---
title: "Functions For Clinical Trial Design"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Functions For Clinical Trial Design}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Introduction 

This vignette walks through functions in {clinfun} that facilitate the design of clinical trials. Examples using published clinical trials are included to illustrate the use of these functions in a real-world context.

To begin, install and load {clinfun}:

```{r setup}
library(clinfun)
```

# Phase II Trial Design

The overall purpose of phase II research is to determine whether a treatment is promising enough to warrant more extensive development. While phase I research assesses safety and clinical pharmacology, phase II research begins assessing the efficacy of a treatment in a well-defined population, while continuing to monitor the safety of the treatment. {clinfun} enables users to construct two common phase II trial designs, Simon's two-stage and single-stage, which are described in more detail below.

## Simon's Two-Stage

The goal of Simon’s two-stage design is to assess the efficacy of a treatment while minimizing the number of patients exposed to an ineffective treatment.

The trial begins by enrolling $n_1$ patients in stage 1. If $\leq r_1$ responses are observed, then early termination occurs, at which point the treatment is not recommended. Otherwise, $n_2$ patients are enrolled in stage 2 and the success of the treatment is based on all data accumulated. At the end of stage 2, if $\leq r$ responses have been observed in total, then the treatment is not recommended; if $>r$ responses have been observed in total, then the treatment is recommended.

When designing a Simon's two-stage trial, the key objective is to determine appropriate response threshold and sample size values, indicated by:

- $r_1$: Response threshold at the end of stage 1

- $r$: Final response threshold

- $n_1$: Stage 1 sample size

- $n$: Total sample size ($n_1 + n_2$)

### Design Parameters and Constraints

In order to determine final design components, the following parameters and constraints must be specified: 

- $p_0$ (`pu`): Treatment response rate considered unacceptable

- $p_1$ (`pa`): Treatment response rate considered desirable

- $\alpha$ (`ep1`): Type I error threshold

- $\beta$ (`ep2`): Type II error threshold

The final design components depend on the response rates considered "unacceptable" ($p_0$) and "desirable" ($p_1$) and must satisfy the desired error rate constraints ($\alpha$, $\beta$) for testing the following hypotheses:

- $H_0$: The true treatment response rate is less than or equal to some unacceptable level ($p \leq p_0$)

- $H_1$: The true treatment response rate is greater than or equal to some desirable level ($p \geq p_1$)

### Optimal vs. Minimax vs. Admissible Designs

Once the design parameters and constraints are provided, {clinfun} determines the final design components based on three different approaches: optimal, minimax, and admissible.

If early termination occurs, the sample size for the trial is $n_1$. If early termination does not occur, the sample size for the trial is $n$ (the maximum sample size for the trial).

Thus, the expected sample size under the null hypothesis is given by

$$ EN(p_0) = n_1 + (1 - PET(p_0)) \cdot n_2 $$

where $PET(p_0)$ is the probability of early termination under the null hypothesis.

The **optimal design** minimizes the expected sample size ($EN(p_0)$), whereas the **minimax design** minimizes the maximum sample size ($n_1 + n_2$). The optimal design typically has a smaller stage 1 sample size but more subjects overall if the trial proceeds to stage 2. The optimal design also typically has a higher $PET(p_0)$ than the minimax design. The minimax design can be preferable in certain scenarios: for example, if $EN(p_0)$ is relatively close to that of the optimal design and the patient accrual rate is slow, or if the patient population is heterogeneous and a small stage 1 is not desirable.

The **admissible design** can be thought of as a compromise between the optimal and minimax designs and is based on a Bayesian decision-theoretic criterion.

### Levatinib Phase II Trial Example

For this example, we will be looking at a phase II trial that assessed the efficacy of lenvatinib in patients with progressive, recurrent or metastatic adenoid cystic carcinoma (Tchekmedyian et al., 2019). Patients received lenvatinib 24 mg/day. Treatment was continued until disease progression, death, unacceptable toxicity, withdrawal of consent, or a decision based on the treating physician's discretion. Treatment response was specifically defined as overall response: achievement of either partial response or complete response, as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1.

The trial employed a Simon's two-stage design. An overall response rate of 5% was considered unacceptable, and an overall response rate of 20% was considered desirable. The type I error threshold used was 10%. The target power used was 90%, equivalent to a 10% type II error threshold.

Summarizing the design parameters and constraints from above, we have:

- $p_0$ (`pu`): 5% overall response rate

- $p_1$ (`pa`): 20% overall response rate

- $\alpha$ (`ep1`): 10% type I error threshold

- $\beta$ (`ep2`): 10% type II error threshold

Thus, the hypotheses used for testing are:

- $H_0$: The true overall response rate is less than or equal to 5% ($p \leq 0.05$)

- $H_1$: The true overall response rate is greater than or equal to 20% ($p \geq 0.2%$)

Now, using `ph2simon()`, specify these parameters and print the resulting object.

```{r}
# Specify the parameters and constraints
trial = ph2simon(0.05, 0.2, 0.1, 0.1)

# Print
trial
```

This output provides the key components ($r_1$, $r$, $n_1$, and $n$) as well as $EN(p_0)$ and $PET(p_0)$ for the selected optimal and minimax designs (shown in the first and second rows, respectively).

As expected, the optimal design has a smaller expected sample size ($EN(p_0)$; about 23 vs. 26), whereas the minimax design has a smaller maximum sample size ($n$; 32 vs. 37).

We can also visualize these results. Using the base `plot()`, plot the trial object.

```{r}
# Plot
plot(trial)
```

This output provides a visual representation of the maximum number of patients ($n$) and expected sample size ($EN(p_0)$) under various scenarios. O represents the optimal design, and M represents the minimax design.

The admissible design can be obtained by applying the `twostage.admissible()` function to the trial object.

```{r}
# Obtain admissible design
twostage.admissible(trial)
```

This output includes the admissible design as well as the optimal and minimax designs. In addition to the design components seen earlier, columns `qLo` and `qHi` give the range of probability values for which the particular design is admissible.

For this particular trial, Tchekmedyian et al. opted for the minimax design. Thus, 18 patients were enrolled in stage 1 ($n_1 = 18$). If no responses were observed ($\leq 0$ reduces to $= 0$ since the number of responses cannot be negative), then early termination would occur. Otherwise, if at least one response was observed, then 14 patients would be enrolled in stage 2 ($n_2 = n - n_1 = 32 - 18 = 14$). At the end of stage 2, if at least 4 responses had been observed in total, then the treatment would be recommended; if 3 or fewer responses had been observed in total, then the treatment would not be recommended ($r = 3$).

Ultimately, Tchekmedyian et al. observed four responses among the 18 patients enrolled in stage 1 and thus proceeded to stage 2, in which an additional 14 patients were enrolled. One response was observed among the stage 2 patients. Thus, at the end of the trial, 5 responses had been observed in total, and lenvatinib was recommended for further development in patients with progressive, recurrent or metastatic adenoid cystic carcinoma.

### Cases Where the Admissible Design Is a Logical Choice

As described above, the optimal design minimizes the expected sample size, the minimax design minimizes the maximum sample size, and the admissible design represents a compromise between the two. Sometimes, the admissible design has only a slightly higher expected sample size than the optimal design and only a slightly higher maximum sample size than the minimax design. In these cases, the admissible design is a logical choice because it achieves the benefits of the both designs with only negligible impacts on the two metrics.

To illustrate this, we can consider a hypothetical example where we specific the following design parameters and constraints:

- $p_0$ (`pu`): 25% overall response rate

- $p_1$ (`pa`): 45% overall response rate

- $\alpha$ (`ep1`): 5% type I error threshold

- $\beta$ (`ep2`): 10% type II error threshold

```{r}
# Specify the parameters and constraints
trial = ph2simon(0.25, 0.45, 0.05, 0.1)

# Obtain admissible design
twostage.admissible(trial)
```

As we can see, the admissible design has an expected sample size that is nearly the same as the optimal design's (34 vs. 33) and a maximum sample size that is nearly the same as the minimax design's (50 vs. 49). It is logical to accept one more patient on whichever metric is of primary interest to obtain a design that performs well across both metrics.

### References

[Simon R. (1989). Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10, 1-10.](https://pubmed.ncbi.nlm.nih.gov/2702835/)

[Jung SH, Carey M and Kim KM. (2001). Graphical Search for Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 22, 367-372.](https://pubmed.ncbi.nlm.nih.gov/11514038/)

[Jung SH, Lee T, Kim K, and George, SL. (2004). Admissible two-stage designs for phase II cancer clinical trials. Statistics in medicine 23(4), 561-569.](https://pubmed.ncbi.nlm.nih.gov/14755389/)

[Tchekmedyian V, Sherman EJ, Dunn L, et al. (2019) Phase II Study of Lenvatinib in Patients With Progressive, Recurrent or Metastatic Adenoid Cystic Carcinoma. J Clin Oncol. 37(18):1529-1537.](https://pubmed.ncbi.nlm.nih.gov/30939095/)

## Single-Stage

A single-stage design may be used instead of a two-stage design when an endpoint requires too much time to evaluate or when early stopping is less of a priority.

The trial enrolls $n$ patients. After all patients have completed the study, if $\leq r$ responses have been observed in total, then the treatment is not recommended; if $>r$ responses have been observed in total, then the treatment is recommended.

When designing a single-stage trial, the key objective is to select appropriate values of $n$ (sample size) and $r$ (response threshold at the end of the study). The response rate considered "unacceptable" (at which point a treatment would not be recommended) and the response rate considered "desirable" must be specified, and the values of $r$ and $n$ must be selected such that the desired error rate constraints are satisfied.

### Design Parameters and Constraints

In order to determine final design components, the following parameters and constraints must be specified:

- $p_0$ (`pu`): Treatment response rate considered unacceptable

- $p_1$ (`pa`): Treatment response rate considered desirable

- $\alpha$ (`ep1`): Type I error threshold

- $\beta$ (`ep2`): Type II error threshold

The final design components depend on the response rates considered "unacceptable" ($p_0$) and "desirable" ($p_1$) and must satisfy the desired error rate constraints ($\alpha$, $\beta$) for testing the following hypotheses:

- $H_0$: The true treatment response rate is less than or equal to some unacceptable level ($p \leq p_0$)

- $H_1$: The true treatment response rate is greater than or equal to some desirable level ($p \geq p_1$)

### Example

For this example, we will be looking at a phase II trial that assessed the efficacy of everolimus in patients with thymoma and thymic carcinoma previously treated with cisplatin-based chemotherapy (Zucali et al., 2018). Patients received everolimus 10 mg/day. Treatment was continued until disease progression, death, unacceptable toxicity, or study discontinuation for any other reason. Treatment response was specifically defined as disease control: achievement of either partial response, complete response, or stable disease, as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1.

The trial employed a single-stage design. A disease control rate of 40% was considered unacceptable, and a disease control rate of 60% was considered desirable. The type I error threshold used was 10%. The target power used was 90%, equivalent to a 10% type II error threshold.

Summarizing the design parameters and constraints from above, we have:

* $p_0$ (`pu`): 40% disease control rate

* $p_1$ (`pa`): 60% disease control rate

* $\alpha$ (`ep1`): 10% type I error threshold

* $\beta$ (`ep2`): 10% type II error threshold

Thus, the hypotheses used for testing are:

* $H_0$: The true disease control rate is less than or equal to 40% ($p \leq 0.4$)

* $H_1$: The true disease control rate is greater than or equal to 60% ($p \geq 0.6%$)

Using `ph2single()`, specify the parameters and print the resulting object.

```{r}
# Specify the parameters and constraints & print
ph2single(0.4, 0.6, 0.1, 0.1)
```

This output provides the key components ($n$ and $r$) for applicable designs as well as the type I and type II error rates corresponding to each of these designs.

Zucali et al. chose the first option listed, the design that enrolls the smallest number of patients ($n = 41$) that still satisfies the error rate constraints. The treatment would be recommended if at least 21 of the first 41 evaluable patients achieved disease control ($r = 20$). Ultimately, greater than 21 of these patients achieved disease control, and everolimus was recommended for further development in patients with thymoma and thymic carcinoma previously treated with cisplatin-based chemotherapy.

### References

[Zucali PA, De Pas T, Palmieri G, et al. (2018). Phase II Study of Everolimus in Patients With Thymoma and Thymic Carcinoma Previously Treated With Cisplatin-Based Chemotherapy. J Clin Oncol. 36(4):342-349.](https://pubmed.ncbi.nlm.nih.gov/29240542/)


# Constructing Stopping Rules

Stopping rules in clinical trials are a useful tool for risk management and can help ensure the protection of participants. Patient outcomes are monitored as the trial progresses, and the trial is terminated if certain conditions are met (eg, if the toxicity rate is too high). Stopping rules define the specific conditions under which the trial should be terminated.

General types of stopping rules include:

- **Stopping for toxicity:** The trial is stopped if the toxicity rate is too high. Terminating the trial minimizes the number of patients exposed to a treatment that is likely too toxic relative to its potential benefits.

- **Stopping for futility:** The trial is stopped if the response rate is too low. Terminating the trial minimizes the number of patients exposed to a potentially ineffective treatment.

A trial may contain one or multiple types of stopping rules. The choice of stopping rules is highly dependent on factors such as disease area, the clinical outcomes of interest, stage of development, and characteristics of the treatment itself.

There are various statistical methods available to construct stopping rules.

This section describes the specific types of stopping rules that can be constructed using {clinfun}.

## Stopping for Toxicity (For Phase II and Phase III Designs)

`toxbdry()` computes a stopping rule for toxicity that can be implemented in phase II and III designs. For example, a stopping rule generated by this function could be layered on top of a design generated by `ph2simon()` or `ph2single()`.

## Stopping for Futility (For Phase II Single-Stage Designs)

A Simon's two-stage design effectively has a stopping rule for futility built in (evaluated at the end of stage 1). However, there are scenarios in which a single-stage design with an added stopping rule for futility is chosen instead of a two-stage design. When the response outcome takes a long time to assess (eg, complete response rate at 30 months) and accrual for a trial is slow, assessing efficacy on a continual basis is preferable to waiting for the completion of stage 1.

`futilbdry()` computes a stopping rule for futility that is specifically intended to be implemented in phase II single-stage designs. Thus, a stopping rule generated by this function would be layered on top of a design generated by `ph2single()`.

# Calculating Sample Size, Effect Size, and  Power

## Fisher's Exact Test

{clinfun} offers a set of functions to determine sample sizes, effect sizes, and power based on Fisher's exact tests. Below is a table summarizing the functions based on Fisher's exact test.

|Function       | Details                                             |
|---------------|-----------------------------------------------------|
|`fe.ssize()`   |Returns a 2x3 matrix with Casagrande, Pike, Smith (CPS) and Fisher's exact sample sizes with power.|
|`fe.mdor()`    |Returns a 3x2 matrix with Schlesselman, CPS and Fisher's exact minimum detectable odds ratios and the corresponding power.|
|`fe.power()`   |Returns a Kx2 matrix with probabilities (p2) and exact power.|
|`CPS.ssize()`  |Returns CPS sample size, which is a very close to the exact. Use this for small differences p2-p1 (hence large sample sizes) to get the result instantaneously.|
|`mdrr()`       |Computes the minimum detectable P(resp given marker+) and P(resp given marker-) configurations when total sample size (n), P(response) (presp) and proportion of subjects who are marker positive (cprob) are specified.|
|`or2pcase()`   |Gives the probability of disease among the cases for a given probability of disease in controls (pcontrol) and odds ratio (OR).|

### Example

```{r}
fe.ssize(p1 = 0.2, p2 = 0.3, power = 0.8)
```

*CAUTION:* As we can see below, when we use base R `power.prop.test()`, we obtain an incorrect result due to normal approximation.

```{r}
power.prop.test(p1 = 0.2, p2 = 0.3, power = 0.8)
```

### References

[Casagrande JT, Pike MC, Smith PG. (1978). An improved approximate formula for calculating sample sizes for comparing two binomial distributions. Biometrics 34, 483-486.](https://pubmed.ncbi.nlm.nih.gov/719125/)

Fleiss, JL. (1981). Statistical Methods for Rates and Proportions.

[Schlesselman JJ. (1987) Re: Smallest detectable relative risk with multiple controls per case. Am J Epidemiol 125(2), 348.](https://pubmed.ncbi.nlm.nih.gov/3812441/)
