Exponential time-to-event endpoint • BATSS

Context

The case study is motivated by the rEECur trial, a seamless phase II/III MAMS trial that investigated the optimal systemic anticancer regimen for recurrent and refractory Ewing sarcoma (McCabe et al. (2019)). As opposed to the original trial, this is designed as a superiority trial where we are interested in demonstrating efficacy of each intervention arm compared to a standard of care. No head-to-head comparisons of interventions are made.

The primary endpoint is event-free survival time (EFS) - the time from randomisation to the first failure event (progression, recurrence, diagnosis of a second malignancy, or death) and is assumed to follow an exponential distribution. In the original trial the 1-year EFS rate was considered as an endpoint, here we look at a time-to-event endpoint.

Design

The chosen design has the following characteristics:

treatment arms: This case study has 4 treatment arms: standard of care (arm “control”), three investigational/novel drug regimen (referred to as arms “A”, “B”, and “C”).
alternative hypotheses: This is a superiority trial where we are interested in demonstrating efficacy compared to the standard of care arm and intervention arms will not be compared to one another.
interim analyses and maximum sample size: We will conduct an interim analyses after 1 year of recruitment and every 1 year (12 months) thereafter until year 6. We will conduct a final analysis after the last participant has completed follow-up if no decision to stop the trial has been made earlier. Our maximum sample size is 800 participants (across all tresatments).
participant accrual and censoring: We assume exponentially distributed time intervals between participants’ enrolment into the study (i.e. a Poisson process) with a rate of 200 participants per year. This will result in an average enrollment period of 4 years. In a second scenario, we will look at a yearly-changing exponential recruitment rate with a slower start. Further, we will follow-up participants until the last participant experienced an event or 5 years of follow-up after the last participant is recruited. No participant attrition will be considered for this example (i.e. no loss-to-follow-up).
endpoint conditional distribution: We will use an exponential model for the data generating process and a Cox proportional hazards model for the analyses.
group allocation: we will consider equal allocation probabilities per group (no response-adaptive randomisation).
efficacy stopping rule: Early stopping of intervention arms for efficacy may occur if there is a high posterior probability of (any) benefit (i.e. $\mathrm{HR}<1$ ) at look $j$ , i.e., when $\begin{equation} \mathrm{Prob}(\beta_k < \delta_e|{y,X}) > b_{e,i} \end{equation}$ where $\beta_k$ denotes the linear predictor of the $k$ th target parameter, $\delta_e = 0$ denotes the (efficacy-related) clinically meaningful treatment effect and $b_{e,i}$ the cut-off values used to declare futility (here, posterior probabilities). We will be using a stricter threshold $b_{e,1}$ during interim analyses and a more lenient value of $b_{e,2}$ during the final analysis.
futility stopping rule: Early stopping of intervention arms for futility may occur if there is a low posterior probability of seeing any difference (i.e., $\mathrm{HR}<1$ ), i.e., when $\begin{equation} \mathrm{Prob}(\beta_k < \delta_f|{y,X}) < b_f \end{equation}$ where $\delta_f=0$ denotes the clinically meaningful treatment effect and $b_f$ is the cut-off value to declare futility.
trial stopping rule: The trial will run until an efficacy decision has been reached for any intervention arm, a futility decision has been reached for each intervention arm or once the maximum sample size has been reached and follow-up period ended.

Self-defined R functions

In the following, we define the group allocation, accrual, efficacy and futility functions corresponding to the design described above.

Group allocation

We will use the function alloc.balanced from the BATSS package, which first allocates the largest possible number of units to the different groups given their exact target probabilities and then assigns randomly the remaining units to the different groups according to multinomial draws.

Participant accrual

In the standard scenario with constant accrual we can just use the base R function rexp to get the exponentially distributed intervals, i.e. accr = rexp, accr.control = list(rate = 200).

For the scenario with a changing accrual rate, we need to generate a function that allows us to change the rates of an exponential distribution after certain time intervals. For simplicity, we will draw from a exponential distribution with rate $\lambda_1$ and change the rates for the first interval after the cumulative sum of previously drawn random numbers exceeds the threshold value. The function batss.surv() will expect an input value n indicating the number of generated numbers as the first element of the function (as for any R base random number generating function). The additional parameters can be passed on by using the accr.control argument in batss.surv, in this example accr.control = list(rates = c(100, 180, 260), changes.at = c(1, 2)).

# function
changing.accrual <- function(n,rates,changes.at){
    i <- 1
    res <- rexp(1,rate=rates[i])
    pos <- 2
    while (i <= length(changes.at)) {
      while(sum(res)<changes.at[i]) {
        res[pos] <- rexp(1,rate=rates[i])
        pos <- pos+1
      }
      i <- i+1
    }
    res <- c(res,rexp(n-(pos-1),rate=rates[i]))
    return(res)
} 

# test
changing.accrual(20, rates = c(5,20), changes.at = 1)

Arm efficacy stopping rule

We need to generate a function that leads to a logical output and takes as input

the ingredients
- posterior for the posterior probability of the target parameter being smaller than delta.eff = 0,
- curr.look and n.look, respectively the number of the current interim analysis and the total number of interim analyses,
the additional parameters (to be added to eff.arm.control in batss.surv)
- $b_e$ that we will name b.

# function
efficacy.fun <- function(posterior, b, curr.look, n.look){
  if (curr.look != n.look) {
    posterior > b[1]
  } else {
    posterior > b[2]
  }
} 

# test
efficacy.fun(0.85, b = c(0.99, 0.95), curr.look = 2, n.look = 4)

Trial efficacy stopping rule

We will use the function eff.trial.any from the BATSS package, which will stop a trial if any target parameter reached efficacy (indicated by the ingredient eff.target).

Arm futility stopping rule

We can select the function fut.arm.simple from the package BATSS to compare the ingredient posterior, the posterior probability of the target parameter being smaller than delta.fut = 0, to the threshold value b = 0.05.

Trial futility stopping

We will stop the trial for futility if all target parameters were declared ineffective. This corresponds to the (default) function fut.trial.all available in the BATSS package.

Monte Carlo Simulations

We will consider two scenarios for both the constant recruitment rate and the changing recruitment rate:

Scenario 1 = ‘global null’: the hazard ratio is equal to 1 for each intervention arm compared to the control.
Scenario 2 = ‘one treatment works’: arms “B” and “C” have a hazard ratio of 1 compared to the control while the hazard ratio for treatment “A” equals 0.75. This corresponds with an increase of the 1-year EFS rate from 0.2 to 0.3 (with our parametrisation of the data generating exponential distribution).

Scenario 1 & 2

# number of trials
R <- 25

# simulation
rEECur.sim <- batss.surv(
  model           = inla.surv(time, status) ~ trt,
  family          = "coxph",
  surv            = simsurv,
  surv.control    = list(lambdas = -log(0.2), gammas = 1,
                         maxt = NULL),
  fup             = 5,
  var             = list(trt = alloc.balanced),
  accr            = rexp,
  accr.control    = list(rate = 200),
  accr.type       = "random",
  prob0           = c(control = 1, A = 1, B = 1, C = 1),
  hr              = c(0.75, 1, 1),
  which           = 1:3,
  alternative     = "less",
  RAR             = NULL,
  interim         = list(time = 1:6),
  eff.arm         = efficacy.fun,
  eff.arm.control = list(b = c(0.99, 0.95)),
  eff.trial       = eff.trial.any,
  delta.eff       = 0,
  fut.arm         = fut.arm.simple,
  fut.arm.control = list(b = 0.05),
  N               = 800,
  R               = R,
  H0              = TRUE,
  extended        = 1,
  computation     = "parallel",
  mc.cores        = parallel::detectCores() - 1,
  control.inla    = list(cmin = 0)
)

You can note

The tuple (time, status) is generated via the simsurv function from the simsurv package. Arguments are passed via the surv.control argument. The default distribution in simsurv is Weibull. We set $\lambda = \mathrm{log}(0.2)$ to get a 1-year EFS rate of 0.2.
the family argument is linked to the analytical model but shares the linear predictor with the data generating model,
participant accrual is set in accr, accr.control and accr.type,
the censoring of participants is done by a combination of maxt = NULL in surv.control (no individual administrative censoring) and fup = 5 (the maximum follow-up time after recruitment of the last participant - ignored if maxt is set),
prob0 provides
- the (equal) allocation probabilities at the start of the trial,
- the names of the groups,
the interim analyses schedule is set to time-based, i.e. interim = list(time = 1:6) for interim analyses after 1, 2, 3,.. , 6 years, respectively, (See the vignette “Definition of the interim schedule in BATSS” to learn how to change this.)
eff.arm is set to the function defined above (i.e., efficacy.fun) and the cut-off values for declaring an intervention efficacious are set to a probability of 0.99 during the trial and 0.95 at the final analysis in eff.arm.control.
fut.arm is set to the standard option fut.arm.simple with additional parameters specified in fut.arm.control,
eff.trial is set to eff.trial.any as discussed above,
fut.trial is not specified and therefore equal to NULL (default) which leads to the desired behaviour in this case,
delta.fut is set to 0 by default after setting delta.eff = 0,
H0 = TRUE will additionally calculate the null hypotheses of all hazard ratios are equal to 1,
control.inla passes information to the INLA routine via ..., setting cmin = 0 will stabilize the numerical optimization by setting the minimum value for the negative Hessian, but may bias estimates!

Attention: The number of repetitions is set to R = 25 for demonstration purposes only.

Scenario 3 & 4 - changing accrual rates

# number of trials
R <- 25

# simulation
rEECur.sim.change <- batss.surv(
  model           = inla.surv(time, status) ~ trt,
  family          = "coxph",
  surv            = simsurv,
  surv.control    = list(lambdas = -log(0.2), gammas = 1,
                         maxt = NULL),
  fup             = 5,
  var             = list(trt = alloc.balanced),
  accr            = changing.accrual,
  accr.control    = list(rates = c(100, 180, 260), changes.at = c(1, 2)),
  accr.type       = "random",
  prob0           = c(control = 1, A = 1, B = 1, C = 1),
  hr              = c(0.75, 1, 1),
  which           = 1:3,
  alternative     = "less",
  RAR             = NULL,
  interim         = list(time = 1:6),
  eff.arm         = efficacy.fun,
  eff.arm.control = list(b = c(0.99, 0.95)),
  eff.trial       = eff.trial.any,
  delta.eff       = 0,
  fut.arm         = fut.arm.simple,
  fut.arm.control = list(b = 0.05),
  N               = 800,
  R               = R,
  H0              = TRUE,
  extended        = 1,
  computation     = "parallel",
  mc.cores        = parallel::detectCores() - 1,
  control.inla    = list(cmin = 0)
)

Same as above except for accr and accr.control.

References

McCabe, Martin G., Veronica Moroz, Maria Khan, Uta Dirksen, Abigail Evans, Nicola Fenwick, Nathalie Gaspar, et al. 2019. “Results of the first interim assessment of rEECur, an international randomized controlled trial of chemotherapy for the treatment of recurrent and primary refractory Ewing sarcoma.” Journal of Clinical Oncology 37 (15_suppl): 11007–7. https://doi.org/10.1200/JCO.2019.37.15\_suppl.11007.