The escalation package by Kristian Brock. Documentation is hosted at https://brockk.github.io/escalation/

## Introduction

The continual reassessment method (CRM) was introduced by O’Quigley, Pepe, and Fisher (1990). It has proved to be a truly seminal dose-finding design, spurring many revisions, variants and imitations.

## Summary of the CRM Design

Pinning a finger on what is the CRM design is complicated because there have been so many versions over the years.

At its simplest, the CRM is a dose-escalation design that seeks a dose with probability of toxicity closest to some pre-specified target toxicity rate, $$p_T$$, in an homogeneous patient group. The hallmark that unifies the CRM variants is the assumption that the probability of toxicity, $$p_i$$, at the $$i$$th dose, $$x_i$$, can be modelled using a smooth mathematical function:

$p_i = F(x_i, \theta),$

where $$\theta$$ is a general vector of parameters. Priors are specified on $$\theta$$, and the dose with posterior estimate of $$p_i$$ closest to $$p_T$$ is iteratively recommended to the next patient(s).

Different variants of the CRM use different forms for $$F$$. We consider those briefly now.

#### Hyperbolic tangent model

O’Quigley, Pepe, and Fisher (1990) first introduced the method using

$p_i = \left( \frac{\tanh{(x_i)} + 1}{2} \right)^\beta,$

with an exponential prior was placed on $$\beta$$.

#### Empiric model (aka power model)

$p_i = x_i^{\exp(\beta)},$

for dose $$x_i \in (0, 1)$$;

#### One-parameter logistic model

$\text{logit} p_i = a_0 + \exp{(\beta)} x_i,$

where $$a_0$$ is a pre-specified constant, and $$x_i \in \mathbb{R}$$.

#### Two-parameter logistic model

$\text{logit} p_i = \alpha + \exp{(\beta)} x_i,$ for $$x_i \in \mathbb{R}$$.

Other model considerations include:

#### Priors

In each of these models, different distributions may be used for the parameter priors.

#### Toxicity skeletons and standardised doses

The $$x_i$$ dose variables in the models above do not reflect the raw dose quantities given to patients. For example, if the dose is 10mg, we do not use x=10. Instead, a skeleton containing estimates of the probabilities of toxicity at the doses is identified. This skeleton could reflect the investigators’ prior expectations of toxicities at all of the doses; or it could reflect their expectations for some of the doses with the others interpolated in some plausible way. The $$x_i$$ are then calculated so that the model-estimated probabilities of toxicity with the parameters taking their prior mean values match the skeleton. This will be much clearer with an example.

In a five dose setting, let the skeleton be $$\pi = (0.05, 0.1, 0.2, 0.4, 0.7)$$. That is, the investigators believe before the trial commences that the probbaility of toxicity at the second dose is 10%, and so on. Let us assume that we are using a one-parameter logistic model with $$a_0 = 3$$ and $$\beta ~ \sim N(0, 1)$$. Then we require that

$\text{logit} \pi_i = 3 + e^0 x_i = 3 + x_i,$

i.e.

$x_i = \text{logit} \pi_i - 3.$

This yields the vector of standardised doses $$x = (-5.94, -5.20, -4.39, -3.41, -2.15)$$. Equivalent transformations can be derived for the other model forms. The $$x_i$$ are then used as covariates in the model-fitting. CRM users specify their skeleton, $$\pi$$, and their parameter priors. From these, the software calculates the $$x_i$$. The actual doses given to patients in SI units do not actually feature in the model.

## Implementation in escalation

escalation simply aims to give a common interface to dose-selection models to facilitate a grammar for specifying dose-finding trial designs. Where possible, it delegates the mathematical model-fitting to existing R packages. There are several R packages that implement CRM models. The two used in escalation are the dfcrm package (Y. K. Cheung 2011; K. Cheung 2013); and the trialr package (Brock 2019, 2020). They have different strengths and weaknesses, so are suitable to different scenarios. We discuss those now.

#### dfcrm

dfcrm offers:

• the empiric model with normal prior on $$\beta$$;
• the one-parameter logistic model with normal prior on $$\beta$$.

dfcrm models are fit in escalation using the get_dfcrm function. Examples are given below.

#### trialr

trialr offers:

• the empiric model with normal prior on $$\beta$$;
• the one-parameter logistic model with normal prior on $$\beta$$;
• the one-parameter logistic model with gamma prior on $$\beta$$;
• the two-parameter logistic model with normal priors on $$\alpha$$ and $$\beta$$.

trialr models are fit in escalation using the get_trialr_crm function.

Let us commence by replicating an example from p.21 of Y. K. Cheung (2011). They choose the following parameters:

skeleton <- c(0.05, 0.12, 0.25, 0.40, 0.55)
target <- 0.25
a0 <- 3
beta_sd <- sqrt(1.34)

Let us define a model fitter using the dfcrm package:

library(escalation)
model1 <- get_dfcrm(skeleton = skeleton, target = target, model = 'logistic',
intcpt = a0, scale = beta_sd)

and a fitter using the trialr package:

model2 <- get_trialr_crm(skeleton = skeleton, target = target, model = 'logistic',
a0 = a0, beta_mean = 0, beta_sd = beta_sd)

Names for the function parameters skeleton, target, and model are standardised by escalation because they are fundamental. Further parameters (i.e. those in the second lines of each of the above examples) are passed onwards to the model-fitting functions in dfcrm and trialr. You can see that some of these parameter names vary between the approaches. E.g., what dfcrm calls the intcpt, trialr calls a0. Refer to the documentation of the crm function in dfcrm and stan_crm in trialr for further information.

We then fit those models to the notional outcomes described in the source text:

outcomes <- '3N 5N 5T 3N 4N'

fit1 <- model1 %>% fit(outcomes)
fit2 <- model2 %>% fit(outcomes)

The dose recommended by each of the models for the next patient is:

fit1 %>% recommended_dose()
#> [1] 4
fit2 %>% recommended_dose()
#> [1] 4

Thankfully, the models agree. They advocate staying at dose 4, wary of the toxicity already seen at dose 5.

If we take a summary of each model fit:

fit1 %>% summary()
#> # A tibble: 6 × 9
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>
#> 1 NoDose     0     0              0         0               0       TRUE
#> 2 1          0     0            NaN         0.00768         0.00768 TRUE
#> 3 2          0     0            NaN         0.0265          0.0265  TRUE
#> 4 3          0     2              0         0.0817          0.0817  TRUE
#> 5 4          0     1              0         0.182           0.182   TRUE
#> 6 5          1     2              0.5       0.331           0.331   TRUE
#> # ℹ 2 more variables: recommended <lgl>, Skeleton <dbl>
fit2 %>% summary()
#> # A tibble: 6 × 9
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>
#> 1 NoDose     0     0              0          0              0       TRUE
#> 2 1          0     0            NaN          0.0300         0.00731 TRUE
#> 3 2          0     0            NaN          0.0633         0.0255  TRUE
#> 4 3          0     2              0          0.128          0.0791  TRUE
#> 5 4          0     1              0          0.219          0.178   TRUE
#> 6 5          1     2              0.5        0.340          0.326   TRUE
#> # ℹ 2 more variables: recommended <lgl>, Skeleton <dbl>

We can see that they closely agree on model estimates of the probability of toxicity at each dose. Note that the median perfectly matches the mean in the dfcrm fit because it assumes a normal posterior distribution on $$\beta$$. In contrast, the trialr class uses Stan to fit the model using Hamiltonian Monte Carlo sampling. The posterior distributions for the probabilities of toxicity are evidently non-normal and positively-skewed because the median estimates are less than the mean estimates.

Let us imagine instead that we want to fit the empiric model. That simply requires we change the model variable and adjust the prior parameters:

model3 <- get_dfcrm(skeleton = skeleton, target = target, model = 'empiric',
scale = beta_sd)

model4 <- get_trialr_crm(skeleton = skeleton, target = target, model = 'empiric',
beta_sd = beta_sd)

Fitting each to the same set of outcomes yields:

fit3 <- model3 %>% fit(outcomes)
fit4 <- model4 %>% fit(outcomes) 
fit3 %>% summary()
#> # A tibble: 6 × 9
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>
#> 1 NoDose     0     0              0         0               0       TRUE
#> 2 1          0     0            NaN         0.00701         0.00701 TRUE
#> 3 2          0     0            NaN         0.0299          0.0299  TRUE
#> 4 3          0     2              0         0.101           0.101   TRUE
#> 5 4          0     1              0         0.219           0.219   TRUE
#> 6 5          1     2              0.5       0.372           0.372   TRUE
#> # ℹ 2 more variables: recommended <lgl>, Skeleton <dbl>
fit4 %>% summary()
#> # A tibble: 6 × 9
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>
#> 1 NoDose     0     0              0          0              0       TRUE
#> 2 1          0     0            NaN          0.0309         0.00766 TRUE
#> 3 2          0     0            NaN          0.0668         0.0318  TRUE
#> 4 3          0     2              0          0.142          0.105   TRUE
#> 5 4          0     1              0          0.249          0.225   TRUE
#> 6 5          1     2              0.5        0.381          0.378   TRUE
#> # ℹ 2 more variables: recommended <lgl>, Skeleton <dbl>

In this example, the model estimates are broadly consistent across methodology and model type. However, this is not the general case. To illustrate this point, let us examine a two parameter logistic model fit using trialr (note: this model is not implemented in dfcrm):

model5 <- get_trialr_crm(skeleton = skeleton, target = target, model = 'logistic2',
alpha_mean = 0, alpha_sd = 2, beta_mean = 0, beta_sd = 1)
fit5 <- model5 %>% fit(outcomes)
fit5 %>% summary()
#> # A tibble: 6 × 9
#>   dose     tox     n empiric_tox_rate mean_prob_tox median_prob_tox admissible
#>   <ord>  <dbl> <dbl>            <dbl>         <dbl>           <dbl> <lgl>
#> 1 NoDose     0     0              0          0              0       TRUE
#> 2 1          0     0            NaN          0.0347         0.00394 TRUE
#> 3 2          0     0            NaN          0.0567         0.0171  TRUE
#> 4 3          0     2              0          0.105          0.0625  TRUE
#> 5 4          0     1              0          0.203          0.164   TRUE
#> 6 5          1     2              0.5        0.411          0.384   TRUE
#> # ℹ 2 more variables: recommended <lgl>, Skeleton <dbl>

Now the estimate of toxicity at the highest dose is high relative to the other models. The extra free parameter in the two-parameter model offers more flexibility. There has been a debate in the literature about one-parameter vs two-parameter models (and possibly more). It is generally accepted that a single parameter model is too simplistic to accurately estimate $$p_i$$ over the entire dose range. However, it will be sufficient to identify the dose closest to $$p_T$$, and if that is the primary objective of the trial, the simplicity of a one-parameter model may be entirely justified. The interested reader is directed to O’Quigley, Pepe, and Fisher (1990) and Neuenschwander, Branson, and Gsponer (2008).

Note that the CRM does not natively implement stopping rules, so these classes on their own will always advocate trial continuance:

fit1 %>% continue()
#> [1] TRUE
fit5 %>% continue()
#> [1] TRUE

and identify each dose as admissible:

fit1 %>% dose_admissible()
#> [1] TRUE TRUE TRUE TRUE TRUE

This behaviour can be altered by appending classes to advocate stopping for consensus:

model6 <- get_trialr_crm(skeleton = skeleton, target = 0.3, model = 'empiric',
beta_sd = 1) %>%
stop_when_n_at_dose(dose = 'recommended', n = 6)

fit6 <- model6 %>% fit('2NNN 3TTT 2NTN')

fit6 %>% continue()
#> [1] FALSE
fit6 %>% recommended_dose()
#> [1] 2

Or for stopping under excess toxicity:

model7 <- get_trialr_crm(skeleton = skeleton, target = 0.3, model = 'empiric',
beta_sd = 1) %>%
stop_when_too_toxic(dose = 1, tox_threshold = 0.3, confidence = 0.8)

fit7 <- model7 %>% fit('1NTT 1TTN')

fit7 %>% continue()
#> [1] FALSE
fit7 %>% recommended_dose()
#> [1] NA
fit7 %>% dose_admissible()
#> [1] FALSE FALSE FALSE FALSE FALSE

Or both:

model8 <- get_trialr_crm(skeleton = skeleton, target = 0.3, model = 'empiric',
beta_sd = 1) %>%
stop_when_n_at_dose(dose = 'recommended', n = 6) %>%
stop_when_too_toxic(dose = 1, tox_threshold = 0.3, confidence = 0.8)

#### dfcrm vs trialr

So which method should you use? The answer depends on how you plan to use the models.

The trialr classes produce posterior samples:

fit7 %>%
prob_tox_samples(tall = TRUE) %>%
#> # A tibble: 6 × 3
#>   .draw dose  prob_tox
#>   <chr> <chr>    <dbl>
#> 1 1     1        0.658
#> 2 2     1        0.645
#> 3 3     1        0.701
#> 4 4     1        0.687
#> 5 5     1        0.674
#> 6 6     1        0.553

and these facilitate flexible visualisation:

library(ggplot2)
library(dplyr)

fit7 %>%
prob_tox_samples(tall = TRUE) %>%
mutate(.draw = .draw %>% as.integer()) %>%
filter(.draw <= 200) %>%
ggplot(aes(dose, prob_tox)) +
geom_line(aes(group = .draw), alpha = 0.2)

However, MCMC sampling is an expensive computational procedure compared to the numerical integration used in dfcrm. If you envisage fitting lots of models, perhaps in simulations or dose-paths (see below) and favour a model offered by dfcrm, we recommend using get_dfcrm. However, if you favour a model only offered by trialr, or if you are willing for calculation to be slow in order to get posterior samples, then use trialr.

### Dose paths

We can use the get_dose_paths function in escalation to calculate exhaustive model recommendations in response to every possible set of outcomes in future cohorts. For instance, at the start of a trial using an empiric CRM, we can examine all possible paths a trial might take in the first two cohorts of three patients, starting at dose 2:

skeleton <- c(0.05, 0.12, 0.25, 0.40, 0.55)
target <- 0.25
beta_sd <- 1

model <- get_dfcrm(skeleton = skeleton, target = target, model = 'empiric',
scale = beta_sd)
paths <- model %>% get_dose_paths(cohort_sizes = c(3, 3), next_dose = 2)
graph_paths(paths)

We see that the design would willingly skip dose 3 if no tox is seen in the first cohort. This might warrant suppressing dose-dkipping by appending a dont_skip_doses(when_escalating = TRUE) selector.

Dose-paths can also be run for in-progress trials where some outcomes have been established. For more information on working with dose-paths, refer to the dose-paths vignette.

### Simulation

We can use the simulate_trials function to calculate operating characteristics for a design. Let us use the example above and tell the design to stop when the lowest dose is too toxic, when 9 patients have already been evaluated at the candidate dose, or when a sample size of $$n=24$$ is reached:

model <- get_dfcrm(skeleton = skeleton, target = target, model = 'empiric',
scale = beta_sd) %>%
stop_when_too_toxic(dose = 1, tox_threshold = target, confidence = 0.8) %>%
stop_when_n_at_dose(dose = 'recommended', n = 9) %>%
stop_at_n(n = 24)

For the sake of speed, we will run just fifty iterations:

num_sims <- 50

In real life, however, we would naturally run many thousands of iterations. Let us investigate under the following true probabilities of toxicity:

sc1 <- c(0.25, 0.5, 0.6, 0.7, 0.8)

The simulated behaviour is:

set.seed(123)
sims <- model %>%
simulate_trials(num_sims = num_sims, true_prob_tox = sc1, next_dose = 1)

sims
#> Number of iterations: 50
#>
#> Number of doses: 5
#>
#> True probability of toxicity:
#>    1    2    3    4    5
#> 0.25 0.50 0.60 0.70 0.80
#>
#> Probability of recommendation:
#> NoDose      1      2      3      4      5
#>   0.32   0.62   0.06   0.00   0.00   0.00
#>
#>      1      2      3      4      5
#> 0.6049 0.2537 0.0293 0.1122 0.0000
#>
#> Sample size:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>     3.0     9.0    12.0    12.3    18.0    21.0
#>
#> Total toxicities:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>    2.00    2.25    4.50    4.82    7.00    9.00
#>
#> Trial duration:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>   1.129   8.723  12.861  12.798  17.758  26.542

We see that the chances of stopping for excess toxicity and recommending no dose is about 1-in-4. Dose 1 is the clear favourite to be identified. Interestingly, the stop_when_n_at_dose class reduces the expected sample size to 12-13 patints. Without it:

get_dfcrm(skeleton = skeleton, target = target, model = 'empiric',
scale = beta_sd) %>%
stop_when_too_toxic(dose = 1, tox_threshold = target, confidence = 0.8) %>%
stop_at_n(n = 24) %>%
simulate_trials(num_sims = num_sims, true_prob_tox = sc1, next_dose = 1)
#> Number of iterations: 50
#>
#> Number of doses: 5
#>
#> True probability of toxicity:
#>    1    2    3    4    5
#> 0.25 0.50 0.60 0.70 0.80
#>
#> Probability of recommendation:
#> NoDose      1      2      3      4      5
#>   0.40   0.50   0.08   0.02   0.00   0.00
#>
#>      1      2      3      4      5
#> 0.6250 0.2297 0.0473 0.0980 0.0000
#>
#> Sample size:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>    3.00    9.00   24.00   17.76   24.00   24.00
#>
#> Total toxicities:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>    2.00    5.25    7.00    6.64    9.00   12.00
#>
#> Trial duration:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>    1.05    8.22   19.40   17.66   25.41   32.88

expected sample size is much higher and the chances of erroneously stopping early are also higher. These phenomena would justify a wider simulation study in a real situation.

For more information on running dose-finding simulations, refer to the simulation vignette.

## References

Brock, Kristian. 2019. trialr: Bayesian Clinical Trial Designs in R and Stan.” arXiv e-Prints, June, arXiv:1907.00161. https://arxiv.org/abs/1907.00161.
———. 2020. Trialr: Clinical Trial Designs in ’Rstan’. https://cran.r-project.org/package=trialr.
Cheung, Ken. 2013. Dfcrm: Dose-Finding by the Continual Reassessment Method. https://CRAN.R-project.org/package=dfcrm.
Cheung, Ying Kuen. 2011. Dose Finding by the Continual Reassessment Method. New York: Chapman & Hall / CRC Press.
Neuenschwander, Beat, Michael Branson, and Thomas Gsponer. 2008. Critical aspects of the Bayesian approach to phase I cancer trials.” Statistics in Medicine 27: 2420–39. https://doi.org/10.1002/sim.3230.
O’Quigley, J, M Pepe, and L Fisher. 1990. “Continual Reassessment Method: A Practical Design for Phase 1 Clinical Trials in Cancer.” Biometrics 46 (1): 33–48. https://doi.org/10.2307/2531628.