# Trial Design: basic sanity checks

> The `tidy()` methods used in this vignette follow the pattern established by the `broom` package and turn objects of various classes into tibbles. Currently, they are defined in a code chunk within this tibble that is not displayed, but they will be added to the main `crmPack` package at a later date.

## Introduction

This vignette contains an example of a very basic and superficial examination of the operating characteristics of a basic CRM trial. The design uses a 2-parameter logistic regression model with a log normal prior distribution, and custom rules for choosing the cohort size, deciding whether to stop or continue the trial, defining the maximum permitted increment and selecting the dose for the nest study.

The first step is to define the trial design in `crmPack`…

## Study definition

    library(crmPack)

    # Define the dose grid and an empty data object
    dose_grid <- c(1, 3, 5, 10, 15, 20, 25, 40, 50, 80, 100)
    empty_data <- Data(doseGrid = dose_grid)

    # Initialize the CRM model.
    initial_model <- LogisticLogNormal(
      mean = c(-0.85, 1),
      cov = matrix(c(1, -0.5, -0.5, 1), nrow = 2),
      ref_dose = 56
    )

    # Choose the rule for selecting the next dose.
    next_best <- NextBestNCRM(
      target = c(0.2, 0.35),
      overdose = c(0.35, 1),
      max_overdose_prob = 0.25
    )

    # Choose the rule for the cohort size.
    cohort_size1 <- CohortSizeRange(
      intervals = c(0, 30),
      cohort_size = c(1, 3)
    )
    cohort_size2 <- CohortSizeDLT(
      intervals = c(0, 1),
      cohort_size = c(1, 3)
    )
    cohort_size <- maxSize(cohort_size1, cohort_size2)

    # Choose the rule for stopping.
    stopping_success1 <- StoppingMinCohorts(nCohorts = 3)
    stopping_success2 <- StoppingTargetProb(
      target = c(0.2, 0.35),
      prob = 0.5
    )
    stopping_futility <- StoppingMinPatients(nPatients = 20)
    stopping_trial <- (stopping_success1 & stopping_success2) | stopping_futility

    # Choose the rule for dose increments.
    increment_rule <- IncrementsRelative(
      intervals = c(0, 20),
      increments = c(1, 0.33)
    )

    # Initialize the design.
    design <- Design(
      model = initial_model,
      nextBest = next_best,
      stopping = stopping_trial,
      increments = increment_rule,
      cohort_size = cohort_size,
      data = empty_data,
      startingDose = 3
    )

## Incoherence and rigidity

The `examine` function lists the dose recommendations made by CRM design after the first DLT is reported. The output shows, for each dose and number of DLTs reported, the dose recommendation (`nextBest` dose) made by the model on the assumption that no earlier DLTs were reported at lower doses.

    examine(design) %>% kable()

dose | DLTs | nextDose | stop | increment
--- | --- | --- | --- | ---
3 | 0 | 5 | FALSE | 67
3 | 1 | NA | FALSE | NA
5 | 0 | 10 | FALSE | 100
5 | 1 | 5 | FALSE | 0
10 | 0 | 20 | FALSE | 100
10 | 1 | 10 | FALSE | 0
20 | 0 | 25 | FALSE | 25
20 | 1 | 20 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | FALSE | 0
25 | 1 | 25 | FALSE | 0
25 | 0 | 25 | TRUE | 0
25 | 1 | 25 | TRUE | 0

Cheung (Cheung 2013) (p146) describes a CRM trial* as rigid if there exists a dose from which escalation is not possible regardless of the number of participants who do not experience when treated at that dose. He then (p164) defines a CRM trial as incoherent if, among other criteria, it recommends a dose escalation immediately following the first report of a DLT.

> *: Strictly, Cheung defines incoherence and rigidity only in terms of a trial’s dose-toxicity model. We extend the terms to include all of a trial’s design features, including its escalation, dose selection, increments and stopping rules.

The output from `examine` shows that our initial design is not incoherent (because there is no recommendation to increase the dose used in the next cohort following the first report of a DLT), but it is rigid. It is rigid because, even when no toxicities are reported at 25 mg, the model is unable to recommend escalation before the trial’s futility stopping rule (defined by `stopping_futility` above) is satisfied.

This is clearly unsatisfactory. The problem lies with the increments rule, which allows a maximum escalation of 33% between one dose and the next for doses above 20 mg. The dose grid being used is 1, 3, 5, 10, 15, 20, 25, 40, 50, 80, 100. The ratio between 20 and 25 is 1.25, but between 25 and 40 is 1.6. We can confirm this by checking the behaviour of `maxDose` with artificial data:

    no_tox_below_25_data <- Data(
      doseGrid = dose_grid,
      x = c(1, 3, 5, 10, 15, 20, 25),
      y = rep(0, 7),
      ID = 1L:7L,
      cohort = 1L:7L
    )

    next_dose <- maxDose(increment_rule, no_tox_below_25_data)
    next_dose
    #> [1] 33.25

Thus, the initial dose rule permanently prevents escalation above 25 mg because the highest permitted dose above 25 mg is 33.25 but the lowest such dose in the dose grid is 40. The problem can be solved in at least two ways: we can either relax the increments rule or we can introduce one or more intermediate doses. (We could, of course, do both.) We choose to relax the increments rule and check that, ceteris paribus, this permits escalation from every dose:

    revised_increment_rule <- IncrementsRelative(
      intervals = c(0, 20, 80),
      increments = c(1, 0.67, 0.33)
    )
    tibble(
      HighestDoseUsed = dose_grid,
      MaxPermittedDose = c(
        NA,
        sapply(
          seq_along(dose_grid[-1]),
          function(n) {
            tmp <- dose_grid[-1]
            d <- Data(
              doseGrid = dose_grid,
              x = tmp[1:n],
              y = rep(0, n),
              ID = as.integer(1:n),
              cohort = as.integer(1:n)
            )
            maxDose(revised_increment_rule, d)
          }
        )
      )
    ) %>%
      mutate(EscalationPermitted = lead(HighestDoseUsed < MaxPermittedDose)) %>%
      kable()

HighestDoseUsed | MaxPermittedDose | EscalationPermitted
--- | --- | ---
1 | NA | TRUE
3 | 6.00 | TRUE
5 | 10.00 | TRUE
10 | 20.00 | TRUE
15 | 30.00 | TRUE
20 | 33.40 | TRUE
25 | 41.75 | TRUE
40 | 66.80 | TRUE
50 | 83.50 | TRUE
80 | 106.40 | TRUE
100 | 133.00 | NA

Yes, it does. So we update the design:

    revised_design <- Design(
      model = initial_model,
      nextBest = next_best,
      stopping = stopping_trial,
      increments = revised_increment_rule,
      cohort_size = cohort_size,
      data = empty_data,
      startingDose = 3
    )

## Does the prior make sense?

    examine(revised_design) %>% kable()

dose | DLTs | nextDose | stop | increment
--- | --- | --- | --- | ---
3 | 0 | 5 | FALSE | 67
3 | 1 | NA | FALSE | NA
5 | 0 | 10 | FALSE | 100
5 | 1 | 3 | FALSE | -40
10 | 0 | 20 | FALSE | 100
10 | 1 | 10 | FALSE | 0
20 | 0 | 25 | FALSE | 25
20 | 1 | 20 | FALSE | 0
25 | 0 | 40 | FALSE | 60
25 | 1 | 25 | FALSE | 0
40 | 0 | 50 | FALSE | 25
40 | 1 | 40 | FALSE | 0
40 | 2 | 25 | FALSE | -38
40 | 3 | 20 | FALSE | -50
50 | 0 | 50 | FALSE | 0
50 | 1 | 50 | FALSE | 0
50 | 2 | 40 | FALSE | -20
50 | 3 | 40 | FALSE | -20
50 | 0 | 50 | FALSE | 0
50 | 1 | 50 | FALSE | 0
50 | 2 | 50 | FALSE | 0
50 | 3 | 40 | FALSE | -20
50 | 0 | 50 | FALSE | 0
50 | 1 | 50 | FALSE | 0
50 | 2 | 50 | FALSE | 0
50 | 3 | 50 | FALSE | 0
50 | 0 | 50 | TRUE | 0
50 | 1 | 50 | TRUE | 0
50 | 2 | 50 | TRUE | 0
50 | 3 | 50 | TRUE | 0

Hmmm. We now appear to be stuck at 50 mg. Why is this? Let’s examine the state of the model after escalation to 50 mg without toxicity…

    no_tox_below_50_data <- Data(
      doseGrid = dose_grid,
      x = c(1, 3, 5, 10, 15, 20, 25, 40, 40, 40, 50, 50, 50),
      y = rep(0, 13),
      ID = 1L:13L,
      cohort = c(1L:7L, rep(8L:9L, each = 3))
    )

    default_mcmc_options <- McmcOptions(burnin = 1000, step = 2, samples = 1000)
    no_tox_50_samples <- mcmc(no_tox_below_50_data, initial_model, default_mcmc_options)
    recommended_dose <- nextBest(
      next_best,
      doselimit = Inf,
      samples = no_tox_50_samples,
      model = initial_model,
      data = no_tox_below_50_data
    )
    recommended_dose$value
    #> [1] 50
    recommended_dose$probs
    #>       dose target overdose
    #>  [1,]    1  0.000    0.000
    #>  [2,]    3  0.000    0.000
    #>  [3,]    5  0.000    0.000
    #>  [4,]   10  0.000    0.000
    #>  [5,]   15  0.000    0.000
    #>  [6,]   20  0.001    0.000
    #>  [7,]   25  0.002    0.000
    #>  [8,]   40  0.027    0.001
    #>  [9,]   50  0.122    0.016
    #> [10,]   80  0.145    0.797
    #> [11,]  100  0.071    0.911

    maxDose(revised_increment_rule, no_tox_below_50_data)
    #> [1] 83.5

Whilst the increments rule allows escalation to 83.5 mg, the toxicity estimates provided by the model do not: the current estimate of toxicity at 80 mg is 0.797, well above the limit of 0.25 defined in the dose recommendation rule (`next_best`).

Again, we have several options. We can introduce intermediate doses, thus slowing escalation and allowing the prior model more time to adapt (but probably requiring a larger trial), or we can modify the prior so that it allows faster, but still reasonable, escalation. Let’s try creating a minimally informative prior that is consistent with our first attempt.

    # Fitting the min_inf_model is slow.
    if (file.exists("minInfModel.Rds")) {
      min_inf_model <- readRDS("minInfModel.Rds")
    } else {
      min_inf_model <- MinimalInformative(
        dose_grid,
        56,
        threshmin = 0.1,
        threshmax = 0.4,
        probmin = 0.05,
        probmax = 0.05
      )

      as_tibble(min_inf_model$required) %>%
        add_column(Dose = dose_grid) %>%
        add_column(Type = "Required") %>%
        bind_rows(
          as_tibble(min_inf_model$quantiles) %>%
            add_column(Dose = dose_grid) %>%
            add_column(Type = "Fitted")
        ) %>%
        ggplot() +
        geom_line(aes(x = Dose, y = median, colour = Type), linetype = "solid") +
        geom_line(aes(x = Dose, y = lower, colour = Type), linetype = "dotted") +
        geom_line(aes(x = Dose, y = upper, colour = Type), linetype = "dotted")

      saveRDS(min_inf_model, "minInfModel.Rds")
    }

    min_inf_model$model@params

The prior for θ is given by\[ \boldsymbol\theta = \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\sim N \left(\begin{bmatrix} 0.98 \\ 1.35\end{bmatrix} , \begin{bmatrix} 1.92 & 0.05 \\ 0.05 & 0.00\end{bmatrix} \right) \]

The minimally informative model has a prior distribution of

\[\binom{\alpha}{log(\beta)} \sim N \left( \left[\begin{array}{rr} 0.979 \\ 1.348 \end{array}\right], \left[\begin{array}{rr} 1.923 & 0.045 \\ 0.045 & 0.002\end{array}\right]\right)\]

Does this fix the rigidity problem?

    revised_model <- min_inf_model$model

    revised_design1 <- Design(
      model = revised_model,
      nextBest = next_best,
      stopping = stopping_trial,
      increments = revised_increment_rule,
      cohort_size = cohort_size,
      data = empty_data,
      startingDose = 3
    )

    examine(revised_design1) %>% kable()

dose | DLTs | nextDose | stop | increment
--- | --- | --- | --- | ---
3 | 0 | 5 | FALSE | 67
3 | 1 | 3 | FALSE | 0
5 | 0 | 10 | FALSE | 100
5 | 1 | 3 | FALSE | -40
10 | 0 | 10 | FALSE | 0
10 | 1 | 5 | FALSE | -50
10 | 0 | 15 | FALSE | 50
10 | 1 | 5 | FALSE | -50
15 | 0 | 20 | FALSE | 33
15 | 1 | 10 | FALSE | -33
20 | 0 | 25 | FALSE | 25
20 | 1 | 10 | FALSE | -50
25 | 0 | 25 | FALSE | 0
25 | 1 | 15 | FALSE | -40
25 | 0 | 25 | FALSE | 0
25 | 1 | 20 | FALSE | -20
25 | 0 | 25 | FALSE | 0
25 | 1 | 20 | FALSE | -20
25 | 0 | 40 | FALSE | 60
25 | 1 | 25 | FALSE | 0
40 | 0 | 50 | FALSE | 25
40 | 1 | 40 | FALSE | 0
40 | 2 | 25 | FALSE | -38
40 | 3 | 20 | FALSE | -50
50 | 0 | 80 | FALSE | 60
50 | 1 | 50 | FALSE | 0
50 | 2 | 40 | FALSE | -20
50 | 3 | 25 | FALSE | -50
80 | 0 | 100 | FALSE | 25
80 | 1 | 50 | FALSE | -38
80 | 2 | 50 | FALSE | -38
80 | 3 | 40 | FALSE | -50

Yes, it does. We can now escalate to dose 80, but there are few plateaux as we do so. First, two toxicity-free participants, rather than just one, are needed to escalate from 20 mg to 25 mg. Similarly, four toxicity-free participants are required to escalate from 25 mg to 40 mg. Further fine tuning of the prior would probably allow these numbers to be reduced, but for the purposes of this vignette, we do not explore this possibility.

Now create some data that represent a trial that escalates without any reports of toxicity, and fit the revised model …

    no_tox_data <- Data(
      doseGrid = c(1, 3, 5, 10, 15, 20, 25, 40, 50, 80, 100),
      x = c(c(1, 3, 5, 10, 15, 20), rep(c(25, 40, 50, 80, 100), each = 3)),
      y = rep(0, 21),
      cohort = as.integer(c(1:6, rep(7:11, each = 3))),
      ID = as.integer(1:21)
    )

    no_tox_samples <- mcmc(no_tox_data, revised_model, default_mcmc_options)

    recommended_dose <- nextBest(
      next_best,
      doselimit = Inf,
      samples = no_tox_samples,
      model = initial_model,
      data = no_tox_data
    )
    recommended_dose$probs
    #>       dose target overdose
    #>  [1,]    1  0.000    0.000
    #>  [2,]    3  0.000    0.000
    #>  [3,]    5  0.000    0.000
    #>  [4,]   10  0.000    0.000
    #>  [5,]   15  0.000    0.000
    #>  [6,]   20  0.000    0.000
    #>  [7,]   25  0.000    0.000
    #>  [8,]   40  0.015    0.000
    #>  [9,]   50  0.048    0.000
    #> [10,]   80  0.356    0.027
    #> [11,]  100  0.465    0.095

    stopTrial(stopping_trial, recommended_dose$value, no_tox_samples, initial_model, no_tox_data)
    #> [1] TRUE
    #> attr(,"message")
    #> attr(,"message")[[1]]
    #> attr(,"message")[[1]][[1]]
    #> [1] "Number of cohorts is 11 and thus reached the prespecified minimum number 3"
    #>
    #> attr(,"message")[[1]][[2]]
    #> [1] "Probability for target toxicity is 46 % for dose 100 and thus below the required 50 %"
    #>
    #>
    #> attr(,"message")[[2]]
    #> [1] "Number of patients is 21 and thus reached the prespecified minimum number 20"
    #>
    #> attr(,"individual")
    #> attr(,"individual")[[1]]
    #> [1] FALSE
    #> attr(,"message")
    #> attr(,"message")[[1]]
    #> [1] "Number of cohorts is 11 and thus reached the prespecified minimum number 3"
    #>
    #> attr(,"message")[[2]]
    #> [1] "Probability for target toxicity is 46 % for dose 100 and thus below the required 50 %"
    #>
    #> attr(,"individual")
    #> attr(,"individual")[[1]]
    #> [1] TRUE
    #> attr(,"message")
    #> [1] "Number of cohorts is 11 and thus reached the prespecified minimum number 3"
    #> attr(,"report_label")
    #> [1] "≥ 3 cohorts dosed"
    #>
    #> attr(,"individual")[[2]]
    #> [1] FALSE
    #> attr(,"message")
    #> [1] "Probability for target toxicity is 46 % for dose 100 and thus below the required 50 %"
    #> attr(,"report_label")
    #> [1] "P(0.2 ≤ prob(DLE | NBD) ≤ 0.35) ≥ 0.5"
    #>
    #> attr(,"report_label")
    #> [1] NA
    #>
    #> attr(,"individual")[[2]]
    #> [1] TRUE
    #> attr(,"message")
    #> [1] "Number of patients is 21 and thus reached the prespecified minimum number 20"
    #> attr(,"report_label")
    #> [1] "≥ 20 patients dosed"
    #>
    #> attr(,"report_label")
    #> [1] NA

More importantly, these results illustrate something that should have been obvious from the outset: expecting the trial to identify the MTD definitively in fewer than 20 participants is unrealistic (unless the MTD is very much towards the left hand - lower - end of the dose grid) because it takes almost 20 participants to travel from the lower end of the dose grid to the upper. Ignoring the plateauing mentioned in the previous paragraph, the increments rule we defined means that we need at least one participant at doses of 1, 3, 5, 10, 15 and 20 mg, and three at each of 25, 40, 50, 80 and 100 mg. That’s a total of 6 x 1 + 5 x 3 = 21 participants. So we hit the futility limit before we finish escalation over the dose grid, even if no DLTs are reported. That’s simply not realistic.

So our final alteration is to increase the futility rule from 20 participants to 40.

    revised_stopping_futility <- StoppingMinPatients(nPatients = 40)
    revised_stopping_trial <- (stopping_success1 & stopping_success2) | revised_stopping_futility

    revised_design2 <- Design(
      model = revised_model,
      nextBest = next_best,
      stopping = revised_stopping_trial,
      increments = revised_increment_rule,
      cohort_size = cohort_size,
      data = empty_data,
      startingDose = 3
    )

This new design allows us to escalate over the full extent of the dose grid before the futility stopping rule kicks in. Further refinement of the prior model may remove the need for additional cohorts before escalation from 20, 25 and 50 mg. We leave that as an exercise for the reader.

We now have a design that is worthy of more detailed investigation of its operating characteristics. We will do this in the next vignette in the series, which is yet to be written.

## Final observation

The simulations used in this vignette have used relatively short chains of 1000 samples, purely in the interests of speed. When investigating the properties of a real trial, much longer chain lengths should be used. For example, to estimate a binomial probability to an accuracy of ±1%, an effective sample size (ESS) of around 40,000 is required.

## References

Cheung, Ken. 2013. : Dose-Finding by the Continual Reassessment Method. https://www.routledge.com/Dose-Finding-by-the-Continual-Reassessment-Method/Cheung/p/book/9781420091519.