Dose-Ranging Study

In this vignette, we implement simulation for a dose-ranging trial under adaptive design. We start from a placebo arm and a treatment arm with highest possible dose. At an interim, if we observe promising results, we add arms with doses in between to inform decision at final analysis.

For the sake of simplicity, we assume a binary endpoint of interest. The response rates at doses 0 (placebo), 0.5, 1.5, 2.5, 4 (highest dose) are given by a sigEmax model below, where the response rate is assume to be 10% in placebo, and 25% in arm of the highest dose (dose = 4). For further background information, refer to vignette of the DoseFinding package.

Simulation Settings

In our simulation, we assume

the response rates of binary endpoint as

Dose	0	0.5	1.5	2.5	4
Response Rate (%)	10.0	11.2	20.8	24.1	25.0

readout time is 1 week.
no dropout.
starting with a placebo arm and a treatment arm of the highest dose (= 4), with a 1:1 randomization ratio.
- enroll 5 patients per week for 7 weeks.
an interim is planed when we have collected 30 readouts in total from the two arms.
- if z statistic of odds ratio is greater than 1
  - add three arms of dose 0.5, 1.5, and 2.5 to trial
  - enroll 20 patients per week with a randomization ratio 1:2:2:2:1, until in total 150 patients are enrolled in the trial
  - final analysis is performed when readouts are available for all enrolled patients.
- otherwise, stop the trial.

Define Placebo and High Dose Arms

ep <- endpoint(name = 'ep', type = 'non-tte', readout = c(ep = 1),
               generator = rbinom, size = 1, prob = 0.1)
pbo <- arm(name = 'dose = 0.0')
pbo$add_endpoints(ep)

ep <- endpoint(name = 'ep', type = 'non-tte', readout = c(ep = 1),
               generator = rbinom, size = 1, prob = 0.25)
trt4 <- arm(name = 'dose = 4.0')
trt4$add_endpoints(ep)

Define a Trial

accrual_rate <- data.frame(end_time = c(7, Inf),
                           piecewise_rate = c(5, 20))

trial <- trial(name = '123', n_patients = 150, duration = 14,
               enroller = StaggeredRecruiter, accrual_rate = accrual_rate, 
               silent = TRUE)

trial$add_arms(sample_ratio = c(1, 1), pbo, trt4)
trial
#>  ⚕⚕         Trial Name:  123  
#>  ⚕⚕        Description:  123  
#>  ⚕⚕     Number of Arms:  2  
#>  ⚕⚕    Registered Arms:  dose = 0.0, dose = 4.0  
#>  ⚕⚕       Sample Ratio:  1, 1  
#>  ⚕⚕ Number of Patients:  150  
#>  ⚕⚕   Planned Duration:  14  
#>  ⚕⚕            Regimen:  not set  
#>  ⚕⚕        Random Seed:  1960520344

Define Milestones and Actions

Two milestones are triggered when 30 and 150 readouts are collected, respectively. Note that the 30 readouts are from the placebo and the highest dose arm, while the 150 readouts are from all five arms.

interim <- milestone(name = 'interim',
                     when = eventNumber(endpoint = 'ep', n = 30),
                     action = action_at_interim)

final <- milestone(name = 'final',
                   when = eventNumber(endpoint = 'ep', n = 150),
                   action = action_at_final)

In the action function action_at_interim(), we don’t actually terminate the trial in simulation but always add dose arms. Instead, we save the z statistic for summary later, e.g., computing the proportion of early termination.

action_at_interim <- function(trial){

  ## get data snapshot
  locked_data <- trial$get_locked_data('interim')

  ## compare two arms
  ## Risk difference = response rate in high dose - response rate in placebo
  fit <- fitLogistic(ep ~ arm, placebo = 'dose = 0.0',
                     data = locked_data, alternative = 'greater',
                     scale = 'risk difference')

  ## for summary of early termination
  trial$save(value = fit$z, name = 'z_value')
  trial$save(value = ifelse(fit$z > 1.64, 'add dose arms', 'stop trial'), 
             name = 'interim_decision')

  ## create three dose arms
  ep <- endpoint(name = 'ep', type = 'non-tte', readout = c(ep = 1),
                 generator = rbinom, size = 1, prob = .112)
  trt1 <- arm(name = 'dose = 0.5')
  trt1$add_endpoints(ep)

  ep <- endpoint(name = 'ep', type = 'non-tte', readout = c(ep = 1),
                 generator = rbinom, size = 1, prob = .208)
  trt2 <- arm(name = 'dose = 1.5')
  trt2$add_endpoints(ep)

  ep <- endpoint(name = 'ep', type = 'non-tte', readout = c(ep = 1),
                 generator = rbinom, size = 1, prob = .241)
  trt3 <- arm(name = 'dose = 2.5')
  trt3$add_endpoints(ep)

  ## add three new arms to trial
  trial$add_arms(sample_ratio = c(2, 2, 2), trt1, trt2, trt3)
  
}

At final analysis, we analyze data from all five arms using MCPMod. It calls a helper function go_nogo() which can be found in the Appendix below.

action_at_final <- function(trial){

  locked_data <- trial$get_locked_data('final')

  trial$save(value = go_nogo(locked_data), name = 'decision')

}

Execute a Trial

After registering the two milestones with a listener, we simulate a single trial using controller$run(). We can see that more patients are randomized to the three newly added arms after interim, reflecting the new randomization ratio 1:2:2:2:1.

listener <- listener()
listener$add_milestones(interim, final)

controller <- controller(trial, listener)
controller$run(n = 1, plot_event = TRUE)

In this trial, the z statistic at interim is relatively low (0.93), thus is early terminated even if in simulation we keep going with three newly added dose arms.

output <- controller$get_output()

output %>% 
  kable(escape = FALSE) %>% 
  kable_styling(bootstrap_options = "striped", 
                full_width = FALSE,
                position = "left") %>%
  scroll_box(width = "100%")

trial	seed	milestone_time_<interim>	n_events_<interim>_<ep>	n_events_<interim>_<patient_id>	n_events_<interim>_<arms>	z_value	interim_decision	milestone_time_<final>	n_events_<final>_<ep>	n_events_<final>_<patient_id>	n_events_<final>_<arms>	decision	error_message
123	1960520344	6.8	30	35	c(“dose ….	0.9258209	stop trial	13.7	150	150	c(“dose ….	no-go

We can call controller$run(n = 10) to simulate additional trials. in the function action_at_final(), the go/no-go decision (decision in output) is determined under the assumption that the dose has been expanded, regardless of the interim evidence—this simplification is made solely for the convenience of simulation. However, the overall probability of a "go" should incorporate the interim results—specifically, the trial must not be stopped at interim (i.e., interim_decision == "add dose arms"), and the function go_nogo() must return "go" at the final analysis.

## important: reset before calling run() again
controller$reset()
controller$run(n = 10, plot_event = FALSE, silent = TRUE)
res <- controller$get_output()

res %>% 
  kable(escape = FALSE) %>% 
  kable_styling(bootstrap_options = "striped", 
                full_width = FALSE,
                position = "left") %>%
  scroll_box(width = "100%")

trial	seed	milestone_time_<interim>	n_events_<interim>_<ep>	n_events_<interim>_<patient_id>	n_events_<interim>_<arms>	z_value	interim_decision	milestone_time_<final>	n_events_<final>_<ep>	n_events_<final>_<patient_id>	n_events_<final>_<arms>	decision
123	1915467752	6.8	30	35	c(“dose ….	0.6123731	stop trial	13.7	150	150	c(“dose ….	go
123	1678009487	6.8	30	35	c(“dose ….	0.9258209	stop trial	13.7	150	150	c(“dose ….	no-go
123	1177293210	6.8	30	35	c(“dose ….	0.4918697	stop trial	13.7	150	150	c(“dose ….	no-go
123	155620919	6.8	30	35	c(“dose ….	1.3327856	stop trial	13.7	150	150	c(“dose ….	go
123	1745672662	6.8	30	35	c(“dose ….	1.2247450	stop trial	13.7	150	150	c(“dose ….	go
123	484372378	6.8	30	35	c(“dose ….	1.9364930	add dose arms	13.7	150	150	c(“dose ….	go
123	2788507	6.8	30	35	c(“dose ….	0.4330129	stop trial	13.7	150	150	c(“dose ….	no-go
123	1945408685	6.8	30	35	c(“dose ….	0.4918697	stop trial	13.7	150	150	c(“dose ….	no-go
123	737764402	6.8	30	35	c(“dose ….	0.9258209	stop trial	13.7	150	150	c(“dose ….	go
123	787086561	6.8	30	35	c(“dose ….	1.0954461	stop trial	13.7	150	150	c(“dose ….	go

## Number of overall Go
sum(res$interim_decision == 'add dose arms' & 'decision' == 'go')
#> [1] 0

## Number of early trial termination
sum(res$interim_decision == 'stop trial')
#> [1] 9

Appendix: Code of Helper Function

For completeness, the full code of the helper functions go_nogo() is included below, which informs go/no-go decision at the end of the trial.

go_nogo <- function(data){

  ## candidate models for MCPMod
  doses <- c(0, 0.5, 1.5, 2.5, 4)
  candidates <- Mods(emax = c(.25, 1),
                     sigEmax = rbind(c(1, 3), c(2.5, 4)),
                     betaMod = c(1.1, 1.1),
                     placEff = log(.1/(1 - .1)),
                     maxEff = log(.25/(1 - .25)) - log(.1/(1 - .1)),
                     doses = doses)

  fit <- glm(ep ~ factor(arm) + 0, data = data, family = binomial)
  mu_hat <- coef(fit)
  S_hat <- vcov(fit)

  ## multiple contrast test
  test <- DoseFinding::MCTtest(dose = doses,
                               mu_hat, S = S_hat,
                               models = candidates, type = "general")

  ## model averaging
  model <- DoseFinding::maFitMod(dose = doses,
                                 mu_hat, S = S_hat,
                                 models = c("emax", "sigEmax", "betaMod"))
  
  ## predict response rate per dose
  prd <- predict(model, summaryFct = median, doseSeq = doses)
  
  ## convert to scale of probability
  prd_rate <- 1 / (1 + exp(-prd))

  ## go/no-go rule: MCP test p-value < 0.05 and estimated effect > 10%
  ifelse(min(attr(test$tStat, 'pVal')) < .05 &
           max(prd_rate - prd_rate[1]) > .1, 'go', 'no-go')

}