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

`escalation`

provides a grammar for dose-finding clinical
trials.

Here is a trivial example. Everyone knows that a 3+3 design will escalate after it sees no toxicities in a cohort of three:

```
library(escalation)
get_three_plus_three(num_doses = 5) %>%
fit("1NNN") %>%
recommended_dose()
```

`## [1] 2`

but that it will stick when it sees one-out-of-three toxicities in a cohort:

```
<- get_three_plus_three(num_doses = 5) %>%
x fit("1NNN 2NNN 3NNT")
recommended_dose(x)
```

`## [1] 3`

`continue(x)`

`## [1] TRUE`

Having escalated through doses 1 and 2, here the design advocates treating another cohort at dose 3.

```
<- get_three_plus_three(num_doses = 5) %>%
x fit("1NNN 2NNN 3NNT 3NTN")
recommended_dose(x)
```

`## [1] 2`

`continue(x)`

`## [1] FALSE`

When it sees two-out-of-six toxicities at dose 3, it concludes that dose 3 is too toxic, dose 2 is the MTD, and the trial should stop.

`escalation`

provides functions to use common
dose-escalation methodologies like the continual reassessment method
(CRM), the Bayesian optimal interval design (BOIN), the TPI suite of
designs, efficacy-toxicity designs like EffTox or Wages & Tait, and
(as we have seen) the perennial 3+3:

`get_dfcrm()`

`get_trialr_crm()`

`get_trialr_nbg()`

`get_tpi()`

`get_mtpi()`

`get_mtpi2()`

`get_boin()`

`get_trialr_efftox()`

`get_wages_and_tait()`

`get_three_plus_three()`

These functions create model fitting objects. Where possible,
technical implementations are imported from existing R packages like
`dfcrm`

, `trialr`

, and `BOIN`

. Where no
external implementations is available however, methods are implemented
natively in `escalation`

.

These dose-finding approaches can then be augmented with extra
behaviours to specialise the dose selection process. For example, we can
add behaviours to prevent skipping doses, or to stop when we reach a
certain sample size. `escalation`

supports the following
behaviours:

`dont_skip_doses()`

`stop_at_n()`

`stop_when_n_at_dose()`

`stop_when_too_toxic()`

`stop_when_tox_ci_covered()`

`demand_n_at_dose()`

`try_rescue_dose()`

`follow_path()`

`select_dose_by_cibp()`

Each of these functions overrides the way doses are selected or when
a design decides to stop the trial. The behaviours can be flexibly
combined using the `%>%`

operator from the
`tidyverse`

suite.

These models are then fit to trial outcomes to produce dose
recommendations. No matter how the dose selection behaviours were
combined, the resulting model fits supports a standard interface. The
two most important methods are `recommended_dose()`

to get
the current dose recommendation, and `continue()`

to learn
whether the design advocates continuing patient recruitment.

Having defined this nomenclature for combining dose selection behaviours and providing a standard interface for the resulting analyses, it is simple to run simulations or calculate dose-pathways for future cohorts of patients.

`escalation`

provides an object-oriented approach to
dose-escalation clinical trials in R. See Usage

`escalation`

uses a succinct syntax for describing
dose-finding outcomes, described in Brock (2019) for the phase I setting
and in Brock et al. (2017) for the phase I/II setting.

In a phase I trial, we use the letters:

`T`

to show that toxicity occurred in a patient;`N`

to show that toxicity did not occur in a patient.

In a joint phase I/II trial, like those supported by EffTox, where we have coincident efficacy and toxicity outcomes, those relevant letters are:

`T`

to show that toxicity without efficacy occurred in a patient;`E`

to show that efficacy without toxicity occurred in a patient;`N`

to show that neither occurred;`B`

to show that both occurred.

These outcome letters are strewn behind integer dose-levels to show the outcomes of patients in cohorts. To show that a cohort a three patients was given dose 2, that the first two patients were without toxicity, but the third patient experienced toxicity, we would use the outcome string:

`<- '2NNT' outcomes `

If that cohort was followed by another cohort of three, all of which were without toxicity, the overall outcome string would be:

`<- '2NNT 2NNN' outcomes `

And so on. These strings are used in the `escalate`

package to make it easy to fit models to observed outcomes. There are
many examples below.

A core class in the `escalation`

package is the
`selector`

. It encapsulates the notion that a general
dose-escalation design is able to recommend doses, keep track of how
many patients have been treated at what doses, what toxicity outcomes
have been seen, and whether a trial should continue. This general
interface is true of model-based methods like the CRM and rule-based
methods like the 3+3. Irrespective the particular approach used, the
interface is consistent.

In this tutorial, we will demonstrate each of the types of
`selector`

implemented in the package and how they can be
combined to tailor behaviour.

To begin, let us load `escalation`

`library(escalation)`

At the core of the dose selection process is an algorithm or a model that selects doses in responses to outcomes. The classes capable of performing this core role are:

`get_dfcrm()`

, using the model-fitting code from`dfcrm`

`get_trialr_crm()`

using the model-fitting code from`trialr`

`get_trialr_nbg()`

using the model-fitting code from`trialr`

`get_boin()`

using the model-fitting code from`BOIN`

`get_tpi()`

`get_mtpi()`

`get_mtpi2()`

`get_trialr_efftox()`

using the model-fitting code from`trialr`

`get_wages_and_tait()`

`get_three_plus_three()`

- and
`follow_path()`

Where indicated these methods rely on external packages. Otherwise,
methods are implemented natively in `escalation`

. We look at
several of these below.

The continual reassessment method (O’Quigley, Pepe, and Fisher 1990)
(CRM) is implemented in the `dfcrm`

package by Cheung (2013). The very least information we need to provide
is a dose-toxicity skeleton, and our target toxicity level. The skeleton
represents our prior beliefs on the probabilities of toxicity at each of
the doses under investigation. The model iteratively seeks a dose with
toxicity probability close to the target.

For illustration, let us say we have

```
<- c(0.05, 0.1, 0.25, 0.4, 0.6)
skeleton <- 0.25 target
```

We create a dose-selection model using:

`<- get_dfcrm(skeleton = skeleton, target = target) model `

and we can fit this to outcomes using code like:

`<- model %>% fit('2NNN') fit `

The `fit`

object will tell you the dose recommended by the
CRM model to be administered next. Depending on your preference for
classic R or tidyverse R, you might run:

`recommended_dose(fit)`

`## [1] 4`

or

`%>% recommended_dose() fit `

`## [1] 4`

Either way, you get the same answer. The model advocates skipping straight to dose 4. Clinicians are unlikely to feel comfortable with this. We can respecify the model to expressly not skip doses in escalation. We will do that later on.

For now, let us return to our model fit. We can ask whether the trial should keep going:

`%>% continue() fit `

`## [1] TRUE`

Naturally it wants to continue because `dfcrm`

does not
implement any stopping rules. Again, we will add various stopping
behaviours in sections below.

The CRM-fitting function in `dfcrm`

accepts many arguments
to customise the model form and these are passed onwards by
`get_dfcrm`

function via the `...`

parameter. For
example, to use the one-parameter logit model in `dfcrm`

(rather than the default empiric model) with the intercept term fixed to
take the value 4, we can specify:

```
<- get_dfcrm(skeleton = skeleton, target = target,
fit intcpt = 4, model = 'logistic') %>%
fit('2NNN 3TNN')
%>% recommended_dose() fit
```

`## [1] 3`

`intcpt`

and `logistic`

are the parameter names
chosen by the authors of `dfcrm`

.

We could instead fit the CRM models above using the `trialr`

package by (Brock 2019, 2020).

Reusing the `skeleton`

and `target`

variables
defined above, we fit the same empiric model

```
<- get_trialr_crm(skeleton = skeleton, target = target, model = 'empiric',
model beta_sd = sqrt(1.34))
<- model %>% fit('2NNN') fit
```

The `dfcrm`

package, unless told otherwise, assumes that
you want an empiric model where the prior variance for \(\beta\) is 1.34. In the `trialr`

package, no such assumptions are made so we had to specify those
variables.

All we have changed is the method of inference. `dfcrm`

uses numerical integration to calculate posterior statistics and plugs
those into the dose-toxicity function. In contrast, `trialr`

fits the model using Hamiltonian MCMC sampling via `Stan`

.
Thankfully, the two models agree on the desired next dose:

`%>% recommended_dose() fit `

`## [1] 4`

and that the trial should continue:

`%>% continue() fit `

`## [1] TRUE`

The added bonus we get from the `trialr`

fit, however, is
those samples from the posterior distribution:

`%>% prob_tox_samples() %>% head(10) fit `

```
## # A tibble: 10 × 6
## .draw `1` `2` `3` `4` `5`
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 4.39e- 2 9.04e- 2 0.235 0.384 0.587
## 2 2 3.39e- 6 6.24e- 5 0.00294 0.0212 0.117
## 3 3 8.03e- 2 1.44e- 1 0.311 0.462 0.651
## 4 4 7.41e- 2 1.35e- 1 0.300 0.451 0.642
## 5 5 5.32e- 2 1.05e- 1 0.257 0.408 0.606
## 6 6 5.60e- 7 1.57e- 5 0.00128 0.0122 0.0859
## 7 7 2.28e-17 1.62e-13 0.0000000199 0.00000812 0.00145
## 8 8 2.14e- 1 3.06e- 1 0.490 0.624 0.769
## 9 9 2.63e- 1 3.58e- 1 0.539 0.664 0.796
## 10 10 1.70e- 1 2.56e- 1 0.441 0.582 0.739
```

That facilitates really flexible inference. For example, what is the probability that toxicity at dose 3 is at least 5% greater than that at dose 2? Simple to answer using the posterior samples:

```
library(dplyr)
%>% prob_tox_samples() %>%
fit summarise(prob = mean(`3` > `2` + 0.05))
```

```
## # A tibble: 1 × 1
## prob
## <dbl>
## 1 0.586
```

‘More likely than not’, is the answer.

See the Continual Reassessment Method vignette for more details.

The two-parameter logistic dose-escalation method of Neuenschwander,
Branson, and Gsponer (2008) (NBG) is implemented in the `trialr`

package by Brock (2020).

The very least information we need to provide is a vector of the doses under investigation, a reference dose-level \(d^*\), our target toxicity level, and priors on the logit model intercept, \(\alpha\), and dose gradient, \(\beta\).

For illustration, let us reproduce the notorious example in Figure 1 of Neuenschwander, Branson, and Gsponer (2008) with 15 doses:

```
<- c(1, 2.5, 5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 150, 200, 250)
dose <- '1NNN 2NNNN 3NNNN 4NNNN 7TT'
outcomes
<- get_trialr_nbg(real_doses = dose, d_star = 250, target = 0.3,
fit alpha_mean = 2.15, alpha_sd = 0.84,
beta_mean = 0.52, beta_sd = 0.8, seed = 2020) %>%
fit(outcomes)
```

Sticking at dose 7 is the recommendation:

`%>% recommended_dose() fit `

`## [1] 7`

However, we see that it is a close call as to which dose is closest to the target toxicity level:

`%>% mean_prob_tox() fit `

```
## [1] 0.01229476 0.03139614 0.06501636 0.13376511 0.20021181 0.26219444
## [7] 0.31900339 0.37057264 0.45907389 0.53075063 0.65691058 0.73549743
## [13] 0.82365045 0.86996457 0.89783331
```

This is perhaps unsurprising in a situation with so many doses.

See the Neuenschwander, Branson & Gsponer vignette for more details.

The Toxicity Probability Interval (TPI) method was introduced by Ji, Li, and Bekele (2007). The model requires a few parameters:

```
<- get_tpi(num_doses = 5, target = 0.25, k1 = 1, k2 = 1.5,
model exclusion_certainty = 0.95)
```

The model can be fit to outcomes in the usual way:

`<- model %>% fit('1NNT') fit `

and the returned model fit obeys the same interface as the other classes described here. For instance, the dose recommended for the next cohort is:

`%>% recommended_dose() fit `

`## [1] 1`

See the Toxicity Probability Interval Design vignette for more information.

The Modified Toxicity Probability Interval (mTPI) method was introduced by Ji et al. (2010). It is generally simpler to implement than TPI because the \(\epsilon1\) and \(\epsilon2\) parameters have the intuitive interpretation of forming the bounds of the interval that we regard as containing doses equivalent to the target dose. For instance, if we target a dose with toxicity probability equal to 25%, but would judge doses in the region (20%, 30%) to be satisfactorily toxic, we run:

```
<- get_mtpi(num_doses = 5, target = 0.25,
model epsilon1 = 0.05, epsilon2 = 0.05, exclusion_certainty = 0.95)
```

In this parameterisation, we exclude doses if we are 95% a-posteriori sure that the associated toxicity rate exceeds the target.

We fit the model to outcomes:

`<- model %>% fit('1NNT') fit `

and learn that the recommended next dose is

`%>% recommended_dose() fit `

`## [1] 1`

dose 1, in accordance with Figure 2 of Ji et al. (2010).

See the Modified Toxicity Probability Interval Design vignette for more information.

mTPI was further updated by Guo et al. (2017) to produce mTPI2. Its parameterisation is similar to mTPI.

```
<- get_mtpi2(num_doses = 5, target = 0.25,
model epsilon1 = 0.05, epsilon2 = 0.05,
exclusion_certainty = 0.95)
<- model %>% fit('1NNT')
fit %>% recommended_dose() fit
```

`## [1] 1`

Once again, see the Modified Toxicity Probability Interval Design vignette for more information.

`escalate`

also implements the Bayesian Optimal Interval
(BOIN) dose-finding design by Liu and Yuan (2015) via the `BOIN`

package (Yuan and Liu 2018).

In contrast to CRM, BOIN does not require a dose-toxicity skeleton. In its simplest case, it requires merely the number of doses under investigation and our target toxicity level:

```
<- 0.25
target
<- get_boin(num_doses = 5, target = target) model
```

As before, we can fit the model to some observed outcomes:

`<- model %>% fit('2NNN') fit `

and ask the recommended dose:

`%>% recommended_dose() fit `

`## [1] 3`

The BOIN dose selector natively implements stopping rules, as described by Liu & Yuan. For instance, if the bottom dose is too toxic, the design will advise the trial halts:

```
<- model %>% fit('2NTN 1TTT')
fit %>% continue() fit
```

`## [1] FALSE`

Notice in this scenario that the recommended dose is
`NA`

:

`%>% recommended_dose() fit `

`## [1] NA`

This clarifies that no dose should be recommended for further study. In this setting, this is because all doses are considered too toxic. This is distinct from scenarios where a design advocates stopping a trial and recommending a dose for further study. We will encounter situations like that below.

Since `escalation`

provides many flexible options for
stopping, we have made it possible to suppress BOIN’s native stopping
rule via `use_stopping_rule = FALSE`

. In this instance, the
user may want to add their own stopping rule, e.g. using
`stop_when_too_toxic`

.

Extra parameters are passed to the `get.boundary`

function
in the `BOIN`

package to customise the escalation procedure.
For instance, the boundaries that guide changes in dose are set to be
60% and 140% of the target toxicity rate, by default. To instead use 30%
and 170%, we could run:

```
get_boin(num_doses = 5, target = target,
p.saf = 0.3 * target, p.tox = 1.7 * target) %>%
fit('1NNN 2NNT') %>%
recommended_dose()
```

`## [1] 2`

To observe the effect of the change, note that the default values suppress escalation in this scenario:

```
get_boin(num_doses = 5, target = target) %>%
fit('1NNN 2NNT') %>%
recommended_dose()
```

`## [1] 1`

The parameter names `p.saf`

and `p.tox`

were
chosen by the authors of the `BOIN`

package.

See the Bayesian Optimal Interval Design vignette for more information.

`escalation`

also supports designs that choose doses
according to co-primary efficacy and toxicity outcomes, such as the
BOIN12 design (Lin et al. 2020).

We provide target toxicity and efficacy thresholds via
`phi_t`

and `phi_e`

, the utility of ‘no efficacy
and no toxicity’ via `u2`

, and the utility of ‘efficacy with
toxicity’ via `u3`

on a (0, 100) scale:

```
# Examples in Lin et al.
<- get_boin12(num_doses = 5,
model phi_t = 0.35, phi_e = 0.25,
u2 = 40, u3 = 60)
```

In contrast to the examples above, outcomes now include efficacy as
well as toxicity. `E`

reflects efficacy only, and
`B`

reflects both efficacy and toxicity. The model-fitting
process is largely the same though:

`<- model %>% fit('1NNN 2ENT 3ETT 2EEN') fit `

We have the usual interface:

`%>% recommended_dose() fit `

`## [1] 2`

`%>% continue() fit `

`## [1] TRUE`

`escalation`

also supports other so-called phase I/II
designs that select doses by efficacy and toxicity like Wages and Tait
(2015) and Thall and Cook (2004). See the help pages for more
information.

The 3+3 method is an old method for dose-escalation that uses fixed cohorts of three and pre-specified rules to govern dose-selection (Korn et al. 1994; Le Tourneau, Lee, and Siu 2009).

To create a 3+3 design, we need no more information than the number of doses under investigation:

`<- get_three_plus_three(num_doses = 5) model `

As usual, we can fit the model to some outcomes and learn the recommended dose:

`%>% fit('2NTN') %>% recommended_dose() model `

`## [1] 2`

Korn et al. (1994) described a variant of 3+3 that permits deescalation to ensure that six patients are treated at a dose before it is recommended. To use that option in our model, we could have run:

`<- get_three_plus_three(num_doses = 5, allow_deescalate = TRUE) model `

The model would then advocate deescalation if at least two toxicities are seen at a dose and the dose below has fewer than 6 treated patients:

`%>% fit('2NTT') %>% recommended_dose() model `

`## [1] 1`

The final dose selector in this section is not really a model at all,
so much as a pre-specified path to follow. Let us say that we would like
to escalate through the doses in the absence of toxicity, treating two
patients at each of the first two doses, and three at the other doses.
We can specify such a path in `escalation`

using:

`<- follow_path('1NN 2NN 3NNN 4NNN 5NNN') model `

When fit to data, the method just returns whatever comes next in the sequence:

`%>% fit('1NN 2N') %>% recommended_dose() model `

`## [1] 2`

`%>% fit('1NN 2NN') %>% recommended_dose() model `

`## [1] 3`

When the outcomes diverge from the pre-specified path, however, this selector does not know what to do:

`%>% fit('1NN 2NT') %>% recommended_dose() model `

`## [1] NA`

That rather seems to limit its value. The point of this class is that
we sometimes want to specify what is occasionally referred to as *an
initial escalation plan*. When trial outcomes diverge from the
initial plan, another method takes over. This is a perfect opportunity
to show how different selectors can be joined together. Let us say that
we wish to follow the initial plan described above, but when the first
toxicity event is seen, we want a CRM model to take over. We simply join
the functions together using the pipe operator from
`magrittr`

:

```
<- follow_path('1NN 2NN 3NNN 4NNN 5NNN') %>%
model get_dfcrm(skeleton = skeleton, target = target)
```

Now, when trial outcomes diverge from the path, the CRM model analyses all of the outcomes and recommends the next dose:

`%>% fit('1NN 2NT') %>% recommended_dose() model `

`## [1] 2`

This concludes our look at the core dose-selecting classes. We now turn our attention to the ways in which these methods can be adapted using extra behaviours.

We saw in the CRM example above that the design undesirably wanted to skip straight to a high dose, without trying some of the lower doses. A simple and very common constraint to impose in dose-finding trials is to avoid skipping untested doses.

Resuming our CRM example, we suppress the skipping of untested doses in escalation with:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model dont_skip_doses(when_escalating = TRUE)
```

We then fit the model as before:

```
<- model %>% fit('2NNN')
fit %>% recommended_dose() fit
```

`## [1] 3`

This time, however, the model advocates dose 3. Previously, it wanted to go straight to dose 4.

We prevented skipping dose in escalation. We could have prevented skipping doses in deescalation with:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model dont_skip_doses(when_deescalating = TRUE)
```

or both with:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model dont_skip_doses(when_escalating = TRUE, when_deescalating = TRUE)
```

Let us now investigate some methods that facilitate stopping. The
simplest condition on which to stop is when the total sample size
reaches some pre-specified level. For instance, we might want to treat a
maximum of 15 patients and then stop. To do this, we call the
`stop_at_n`

function and append it onto the end of a core
dose selector, like this:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_at_n(n = 15)
```

When this design has seen fewer than 15 patients, it will select doses and advocate that the trial continues. For instance:

```
<- model %>% fit('1NNN 2TNN 2NNN 3NNN')
fit %>% continue() fit
```

`## [1] TRUE`

The design advocates continuing at dose:

`%>% recommended_dose() fit `

`## [1] 3`

In contrast, once 15 patients are seen,

```
<- model %>% fit('1NNN 2TNN 2NNN 3NNN 3NTN')
fit %>% continue() fit
```

`## [1] FALSE`

the design advocates stopping. It is important to note that, even though the design has stopped, it still recommends that a dose be studied at the next trial phase:

`%>% recommended_dose() fit `

`## [1] 3`

This is in contrast to the scenario where a trial is stopped because
all doses are inappropriate. In this scenario, the dose recommendation
would be `NA`

. We will encounter this in examples below.

Another common approach is to stop a dose-finding experiment when a given number of patients have been treated at a particular dose.

Continuing with our CRM model, to stop when nine patients have been treated at the dose that is about to be recommended again, we use:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_n_at_dose(n = 9, dose = 'recommended')
```

We can observe how this alters the dose-selection model. Here we see six patients treated at dose 2:

`<- model %>% fit('1NNN 2TNN 2NTN') fit `

The model recommends that dose 2 should be given to more patients:

`%>% continue() fit `

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 2`

If the next cohort results in dose 2 being recommended yet again, i.e. to bring the total number of patients at dose 2 to nine or more, the model stops:

```
<- model %>% fit('1NNN 2TNN 2NTN 2NNN')
fit %>% continue() fit
```

`## [1] FALSE`

`%>% recommended_dose() fit `

`## [1] 2`

In this scenario, dose 2 is the final recommended dose and the trial stops gracefully at a pre-specified stopping rule.

This behaviour can also be configured to stop when any dose has been
given *n* times:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_n_at_dose(n = 9, dose = 'any')
```

or when a particular dose-level has been given *n* times:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_n_at_dose(n = 9, dose = 3)
```

Naturally, you can combine this behaviour with other behaviours. The following model stops the trial when nine patients have been evaluated at the recommended dose or when 21 patients have been treated in total, whichever occurs first:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_n_at_dose(n = 9, dose = 'recommended') %>%
stop_at_n(n = 21)
```

The two stopping mechanisms above scrutinise the number of patients treated. In many situations, this will be valuable. However, in other situations, we might want to stop when a threshold amount of statistical information is obtained. One way to achieve this is to stop when the confidence interval or credible interval for the probability of toxicity at a dose is covered by a specified range.

For instance, we know that the BOIN design seeks a target toxicity level, and we have used a target of 25% in our examples. We might say that we are sure enough about the recommended dose when the associated 90% credible interval (because BOIN is a Bayesian design) of the toxicity probability falls in the region 10% - 40%.

```
<- get_boin(target = target, num_doses = 5) %>%
model stop_when_tox_ci_covered(dose = 'recommended', lower = 0.10, upper = 0.4)
```

Say that we observe the following trial path:

```
<- model %>%
fit fit('1NNN 2NTN 2TNN 2NNN 2NNT 2NTN 2NNN 2TNN')
```

The design recommends dose 2 and it also advocates stopping:

`%>% recommended_dose() fit `

`## [1] 2`

`%>% continue() fit `

`## [1] FALSE`

This is because the lower bound of the 90% interval for the probability of toxicity at dose 2 is at least 10%:

`%>% prob_tox_quantile(p = 0.05) fit `

```
## 1 2 3 4 5
## 0.0 0.1 NA NA NA
```

and the upper bound is no more than 40%:

`%>% prob_tox_quantile(p = 0.95) fit `

```
## 1 2 3 4 5
## 0.1 0.4 NA NA NA
```

It may be intersting to note that our CRM model would not stop in this scenario:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_tox_ci_covered(dose = 'recommended', lower = 0.10, upper = 0.4)
<- model %>%
fit fit('1NNN 2NTN 2TNN 2NNN 2NNT 2NTN 2NNN 2TNN')
%>% continue() fit
```

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 2`

This is because the lower bound of the 90% CI falls slightly outside the sought range:

`%>% prob_tox_quantile(p = 0.05) fit `

`## [1] 0.04876626 0.09809797 0.24712623 0.39695491 0.59744927`

As before, we can specify `dose = 'recommended'`

,
`dose = 'any'`

, or a particular numerical dose-level with
`dose = 3`

, for example.

It should be appreciated that this approach only works when the underlying model extends a way of calculating quantiles and uncertainty intervals. The 3+3 lacks a statistical foundation and does not offer quantiles:

```
get_three_plus_three(num_doses = 5) %>%
fit('1NNN 2NTN') %>%
prob_tox_quantile(p = 0.05)
```

`## [1] NA NA NA NA NA`

The stopping rules considered so far stop a trial and recommend a dose once some critical threshold of information is obtained. We will naturaly want to stop if all doses are too toxic.

We saw above that some model-based dose-finding approaches can calculate quantiles. We can take this idea further and advocate stopping when there is sufficient evidence that the toxicity probability at some dose exceeds a critical threshold. In such circumstances, no dose will be recommended because all doses of the treatment will be deemed to be excessively toxic.

Let us set up a rule to stop and recommend no dose if the probability of toxicity at the lowest dose is too high:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.7)
```

The above examples stops when 70% of the probability mass or posterior distribution of the probability of toxicity at dose 1 exceeds 35%. With an isolated toxicity incidence at dose 1, the model advocates continuing at dose 1:

```
<- model %>% fit('1NTN')
fit %>% continue() fit
```

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 1`

This is because the probability that the toxicity rate exceeds 35% is less than 70%:

`%>% prob_tox_exceeds(0.35) %>% round(2) fit `

`## [1] 0.35 0.53 0.82 0.95 1.00`

However, with material additional toxicity at dose 1, the design now advocates stopping:

```
<- model %>% fit('1NTN 1TTT')
fit %>% continue() fit
```

`## [1] FALSE`

Furthermore, no dose is recommended:

`%>% recommended_dose() fit `

`## [1] NA`

This is because we are now at least 70% sure that the lowest dose is too toxic:

`%>% prob_tox_exceeds(0.35) %>% round(2) fit `

`## [1] 0.87 0.95 1.00 1.00 1.00`

Once again, we can specify `dose = 'recommended'`

,
`dose = 'any'`

, or a particular numerical dose-level with
`dose = 3`

, for example. We also require that the underlying
model supports the calculation of quantiles. BOIN supports this
fucntionality:

```
<- get_boin(target = target, num_doses = 5) %>%
model stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.7)
<- model %>% fit('1NTN 1TTT')
fit %>% continue() fit
```

`## [1] FALSE`

`%>% recommended_dose() fit `

`## [1] NA`

`%>% prob_tox_exceeds(0.35) %>% round(2) fit `

`## [1] 0.95 NA NA NA NA`

but a non-statistical method like 3+3 does not.

We have looked at many behaviours that provide stopping. We can also look at some behaviours that delay stopping.

We might want to guarantee that we treat at least *n* patients
at a dose before we permit a dose-finding trial to stop. For instance,
we might not feel comfortable recommending a dose for the next phase of
study if it has only been evaluated in a small number of patients.

It makes sense for this behaviour to be used with a design that would otherwise stop. Let us say that we would normally like to stop after 18 patients have been treated. However, we will also demand that at least 6 patients be treated at the recommended dose before stopping is allowed, irrespective the overall sample size. We specify:

```
<- get_boin(target = target, num_doses = 5) %>%
model stop_at_n(n = 18) %>%
demand_n_at_dose(n = 6, dose = 'recommended')
```

In the following situation:

```
<- model %>% fit('1NNN 2NNT 3NTN 3NNN 4TTN 3NTT')
fit %>% continue() fit
```

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 2`

the design advocates continuing at dose 2 even though 18 patients
have been evaluated. This is because the `demand_n_at_dose`

function is overriding the stopping behaviour of `stop_at_n`

.
It is requesting that the trial continue at dose 2 instead of stopping
with only three patients treated at the nominal recommended dose.

It is important to recognise that the order of the functions matters. If we flip the order of the constraints in the example above, the outcome is different:

```
<- get_boin(target = target, num_doses = 5) %>%
model demand_n_at_dose(n = 6, dose = 'recommended') %>%
stop_at_n(n = 18)
<- model %>% fit('1NNN 2NNT 3NTN 3NNN 4TTN 3NTT')
fit %>% continue() fit
```

`## [1] FALSE`

`%>% recommended_dose() fit `

`## [1] 2`

Now the `stop_at_n`

constraint overrides the action of
`demand_n_at_dose`

to halt the trial when *n=18*, even
though only three patients have been evaluated at dose 2. It overrides
because it comes later in the decision chain. Users should be aware that
commands that come later take precedence.

Once again, we can specify `dose = 'recommended'`

,
`dose = 'any'`

, or a particular numerical dose-level with
`dose = 3`

, for example.

In summary, the `demand_n_at_dose`

function delays
stopping in a scenario when a dose is being selected.

In contrast to `demand_n_at_dose`

, the
`try_rescue_dose`

function delays stopping in a scenario
where no dose is going to be selected. It overrides a decision to stop
and recommend no dose when fewer than *n* patients have been
evaluated at a given dose. Thus, it provides a facility to ensure that
some “rescue” dose has been tried before stopping is allowed.

This is another function where effective demonstration requires a design that would normally stop. Let us say that we will stop if we are 80% sure that the toxicity rate at the lowest dose exceeds 35%. But before we stop, we want to ensure that at least two patients have been evaluated at the lowest dose. We write:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model stop_when_too_toxic(dose = 1, tox_threshold = 0.35, confidence = 0.8) %>%
try_rescue_dose(dose = 1, n = 2)
```

Then, even when this design sees some major toxicity at dose 2:

```
<- model %>% fit('2TTT')
fit %>% continue() fit
```

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 1`

the design will not advocate stopping, even though the posterior confidence that the tox rate at dose 1 exceeds 35% is greater than 80%:

`%>% prob_tox_exceeds(0.35) fit `

`## [1] 0.8673669 0.9307674 0.9857421 0.9971830 0.9998310`

Once two patients are seen at dose 1, stopping can be countenanced. If those two patients tolerate treatment at dose 1:

```
<- model %>% fit('2TTT 1NN')
fit %>% continue() fit
```

`## [1] TRUE`

`%>% recommended_dose() fit `

`## [1] 1`

then stopping is not advocated because the posterior belief is now that dose 1 is not excessively toxic:

`%>% prob_tox_exceeds(0.35) fit `

`## [1] 0.6683818 0.8195981 0.9668375 0.9951862 0.9998694`

However, if even one of those patients at dose 1 experiences toxicity:

```
<- model %>% fit('2TTT 1NT')
fit %>% continue() fit
```

`## [1] FALSE`

`%>% recommended_dose() fit `

`## [1] NA`

Then the trial stops and no dose is recommended.

The `try_rescue_dose`

function allows researchers to
rescue situations where otherwise sensible stopping criteria may prove
too sensitive to chance events in very small sample sizes.

This function implements the convex infinite bounds penalisation
(CIBP) criterion of Mozgunov and Jaki (2020) that adjusts the way doses
are selected in CRM trials. Their method is mindful of the uncertainty
in the estimates of the probability of toxicity and uses an asymmetry
parameter, 0 < *a* < 2, to penalise escalation to risky
doses. The method alters the way doses are selected but not when the
trial should stop. For *a < 1*, the criterion penalises toxic
doses more heavily, making escalation decisions more conservative.

To add the behaviour to a dose-finding design, we run:

```
<- get_dfcrm(skeleton = skeleton, target = target) %>%
model select_dose_by_cibp(a = 0.3)
```

The model is then fit to outcomes in the usual way:

```
%>%
model fit('1NTN') %>%
recommended_dose()
```

`## [1] 1`

We have described at length above the flexible methods that
`escalation`

provides to specify dose-escalation designs and
tailor trial behaviour. Once designs are specified, we can investigate
their operating characteristics by simulation using the
`simulate_trials`

function, and efficiently compare designs
using Sweeting et al. (2023)’s method in `simulate_compare`

.
We can also exhaustively calculate dose recommendations for future
cohorts using the `get_dose_paths`

function. Both of these
topics are the topics of full vignettes so please check them out.

```
# Install the latest official version from CRAN with:
install.packages("escalation")
# Alternatively, install the latest code from GitHub:
::install_github("brockk/escalation") devtools
```

I plan to add model-fitting functions for EWOC via ewoc, further methods for phase I/II designs, and perhaps also methods for dual agents.

I want to investigate adding some further stopping functions like those researched by Zohar and Chevret (2001).

Finally, I will investigate adding time-to-event versions of the designs presented here, the so-called TITE designs. These will require a different approach to simulation because cohorts no longer apply.

This package is still in active development. There are thousands of unit tests run each time the package code is updated. However, that certainly does not mean that the code is bug free. You should always be on the defensive. This software is offered with no guarantee at all. If you have found a bug, please drop me a line and also log it here:

https://github.com/brockk/escalation/issues

If you want help using the package, feel free to contact me by email.

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.

Brock, Kristian, Lucinda Billingham, Mhairi Copland, Shamyla Siddique,
Mirjana Sirovica, and Christina Yap. 2017. “Implementing the EffTox
Dose-Finding Design in the Matchpoint Trial.” *BMC Medical Research
Methodology* 17 (1): 112. https://doi.org/10.1186/s12874-017-0381-x.

Cheung, Ken. 2013. *Dfcrm: Dose-Finding by the Continual Reassessment
Method*. https://CRAN.R-project.org/package=dfcrm.

Guo, Wentian, Sue-Jane Wang, Shengjie Yang, Henry Lynn, and Yuan Ji.
2017. “A Bayesian Interval Dose-Finding Design addressingOckham’s Razor:
mTPI-2.” *Contemporary Clinical Trials* 58: 23–33.

Ji, Yuan, Yisheng Li, and B. Nebiyou Bekele. 2007. “Dose-finding in phase I clinical trials based on toxicity
probability intervals.” *Clinical Trials* 4 (3): 235–44.
https://doi.org/10.1177/1740774507079442.

Ji, Yuan, Ping Liu, Yisheng Li, and B. Nebiyou Bekele. 2010. “A modified toxicity probability interval method for
dose-finding trials.” *Clinical Trials* 7 (6): 653–63. https://doi.org/10.1177/1740774510382799.

Korn, Edward L., Douglas Midthune, T. Timothy Chen, Lawrence V.
Rubinstein, Michaele C. Christian, and Richard M. Simon. 1994. “A
Comparison of Two Phase I Trial Designs.” *Statistics in
Medicine* 13 (18): 1799–1806. https://doi.org/10.1002/sim.4780131802.

Le Tourneau, Christophe, J. Jack Lee, and Lillian L. Siu. 2009. “Dose
Escalation Methods in Phase i Cancer Clinical Trials.” *Journal of
the National Cancer Institute* 101 (10): 708–20. https://doi.org/10.1093/jnci/djp079.

Lin, Ruitao, Yanhong Zhou, Fangrong Yan, Daniel Li, and Ying Yuan. 2020.
“BOIN12: Bayesian Optimal Interval Phase i/II Trial Design for
Utility-Based Dose Finding in Immunotherapy and Targeted Therapies.”
*JCO Precision Oncology* 4: 1393–1402.

Liu, Suyu, and Ying Yuan. 2015. “Bayesian Optimal Interval Designs for
Phase I Clinical Trials.” *Journal of the Royal Statistical Society:
Series C (Applied Statistics)* 64 (3): 507–23. https://doi.org/10.1111/rssc.12089.

Mozgunov, Pavel, and Thomas Jaki. 2020. “Improving Safety of the
Continual Reassessment Method via a Modified Allocation Rule.”
*Statistics in Medicine* 39 (7): 906–22. https://doi.org/10.1002/sim.8450.

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.

Sweeting, Michael, Daniel Slade, Daniel Jackson, and Kristian Brock.
2023. “Potential Outcome Simulation for Efficient Head-to-Head
Comparison of Adaptive Dose-Finding Designs.” *Preprint*.

Thall, Peter F, and John D Cook. 2004. “Dose-Finding Based on
Efficacy–Toxicity Trade-Offs.” *Biometrics* 60 (3): 684–93.

Wages, Nolan A, and Christopher Tait. 2015. “Seamless Phase i/II
Adaptive Design for Oncology Trials of Molecularly Targeted Agents.”
*Journal of Biopharmaceutical Statistics* 25 (5): 903–20.

Yuan, Ying, and Suyu Liu. 2018. *BOIN: Bayesian Optimal INterval
(BOIN) Design for Single-Agent and Drug- Combination Phase i Clinical
Trials*. https://CRAN.R-project.org/package=BOIN.

Zohar, Sarah, and Sylvie Chevret. 2001. “The Continual Reassessment
Method: Comparison of Bayesian Stopping Rules for Dose-Ranging Studies.”
*Statistics in Medicine* 20 (19): 2827–43. https://doi.org/10.1002/sim.920.