,  alternative = two.sided
#> 
#>           survival std.err lower 95% upper 95%
#> control       0.42  0.0698    0.2829    0.5510
#> treatment     0.56  0.0702    0.4124    0.6842
#> 
#>                                      Est. lower 95% upper 95%      z Pr(>|z|)
#> difference (treatment - control)  0.14000  -0.05738   0.32500 -1.395    0.163

# Formula interface
d <- data.frame(time, event, group)
milestone_fast(Surv(time, event) ~ group, data = d, tau = 10)
#> Milestone survival (two-group)
#> 
#>   tau = 10,  control = 0
#>   method = wald,  alternative = two.sided
#> 
#>           survival std.err lower 95% upper 95%
#> control       0.42  0.0698    0.2832    0.5568
#> treatment     0.56  0.0702    0.4224    0.6976
#> 
#>                                      Est. lower 95% upper 95%     z Pr(>|z|)
#> difference (treatment - control)  0.14000  -0.05403   0.33403 1.414    0.157