Example 1: Optimal randomization ratio

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Godwin Yung

2024-05-08

We provide here code to replicate Figures 1C and 2 from Example 1 of our manuscript. We rely on R packages tidyverse and ggplot2 for cleaner coding and advanced visualizations.

Preliminaries

As described in Web Appendix C, we define ratio of events to patients, control median survival, and accrual duration as increasing functions of HR:

fun_hr <- function(HR, k, range.min, range.max) {
  (HR-0.3) / (0.9-0.3) * exp(k*(HR-0.3)) / exp(k*(0.9-0.3)) * (range.max-range.min) + range.min
}

The actual assumptions are visualized below:

HR.vec <- seq(0.3, 0.9, 0.01)
plot(HR.vec, fun_hr(HR=HR.vec, k=5, range.min=0.6, range.max=0.7),
     xlab="Hazard ratio",
     ylab="d/n ratio",
     type="l")
plot(HR.vec, fun_hr(HR=HR.vec, k=0, range.min=6, range.max=24),
     xlab="Hazard ratio",
     ylab="Control median survival (months)",
     type="l")
plot(HR.vec, fun_hr(HR=HR.vec, k=2.3, range.min=12, range.max=48),
     xlab="Hazard ratio",
     ylab="Accrual duration (months)",
     type="l")

Figure 1C

To calculate power across a vector of hazard ratios (HR.vec) and randomization ratio (p1.vec), we begin by initializing the output table. Note that, in addition to keeping track of the power, the table also keeps track of the number of expected events contributed by each arm. Also, for the sake of efficency, the range of scenarios covered in this exercise will be less than that in the manuscript. The resulting figure will therefore be coarser:

p1.vec <- seq(0.5, 0.7, 0.05)
optimal <- tibble(
  p1 = rep(p1.vec, each=length(HR.vec)),
  HR = rep(HR.vec, length(p1.vec)),
  accr_time = fun_hr(HR=HR, k=2.3, range.min=12, range.max=48),
  d = (qnorm(0.975)+qnorm(0.9))^2/0.5/0.5/log(HR)^2, # schoenfeld 1981
  n = d / fun_hr(HR, k=5, range.min=0.6, range.max=0.7),
  m = fun_hr(HR=HR, k=0, range.min=6, range.max=24),
  total_time = 0,
  power=0,
  events0=0,
  events1=0
)

We then fill in the table using the functions power_two_arm and exp_events described in the vignette basic_functionalities:

R   <- dim(optimal)[1]
for (r in 1:R) {
  
  # scenario specific parameters
  p1        <- optimal$p1[r]
  HR        <- optimal$HR[r]
  accr_time <- optimal$accr_time[r]
  d         <- optimal$d[r]
  n         <- optimal$n[r]
  m         <- optimal$m[r]
  
  # create arm objects
  arm0 <- create_arm(size=n*(1-p1),
                     accr_time=accr_time,
                     accr_interval=accr_time*c(0,0.25,0.5,1), # piecewise-uniform accrual
                     accr_param=c(0.05,0.25,0.7),
                     surv_scale=per2haz(m),
                     loss_scale=per2haz(m)*0.05,
                     follow_time=12)
  arm1 <- create_arm(size=n*p1,
                     accr_time=accr_time,
                     accr_interval=accr_time*c(0,0.25,0.5,1),
                     accr_param=c(0.05,0.25,0.7),
                     surv_scale=per2haz(m/HR),
                     loss_scale=arm0$loss_scale,
                     follow_time=12)
  
  # update total_time and follow_time
  duration                <- exp_duration(arm0, arm1, d=d)
  arm0$total_time         <- duration
  arm0$follow_time        <- duration - arm0$accr_time
  arm1$total_time         <- duration
  arm1$follow_time        <- duration - arm1$accr_time
  
  # record results
  optimal$total_time[r]   <- duration
  optimal$power[r]        <- power_two_arm(arm0, arm1)
  optimal$events0[r]      <- exp_events(arm0)
  optimal$events1[r]      <- exp_events(arm1)
  
}

head(optimal, 5)
#> # A tibble: 5 × 10
#>      p1    HR accr_time     d     n     m total_time power events0 events1
#>   <dbl> <dbl>     <dbl> <dbl> <dbl> <dbl>      <dbl> <dbl>   <dbl>   <dbl>
#> 1   0.5  0.3       12    29.0  48.3   6         23.1 0.888    19.4    9.63
#> 2   0.5  0.31      12.2  30.6  51.1   6.3       23.7 0.889    20.3   10.3 
#> 3   0.5  0.32      12.3  32.4  53.9   6.6       24.3 0.889    21.4   11.0 
#> 4   0.5  0.33      12.5  34.2  57.0   6.9       24.9 0.890    22.4   11.8 
#> 5   0.5  0.34      12.7  36.1  60.1   7.2       25.4 0.891    23.6   12.6

The following code produces Figure 1C:

group_by(optimal, HR) %>%
  filter(power==max(power)) %>% # identify p1 that maximizes power
  mutate(eprop=events1/d) %>%
  select(HR, p1, eprop) %>%
  gather(category, prop, 2:3) %>%
  mutate(category=ifelse(category=="p1", "Patients", "Events")) %>%
  ggplot(aes(x=HR,y=prop)) +
  geom_line(aes(col=category, lty=category), lwd=0.8) +
  labs(x="Hazard ratio", 
       y="Proportion contributed by active arm",
       col="",
       lty="")

Figure 2

Figure 2 can be produced via similar steps. Again, for sake of efficiency, a coarser grid of HRs will be considered here. Also, the empirical power based on simulations will not be calculated. First, we initialize the output table:

p1.vec <- c(0.5, 3/5, 2/3)
HR.vec <- seq(0.3, 0.9, 0.05)
fixed <- tibble(
  p1 = rep(p1.vec, each=length(HR.vec)),
  HR = rep(HR.vec, length(p1.vec)),
  accr_time = fun_hr(HR=HR, k=2.3, range.min=12, range.max=48),
  d = (qnorm(0.975)+qnorm(0.9))^2/0.5/0.5/log(HR)^2, # schoenfeld 1981
  n = d / fun_hr(HR, k=5, range.min=0.6, range.max=0.7),
  m = fun_hr(HR=HR, k=0, range.min=6, range.max=24),
  ed = 0, # power, schoenfeld
  mu = 0, # power, recommended asymptotic approximation
  mu_b = 0, # power, block randomization
  mu_s = 0  # power, simple randomization
)

Then we populate the table:

R   <- dim(fixed)[1]
for (r in 1:R) {
  
  # scenario specific parameters
  p1        <- fixed$p1[r]
  HR        <- fixed$HR[r]
  accr_time <- fixed$accr_time[r]
  d         <- fixed$d[r]
  n         <- fixed$n[r]
  m         <- fixed$m[r]
  
  # create arm objects
  arm0 <- create_arm(size=n*(1-p1),
                     accr_time=accr_time,
                     accr_interval=accr_time*c(0,0.25,0.5,1), # piecewise-uniform accrual
                     accr_param=c(0.05,0.25,0.7),
                     surv_scale=per2haz(m),
                     loss_scale=per2haz(m)*0.05,
                     follow_time=12)
  arm1 <- create_arm(size=n*p1,
                     accr_time=accr_time,
                     accr_interval=accr_time*c(0,0.25,0.5,1),
                     accr_param=c(0.05,0.25,0.7),
                     surv_scale=per2haz(m/HR),
                     loss_scale=arm0$loss_scale,
                     follow_time=12)
  
  # update total_time and follow_time
  duration                <- exp_duration(arm0, arm1, d=d)
  arm0$total_time         <- duration
  arm0$follow_time        <- duration - arm0$accr_time
  arm1$total_time         <- duration
  arm1$follow_time        <- duration - arm1$accr_time
  
  # record results
  fixed$ed[r]       <- power_two_arm(arm0, arm1, 
                                     test=list(test="weighted logrank", 
                                               mean.approx="event driven"))
  fixed$mu[r]       <- power_two_arm(arm0, arm1)
  fixed$mu_b[r]     <- power_two_arm(arm0, arm1, 
                                     test=list(test="weighted logrank", 
                                               var.approx="block"))
  fixed$mu_s[r]     <- power_two_arm(arm0, arm1, 
                                     test=list(test="weighted logrank", 
                                               var.approx="simple"))
  
}

The following code produces Figure 2:

gather(fixed, key="approximation", value="power", 7:10) %>%
  ggplot(aes(x=HR, y=power)) +
  geom_line(aes(col=approximation, lty=approximation)) +
  facet_wrap(~round(p1,2)) +
  labs(x="Hazard ratio",
       y="Power",
       col="",
       lty="")

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They may not be fully stable and should be used with caution. We make no claims about them.