# 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.

    mods <- DoseFinding::Mods(sigEmax = rbind(c(1, 3)),
                              placEff = log(.1/(1 - .1)),
                              maxEff = log(.25/(1 - .25)) - log(.1/(1 - .1)),
                              doses = c(0, 0.5, 1.5, 2.5, 4))

    DoseFinding::plotMods(mods, trafo = function(x) 1/(1+exp(-x)))

    ## response rates on curve
    x <- DoseFinding::getResp(mods, doses = c(0, 0.5, 1.5, 2.5, 4))
    1 / (1 + exp(-unclass(x)))
    #>      sigEmax1
    #> 0   0.1000000
    #> 0.5 0.1117242
    #> 1.5 0.2080893
    #> 2.5 0.2407520
    #> 4   0.2500000
    #> attr(,"parList")
    #> attr(,"parList")$sigEmax1
    #>        e0      eMax      ed50         h
    #> -2.197225  1.115778  1.000000  3.000000

## 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 | error_message
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
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')

    }