Figure 3
Suppose 240 active and 120 control patients are enrolled over the course of 14 months and followed for 11 months. Assume that accrual follows a truncated exponential distribution with shape parameter 0.1; loss to follow-up follows a Weibull distribution with shape parameter 2 and scale parameter \(log(1/0.99)^{1/2}/25\), so that 1% of patients are permanently censored over 25 months of follow-up; and survivals follow exponential distributions with 9 and 6 month medians. All these information are captured in the following “arm” objects:
# Active
arm1 <- create_arm(size=240,
accr_time=14,
accr_dist="truncexp",
accr_param=0.1,
surv_scale=per2haz(9),
loss_scale=log(1/0.99)^(1/2)/25,
loss_shape=2,
total_time=25)
# Control
arm0 <- create_arm(size=120,
accr_time=14,
accr_dist="truncexp",
accr_param=0.1,
surv_scale=per2haz(6),
loss_scale=log(1/0.99)^(1/2)/25,
loss_shape=2,
total_time=25)To visualize accrual and loss-to-follow-up:
# Accrual
tibble(
x = seq(0, 14, 0.1),
y = paccr(x, arm0)
) %>%
ggplot(aes(x, y)) +
geom_line() +
labs(x = "Time from first patient in (months)",
y = "Accrual CDF")
# Loss to follow-up
tibble(
x = seq(0, 25, 0.1),
y = ploss(x, arm0)
) %>%
ggplot(aes(x, y)) +
geom_line() +
labs(x = "Time from study entry (months)",
y = "Loss to follow-up CDF")
In the manuscript, we explore the impact on power when survival in the active arm follows instead a 2-piece exponential distribution, where the first piece overlaps with the control arm and the second piece deviates in such a way that median survival remains at 9 months. Denoting the changepoint by \(\tau_1\) and the arm-specific hazard rates by \(\lambda_0\) and \((\lambda_{11}, \lambda_{12})=(\lambda_0, \lambda_{12})\), simple algebra dictates that \(\lambda_{12}=\{\lambda: e^{-\lambda(9-\tau_1)}=\frac{0.5}{e^{-\lambda_0 \tau_1}}\}\). We can utilize functions in npsurvSS to calculate and visualize survival in the active arm under various changepoints:
# Calculate survival curves
x.vec <- seq(0, 25, 0.1) # vector of unique x-coordinates
tau1.vec <- seq(0, 4.5, 1.5) # vector of unique changepoints
arm1t <- arm1 # initialize active arm
y <- c() # vector or all y-coordinates
for (tau1 in tau1.vec) {
# Update scale and interval parameters for the active arm
arm1t$surv_scale <- c(arm0$surv_scale[1], per2haz(9-tau1, 1-0.5/psurv(tau1, arm0, lower.tail=F)))
arm1t$surv_interval <- c(0, tau1, Inf)
# Calculate y-coordinates
y <- c(y, psurv(x.vec, arm1t, lower.tail=F))
}
# Visualize survival curves
tibble(
tau1 = rep(tau1.vec, each=length(x.vec)),
x = rep(x.vec, length(tau1.vec)),
y = y
) %>%
ggplot(aes(x, y)) +
geom_line(aes(color=factor(tau1), lty=factor(tau1))) +
labs(x = "Time from study entry (months)",
y = "Survival function",
color = "Changepoint",
lty = "Changepoint")
