---
title: "Toxicity Probability Interval Design"
output: 
  rmarkdown::html_vignette:
    df_print: tibble
bibliography: library.bib
vignette: >
  %\VignetteIndexEntry{Toxicity Probability Interval Design}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7, fig.height = 5
)

options(rmarkdown.html_vignette.check_title = FALSE)
```

# Introduction
The toxicity probability interval (TPI) design was introduced by @Ji2007.
It is one of a series of dose-finding trial designs that works by partitioning the probability of toxicity into a set of intervals.
These designs make dose-selection decisions that are determined by the interval in which the probability of toxicity for the current dose is believed to reside.

# Summary of the TPI Design
Core to this design is a beta-binomial Bayesian conjugate model.
For hyperparameters $\alpha$ and $\beta$, let the probability of toxicity at dose $i$ be $p_i$, with prior distribution

$$p_i \sim Beta(\alpha, \beta).$$

If $n_i$ patients have been treated at dose $i$, yielding $x_i$ toxicity events, the posterior distribution is 

$$ p_i | data \sim Beta(\alpha + x_{i}, \beta + n_{i} - x_{i}).$$

Using this distribution, let the standard deviation of $p_i$ be denoted by $\sigma_i$.
The design seeks a dose with probability of toxicity close to some pre-specified target level, $p_T$.
The entire range of possible values for $p_i$ can be broken up into the following intervals: 

* The underdosing interval (UI), defined as $(0, p_{T} - K_{2} \sigma_{i})$;
* The equivalence interval (EI), defined as $(p_{T} - K_{2} \sigma_{i}, p_{T} + K_{1} \sigma_{i})$;
* The overdosing interval (OI), defined as $(p_{T} + K_{1} \sigma_{i}, 1)$;

for pre-specified model constants, $K_1, K_2$.
These intervals are  mutally-exclusive and mutually-exhaustive, meaning that every possible probability belongs to precisely one of them.
In other words, these intervals form a partition of the probability space, $(0, 1)$.

Using the posterior distribution, we can calculate the three probabilities

$$p_{UI} = Pr(p_i \in \text{UI}), \enspace p_{EI} = Pr(p_i \in \text{EI}), \enspace p_{OI} = Pr(p_i \in \text{OI}).$$

By definition, $p_{UI} + p_{EI} + p_{OI} = 1$.
The logical action in the dose-finding trial depends on which of these three probabilities is the greatest. 
If $p_{UI} > p_{EI}, p_{OI}$, then the current dose is likely an underdose, so our desire should be to escalate dose to $i+1$.
In contrast, if $p_{OI} > p_{UI}, p_{EI}$, then the current dose is likely an overdose and we will want to de-escalate dose to $i-1$ for the next patient.
If $p_{EI} > p_{UI}, p_{OI}$, then the current dose is deemed sufficiently close to $p_T$ and we will want to stay at dose-level $i$.

Further to these rules regarding dose-selection, the following rule is used to avoid recommending dangerous doses.
A dose is deemed inadmissible for being excessively toxic if

$$ Pr(p_{i} > p_{T} | data) > \xi,$$

for a certainty threshold, $\xi$.
If a dose is excluded by this rule, it should not be recommended by the model.
Irrespective the probabilities $p_{UI}, p_{EI}, p_{OI}$, the design will recommend to stay at dose $i$ rather than escalate to a dose previously identified as being inadmissible.
Furthermore, the design will advocate stopping if the lowest dose is inferred to be inadmissible.

In their paper, the authors demonstrate acceptable operating performance using $\alpha = \beta = 0.005$, $K_{1} = 1$, $K_{2} = 1.5$ and $\xi = 0.95$.
See @Ji2007 and @Ji2017 for full details.


# Implementation in `escalation`

To demonstrate the method, let us fit the design to a cohort of three patients treated at the first of five doses, one of whom experienced toxicity.
For illustration, use the parameters chosen in @Ji2007:

```{r, message=FALSE}
library(escalation)

model <- get_tpi(num_doses = 5, target = 0.3, alpha = 0.005, beta = 0.005, 
                 k1 = 1, k2 = 1.5, exclusion_certainty = 0.95)

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

The dose recommended for the next cohort is

```{r}
fit %>% recommended_dose()
```

Unsurprisingly, the design does not advocate escalation.
Importantly, the modest toxicity seen so far is not enough to render dose 1 inadmissible:

```{r}
fit %>% dose_admissible()
```

Let us imagine that we treat another two cohorts at dose 1, and see no toxicity:

```{r}
fit <- model %>% fit('1NNT 1NNN 1NNN') 
```

Now, the design is happy to escalate:

```{r}
fit
```

Let us imagine, however, that dose 2 is surprisingly toxic, yielding three toxicities:

```{r}
fit <- model %>% fit('1NNT 1NNN 1NNN 2TTT') 
```

Despite the low sample size, the statistical model believes that dose 2 is excessively toxic:

```{r}
fit %>% prob_tox_exceeds(threshold = 0.25)
```

and thus inadmissible:

```{r}
fit %>% dose_admissible()
```

Note that since dose 2 is believed to be inadmissible, the assumption of monotonically increasing toxicity means that the doses higher than dose 2 are excessively toxic too.


## Dose paths

In Table 1 of their publication, @Ji2007 list some model recommendations conditional on hypothesised numbers of toxicities in cohorts of varying size.
We can use the `get_dose_paths` function, for instance, to calculate exhaustive model recommendations after a single cohort of three is evaluated at dose 2:

```{r, message=FALSE}
paths <- model %>% get_dose_paths(cohort_sizes = c(3), next_dose = 2)

library(dplyr)
as_tibble(paths) %>% select(outcomes, next_dose) %>% print(n = 100)
```

This table confirms the advice following a cohort of three to de-escalate if 2 or 3 toxicities are seen, to escalate if no toxicity is seen, otherwise to remain.
Note that the recommendations would actually have been the same if `next_dose = 3` or `next_dose = 4`.
In this five-dose setting, they would naturally have been slightly different if `next_dose = 1` or `next_dose = 5` because we cannot de-escalate below dose 1 or escalate above dose 5.

We can easily visualise those paths with:
```{r}
graph_paths(paths)
```

For more information on working with dose-paths, refer to the dose-paths vignette.


## Simulation

@Ji2007 present simulations in their Table 2, comparing the performance of their TPI method to other designs.
We can use the `simulate_trials` function to reproduce the operating characteristics.

Their example concerns a clinical trial of eight doses that targets 25% toxicity.
We must respecify the `model` object to reflect this.
They also elect to limit the trial to a sample size of $n=30$:

```{r}
model <- get_tpi(num_doses = 8, target = 0.25, k1 = 1, k2 = 1.5,
                 exclusion_certainty = 0.95) %>%
  stop_at_n(n = 30)
```

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

```{r}
num_sims <- 50
```

In real life, however, we would naturally run many thousands of iterations.
Their scenario 1 assumes true probability of toxicity:

```{r}
sc1 <- c(0.05, 0.25, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95)
```

at which the simulated behaviour is:
```{r}
set.seed(123)
sims <- model %>%
  simulate_trials(num_sims = num_sims, true_prob_tox = sc1, next_dose = 1)

sims
```

This reproduces their finding that dose 2 is overwhelmingly likely to be recommended, and that the sample size is virtually guaranteed to be 30, i.e. early stopping is unlikely.

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


# References
