---
title: "UnplanSimon"
author:
- Yunhe Liu
- Haitao Pan
output:
  html_document:
    toc: true
    toc_depth: '3'
    df_print: paged
vignette: >
  %\VignetteIndexEntry{UnplanSimon}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

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

## Introduction
The `UnplanSimon` package serves three purposes, including:

* Providing an Adaptive Threshold Simon Design (ATS Simon) method for Simon's 
two-stage design in oncology trials when the realized sample sizes in the $1^{st}$ 
and/or $2^{nd}$ stage(s) are different from the planned sample sizes in the $1^{st}$ 
and/or $2^{nd}$ stage(s). The proposed ATS Simon design tries to follow 
sample sizes of the original design, to that end, this design updates the original
thresholds of $(r_1, r)$ in the $1^{st}$ and/or the $2^{nd}$ stage(s) to satisfy the 
type I error rate as the original planned design (note: power will decrease if the
realized sample size is smaller than the original one).

* Providing an Adaptive Threshold and Sample Size Simon Design (ATSS Simon) method 
for Simon's two-stage design in oncology trials when the realized sample sizes 
in the $1^{st}$ and/or $2^{nd}$ stage(s) are different from the planned 
sample sizes in the $1^{st}$ and/or $2^{nd}$ stage(s). The proposed ATSS 
Simon method updates not only the original threshold of $(r_{1}^{*}, r^*)$ 
but the original sample sizes of $(n_{1}^{*}, n^*)$ to satisfy the type I error 
rate and power requirements as the original planned design (note:	unlike the 
ATS Simon design, the sample size here will also be updated to satisfy the 
original power). In addition, the ATSS Simon design also satisfies the other 
criteria as in the originally planned design, such as minimizing the average 
sample size under the null hypothesis *$H_0$*.

* Providing	comprehensive	post-trial inference tools at the end of the trial 
for Simon's two-stage design when under- or over-enrollment occurs. This 
includes the computation of point estimates, confidence intervals, and 
p-values for the proposed ATS and ATSS Simon design methods.

This vignette introduces functionalities in {UnplanSimon} tailored for	
designing	single-arm clinical trials using the proposed method. Examples featuring
Simon's	two-stage designs from {clinfun} demonstrate the application of these 
functions within our package.

To begin, install and load {UnplanSimon} and {clinfun}:

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

## Overview of Simon's Two-Stage Design
Simon's two-stage design is a statistical method used in clinical trials, particularly in Phase II studies, to evaluate the efficacy of a new treatment while minimizing the number of patients exposed
to potentially ineffective treatments. It consists of two stages:  

Stage 1:  
\-A predetermined number of patients ($n_1$) are enrolled and treated.  
\-If the number of responses (positive outcomes) in these patients meets or exceeds a
certain threshold ($r_1$), the trial continues to the second stage.  
\-If the number of responses is equal to or below this threshold, the trial is stopped early 
for futility, concluding that the treatment is ineffective.  

Stage 2:  
\-An additional number of patients ($n_2$) are enrolled and treated.  
\-The total number of responses from both stages is then evaluated against a 
final threshold ($r$).  
\-If the total number of responses exceeds the final threshold, the 
treatment is considered promising for further study.  
\-If not, the treatment is considered ineffective.  

Simon's two-stage design aims to reduce the number of patients receiving ineffective
treatment while ensuring that promising treatments are identified efficiently. It balances
the need for early stopping in case of futility with the need for sufficient data to make a
reliable decision about the treatment's efficacy (Simon, 1989).

From the above description, when designing a trial using Simon's two-stage design, 
there are four key design parameters that need to be specified:

- $r_1$: Threshold in stage 1

- $r$: Threshold in stage 2

- $n_1$: Number of patients in stage 1

- $n$: Number of patients in stages 1 and 2, e.g., ($n_1 + n_2$)

### Design Parameters and Constraints
In order to determine the design parameters for Simon's two-stage design, the followings 
are required:  

- $p_0$: Unacceptable efficacy rate

- $p_1$: Desirable efficacy rate

- $\alpha$: Type I error rate

- $\beta$: Type II error rate

That is,  

- $H_0$: 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 1 (Designing a Study Using Simon's Two-Stage Design)

One trial employed Simon's two-stage design, considering an overall response rate
of 25\% as unacceptable and 45\% as desirable. The trial set a type I error rate at
10\% and aimed for a target power of 90\%, corresponding to a 10\% type II error rate.

From the above, we have:

- $p_0$: 25% overall response rate

- $p_1$: 45% overall response rate

- $\alpha$: 10% type I error rate

- $\beta$: 10% type II error rate

Thus, the hypotheses used for testing are:

- $H_0$: The true overall response rate is less than or equal to 25% ($p \leq 0.25$)

- $H_1$: The true overall response rate is greater than or equal to 45% ($p \geq 0.45%$)

Now, using `ph2simon()` function from clinfun R package, specify these 
parameters and print the resulting object.

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

This output provides the computed design parameters ($r_1$, $r$, $n_1$, $n$) as 
well as $EN(p_0)$ and $PET(p_0)$ for three designs, optimal, minimax, and admissible designs.

As expected, the optimal design has the smallest expected sample size under the
Null ($EN(p_0)$ = 28.36), whereas the minimax design has the smallest maximum 
sample size ($n$ = 39). 

## Adaptive Threshold Simon Design (ATS Simon)
In single arm phase II studies, under-enrollment or over-enrollment can be 
common issues when using Simon’s two-stage design. For rare diseases, it can be 
particularly challenging to recruit additional patients quickly in cases of 
under-enrollment. Therefore, we propose an Adaptive Threshold Simon design to 
assist clinical investigators in making decisions based on the actual sample size 
instead of the planned one while still adhering to the original design framework 
when under- or over-enrollment occurs.  

A thorough literature review identified numerous clinical trials 
utilizing Simon's two-stage designs, where the proportion of patients 
classified as inevaluable for response surpassed thresholds of 20%, 30%, and even 40%. 
The factors contributing to inevaluability included patients failing to complete 
the requisite number of cycles for response evaluation, being deemed ineligible 
upon central review, early withdrawal due to noncompliance, among other 
considerations (Ji, 2022), leading to under-enrollment during at Go/NoGo decision-making	
timepoint in studies. Over-enrollment in multi-site trials often occurs due
to variations in patient recruitment rates across	different	locations.

It is important to note that deviations our	methods	address, such as 
under-enrollment and over-enrollment,	are *incidental* or	*non-informative*, 
indicating they arise from unforeseen	circumstances	or factors rather than 
deliberate bias or systematic error. 

We identify an updated $1^{st}$ stage threshold $r_1^{*}$ based on the actual 
$1^{st}$ stage sample size $n_{1}^*$ such that the updated design's probability of 
early termination (PET) under the null hypothesis is the closest to the original 
PET under the null hypothesis.
$$ P\left(Y_{1} \leq r_1^* \mid n_{1}^{*}, p_{0}\right) \approx P\left(Y_{1} \leq r_1 \mid n_{1}, p_{0}\right)$$
The primary objective of Simon's two-stage design, as well as our proposed 
ATS or ATSS Simon design, in single arm phase II studies, is to identify and 
eliminate ineffective drugs as early as possible, Therefore, in cases of 
deviations in sample size, the updated type I error rate ($\alpha$) of a new 
method (including the operating characteristic, like probability of early 
termination) should still be close to the original one.  

In the field of group sequential design, the alpha-spending function is a natural
way to reallocate the type I error rate ($\alpha$). We have adopted this method 
in our proposed ATS Simon's design.  

Based on the identified $r_1^{*}$, we then define the alpha-spending function based
on the actual total sample size ($n^{*}$). Specifically, we use the Lan-DeMets 
spending function (Lan-DeMets et al., 1983).

$$
\alpha(n^{*}) = 
\begin{cases}
{2-2 \Phi\left(\frac{z_{1-\alpha/2}}{(n^*/n)^{1/2}}\right)} & \text{if } n^{*} \leq n \\
\alpha                                                            & \text{if } n^{*} > n
\end{cases}
$$  
where $\alpha$ here is the type I error rate in the original Simon 
two-stage design.  

We can see, if the actual sample size $n^{*} > n$, the updated alpha $\alpha(n^{*})$
will be at most $\alpha$ (the original one) and if the actual total sample size
$n^{*} < n$, the $\alpha(n^{*})$ may be smaller than $\alpha$ due to smaller sample
size. Therefore, ATS Simon's design can control the type I error rate at or below 
the original level. Based on the above information, we identify a smallest 
integer $r^{*}$ as the threshold at $2^{nd}$ stage such that
$$ P\left(Y_{1} > r_1^{*}, Y > r^{*} \mid n_{1}^{*}, n^{*}, p_{0}\right) 
\approx \alpha(n^{*})$$

Here, $n_1^{*}$ denotes as the actual sample size at the $1^{st}$ stage;
$n^{*}$ denotes as the actual sample size of the two stages.  

Rationale of identifying a smallest integer $r^{*}$ is that probability
$P\left(Y_{1} > r_1^{*}, Y > r^{*} \mid n_{1}^{*}, n^{*}, p\right)$ is a 
decreasing function of the threshold $r^{*}$ in stage 2. So, if we find such a 
$r^{*}$ under $p_0$ and based on the fact of $\alpha(n^{*}) \leq \alpha$, the 
type I error rate is well controlled. Since the trends of alpha and power are 
the same (i.e., if alpha increases, power also increases), if alpha is close to 
the original value, the power will also be close to the original value. This 
means we can simultaneously maintain the original power or minimize power loss. 
That is, if we select a larger $r^{*}$ instead of the smallest one, we cannot
minimize the power loss.  

### Example 2 (ATS Simon design for Addressing Under-or Over-Enrollment in Simon's Two-Stage Design based on Alpha Spending Function)

#### Example 2.1 (Under-enrollment in $1^{st}$ stage)
Firstly, let's introduce the usage of `ATS_Design()`. Under-enrollment in the 
first stage is more critical based on our understanding, so we will focus on this 
scenario for illustration purposes. In the optimal Simon’s two-stage design 
described in Example 1, the planned sample size for the $1^{st}$ and $2^{nd}$ 
stages are 14 and 30, respectively.  

Suppose we currently have outcome data for only 11 patients in the $1^{st}$ stage
and assume the $2^{nd}$ stage sample size for evaluable patients remains as 
planned, We can use `ATS_Design()` to provide updated thresholds for the 
interim analysis, without	needing	to wait for the	number of evaluable	patients 
to reach 14. This means the input *`n1_star`* is 11 and *`n_star`* is 41 (=11+30 
instead of the original 14+30 as the total sample size), as the original optimal
design specifies a sample size of 30 for the second stage.  
```{r}
ATS_Design(n1=14,n=44,n1_star=11,n_star=41,r1=3,r=14,p0=0.25,p1=0.45,alpha=0.1)
```

The updated design parameters, $(r_1^{*},r^{*})$, by ATS Simon design method are
(2, 14). We also output $(n_1^{*},n^{*})$ and they are just actual sample sizes
of stage 1 and the total sample size. The type I error rate of 
this updated design is 0.06, which is below the original type I error constraint 
of 0.1. As we know, if alpha decreases, power will also decrease. Therefore, the
current power of 85.4% is lower than the original design's 90% as expected, but
still close to the original power. Additionally, the probability of early 
termination is 0.455, which is close to the original design's 0.521.  

Now, suppose the number of patients responding to the new treatment in the first 
stage is > 2. In that case, we will make a “Go” 
decision and recruit an additional 30 patients for the second stage of the study.  

#### Example 2.2 (Under-enrollment in both $1^{st}$ and $2^{nd}$ stages)
We now consider an under-enrollment  scenario in the $2^{nd}$ stage, 
in addition to the $1^{st}$ stage's under-enrollment.  

For example, if the actual sample size in the $2^{nd}$ stage is 28 (the planned
one is 30), indicating under-enrollment. Now the total sample size is 
39, e.g., the input *`n_satr`* = 39.  

Using the updated design parameters with sample sizes at stages 1 and 2 of 
$(n_1^{*},n^{*})$ = (11, 39), and the other original design parameters in the
following code, we can determine the updated design parameters and operating 
characteristics.  
```{r}
ATS_Design(n1=14,n=44,n1_star=11,n_star=39,r1=3,r=14,p0=0.25,p1=0.45,alpha=0.1)
```
Our updated design parameters $(r_1^{*},r^{*})$ are now (2, 13). We also output 
$(n_1^{*},n^{*})$ and they are just actual sample sizes of stage 1 and the total 
sample size. The updated type I error rate is 0.077, which is below the constraint
of 0.1.  

#### Example 2.3 (Under-enrollment in $1^{st}$ and Over-enrollment in $2^{nd}$ stage)
As another example, suppose the actual sample size in the $2^{nd}$ stage is now 
31, indicating over-enrollment. Therefore, the input *`n_satr`* would be 42.
```{r}
ATS_Design(n1=14,n=44,n1_star=11,n_star=42,r1=3,r=14,p0=0.25,p1=0.45,alpha=0.1)
```
The updated design parameters $(r_1^{*},r^{*})$ are now (2, 14). We also output 
$(n_1^{*},n^{*})$ and they are just actual sample sizes of stage 1 and the total 
sample size.The type I error for our ATS Simon design is 0.071, which is below 
the constraint of 0.1.  

## Adaptive	Threshold	and	Sample Size Simon Design (ATSS Simon)  
Rather than	offering the above ATS design	by merely	adjusting	thresholds based 
on the realized	sample size, we	present	another	option that follows the original
Simon's two-stage design algorithm:	the	Adaptive Threshold and Sample Size Simon 
(ATSS Simon)	method.	This approach extends	Simon’s	two-stage design by 
simultaneously adjusting both thresholds and sample size.  

The	overall strategy of this method is detailed	as below:  

$\bullet$ Scenario 1: When under-enrollment or over-enrollment occurs at the $1^{st}$ 
stage, we identify the design parameters $(r_{1}^{*}, r^{*}, n^{*})$ based 
on the actual sample size $n_{1}^*$ at the $1^{st}$ to satisfy the significance 
level $\alpha$ and power $1-\beta$:
$$P\left(Y_{1} > r_1^*, Y > r^* \mid n_{1}^{*}, n^{*}, p_{0}\right) \leq \alpha$$
$$P\left(Y_{1} > r_1^*, Y > r^* \mid n_{1}^{*}, n^{*}, p_{1}\right) \geq 1-\beta$$
In addition, the design parameters $(r_{1}^{*}, r^{*}, n^{*})$ satisfies the 
same criteria as in the Optimal Simon's Two Stage: minimizing the average sample 
size under the null hypothesis.  

$\bullet$ Scenario 2 ($n^{*}$ changes again): When a Go decision has been made, 
the realized sample size $n^{**}$ may be again different from $n^{*}$. 
Further adjustment of the threshold at the $2^{nd}$ stage is needed. So, we 
update again this threshold $r^{*}$ such that we identify a smallest integer
$r^{**}$ given the design parameters $(r_{1}^{*}, n_{1}^{*}, n^{**})$ satisfying
$$P\left(Y_{1} > r_1^*, Y > r^{**} \mid n_{1}^{*}, n^{**}, p_{0}\right) \leq \alpha $$
Here, $n_1^{*}$ denotes as the actual sample size at the $1^{st}$ stage;
$n^{*}$ denotes as the actual total sample size of the two stages after
the interim analysis based on $n_1^{*}$；
$n^{**}$ denotes as the actual total sample size of the two stages if $n^{*}$ is
again updated.

### Example 3 (Adaptive Threshold and Sample Size Simon Design (ATSS Simon) for Under- and/or Over-enrollment)
In the optimal Simon's two-stage design described in Example 1, 
the planned sample size for the $1^{st}$ and $2^{nd}$ stages are 14 and 30, 
respectively.  

#### Example 3.1 (Under-enrollment at $1^{st}$ stage)
Suppose we currently have outcome data for only 11 patients in the $1^{st}$ 
stage and assume the $2^{nd}$ stage sample size remains as planned. 
We can use `ATSS_Design_Stage1()` to implement	the	introduced ATSS	Simon	algorithm. 
Here, the parameter *`n1_satr`* is 11.  

Note: Different from the previous ATS examples, we now should be noted that it 
is unnecessary to input *`n_star`* as a parameter, since the current ATSS method 
will compute an updated total sample size based on the sample size deviation of 
the stage 1 from the original design.
```{r}
ATSS_Design_Stage1(p0=0.25,p1=0.45,n1_star=11,alpha=0.1,beta=0.1)
```
The updated design parameters $(r_1^{*},r^{*},n_1^{*},n^{*})$ are (2, 15, 11, 47).
The type I error for our redesign is 0.09, which is controlled well and < 0.1.
The	updated	power	for our	redesigned study is 0.901, surpassing	the	constraint 
of 0.9. This increase is due to	the	ATSS Simon design method, which now provides 
a larger total sample	size of 47,	compared to 44 in the original design.  

#### Example 3.2 (Under-enrollment at	$1^{st}$ and $2^{nd}$ stages)
If more than 2 patients respond	to the new treatment in	the	first	stage	of our 
redesigned study,	we will proceed	to recruit 36 additional patients 
(47 total minus the 11 from the $1^{st}$ stage). However, if now the actual sample 
size in	the second stage is	34,	indicating	under-enrollment,	we can use the 
function `ATSS_Design_Stage2()` to update the design parameters based on the 
realized total sample	size. This adjustment	means	the	parameter	*`n_double_star`* 
is now	45.  

Note: In the following code, *`n1_star`* and *`n_double_star`* are fixed since 
the trial is complete. Essentially, the ATSS algorithm only updates the threshold 
for stage 2, *`r_star`*. In this example, however, *`r_star`* remains unchanged
compared to Example 3.1.
```{r}
ATSS_Design_Stage2(p0=0.25,p1=0.45,r1_star=2,n1_star=11,n_double_star=45,alpha=0.1)
```
Our updated design parameters $(r_1^{*},r^{*},n_1^{*},n^{*})$ are (2, 15, 11, 45).
And the updated type I error rate is 0.066 smaller than the original one of 0.1 
and the updated power is 0.878, both are due to the realized smaller total sample
size, though in this example, the updated *`r_star`* is still 15.

#### Example 3.3 (Under-enrollment at	$1^{st}$ and Over-enrollment at $2^{nd}$ stage)
Suppose	more than 2	patients respond to	the	new	treatment	at the end of	the	
$1^{st}$ stage of	our redesigned study.	In that case,	we will make a “Go” 
decision to	recruit 36 more	patients into our study. However, if the actual 
sample size in the $2^{nd}$ stage is 37, indicating over-enrollment, we can 
use	the function `ATSS_Design_Stage2()` to update the design parameters 
based	on the actual total	sample size. In	the	following	code,	this means the 
parameter	*`n_double_star`* is now 48.
```{r}
ATSS_Design_Stage2(p0=0.25,p1=0.45,r1_star=2,n1_star=11,n_double_star=48,alpha=0.1)
```
Our updated design parameters $(r_1^{*},r^{*},n_1^{*},n^{*})$ are (2, 16, 11, 48). 
And updated type I error rate is 0.061 and updated power is 0.884.  

Note: Compared to Example 3.2, although the current total sample size is larger
(48 vs. 45), we expect to see an increase in power and type I error rate larger
than those in Example 3.2. However, while the power is as expected, the type I
error rate is not. This is because the optimally searched threshold at stage 2, 
*`r_star`*, is 16, compared to 15 in Example 3.2. Our algorithm dictates controlling
the type I error rate and could only identify 16 as the threshold.
```{r}
## Our algorithm searching process
result <- data.frame(
  r_star = round(c(2.0000000, 3.0000000, 4.0000000, 5.0000000, 6.0000000, 
                   7.0000000, 8.0000000, 9.0000000, 10.0000000, 11.0000000, 
                   12.0000000, 13.0000000, 14.0000000, 15.0000000, 16.0000000)),
  alpha = round(c(0.5447991, 0.5447929, 0.5447130, 0.5442052, 0.5421068, 
                  0.5357601, 0.5207790, 0.4920445, 0.4460005, 0.3831060, 
                  0.3087428, 0.2317242, 0.1611787, 0.1035884, 0.0614173), 3),
  power = round(c(0.9347765, 0.9347765, 0.9347765, 0.9347765, 0.9347763, 
                  0.9347752, 0.9347684, 0.9347366, 0.9346115, 0.9341919, 
                  0.9329744, 0.9298792, 0.9229204, 0.9089765, 0.8839142), 3)
)
print(result)
```
### Summary
Both the proposed	ATS	and	ATSS Simon design	methods	can	address under- and 
over-enrollment. The ATS algorithm updates the thresholds	based	on the actual 
sample sizes,	maintaining	control	over the type I error rate $\alpha$ but 
potentially	resulting	in lower power than the original design	during 
under-enrollment.	In contrast, the ATSS	algorithm	updates both the thresholds 
and	sample sizes, thereby	controlling	the $\alpha$ while maintaining power 
similar to the original	design,	though it	may	require	a	larger sample	size.



## Post-Trial Inference for ATS and ATSS Simon Designs with Under- and Over-Enrollment

Key design details of two-stage single-arm trials are frequently left unreported,
and their statistical inference is often not conducted in a way that mitigates the 
bias introduced by interim analyses. This issue is particularly concerning given 
the increasing reliance on non-randomized trials, which now represent a significant 
portion of the evidence on treatment effectiveness in rare, biomarker-defined 
patient subgroups (Grayling, 2021). Motivated by this problem, we inherit some 
widely used post-trial inference methods for ATS and ATSS Simon Designs with 
Under- and Over-Enrollment.

###  Point Estimate
When a multistage trial is ended, we also want to estimate the true response 
probability ${\pi}_{p}$ of the new therapy. The most commonly used estimator is 
the sample response rate, i.e. the maximum likelihood estimator (MLE):
$$
\hat{\pi}_{p} = \frac{S}{N}
$$
However, in multi-stage designs like Simon's Two-Stage design, we observe only 
extreme cases by crossing the threshold in the first stage, and hence the MLE 
is biased. In other words, the MLE is biased due to the sequential nature of 
the trial. This is known as the *optional sampling effect*. Here, we adopt Jung's 
method for the estimation of the binomial probability in  multistage clinical 
trials (Jung, 2004). Based on the Rao-Blackwell theorem, they derived the 
uniformly minimum variance unbiased estimator (UMVUE) as the conditional 
expectation of an unbiased estimator, which in this case is simply the maximum 
likelihood estimator based only on the first stage data, given the sufficient 
statistic. Let $M$ denote the stopping stage ($2^{nd}$ stage in our context) and 
let $S=S_M$ denote the total number of responders accumulated up to the stopping 
stage. For observation $(m,s)$, the UMVUE of the response rate $p$ is given by:
$$
\hat{\pi}_{p} = 
\begin{cases}
\frac{S}{n_1} & \text{if } m = 1 \\
\frac{\sum_{x_{1}=\left(r_{1}+1\right) \vee\left(S-n_{2}\right)}^{S \wedge n_{1}}
      {n_1 \choose x_1}
      {n_2-1 \choose S-x_1-1}}{
      \sum_{x_{1}=\left(r_{1}+1\right) \vee\left(S-n_{2}\right)}^{S \wedge n_{1}}
      {n_1 \choose x_1}
      {n_2 \choose S-x_1}}
              & \text{if } m = 2
\end{cases}
$$  
At stage $m$, we may accrue slightly more (or possibly less) patients than 
planned sample size as we introduced previously, especially in multicenter 
trials. UMVUE also provides an unbiased estimator for the conditional expectation
by using all realized sample size.  


###  Confidence intervals:  
A conventional approach for constructing confidence intervals is	to use 
the	Clopper-Pearson	exact	confidence interval, disregarding	the	group	sequential 
nature of the	trial (Clopper et al., 1934).
$$P(Y \geq y \mid p_{L}) = \sum_{k=y}^{n}{n \choose k}p_{L}^{k}(1-p_{L})^{n-k}=\frac{\alpha}{2}$$
$$P(Y \leq y \mid p_{U}) = \sum_{k=y}^{n}{n \choose k}p_{U}^{k}(1-p_{U})^{n-k}=\frac{\alpha}{2}$$
We now focus on	constructing confidence intervals	by considering deviations	in 
sample sizes from	the	planned	ones in	the $1^{st}$ and/or $2^{nd}$ stages. 
Jung (2004)	proposed a method	especially when the treatment successfully goes to 
the second stage. Let $M$ denote the stage at which a trial is terminated, 
and $S$ denote the number of responders at stage $M$. This method	constructs 
confidence intervals based on the stochastic ordering	of the distribution	of 
$(M,S)$ with respect to the response rate $p$.

$$P(\hat{p_{u}}(M,S) \geq \hat{p_{u}}(m,s) \mid p_{L}) = \frac{\alpha}{2}$$
$$P(\hat{p_{u}}(M,S) \leq \hat{p_{u}}(m,s) \mid p_{U}) = \frac{\alpha}{2}$$
However, the Clopper-Pearson confidence	interval is	known	to be conservative, 
with the actual	confidence level being bounded below by	$(1-\alpha)$.	Jung’s 
method also	inherits this	conservatism (Porcher et al., 2012).   

To correct for this conservative nature, Porcher extended the Jung's confidence 
interval with a mid-$p$ approach (Porcher et al., 2012). This	method is	our 
recommended	approach,	even though	all	the	above	methods	have also been integrated
into the {UnplanSimon} R package.	
$$P(\hat{p_{u}}(M,S) > \hat{p_{u}}(m,s) \mid p_{L}) + \frac{1}{2}P(P(\hat{p_{u}}(M,S) = \hat{p_{u}}(m,s) \mid p_{L}))
  = \frac{\alpha}{2}$$
$$P(\hat{p_{u}}(M,S) < \hat{p_{u}}(m,s) \mid p_{U}) + \frac{1}{2}P(P(\hat{p_{u}}(M,S) = \hat{p_{u}}(m,s) \mid p_{U}))
  = \frac{\alpha}{2}$$


###  $p$-Value:
Kunzmann et al. (2024) raised a question whether a frequentist inferential 
framework for two-stage designs has a consistent test decision among $p$-values, 
point estimates, and confidence intervals. In practice, however, the consistency 
in those test decisions is often presumed by the non-statistical readership of 
published trial results and ambiguous situation can be avoided by using a 
consistent framework.

So here, we used the $p$-value defined as the probability of obtaining more 
extreme estimates toward $H_1$ than the observed one when $H_0$ is true based 
on the UMVUE ordering (Jung, 2006). Hence, for testing $H_0:p=p_0$ against 
$H_0:p=p_0 (p_0<p_1)$, the $p$-value for an estimate $\hat{p}(m,s)$ will be 
given as
$$
p_s = 
\begin{cases}
1-\sum_{(i,j):\hat{p_u}(i,j) < \hat{p_u}(m,s)}f_{p_0}(i,j) & \text{if } m = 1 \\
\sum_{(i,j):\hat{p_u}(i,j)\geq \hat{p_u}(m,s)}f_{p_0}(i,j) & \text{if } m = 2
\end{cases}
$$
It can be rewritten as 
$$
p_s = 
\begin{cases}
P(Y_1 \geq s \mid p_0) & \text{if } m = 1 \\
\sum_{y_1=r_1+1}^{n_1} P(Y_1 =y_1 \mid p_0) P(Y_2 \leq s-y_1 \mid p_0)          
                       & \text{if } m = 2
\end{cases}
$$

### Example 4 (Post-Trial Inference usage for ATS and ATSS Simon Designs)


#### Example 4.1 (Post-Trial Inference for ATS Simon Design with Under-Enrollment)
Suppose we use the ATS Simon's design method to address the problem
of under-enrollment. The original Simon's Two-Stage design is $(r_1,r,n_1,n) = (3,14,14,44)$.
Considering the under-enrollment problem only at the $1^{st}$ stage like Example 2.1, 
the new design is $(r_1^{*},r^*,n_1^{*},n^{*}) = (2,14,11,41)$. Especially, it 
should be noted that the input *`alpha`* here is the updated type I error 
constraint $\alpha(n^{*})$ if we used the ATS Simon's design. In Example 2.1, the
updated type I error constraint $\alpha(n^{*})$ is 0.088. So it means the input 
'*`alpha`* here is 0.088. If the new treatment successfully enters the second stage 
and 20 patients respond to it, this means the parameter *`m`* is 2 and *`s`* is 20 
under this context.  

We now compute the point estimate (UMVUE), confidence interval and $p$-value 
based on the above setting. Note here, the option *`"CP"`* means Clopper-Pearson 
exact confidence interval，option *`"Jung"`* means Jung’s confidence interval and 
option *`"MIDp"`* means mid-$p$ approach confidence interval. 

```{r}
SimonAnalysis(m=2, s=20, n1=11, n2=30, r1=2, r=14, alpha=0.088, quantile=c(0.025,0.975), CI_option = "CP", p0=0.25)
SimonAnalysis(m=2, s=20, n1=11, n2=30, r1=2, r=14, alpha=0.088, quantile=c(0.025,0.975), CI_option = "Jung", p0=0.25)
SimonAnalysis(m=2, s=20, n1=11, n2=30, r1=2, r=14, alpha=0.088, quantile=c(0.025,0.975), CI_option = "MIDp", p0=0.25)
```
Note: the computed UMVUEs and p values are the same since the above introduced
same UMVUE method and approach for computing p value applied to all the three
ways of computing the CIs. And the input *`alpha`* here is the updated type I
error constraint $\alpha(n^{*})$ if we used the ATS Simon's design.

We can see the point estimate (UMVUE) for the response rate is 0.494; 
Clopper-Pearson exact confidence interval is (0.347, 0.630), 
Jung’s confidence interval is (0.329, 0.629), mid-$p$ approach confidence interval 
is (0.339, 0.641). Also, we see that $p$-value is 0.001 smaller than the 
updated type I error constraint $\alpha(n^{*})$ of 0.088. So we can reject 
the null hypothesis $H_0$: The true treatment response rate is less than or 
equal to some unacceptable level ($p \leq p_0=0.25$).


#### Example 4.2 (Post-Trial Inference for ATSS Simon Design with Under-Enrollment)
Suppose we use the ATSS Simon's design method to address the problem
of under-enrollment. The original Simon's Two-Stage design is $(r_1,r,n_1,n) = (3,14,14,44)$. 
Considering the under-enrollment problem only at the $1^{st}$ stage like 
Example 3.1, the new design is $(r_1^{*},r^*,n_1^{*},n^{*}) = (2,15,11,47)$. 
If the new treatment successfully enters the second stage and 22 patients respond 
to it, this means the parameter *`m`* is 2 and *`s`* is 22 under this context.  

We now compute the point estimate (UMVUE), confidence interval and $p$-value 
based on the above setting. Note here, the option *`"CP"`* means Clopper-Pearson 
exact confidence interval，option *`"Jung"`* means Jung’s confidence interval and 
option *`"MIDp"`* means mid-$p$ approach confidence interval.

```{r}
SimonAnalysis(m=2, s=22, n1=11, n2=36, r1=2, r=15, alpha=0.1, quantile=c(0.025,0.975), CI_option = "CP", p0=0.25)
SimonAnalysis(m=2, s=22, n1=11, n2=36, r1=2, r=15, alpha=0.1, quantile=c(0.025,0.975), CI_option = "Jung", p0=0.25)
SimonAnalysis(m=2, s=22, n1=11, n2=36, r1=2, r=15, alpha=0.1, quantile=c(0.025,0.975), CI_option = "MIDp", p0=0.25)
```

We can see the point estimate (UMVUE) for the response rate is 0.478; 
Clopper-Pearson exact confidence interval is (0.342, 0.597), Jung’s confidence 
interval is (0.322, 0.604), mid-$p$ approach confidence interval is (0.330, 0.615). 
Also, we see that $p$-value is 0.001 smaller than the type I error constraint 
$\alpha$ 0.01. So we can reject the null hypothesis $H_0$: 
The true treatment response rate is less than or equal to some unacceptable level 
($p \leq p_0=0.25$).







## References
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Jung, S. H., Owzar, K., George, S. L., & Lee, T. (2006). P-value calculation for multistage phase II cancer clinical trials. Journal of Biopharmaceutical Statistics, 16(6), 765-775. <https://doi.org/10.1080/10543400600825645>


















