# A multi-arm multi-stage platform design that allows pre-planned addition of
arms while still controlling the family-wise error

Peter Greenstreet, Thomas Jaki, Alun Bedding, Chris Harbron, Pavel Mozgunov

###### Abstract

There is growing interest in platform trials that allow for adding of new
treatment arms as the trial progresses as well as being able to stop
treatments part way through the trial for either lack of benefit/futility or
for superiority. In some situations, platform trials need to guarantee that
error rates are controlled. This paper presents a multi-stage design that
allows additional arms to be added in a platform trial in a pre-planned
fashion, while still controlling the family wise error rate. A method is given
to compute the sample size required to achieve a desired level of power and we
show how the distribution of the sample size and the expected sample size can
be found. A motivating trial is presented which focuses on two settings, with
the first being a set number of stages per active treatment arm and the second
being a set total number of stages, with treatments that are added later
getting fewer stages. Through this example we show that the proposed method
results in a smaller sample size while still controlling the errors compared
to running multiple separate trials.

##  1 Introduction

Clinical trials take many years to run and during this time it is not uncommon
for new promising treatments to emerge that warrant evaluation. It may be
advantageous to include these treatments into an ongoing trial, due to the
shared trial infrastructure and the possibility to use a shared control group.
This can result in useful therapies being identified faster while reducing
cost and time (Cohen et al.,, 2015). The trial potentially requires less
administrative and logistical effort than setting up separate trials, so can
noticeably speed up the development process (Burnett et al.,, 2020). The
addition of more arms may also enhance the recruitment, as patients have a
higher chance of receiving an experimental treatment, therefore, making them
potentially more likely to join a trial (Meurer et al.,, 2012).

There is an ongoing discussion about how to add new treatments to clinical
trials (Lee et al.,, 2021). Cohen et al., (2015) conclude in their literature
review that there is no systematic or comprehensive method for adding new arms
to clinical studies. Recently, several approaches to adding treatment arms
have been proposed which aim to help tackle this issue. Bennett and Mander,
(2020) and Choodari-Oskooei et al., (2020) propose approaches which extend the
Dunnett test (Dunnett,, 1955) to allow for unplanned additional arms to be
included into multi-arm trials while still controlling the family wise error
rates (FWER). This methodology does not incorporate the possibility of interim
analyses.

Interim analyses are a further way to potentially improve the design of a
clinical trial (Wason et al.,, 2016). They allow for ineffective treatments to
be dropped for futility (or lack of benefit) earlier, as well as allowing the
trial to stop early if a superior treatment is found. Both of these can result
in the reduction of the expected sample sizes and costs of a trial. Multi-arm
multi-stage (MAMS) design (e.g. Magirr et al.,, 2012) allows for several
treatments to be evaluated within one study and incorporate interim analyses
for efficiency, but does not allow for additional arms to be added throughout
the trial. Burnett et al., (2020) developed an approach that incorporates
unplanned additional treatment arms to be added to a trial already in progress
using the conditional error principle (Proschan and Hunsberger,, 1995), to
allow for modifications during the course of a trial. The unplanned nature of
the adaptation, however, means that type I error and power for different arms
may be different. As a result, the additional treatments can be underpowered.

In this work, we provide an analytical method for adding of treatments to a
multi-arm multi-stage (MAMS) trial in a pre-planned manner, while still
controlling the statistical errors. The design assumes that, at the design
stage, it is known that new treatments will be added later into the trial.
Unlike the currently developed approaches, the approach proposed allows for a
trial with multiple stages and multiple arms to be designed, so that pre-
planned additional treatments can be added to the trial while still
controlling the family wise error rate (FWER) in the strong sense (Dmitrienko
et al.,, 2009) and achieve suitable power for the trial.

We focus our investigation on two settings: (i) each active treatment has the
same number of stages; (ii) there is a fixed total number of stages. Using
these two settings, we revisit the design of FLAIR (Howard et al.,, 2021) that
had a treatment arm added during the trial. We derive the critical boundaries,
the sample sizes and the expected sample sizes for trials based on FLAIR, and
study the effect on errors when deviating from the planned additions. These
two settings are compared to running separate trials, using the MAMS design by
Magirr et al., (2012) and using the original trial design without adjusting
for additional treatments.

##  2 Method

###  2.1 Setting

Consider a clinical trial with up to $K$ experimental arms that will be tested
against one common control arm with $K^{\star}$ experimental arms starting at
the beginning of the trial, where $K^{\star}\geq 1$, and $K-K^{\star}$ arms
being added later. The primary outcome on each patient is independent and
normally distributed with known variance $\sigma^{2}$. In total, the control
treatment is recruited for a maximum of $J_{0}$ stages with there being a
maximum of $J_{0}-1$ interim analyses, with an analysis taking place at the
end of each stage. Each of the active treatments can have any number of stages
(provided it is pre-specified and their total number is less or equal to
$J_{0}-1$) which coincide with the analysis for the other treatments. Each
additional treatment can be added at any of the interim analyses as long as
this is pre-planned at the design stage of the trial development. When
comparing the control to the active treatments only the concurrent controls
are used in the comparisons. This means only participants recruited to the
control arm at the same time as the active arm are used in the comparisons.

The null hypotheses of interest are
$H_{01}:\mu_{1}\leq\mu_{0},H_{02}:\mu_{2}\leq\mu_{0},...,H_{0K}:\mu_{K}\leq\mu_{0},$
where $\mu_{1},\ldots,\mu_{K}$ are the mean responses on the $K$ experimental
treatments and $\mu_{0}$ is the mean response of the control group. The global
null hypothesis, $\mu_{0}=\mu_{1}=\mu_{2}=\ldots=\mu_{K}$, is denoted by
$H_{G}$. Each of the $K$ hypotheses is potentially tested at a series of
analyses indexed by $j=1,\ldots,J_{k}$ where $J_{k}$ is the maximum number of
analyses for a given treatment $k=1,\ldots,K$. Let $J$ denote the maximum
number of planned analyses of any of the active treatments,
$J=\max_{k}(J_{k})$. The total number of stages of the control treatment is
$J_{0}$. Let $s(k)$ be the stage when treatment $k$ is added to the trial and
define the vector of adding times by $S=(s(1),\ldots,s(K))$. We denote the
ratio of patients recruited to treatment $k$ by the end of its $j^{\text{th}}$
stage by $r_{k,j}$ and denote $n_{k,j}$ as the number of patients recruited to
treatment $k$ by the end of its $j^{\text{th}}$ stage with $k=0$ denoting the
control treatment. The number of patients recruited to the first stage of
treatment $k$ is defined as $n_{k}$ so that
$n_{k,j}=n_{k}\frac{r_{k,j}}{r_{k,1}}$. The total sample size of a trial is
denoted by $N$, where the maximum total planned sample size,
$\max(N)=\sum_{k=0}^{K}n_{k,J_{k}}$. At analysis $j$ for treatment $k$, to
test $H_{0k}$ it is assumed that responses, $X_{k,i}$, from patients
$i=1,\ldots,n_{k,j}$ are observed, as well as the responses $X_{0,i}$ from
patients $i=n_{0,s(k)}+1,\ldots,n_{0,s(k)+j}$, which are the outcome of the
patients allocated to the control which have been recruited since treatment
$k$ has been added into the trial up to the $j$th analysis of treatment $k$.
The test statistics

$Z_{k,j}=\frac{n_{k,j}^{-1}\sum^{n_{k,j}}_{i=1}X_{k,i}-(n_{0,s(k)+j}-n_{0,s(k)})^{-1}\sum^{n_{0,s(k)+j}}_{i=n_{0,s(k)}+1}X_{0,i}}{\sigma\sqrt{(n_{k,j})^{-1}+(n_{0,s(k)+j}-n_{0,s(k)})^{-1}}},$

are used to test hypothesis $H_{0k}$. Upper and lower stopping boundaries,
$U_{k}=(u_{k,1},\ldots,u_{k,J_{k}})$ and $L_{k}=(l_{k,1},\ldots,l_{k,J_{k}})$,
are used for the decision-making as follows. If $Z_{k,j}>u_{k,j}$ then
$H_{0k}$ is rejected and the trial stops with the conclusion that treatment
$k$ is superior to control. If $Z_{k,j}<l_{k,j}$ then treatment $k$ is dropped
from all subsequent stages of the trial. If the $Z$ statistics for all the
treatments fall below their lower boundary, the trial stops for futility.
Treatment $k$ and control continues to its next stage $j+1$ if neither of
these conditions are met, so $l_{k,j}\leq Z_{k,j}\leq u_{k,j}$. The boundaries
are found to control the family wise error rate (FWER) in the strong sense at
a specified desired level $\alpha$ which is defined as

$\displaystyle P(\text{reject at least one true }H_{0k}\text{ under any null
configuation},k=1,\ldots,K)\leq\alpha.$

While this work concerns the general procedure of adding, we focus on two
special cases - see Figure 1. Setting 1 is the case that each active treatment
is planned to have the same number of stages regardless of when it is added.
Setting 2 is the case with a set total number of stages with the later a
treatment is added the fewer stages are planned for it. Note in Setting 1,
$J_{1}=\ldots=J_{K}=J$ and $J_{0}=\max(S)+J$ and in Setting 2, $J_{0}=J$ and
$J_{k}=J-s(k)$.

Figure 1: Examples of the two settings for a two active arm setting with
$J_{0}=3$ and Treatment 2 being added at stage 1. For Setting 1, both active
treatments get 2 interim analyses, and for Setting 2, Treatment 2 is added one
stage and, hence, has one fewer stage. The grey represents areas of possible
shared control group. The dashed black line represents an interim analysis.

###  2.2 Strong control of FWER

Following the method of Dunnett, (1955), one can exploit the correlation
between the test statistics arising from the common control responses. This
description follows Magirr et al., (2012). For any vector of constants
$\Theta=(\theta_{1},\ldots,\theta_{K})$ and $k=1,\ldots,K,$
$j=1,\ldots,J_{k}$, letting
$I_{k,j}=\sigma^{2}(n_{k,j}^{-1}+(n_{0,j+s(k)}-n_{0,s(k)})^{-1})$, define the
events,

$\displaystyle A_{k,j}(\theta_{k})=$
$\displaystyle[Z_{k,j}<l_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}],$
$\displaystyle B_{k,j}(\theta_{k})=$
$\displaystyle[l_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}<Z_{k,j}<u_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}].$

If $\mu_{k}-\mu_{0}=\theta_{k}$ for $k=1,\ldots,K$, the event that
$H_{1},\ldots,H_{K}$ all fail to be rejected is equivalent to

$\displaystyle\bar{R}_{K}(\Theta)$
$\displaystyle=\bigcap_{k\in\\{m_{1},\ldots,m_{K}\\}}\Bigg{(}\bigcup^{J_{k}}_{j=1}\Bigg{[}\bigg{(}\bigcap^{j-1}_{i=1}B_{k,i}(\theta_{k})\bigg{)}\cap
A_{k,j}(\theta_{k})\Bigg{]}\Bigg{)},$

with the convention that $\bigcap^{0}_{i=1}=\Omega$ where $\Omega$ is the
whole sample space, $m_{1}\in\\{1,\ldots,K\\}$ and
$m_{k}\in\\{1,\ldots,K\\}\backslash\\{m_{1},\ldots,m_{k-1}\\}$. The notation
$m_{k}$ is used to reflect the fact that the order in which treatments are
added affects the FWER. The events $A_{k,j}(\theta_{k})$ and
$B_{k,i}(\theta_{k})$ can be rearranged so the following holds

$P(\bar{R}_{K}(\Theta))=\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\Bigg{[}\prod_{k=1}^{K}\Bigg{(}\sum_{j=1}^{J_{k}}\Phi_{j}(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})\Bigg{)}\Bigg{]}d\Phi(t_{1})\ldots
d\Phi(t_{J_{0}}),$ (2.1)

where

$t_{j}=\frac{\sum_{i=n_{0,(j-1)}+1}^{n_{0,j}}(X_{0,i}-\mu_{0})}{\sigma\sqrt{n_{0,j}-n_{0,(j-1)}}}.$
(2.2)

Here $\Phi(\cdot)$ denotes the standard normal distribution function, and
$\Phi_{j}(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})$ denotes the
result of integrating the $j$-dimensional normal density with mean zero and
correlation matrix, $\Sigma_{k,j}$ with the $(i,i^{\star})$th element $(i\leq
i^{\star})$ of $\Sigma_{k,j}$ is $\sqrt{\frac{r_{k,i}}{r_{k,i^{\star}}}}$. The
integration is over the region defined by a vector of lower limits
$L_{k,j}(\theta_{k})=(l_{k,1}(\theta_{k}),\ldots
l_{k,j-1}(\theta_{k}),-\infty)$, and upper limits,
$U_{k,j}(\theta_{k})=(u_{k,1}(\theta_{k}),\ldots
u_{k,j-1}(\theta_{k}),l_{k,j}(\theta_{k}))$, where

$\displaystyle l_{k,j}(\theta_{k})=$ $\displaystyle
l_{k,j}\sqrt{1+\frac{r_{k,j}}{r_{0,s(k)+j}-r_{0,s(k)}}}+\frac{\sqrt{r_{k,j}}}{r_{0,s(k)+j}-r_{0,s(k)}}$
$\displaystyle\bigg{(}\sum^{j}_{i=1}t_{s(k)+i}\sqrt{r_{0,s(k)+i}-r_{0,s(k)+(i-1)}}\bigg{)}-\frac{\theta_{k}\sqrt{n_{k,j}}}{\sigma},$
$\displaystyle u_{k,j}(\theta_{k})=$ $\displaystyle
u_{k,j}\sqrt{1+\frac{r_{k,j}}{r_{0,s(k)+j}-r_{0,s(k)}}}+\frac{\sqrt{r_{k,j}}}{r_{0,s(k)+j}-r_{0,s(k)}}$
$\displaystyle\bigg{(}\sum^{j}_{i=1}t_{s(k)+i}\sqrt{r_{0,s(k)+i}-r_{0,s(k)+(i-1)}}\bigg{)}-\frac{\theta_{k}\sqrt{n_{k,j}}}{\sigma}.$

Note that, in contrast to Magirr et al., (2012), the proposed approach
accounts for the fact that treatments can be added at different points and
hence $l_{k,j}(\theta_{k})$ and $u_{k,j}(\theta_{k})$ depend on $s(k)$. It
also allows for different stopping boundaries per treatment:
$A_{k,j}(\theta_{k})$ and $B_{k,j}(\theta_{k})$ depend on $l_{k,j}$ and
$u_{k,j}$ and there are different maximum numbers of stages per treatment.
Then, one can obtain the following result.

######  Theorem 2.1.

For any $\Theta$, under the conditions above, $P(\text{reject at least one
true }H_{0k}|\Theta)\leq P(\text{reject at least one true }H_{0k}|H_{G})$.

The proof of Theorem 2.1 is given in the Supporting Information in Web
Appendix A. It follows from Theorem 2.1 that the FWER is maximized under the
global null hypothesis.

######  Corollary 2.1.1.

Setting $\Theta=\mathbf{0}$ and finding $P(\bar{R}_{K}(\Theta))$ such that
$P(\bar{R}_{K}(\Theta))=1-\alpha$ controls FWER in the strong sense at level
$\alpha$.

###### Proof.

Under the global null hypothesis $\mu_{0}=\mu_{k}$ for all $k\in 1,\ldots K$
so that $\Theta=(0,\ldots,0)=\mathbf{0}$. Using Theorem 2.1 FWER is controlled
in the strong sense at level $\alpha$ if
$P(\bar{R}_{K}(\mathbf{0}))=1-\alpha$. ∎

As a result of Corollary 2.1.1, the stopping boundaries under the global null
hypothesis, which result in $P(\bar{R}_{K}(\mathbf{0}))=1-\alpha$, will
guarantee strong control of FWER at level $\alpha$.

As mentioned above, the proposed methodology allows for different critical
boundaries to be used for each treatment $k$ as seen in Equation (2.1). To
find the boundaries one can use the functions $L_{k}=f_{k}(a_{k})$ and
$U_{k}=g_{k}(a_{k})$ to reduce the number of unknowns, where $f_{k}$ and
$g_{k}$ are the functions for the shape of the upper and lower boundaries
respectively and $a_{k}$ are scalar parameters specific to each active
treatment. One can use a single parameter $a$ to find the boundaries so
$f_{k}=f_{k^{\prime}}$, $g_{k}=g_{k^{\prime}}$ and $a_{k}=a_{k^{\prime}}$
which is similar to the method presented in Magirr et al., (2012), with the
advantage of there being an equal number of unknowns to equations. However,
using the same boundaries for each treatment arm, regardless of when it was
added, can result in different probabilities of dropping each treatment which
might be undesirable. It may be of interest in having a different stopping
boundary shapes for each treatment, as the same shape for each treatment may
not be optimal as the trial may need greater sample size compared to a design
with different stopping boundary shapes as seen in the Supporting Information
(Web Appendix D). This requires using $L_{k}=f_{k}(a_{k})$ and
$U_{k}=g_{k}(a_{k})$ which results in $K$ scaler parameters to be found,
$\mathbf{a}=(a_{1},\ldots,a_{K})$.

To calculate $a_{k}$ for all $k=1,\ldots,K$, we introduce the requirements on
the pairwise error rate (PWER) being the same for all active treatments, where
PWER is the probability of rejecting the null hypothesis $H_{0.k}$
incorrectly. For the PWER for treatment $k$, it is assumed that no active
treatment other than $k$ can stop the trial. This assumption eliminates the
possibility of stopping the trial without finishing testing treatment $k$,
hence, maximises the probability of the corresponding error. The PWER denoted
by $\alpha^{\star}_{k}$ for treatment $k$ is

$\alpha^{\star}_{k}=1-\sum^{J_{k}}_{j=1}\Phi(U^{\star}_{k,j},L^{\star}_{k,j},\ddot{\Sigma}_{k,j}),$
(2.3)

with $L^{\star}_{k,j}=(l_{k,1},\ldots,l_{k,j-1},-\infty)$,
$U^{\star}_{k,j}=(u_{k,1},\ldots,u_{k,j-1},l_{k,j})$, and covariance matrix
$\ddot{\Sigma}_{k,j}$. The $(i,i^{\star})$th element $(i\leq i^{\star})$ of
$\ddot{\Sigma}_{k,j}$ is

$\displaystyle\Bigg{(}\sqrt{r_{k,i}^{-1}+(r_{0,s(k)+i}-r_{0,s(k)})^{-1}}\sqrt{r_{k,i^{\star}}^{-1}+(r_{0,s(k)+i^{\star}}-r_{0,s(k)})^{-1}}\Bigg{)}^{-1}\Bigg{(}\frac{1}{r_{k,i^{\star}}}+\frac{1}{r_{0,s(k)+i^{\star}}-r_{0,s(k)}}\Bigg{)}.$

To ensure equal PWER across all the treatments and ensure FWER is controlled
the iterative approach in Algorithm 1 is proposed. This approach yields the
desired properties as with each iteration we update every $a_{k}$ so that PWER
is equal for all the active treatments and then using step H we ensure that
the FWER is controlled by using Corollary 2.1.1.

Algorithm 1 Iterative approach to compute the stopping boundaries

  * 0

Begin by assuming $\mathbf{a}=(a_{1},a_{1},\ldots,a_{1})$ and find $a_{1}$
such that $\mathbf{a}$ controls FWER at a specified level, $\alpha$, using
Equation (2.1) with $\Theta=0$. Then repeat the following iterative steps
until each element of $\mathbf{a}$ no longer changes between iterations within
some small $\epsilon$:

  * 1

Find $a_{2}$ such that $\alpha^{\star}_{2}=\alpha^{\star}_{1}$.

$\vdots$

  * H-1

Find $a_{K}$ such that $\alpha^{\star}_{K}=\alpha^{\star}_{1}$.

  * H

Find $a^{\prime}$ such that $\mathbf{a}=a^{\prime}(a_{1},a_{2},\ldots,a_{K})$
results in Equation (2.1) with $\Theta=0$ equalling $\alpha$.

###  2.3 Power and sample size for each treatment

The aim of the trial design in question is to find the required sample size to
achieve the power for every hypothesis greater than a pre-specified level
$(1-\beta)$. We assume that any given treatment $k^{\prime}$, is recommended
when (i) its test statistic crossed the corresponding upper boundary, and (ii)
its test statistic is the largest one, where $k^{\prime}=1,\ldots,K$. The
power is defined as the probability of rejecting $H_{0k^{\prime}}$ for
$k^{\prime}$, such that $\mu_{k^{\prime}}-\mu_{0}=\theta^{\prime}$ and
$\mu_{k}-\mu_{0}=\theta_{0}$ for $k\neq k^{\prime}$ where $\theta^{\prime}$ is
the clinically interesting treatment effect and $\theta_{0}$ is the highest
uninteresting treatment effect. This setting is known as the least favourable
configuration for treatment $k^{\prime}$ (which is denoted as
LFC${}_{k^{\prime}}$, Thall et al.,, 1988).

Let $\Pi_{k^{\prime},J^{\prime}}$, denote the probability that under the
LFC${}_{k^{\prime}}$, no null hypotheses are rejected before the
$J^{\prime}$th analysis for treatment $k^{\prime}$, with treatment
$k^{\prime}$ not being stopped for futility at any of these analyses, and at
analysis $J^{\prime}$, $H_{0.k^{\prime}}$ being rejected and treatment
$k^{\prime}$ being recommended, where $J^{\prime}=1,\ldots,J_{k^{\prime}}$.
The power for rejecting treatment $k^{\prime}$ is then given by
$\Pi_{k^{\prime},1}+\Pi_{k^{\prime},2}\ldots+\Pi_{k^{\prime},J_{k^{\prime}}}$.
To obtain $\Pi_{k^{\prime},J^{\prime}}$, we find the probability that
$H_{0k^{\prime}}$ is not rejected before analysis $J^{\prime}$ and treatment
$k^{\prime}$ is not dropped for futility before analysis $J^{\prime}$ assuming
$t_{1},\ldots,t_{s(k^{\prime})+J^{\prime}}$ and $v_{J^{\prime}}$ are known,
where $t$ is defined in Equation (2.2) and
$v_{J^{\prime}}=\frac{\bar{X}_{k^{\prime},J^{\prime}}-\mu_{k^{\prime}}}{\frac{\sigma}{\sqrt{n_{k^{\prime},J^{\prime}}}}}$,
with $t_{1},\ldots,t_{s(k^{\prime})+J^{\prime}}$ and $v_{J^{\prime}}$ and then
integrate over every possible value as can be seen below. This probability is
$\Phi_{J^{\prime}-1}(\tilde{L}_{k^{\prime},J^{\prime}-1}(\theta^{\prime}),\tilde{U}_{k^{\prime},J^{\prime}-1}(\theta^{\prime}),\tilde{\Sigma}_{k^{\prime},J^{\prime}-1}),$
where
$\tilde{L}_{k^{\prime},J^{\prime}-1}(\theta)=(\tilde{l}_{k^{\prime},1}(\theta),\ldots\tilde{l}_{k^{\prime},J^{\prime}-1}(\theta))$
and
$\tilde{U}_{k^{\prime},J^{\prime}-1}(\theta)=(\tilde{u}_{k^{\prime},1}(\theta),\ldots\tilde{u}_{k^{\prime},J^{\prime}-1}(\theta))$.
The $(i,i^{\star})$th element $(i\leq i^{\star})$ of the correlation matrix
$\tilde{\Sigma}_{k^{\prime},j}$ is
$\sqrt{\frac{r_{k^{\prime},i}(r_{k^{\prime},J^{\prime}}-r_{k^{\prime},i^{\star}})}{r_{k^{\prime},i^{\star}}(r_{k^{\prime},J^{\prime}}-r_{k^{\prime},i})}}$
and

$\displaystyle\tilde{l}_{k^{\prime},j}(\theta)=$
$\displaystyle\sqrt{\frac{r_{k^{\prime},J^{\prime}}}{r_{k^{\prime},J^{\prime}}-r_{k^{\prime},j}}}\bigg{(}l_{k^{\prime},j}(\theta)-v_{J^{\prime}}\sqrt{\frac{r_{k^{\prime},j}}{r_{k^{\prime},J^{\prime}}}}\bigg{)},$
$\displaystyle\tilde{u}_{k^{\prime},j}(\theta)=$
$\displaystyle\sqrt{\frac{r_{k^{\prime},J^{\prime}}}{r_{k^{\prime},J^{\prime}}-r_{k^{\prime},j}}}\bigg{(}u_{k^{\prime},j}(\theta)-v_{J^{\prime}}\sqrt{\frac{r_{k^{\prime},j}}{r_{k^{\prime},J^{\prime}}}}\bigg{)}.$

The event that $H_{0k^{\prime}}$ is rejected at analysis $J^{\prime}$ assuming
that $t_{1},\ldots,t_{s(k^{\prime})+J^{\prime}}$ and $v_{J^{\prime}}$ are
known and where $\mathbbm{1}\\{\cdot\\}$ is an indicator function, is

$\displaystyle\xi_{J^{\prime}}=$
$\displaystyle\mathbbm{1}\\{\frac{v_{J^{\prime}}}{\sqrt{n_{k^{\prime},J^{\prime}}}}-\frac{\sum_{i=1}^{J^{\prime}}t_{s(k^{\prime})+i}\sqrt{n_{0,s(k^{\prime})+i}-n_{0,s(k^{\prime})+i-1}}}{n_{0,s(k^{\prime})+J}-n_{0,s(k^{\prime})}}+\frac{\theta^{\prime}}{\sigma}>u_{J^{\prime}}\sqrt{n^{-1}_{k^{\prime},J^{\prime}}+(n_{0,s(k^{\prime})+J^{\prime}}-n_{0,s(k^{\prime})})^{-1}}\\},$

The probability that $H_{0k}$ is not rejected before analysis $J^{\prime}$ for
treatment $k^{\prime}$ for all $k\in 1,\ldots,k-1,k+1,\ldots K$ assuming
$t_{1},\ldots,t_{s(k^{\prime})+J^{\prime}}$ and $v_{J^{\prime}}$ are known is

$\displaystyle\gamma_{k,J^{\prime}}=$
$\displaystyle\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)>0\\}\bigg{[}\sum_{j=1}^{\min(J_{k},s(k^{\prime})+J^{\prime}-s(k))}\mathbbm{1}\\{s(k^{\prime})+J^{\prime}>s(k)+j\\}\Phi(L_{k,j}(\theta_{0}),U_{k,j}(\theta_{0}),\Sigma_{k,j})$
$\displaystyle+\mathbbm{1}\\{s(f)+J^{\prime}=s(k)+j\\}\Phi(L_{k,j}(\theta_{0}),\dot{U}_{k,j}(\theta_{0}),\Sigma_{k,j})\bigg{]}+\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)\leq
0\\},$

where $\dot{U}_{k,j}(\theta_{0})=(u_{k,1}(\theta_{0}),\ldots
u_{k,j-1}(\theta_{0}),\dot{u}_{k,j}(\theta_{0})),$ with

$\displaystyle\dot{u}_{k,j}(\theta_{0})=$
$\displaystyle\max\bigg{[}u_{k,s(k^{\prime})-s(k)+J^{\prime}}(\theta_{0}),\frac{\sqrt{n_{k,s(k^{\prime})-s(k)+J^{\prime}}}}{\sqrt{n_{k^{\prime},J^{\prime}}}}v_{Y}+\frac{n_{k,s(k^{\prime})-s(k)+J^{\prime}}(\theta^{\prime}-\theta_{0})}{\sigma}\bigg{]}.$

One can then find $\Pi_{k^{\prime},J^{\prime}}$ as

$\displaystyle\begin{split}\Pi_{k^{\prime},J^{\prime}}=&\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\Phi(\tilde{L}_{k^{\prime},J^{\prime}-1}(\theta^{\prime}),\tilde{U}_{k^{\prime},J^{\prime}-1}(\theta_{1}),\tilde{\Sigma}_{k^{\prime},J^{\prime}-1})\\\
&(\prod_{k=1,k\neq
k^{\prime}}^{K}\gamma_{k,J^{\prime}})(\xi_{J^{\prime}})\;d\Phi(t_{1})\ldots
d\Phi(t_{s(k^{\prime})+J^{\prime}})\;d\Phi(v_{J^{\prime}}).\end{split}$ (2.4)

To ensure that all the experimental treatments achieve the pre-specified power
under the corresponding LFC${}_{k^{\prime}}$, the sample size must be found in
order for
$\Pi_{k^{\prime},1}+\Pi_{k^{\prime},2}\ldots+\Pi_{k^{\prime},J_{k^{\prime}}}\geq
1-\beta$ for all $k^{\prime}$. Due to treatments potentially starting at
different times and having different number of stages, each treatment may
require a different number of patients to achieve the same power. As changing
the sample size of one treatment effects the power of another, an iterative
approach is proposed to calculate the required sample size per treatment per
stage. Specifically, to have all the treatments controlled at the same
specified power, $1-\beta$ under their LFC${}_{k^{\prime}}$, one needs to
define $\mathbf{n}=(n_{0},\ldots,n_{K})$, then by assuming each $n_{k}$ can
take any real value, use Algorithm 2. Once $\mathbf{n}$ is found using this
Algorithm 2, round up each value of $\mathbf{n}$ to its nearest integer, then
recalculate $\mathbf{a}$ with these new values for $\mathbf{n}$ to account for
the fact that the ratios have now also changed between treatments.

Algorithm 2 Iterative approach to compute the sample size

  * 0

Begin by assuming $n_{k}=1$ for all $k,\in 0,\ldots,K$ then calculate
$\mathbf{a}$ using Algorithm 1. Find $n_{1}$ such that
$\Pi_{1,1}+\Pi_{1,2}\ldots+\Pi_{1,J_{1}}=1-\beta$ with $n_{k}=n_{1}$ for all
$k\in 0,\ldots,K$ and update $\mathbf{n}$. Then repeat the following iterative
steps until each element of $\mathbf{n}$ no longer changes between iterations
within some small $\epsilon$:

  * 1

Find $n_{1}$ such that $\Pi_{1,1}+\Pi_{1,2}\ldots+\Pi_{1,J_{1}}=1-\beta$.

  * 2

Find $n_{2}$ such that $\Pi_{2,1}+\Pi_{2,2}\ldots+\Pi_{2,J_{2}}=1-\beta$.

$\vdots$

  * H

Find $n_{K}$ such that $\Pi_{K,1}+\Pi_{K,2}\ldots+\Pi_{K,J_{K}}=1-\beta$.

  * H+1

Find $n_{0}$ and $r_{0,1},\ldots r_{0,J_{0}}$ based on $n_{1},\ldots,n_{K}$
then recalculate $\mathbf{a}$ using Algorithm 1.

###  2.4 Sample size distribution and Expected sample size

The distribution of the sample size and expected sample size can both be
calculated by finding the probability of every possible outcome of the trial
denoted by $P_{\tilde{J},Q}$. Define
$\tilde{J}=(\tilde{j}(1),\ldots,\tilde{j}(K))$ with
$\tilde{j}(k)={1,\ldots,J_{k}}$ as the point in which treatment $k$ would
finish being tested, ignoring the possibility that the trial has already
stopped early as a different treatment is found which is superior to the
control. This is done in order to remove the dependence between each active
arm. We define $Q=(q(1),\ldots,q(K))$ with $q(k)=\infty$ if treatment $k$ goes
below the lower stopping boundary at point $\tilde{j}(k)$ and $q(k)=1$, if
treatment $k$ goes above the upper stopping boundary at point $\tilde{j}(k)$.
Due to ignoring the possibility of the trial already stopping early, every
active treatment will either stop for futility or efficacy therefore $q(k)$
can only take one of two values. We find

$\displaystyle P_{\tilde{J},Q}=$
$\displaystyle\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\prod_{k=1}^{K}\Bigg{[}\mathbbm{1}\\{q(k)=1\\}\Phi(L^{+}_{k,\tilde{j}(k)}(\theta_{k}),U^{+}_{k,\tilde{j}(k)}(\theta_{k}),\Sigma_{k,\tilde{j}(k)})$
$\displaystyle+\mathbbm{1}\\{q(k)=\infty\\}\Phi(L_{k,\tilde{j}(k)}(\theta_{k}),U_{k,\tilde{j}(k)}(\theta_{k}),\Sigma_{k,\tilde{j}(k)})\Bigg{]}d\Phi(t_{1}),\ldots,d\Phi(t_{\max(\tilde{J}+S)}),$

where $U^{+}_{k,j}=(u_{k,1}(\theta_{k}),\ldots u_{k,j-1}(\theta_{k}),\infty)$
and $L^{+}_{k,j}=(l_{k,1}(\theta_{k}),\ldots
l_{k,j-1}(\theta_{k}),u_{k,j}(\theta_{k}))$. The $P_{\tilde{J},Q}$ are then
associated with their given total sample size $N_{\tilde{J},Q}$ for that given
$\tilde{J}$ and $Q$.

$\displaystyle N_{\tilde{J},Q}=$
$\displaystyle\bigg{(}\sum^{K}_{k=1}n_{k,\max(\min(\tilde{j}(k)+s(k),(\tilde{J}+S)\circ
Q)-s(k),0)}\bigg{)}+n_{0,\min(\max(\tilde{J}+S),(\tilde{J}+S)\circ Q)},$

where $\circ$ is the scalar product. To obtain the sample size distribution
each value of $\tilde{J}$ and $Q$ which result in the same value of
$N_{\tilde{J},Q}$ is associated with its corresponding $P_{\tilde{J},Q}$. This
set of $P_{\tilde{J},Q}$ is then summed together to give the probability of
the realisation of this sample size. To find the sample size distribution for
each active arm one can associate
$n_{k,\max(\min(\tilde{j}(k)+s(k),(\tilde{J}+S)\circ Q)-s(k),0)}$ with its
corresponding $P_{\tilde{J},Q}$, and this can similarly be done for the
control treatment. The expected sample size for N for a given $\Theta$,
$E(N|\Theta)$, can be found by summing up every possible combination of
$\tilde{J}$ and $Q$,

$E(N|\Theta)=\sum_{\begin{subarray}{c}\tilde{j}(k)=1\\\
k=1,2,\ldots,K\end{subarray}}^{J_{k}}\sum_{\begin{subarray}{c}q(k)\in\\{1,\infty\\}\\\
k=1,2,\ldots,K\end{subarray}}P_{\tilde{J},Q}N_{\tilde{J},Q}.$

##  3 Numerical Evaluations

###  3.1 Motivating trial

In recent years, there have been several platform trials conducted and their
use appears to be increasing during the COVID-19 pandemic (Stallard et al.,,
2020). One platform trial example is FLAIR (Howard et al.,, 2021). It is a
randomised, controlled, open-label, confirmatory trial in chronic lymphocyte
leukaemia. When designing FLAIR there was the plan to add an active treatment
during the trial as well as an interim analysis halfway through the planned
sample size for each treatment. In the actual trial, two additional arms were
added, one being an additional control arm. The original design of the study
considered pairwise type I error control only and hence did not provide the
FWER control.

We revisit the design of the FLAIR trial accounting for one additional active
treatment arm being planned to be added mid-trial. One active and a control
arm begin the trial, and apply the proposed methodology to control the FWER in
the strong sense. One can argue that the controlling FWER in this trial is
essential as the trial aimed to test different combinations of treatments with
the same common compound - ibrutinib for all the active treatments whilst the
main control was not based on ibrutinib (Wason et al.,, 2014). This might be a
regulatory requirement to control the FWER while adding an arm during the
trial (FDA,, 2019).

Based on the planned effect given in FLAIR, the interesting treatment
difference is $\theta^{\prime}=-\log(0.69)$, $\sigma=1$, and the uninteresting
treatment effect is assumed (as not being used in the original design) to be
$\theta_{0}=-\log(0.99)$. The desired power in FLAIR was 80%, while the type-I
error of each treatment comparison was 2.5% (one-sided). While still targeting
the same power, we will use a more stringent target of 2.5% FWER (one-sided).
In line with FLAIR, we use a total number of stages for both settings to be
three. Therefore, in Setting 1, treatments 1 and 2 will both have two stages,
whereas, in Setting 2, Treatment 1 will have three stages and Treatment 2 will
have two (see Figure 1). The interims are equally spaced for the active
treatments across all stages so $r_{k,j}=j$ for $k>0$. Informed by the
recruitment to FLAIR, we assume a constant recruitment rate of 21 patients per
month.

The operating characteristics that will be studied for both methods include
the FWER and power under LFCk. These two are studied to ensure that the trial
design meets the required error control. Other operating characteristics
stated include the maximum number of stages per active treatment arm, denoted
by NSk, as well as the number of patients per arm per stage. Also shown for
each setting is the maximum sample size and duration until the trial is
complete as well as the expected sample size and duration. The duration of the
trial is denoted by $T$. These values are found in order to compare the
different designs.

###  3.2 Competing designs

We compare the proposed designs to three alternative methods. All of these are
in the frequentest framework. For brevity, the main focus of the competing
designs comparison will be on the Setting 2 (with the results for Setting 1
provided in Supporting Information).

The first competing approach is to evaluate each active treatment in separate
trials. In line with Setting 2, the first study uses the 3-stage design and
the second one - the 2-stage one. This approach will be referred to as
“Separate trials”. Two variations on running separate trials are studied. The
first controls the FWER across both trials using $\alpha^{\prime}$, the error
rate for each trial, chosen so that $(1-\alpha^{\prime})^{2}=0.975$. This
results in a type I error for each trial of 1.26%. The second variation does
not control the FWER across the two trials.

The second competing design is the MAMS approach proposed by Magirr et al.,
(2012). We will refer to these approach as “MAMS trial”. Note, that under this
MAMS approach, the trial cannot start until all treatments are ready. As this
approach requires equal numbers of stages per treatment, both the results for
running a 2 stage and 3 stage trial are presented.

The third competing method uses the same design parameters as originally
planned for the 2-arm 3-stage trial and then also uses them for the additional
treatment. This approach will be referred to as the “Naive MAMS” as it does
not adjust the design parameters for the added arm. This will also allow to
demonstrate the effect of not adjusting a design for additional arms. We
provide the results for both the same maximum sample size as originally
planned, and for the same sample size per arm per stage, $n_{j,k}$.

The operating characteristics of the different design options are provided in
Table 1. The calculations were carried out using R (R Core Team,, 2021) with
the method given here having the multivariate normal probabilities being
calculated using the package mvtnorm (Genz et al.,, 2021) and the outer
integrals being calculated using the quadrature rule with the packages gtools
(Warnes et al.,, 2021) and statmod (Smyth et al.,, 2021). Code is available at
https://github.com/pgreenstreet/platform-design-with-addition-of-arms. The
comparison results were obtained using the R package MAMS (Jaki et al.,,
2019).

###  3.3 Results

All the results can be seen in Table 1. For all the designs the triangular
shaped stopping boundaries are used (Whitehead,, 1997; Wason and Jaki,, 2012).
The first two rows of Table 1 shows the proposed design under Setting 1 and 2,
respectively with the Supporting Information containing the results of using
other stopping boundary shapes.

Table 1: Operating characteristics of the proposed design under two settings
and competing approaches: Running trials separate (“Separate Trials”), MAMS
design by (Magirr et al.,, 2012) (“MAMS”), and using a “naive” MAMS approach
for the FLAIR trial.

| FWER | LFC1 | NS1 | $n_{1}$ | $\max(N)$ | $E(N|H_{G})$ | $E(N|\text{LFC}_{1})$ | $E(N|\text{LFC}_{2})$
---|---|---|---|---|---|---|---|---
|  | LFC2 | NS2 | $n_{2}$ | $\max(T)$ | $E(T|H_{G})$ | $E(T|\text{LFC}_{1})$ | $E(T|\text{LFC}_{2})$
Setting 1 | 0.025 | 0.802 | 2 | 76 | 540 | 351.8 | 285.8 | 400.8
|  | 0.804 | 2 | 78 | (25.7) | (16.8) | (13.6) | (19.1)
Setting 2 | 0.025 | 0.802 | 3 | 46 | 492 | 303.3 | 296.6 | 347.8
|  | 0.803 | 2 | 77 | (23.4) | (14.4) | (14.1) | (16.8)
Separate trials | 0.025 | 0.801 | 3 | 53 | 626 | 349.1 | 405.0 | 400.6
FWER control |  | 0.805 | 2 | 77 | (29.8) | (16.6) | (19.3) | (19.1)
Separate trials | 0.049 | 0.807 | 3 | 46 | 536 | 302.2 | 340.1 | 336.1
no FWER control |  | 0.803 | 2 | 65 | (25.5) | (14.4) | (16.2) | (16.0)
MAMS trial | 0.025 | 0.804 | 2 | 76 | 456 | 280.7 | 309.8 | 309.8
2 Stage |  | 0.804 | 2 | 76 | (26.1) | (17.7) | (19.1) | (19.1)
MAMS trial | 0.025 | 0.805 | 3 | 53 | 477 | 258.0 | 289.4 | 289.4
3 stage |  | 0.805 | 3 | 53 | (27.1) | (16.7) | (18.2) | (18.2)
Naive MAMS | 0.044 | 0.804 | 3 | 46 | 368 | 219.0 | 217.3 | 239.7
same $n_{j,k}$ |  | 0.564 | 2 | 46 | (17.5) | (10.4) | (10.3) | (11.4)
Naive MAMS | 0.044 | 0.716 | 3 | 46/30 | 276 | 177.2 | 178.2 | 190.8
same $\max(N)$ |  | 0.408 | 2 | 31 | (13.1) | (8.4) | (8.5) | (9.1)

Key: $E(N|H_{0.G})$, $E(N|\text{LFC}_{k})$, $E(T|H_{0.G})$,
$E(T|\text{LFC}_{k})$ is the expected sample size and trial duration under the
null and under the LFC for treatment $k$, respectively.

Under Setting 1, which has two stages for each active treatment, the design
requires 42 and 44 patients per stage for Treatment 1 and Treatment 2,
respectively. As a result, the maximum sample size is 540, and the expected
sample sizes varies between approximately 286 and 401. The maximum duration of
the trial is 25.7 months, and the expected one varies between 13.6 and 19.1
months. Under Setting 2, in which the first active treatment has 3 stages and
the second has only 2, the required sample size per stage are is 46 and 77,
respectively. This results in the maximum sample size of 492 and expected
sample size between 296.6 and 347.8 depending on the configuration. This
resulted in a maximum duration of 23.4 months and the expected duration varies
between 14.1-16.8 months.

Comparing the sample size and trial duration for the proposed designs under
Settings 1 and 2, under most configurations Setting 2 has lower expected
sample sizes, and requires nearly 50 fewer patients in the maximum sample size
to achieve 80% power while controlling the FWER at 2.5%. This also translates
into the shorter duration. Setting 1 has advantages over Setting 2 under the
case when Treatment 1 is superior, and Treatment 2 has the uninteresting
effect. However, the difference in the expected sample size is around 10
patients and the difference in the expected duration is 0.5 months. For this
reason, we will focus on the comparisons with Setting 2 with the results for
Setting 1 provided in Supporting Information.

The next two rows of Table 1 show the operating characteristics of running two
separate trials. To match the setting of the FLAIR (and Setting 2), it is
assumed that the first treatment has three stages while the second has two.
Under the FWER across these separate trials controlled, the maximum and
expected sample sizes are noticeably larger - with the difference of 134
patients required to achieve 80% power. Running two separate trials with the
FWER also increases the maximum duration by 6 months, and the expected
duration by 2-5 months, on average, depending on the configuration. Under two
trials not controlling the FWER, the advantage of the proposed design under
Setting 2 still persists. Two separate trials would require 44 more patients,
and the expected sample size are only nearly 11 patients lower under LFC2
which results in less than 1 month in recruitment time. Comparing this to the
expected sample size under LFC1 which is around 44 patients lower for Setting
2 which results in a saving of time of over 2 months.

The second competing method is the MAMS approach proposed by Magirr et al.,
(2012) that requires all treatments to start at the same time, and uses the
critical values controlling the FWER, and the sample size achieving 80% power.
We consider both 2- and 3-stage variants for a fairer comparison. The duration
until the trial finishes for both includes the time before the trial would be
able to start. This is calculated by assuming that Treatment 2 is ready when
planned for Setting 2 and then calculating the time before this treatment is
added. Therefore, this is the time for the first 92 patients to be recruited -
4.4 months. Under this MAMS design, the maximum sample size is lower than for
the proposed one under Setting 2 (36 and 15 patients for the 2- and 3-stage
designs, respectively). The maximum duration, however, is increased by 3-4
months. The expected duration is also increase for all the configuration
studied with an increase of between 1.4-5 months.

The final comparison method of naively using the original design for a 2 arms
3 stage trial is shown. The original design is to have 46 patients per arm per
stage which results in 276 patients. Therefore when $n_{j,k}$ is kept the same
this results in a maximum sample size of 368. For this approach the FWER is
inflated by over 75%. The power under LFC1 is still above the desired level
however under LFC2 the power decreases to 56.4% so is well below the target of
80%. For the naive approach where the maximum sample size remains the same
there is therefore a change in sample size for the first treatment’s
2${}^{\text{nd}}$ and 3${}^{\text{rd}}$ stages to accommodate the addition of
the new treatment. As a result there is 46 patients on Treatment 1 at stage 1
then this decreases to 30 for Treatment 1’s final two stages. In order to keep
$\max(N)=276$ then $n_{2}$ was set to equal 31 patients. In this case the FWER
is inflated by over 75% and neither the power under the LFC1 or LFC2 is
controlled at the desired level. The drop in power for the LFC2 between
Setting 2 and this naive approach is over 35% which is a dramatic loss in
power. This poor result is to be predicted for this naive approach as it does
not have bounds designed to control FWER or the required number of patients to
get the desired power for either treatment.

The results of other common stopping boundary shapes and combinations of these
can be seen in the Supporting Information in Web Appendix D for Setting 1 and
Setting 2. In Web Appendix C of the Supporting Information the comparison
results to Setting 1 can be seen when using the triangular stopping
boundaries.

Overall this section has shown how the methods proposed in this paper could
work in order to design a clinical trial in which an additional treatment is
added later. In addition, competing frequentest approaches are studied to see
how they compare. It can be seen that there is benefit to using the method
proposed here compared to using these other methods either with regards to
sample size or trial duration.

###  3.4 Sample Size Distribution

The distribution of the total sample size and the sample size of each
treatment for Setting 2 under the global null is given in Figure 2. Analogous
results for Setting 1 can be seen in the Supporting Information (Web Appendix
E) along with the expression for the probability mass function for the total
sample size for Setting 2. The design under Setting 2 results in the
interquartile range of 246 to 292 and median of 292 under the global null for
the total sample size. These figures can be used by the trial team and given
to funders and regulators to help with the communication of how many patients
are likely to be required for the trial.

Figure 2: Cumulative distribution functions (CDF) of the number of treatments
needed for the trial in Setting 2 of the total sample size and for each arm
individually

##  4 Robustness to deviations in the planned adding

In the FLAIR trial, the second active treatment was not added until about
three quarters of the way through the recruitment for the first treatment. In
this section, the effect of adding the treatments earlier or later than
planned (i.e. at the first interim for the considered example) will be studied
using simulations. We consider three approaches of how a treatment could be
added later or earlier to a trial are studied. In all approaches the total
maximum sample size is fixed to be the same. Below, we focus on Setting 2, and
similar results for Setting 1 are given in the Supporting Information Web
Appendix F.

Approach 1 is to change the timing of the interim analysis for Treatment 1 so
it is conducted when Treatment 2 is added. Once the second active treatment is
added, the allocation ratio for Treatment 1 to control changes as in the
original proposed design. The patients remaining from the total sample size
are then shared out across the phases with respect to the pre-set allocation
ratio between each treatment. The pre-set stopping boundaries are used.
Approach 2 follows Approach 1, but instead of keeping the original boundaries
the bounds are recalculated using Algorithm 1 with the allocation ratios of
each treatment at the time the additional treatment is actually added.
Approach 3 keeps the timing of the interim analysis for Treatment 1 unchanged,
and, at this point, the allocation ratio changes.

The effect of adding the second active treatment to the trial after only
recruiting 1 patient to the control, up to recruiting 189 patients to the
control, is studied. With the first treatment receiving the correct number of
patients based on its recruitment rate relative to the controls recruitment
rate. i.e 1 patient will have also been recruited to Treatment 1 before
Treatment 2 is added in the earliest example.

Figure 3 shows the resulting FWER and Power for different times when the new
treatment is added based on 10 million simulations for each case.

(a)

(b)

Figure 3: The effect of adding the treatment later or earlier than planned
using three different approaches for Setting 2 on power and FWER. With sub-
figure (a) showing the FWER under the global null, and in sub-figure (b)
showing the power under the LFC for Treatment 1 and the power under the LFC
for Treatment 2.

Figure 3(a) shows how the FWER varies for each one of the approaches with
Approach 2 having the least variation and Approach 1 having the most variation
in FWER under the global null. The maximum inflation happens for Approach 3 of
an increase in FWER under the global null to 2.54% when the second treatment
is added after 100 patients are recruited to the control. For Approach 1 the
FWER increases until the planned adding time and then decreases. Approach 2
stays constantly around the planned FWER and for Approach 3 the FWER starts
below 2.5% and increases until 100 patients, before starting to decrease.

The changing FWER for Approach 1 and 3 are caused by two opposing forces. The
first of which is when Treatment 1 is added earlier there is a decrease in
correlation between the first interim for Treatment 1 and the rest of its
analyses, this is caused by a decrease in $\sqrt{r_{1,1}/r_{1,2}}$ and
$\sqrt{r_{1,1}/r_{1,3}}$. The second, is there is an increase in correlation
between the Z-statistics for Treatment 1 and 2 as there is now an increased
number of shared control patients. These two opposing forces make it difficult
to predict what effect any change will have on the FWER without running the
calculations or using simulations. In order to guarantee that the FWER is
controlled then either the second treatment needs to be added when it was
planned to be or recalculate the stopping boundaries for each point as done in
Approach 2.

Considering the power, for the all considered approaches the later the
additional arm is added the lower the power for this arm is. For Approaches 1
and 2, the power for the Treatment 1 increases the later the additional arm is
added whereas the power remains almost constant for Approach 3. One issue
therefore with Approaches 1 and 2 is that power for all the treatments under
the LFC is no longer controlled unless the treatment is added at the
preplanned time. However, when using Approach 3, the power is controlled for
both treatments when the treatment is added earlier as well as the FWER being
controlled. This is not always the case as can be seen in the Supporting
Information (Web Appendix G) which provides an example with higher
uninteresting treatment effect. Higher $\theta_{0}$ results in a greatly
increased chance of taking Treatment 2 forward before the second analysis for
Treatment 1 due to $\theta_{0}$ effect on the sample size of the second active
treatment. This reiterates why using the pre-planned design and assessing the
impact of deviations of the plan are crucial. One potential solution to this
problem of controlling the errors when adding the treatment at a different
time to when it was planned is to recalculate all design parameters (including
the sample size).

This section has highlighted the importance of using the original plan in
order to control the FWER and power under the LFC. Therefore it is important
when using this design to ensure that the additional treatment will be ready
for the pre-planned addition time. One key point is that if the new treatment
is not added to the trial at all, the design will still guarantee control of
FWER and power for the treatments already in the trial. Furthermore by using
the original plan this removes the bias potentially caused by changing when
the additional treatments are added, in order to benefit treatments already in
the trial.

##  5 Discussion

In this paper, a general design for adding additional treatments in a pre-
planned manner is developed and explored. This design ensures strong control
of the FWER and power under the least favourable configuration and allows for
interim analyses. Both sample size distribution and expected sample size can
be calculated. Two iterative approaches are given to allow for multiple
stopping boundary shapes and to allow for different numbers of patients on
each treatment depending on when the treatment is added to the trial. Two
different designs based on FLAIR for the two settings are presented. These
designs are then further explored where the effect of adding treatments later
or earlier than planned is studied.

Overall the method proposed here, which builds on the work of Magirr et al.,
(2012), has shown that running a platform trial where treatments are added at
later points can result in a considerably more favourable design to running
separate trials with respect to maximum and expected sample size. This
approach has shown that it can be worthwhile starting a trial earlier with the
available treatments and then planning to add treatments later, compared to
waiting until all are ready then beginning the trial with respect to the time
it takes before the trial concludes. This is true both when there is either a
constant or increasing recruitment rate. In Section 3.3 it was seen that using
Setting 2 compared to 1 can be potentially beneficial with regards to the
trials sample size and duration. This makes intuitive sense as this results in
increased correlation between the test statistics as there are more shared
controls which results in a reduction in the FWER of the trial for the given
boundaries.

Throughout this paper only concurrent controls are used, as argued by Lee and
Wason, (2020). While it would be possible to extend the design to allow for
non-concurrent controls, error control will only be guaranteed if there is no
difference between the concurrent and the non-concurrent controls, but this is
impossible to know before the trial. In this paper we assumed that an interim
analysis is conducted at the time that a new treatment is added. This not only
simplifies calculations, but is also sensible as if a new treatment is being
added to the trial then the other treatments in the trial may as well also be
studied at this point. This has two benefits, the first is there is the
opportunity for a treatment to be declared superior to the control before
recruitment of the new treatment starts. The second is it allows the study of
all the patients on the control treatment from before the additional treatment
is added, so potentially making it easier at later stages to know which
controls are concurrent.

PWER was used in order to calculate $a_{1},a_{2},\ldots,a_{K}$ in order to
share the FWER out among the treatments. This was used as it ensures the
highest possible probability of rejecting a null hypothesis is the same for
all treatments. However there are a multitude of different ways the FWER could
be shared such as having: the probability of rejecting a null hypothesis under
the global null the same; or the probability that a treatment is taken forward
to the next phase the same. One may also want to consider one of these
approaches. To do this the same iterative approach as given in Section 2.2
with a different Equation (2.3) can be used.

When calculating the expected sample size every possible outcome of the trial
was enumerated resulting in a very computationally costly procedure. In
Section Web Appendix B of the Supporting Information a more efficient approach
is provided for the computation of the expected sample size. The cost of this
efficiency is that the algorithm does not yield the full sample size
distribution in addition to the expected sample size.

An area for further research is deciding whether to wait for all the
treatments to be ready or to start the trial with the ability to add
preplanned treatments later. A lot of factors need to be considered when
choosing this such as: recruitment time, recruitment cost, time left before
all the treatments are ready, and the cost of delaying development of existing
treatments. Therefore an area for further research from this paper is looking
at how a decision framework, such as the one discussed in Lee et al., (2019),
could be used.

Some of the potential limitations with this approach are that it assumes
normally distributed data. By using asymptotic normality as discussed in Jaki
and Magirr, (2013), however, other endpoints can also be used. Another area is
the fact it is assumed that the common variance is known. However using an ad
hoc approach such as the one in Magirr et al., (2012) can also be used to
transform individual test statistics to combat this issue.

## Acknowledgements

This report is independent research supported by the National Institute for
Health Research (NIHR300576). The views expressed in this publication are
those of the authors and not necessarily those of the NHS, the National
Institute for Health Research or the Department of Health and Social Care
(DHSC). PM and TJ also received funding from UK Medical Research Council
(MC_UU_00002/14). This paper is based on work completed while PG was part of
the EPSRC funded STOR-i centre for doctoral training (EP/S022252/1).

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## Supporting Information

##  Web Appendix A Proof of strong control of FWER

Recall that
$A_{k,j}(\theta_{k})=[Z_{k,j}<l_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}]$
and
$B_{k,j}(\theta_{k})=[l_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}<Z_{k,j}<u_{k,j}+(\mu_{k}-\mu_{0}-\theta_{k})I^{1/2}_{k,j}]$.

###### Proof.

For any $\epsilon_{k}>0$,

$\displaystyle\bigcup^{J_{k}}_{j=1}\Bigg{[}\bigg{(}\bigcap^{j-1}_{i=1}B_{k,i}(\theta_{k}+\epsilon_{k})\bigg{)}\cap
A_{k,j}(\theta_{k}+\epsilon_{k})\Bigg{]}\subseteq\bigcup^{J_{k}}_{j=1}\Bigg{[}\bigg{(}\bigcap^{j-1}_{i=1}B_{k,i}(\theta_{k})\bigg{)}\cap
A_{k,j}(\theta_{k})\Bigg{]}.$

Take any

$w=(Z_{k,1},\ldots,Z_{k,J})\in\bigcup^{J_{k}}_{j=1}\Bigg{[}\bigg{(}\bigcap^{j-1}_{i=1}B_{k,i}(\theta_{k}+\epsilon_{k})\bigg{)}\cap
A_{k,j}(\theta_{k}+\epsilon_{k})\Bigg{]}.$

For some $q\in\\{1,\ldots,J_{k}\\},$ for which $Z_{k,q}\in
A_{k,q}(\theta_{k}+\epsilon_{k})$ and $Z_{k,j}\in
B_{k,j}(\theta_{k}+\epsilon_{k})$ for $j=1,\ldots,q-1$. $Z_{k,q}\in
A_{k,q}(\theta_{k}+\epsilon_{k})$ implies that $Z_{k,q}\in
A_{k,q}(\theta_{k})$. Furthermore $Z_{k,q}\in
B_{k,q}(\theta_{k}+\epsilon_{k})$ implies that $Z_{k,q}\in
B_{k,q}(\theta_{k})\cup A_{k,q}(\theta_{k})$ for some $j=1,\ldots,q-1$.
Therefore,

$w\in\bigcup^{J_{k}}_{j=1}\Bigg{[}\bigg{(}\bigcap^{j-1}_{i=1}B_{k,i}(\theta_{k})\bigg{)}\cap
A_{k,j}(\theta_{k})\Bigg{]}.$

Next suppose for any $m_{1},\ldots,m_{K}$ where $m_{1}\in\\{1,\ldots,K\\}$ and
$m_{k}\in\\{1,\ldots,K\\}\backslash\\{m_{1},\ldots,m_{k-1}\\}$ with
$\theta_{m_{1}},\ldots,\theta_{m_{l}}\leq 0$ and
$\theta_{m_{l+1}},\ldots,\theta_{m_{K}}>0$. Let
$\Theta_{l}=(\theta_{m_{1}},\ldots,\theta_{m_{l}})$. Then

$\displaystyle P($ $\displaystyle\text{reject at least one true
}H_{0.k}|\Theta)$ $\displaystyle\leq P(Z_{k,j}>u_{k,j}\text{ for some
}(k,j)\in\\{(m_{1},1)\ldots,(m_{1},J_{m_{1}}),(m_{2},1)$
$\displaystyle\ldots,(m_{l},1),\ldots,(m_{l},J_{m_{l}})\\}|\Theta)$
$\displaystyle=1-P(\bar{R}_{l}(\Theta_{l}))$ $\displaystyle\leq
1-P(\bar{R}_{l}(0))$ $\displaystyle\leq 1-P(\bar{R}_{K}(0))$
$\displaystyle=P(Z_{k,j}>u_{k,j}\text{ for some
}(k,j)\in\\{(m_{1},1)\ldots,(m_{1},J_{m_{1}}),(m_{2},1)$
$\displaystyle\ldots,(m_{K},1),\ldots,(m_{K},J_{m_{K}})\\}|H_{0.G})$
$\displaystyle=P(\text{reject at least one true }H_{0.k}|H_{0.G}).$

∎

##  Web Appendix B  Efficient computation of expected sample size

The sample size calculation can be split into four sections. Two sections that
focus on the control treatment and two sections that focus on the active
treatments. These sections are:

  1. 1.

The probability the control treatment finishes at each stage $j_{0}$ as the
trial is stopped, given no null hypotheses are rejected, where $j_{0}\in
1,\ldots,J_{0}$. This is calculated by taking the difference between the
probability that every treatment is stopped for futility by the control’s
$j_{0}$th stage, denoted by $\Psi_{j}$ and every treatment is stopped for
futility by the control’s stage $j_{0}-1$. The control treatment cannot stop
being recruited until either: one or more null hypotheses is rejected, or
until all the active treatments have had at least one stage. Therefore, in
this calculation only the stages after every treatment has been added to the
trial need to be considered, so $s^{\star}$ is defined as $s^{\star}=\max(S)$.
Using this gives for $j_{0}>s^{\star}$,

$\Psi_{j_{0}}=\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\Bigg{[}\prod_{k=1}^{K}\Bigg{(}\sum_{j=1}^{\min[J_{k},j_{0}-s(k)]}\Phi(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})\Bigg{)}\Bigg{]}d\Phi(t_{1})\ldots
d\Phi(t_{j_{0}}),$

and for $j_{0}\leq s^{\star}$ gives $\Psi_{j_{0}}=0$ and $\Psi_{0}=0$.

  2. 2.

The probability the control treatment finishes at each stage as the trial is
stopped, given that a null hypothesis is rejected. This is calculated by
taking the difference between the probability that at least one null
hypothesis is rejected by the control treatments ${j_{0}}$th stage, denoted by
$\Upsilon_{j_{0}}$, and that at least one null hypothesis is rejected by the
control’s ${j_{0}}-1$ stage.

$\displaystyle\Upsilon_{j_{0}}=1-$
$\displaystyle\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\prod_{k=1}^{K}\Bigg{[}\mathbbm{1}\\{j_{0}-s(k)>0\\}\bigg{[}\mathbbm{1}\\{j_{0}-s(k)>1\\}$
$\displaystyle\;\sum_{j=1}^{\min(J_{k},j_{0}-s(k)-1)}\Phi(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})+\mathbbm{1}\\{j_{0}-s(k)\leq
J_{k}\\}$
$\displaystyle\;\;\Phi(L_{h,j_{0}-s(k)}(\theta_{k}),\ddot{U}_{h,j_{0}-s(k)}(\theta_{k}),\Sigma_{h,j_{0}-s(k)})\bigg{]}+\mathbbm{1}\\{j_{0}-s(k)\leq
0\\}\Bigg{]}d\Phi(t_{1})\ldots d\Phi(t_{j_{0}}),$

where $\ddot{U}_{k,j}(\theta_{k})=(u_{k,1}(\theta_{k}),\ldots
u_{k,j-1}(\theta_{k}),u_{k,j}(\theta_{k})).$ With $\Upsilon_{0}=0$.

  3. 3.

The probability treatment $k^{\prime}$ stops at each of its $J^{\prime}$th
stages because at least one other treatments null hypothesis has been rejected
at this stage, denoted by $\Lambda_{k^{\prime},J^{\prime}}$ where,

$\displaystyle\Lambda_{k^{\prime},J^{\prime}}=$
$\displaystyle\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\bigg{[}\varpi_{k^{\prime},s(k^{\prime})+J^{\prime}-1}-\varpi_{k^{\prime},s(k^{\prime})+J^{\prime}}\bigg{]}\bigg{[}\mathbbm{1}\\{J^{\prime}-1>0\\}$
$\displaystyle\;\Phi(\ddot{L}_{k^{\prime},J^{\prime}-1}(\theta_{k}^{\prime}),\ddot{U}_{k^{\prime},J^{\prime}-1}(\theta_{k}^{\prime}),\Sigma_{k^{\prime},J^{\prime}-1})+\mathbbm{1}\\{J^{\prime}-1\leq
0\\}\bigg{]}d\Phi(t_{1})\ldots d\Phi(t_{s(k^{\prime})+J^{\prime}}),$

where $\ddot{L}_{k,j}(\theta_{k})=(l_{k,1}(\theta_{k}),\ldots
l_{k,j-1}(\theta_{k}),l_{k,j}(\theta_{k}))$ and

$\displaystyle\varpi_{k^{\prime},j_{0}}=$ $\displaystyle\prod_{k=1,k\neq
k^{\prime}}^{K}\Bigg{[}\mathbbm{1}\\{j_{0}-s(k)>0\\}\bigg{[}\mathbbm{1}\\{j_{0}-s(k)>1\\}\sum_{j=1}^{\min(J_{k},j_{0}-s(k)-1)}\Phi(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})+$
$\displaystyle\;\mathbbm{1}\\{j_{0}-s(k)\leq
J_{k}\\}\Phi(L_{k,j_{0}-s(k)}(\theta_{k}),\ddot{U}_{k,j_{0}-s(k)}(\theta_{k}),\Sigma_{k,j_{0}-s(k)})\bigg{]}+\mathbbm{1}\\{j_{0}-s(k)\leq
0\\}\Bigg{]},$

with $\varpi_{k^{\prime},0}=1$.

  4. 4.

The probability treatment $k^{\prime}$ stops at each of its $J^{\prime}$th
stages because only $H_{0k^{\prime}}$ is rejected, or no null hypotheses are
dropped at this stage and treatment $k^{\prime}$ is stopped as its test
statistic drops below its lower boundary for that stage
($\Xi_{k^{\prime},J^{\prime}}$).

$\displaystyle\Xi_{k^{\prime},J^{\prime}}=$
$\displaystyle\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\Bigg{(}\prod_{k=1,k\neq
k^{\prime}}^{K}\Bigg{[}\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)>0\\}\bigg{(}\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)>1\\}$
$\displaystyle\;\bigg{[}\sum_{j=1}^{\min(J_{k},s(k^{\prime})+J^{\prime}-s(k)-1)}\Phi(L_{k,j}(\theta_{k}),U_{k,j}(\theta_{k}),\Sigma_{k,j})\bigg{]}+\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)\leq
J_{k}\\}$
$\displaystyle\;\;\Phi(L_{k,s(k^{\prime})+J^{\prime}-s(k)}(\theta_{k}),\ddot{U}_{h,s(k^{\prime})+J^{\prime}-s(k)}(\theta_{k}),\Sigma_{k,s(k^{\prime})+J^{\prime}-s(k)})\bigg{)}+\mathbbm{1}\\{s(k^{\prime})+J^{\prime}-s(k)\leq
0\\}\Bigg{]}\Bigg{)}$
$\displaystyle\;\;\;\Bigg{[}\Phi(L_{k^{\prime},J^{\prime}}(\theta_{k^{\prime}}),U_{k^{\prime},J^{\prime}}(\theta_{k^{\prime}}),\Sigma_{k^{\prime},J^{\prime}})+\Phi(L^{+}_{k^{\prime},J^{\prime}}(\theta_{k^{\prime}}),U^{+}_{k^{\prime},J^{\prime}}(\theta_{k}^{\prime}),\Sigma_{k^{\prime},J^{\prime}})\bigg{]}d\Phi(t_{1})\ldots
d\Phi(t_{s(k^{\prime})+J^{\prime}}),$

where

$\displaystyle U^{+}_{k,j}(\theta_{k^{\prime}})=$
$\displaystyle(u_{k,1}(\theta_{k^{\prime}}),\ldots
u_{h,j-1}(\theta_{k^{\prime}}),\infty)$ $\displaystyle
L^{+}_{k,j}(\theta_{k^{\prime}})=$
$\displaystyle(l_{k,1}(\theta_{k^{\prime}}),\ldots
l_{h,j-1}(\theta_{k^{\prime}}),u_{h,j}(\theta_{k^{\prime}}))$

Using the probabilities calculated above the expected sample size is:

$\displaystyle
N_{E}=\sum^{W}_{j_{0}=1}(\Psi_{j_{0}}+\Upsilon_{j_{0}}-\Psi_{j_{0}-1}-\Upsilon_{j_{0}-1})n_{0,j_{0}}+\sum^{K}_{k^{\prime}=1}\sum^{J_{k^{\prime}}}_{J^{\prime}=1}(\Xi_{k^{\prime},J^{\prime}}+\Lambda_{k^{\prime},J^{\prime}})n_{k^{\prime},J^{\prime}}.$

##  Web Appendix C Table of results for Setting 1 based on the motivating
trial

As done for Setting 2 in Table 1 of the main paper in Web Table 1 the results
of the different comparison approach given in Section 3.2 is shown. Now for
the expected duration until the MAMS trial finishes it is now assumed that the
trial does not begin until the beginning of the second stage for Setting 1.
Therefore 152 patients have already been recruited which equals 7.2 months.

Table 1: The results of the triangular stopping boundary shape on the design
configuration for the motivating FLAIR trial under Setting 1 and three
competing approaches: when running each trial separately, when using the MAMS
design by (Magirr et al.,, 2012) and when using the naive MAMS approach

| FWER | LFC1 | NS1 | $n_{1}$ | $\max(N)$ | $E(N|H_{G})$ | $E(N|\text{LFC}_{1})$ | $E(N|\text{LFC}_{2})$
---|---|---|---|---|---|---|---|---
|  | LFC2 | NS2 | $n_{2}$ | $\max(T)$ | $E(T|H_{G})$ | $E(T|\text{LFC}_{1})$ | $E(T|\text{LFC}_{2})$
Setting 1 | 0.025 | 0.802 | 2 | 76 | 540 | 351.8 | 285.8 | 400.8
|  | 0.804 | 2 | 78 | (25.7) | (16.8) | (13.6) | (19.1)
Separate trials | 0.025 | 0.805 | 2 | 77 | 616 | 368.2 | 419.3 | 419.3
FWER control |  | 0.805 | 2 | 77 | (29.3) | (17.5) | (20.0) | (20.0)
Separate trials | 0.049 | 0.803 | 2 | 65 | 520 | 316.2 | 349.7 | 349.7
no FWER control |  | 0.803 | 2 | 65 | (24.8) | (15.1) | (16.7) | (16.7)
MAMS trial | 0.025 | 0.804 | 2 | 76 | 456 | 280.7 | 309.8 | 309.8
2 Stage |  | 0.804 | 2 | 76 | (29.0) | (20.6) | (22.0) | (22.0)
Naive MAMS | 0.048 | 0.802 | 3 | 65 | 455 | 299.3 | 234.0 | 331.4
same $n_{j,k}$ |  | 0.787 | 2 | 65 | (21.7) | (14.3) | (11.1) | (15.8)
Naive MAMS | 0.047 | 0.676 | 2 | 65/26 | 260 | 197.8 | 172.8 | 214.4
same $\max(N)$ |  | 0.414 | 2 | 26 | (12.4) | (9.4) | (8.2) | (10.2)

Key: $E(N|H_{0.G})$, $E(N|\text{LFC}_{k})$, $E(T|H_{0.G})$,
$E(T|\text{LFC}_{k})$ is the expected sample size and trial duration under the
null and under the LFC for treatment $k$, respectively.

##  Web Appendix D Tables of results based on the motivating trial for
different stopping boundaries

In Web Table 2 and Web Table 3 the results for the different combinations of
stopping boundary shapes are shown for Setting 1 and 2 respectively. The
stopping boundary shapes which are considered here are Pocock (Pocock,, 1977),
O’Brien and Fleming (O’Brien and Fleming,, 1979) and Triangular stopping
boundaries (Whitehead,, 1997). However for both the Pocock boundary shape and
the O’brien and Flemming boundary shape the symmetric futility boundary may be
too stringent of a requirement to be able to drop ineffective treatments,
therefore a simple alternative is $l_{k,j}=0$ for $j<J_{k}$ is used. As can be
seen in these tables the upper and lower stopping boundaries are given, with
the top row being the boundaries for the first active treatment added and the
second row being for the second active treatment.

Table 2: The results of different stopping boundary shapes on the design
configuration for the example trial under Setting 1.

$\text{SB}_{1}$ | FWER | LFC1 | NS1 | $L$ | $U$ | $n_{1}$ | $\max(N)$ | $E(N|H_{G})$ | $E(N|\text{LFC}_{1})$ | $E(N|\text{LFC}_{2})$
---|---|---|---|---|---|---|---|---|---|---
$\text{SB}_{2}$ |  | LFC2 | NS2 |  |  | $n_{2}$ | $\max(T)$ | $E(T|H_{G})$ | $E(T|\text{LFC}_{1})$ | $E(T|\text{LFC}_{2})$
OBF | 0.025 | 0.803 | 2 | $\begin{pmatrix}0&2.238\\\ 0&2.238\\\ \end{pmatrix}$ | $\begin{pmatrix}3.165&2.238\\\ 3.165&2.238\\\ \end{pmatrix}$ | 69 | 491 | 383.7 | 322.9 | 430.6
OBF |  | 0.804 | 2 | 71 | 23.4 | 18.3 | 15.4 | 20.5
OBF | 0.025 | 0.803 | 2 | $\begin{pmatrix}0&2.235\\\ 0&2.435\\\ \end{pmatrix}$ | $\begin{pmatrix}3.160&2.235\\\ 2.435&2.435\\\ \end{pmatrix}$ | 67 | 509 | 395.6 | 329.4 | 405.8
Po |  | 0.805 | 2 | 77 | 24.2 | 18.8 | 15.7 | 19.3
OBF | 0.025 | 0.803 | 2 | $\begin{pmatrix}0&2.235\\\ 0.832&2.354\\\ \end{pmatrix}$ | $\begin{pmatrix}3.161&2.235\\\ 2.496&2.354\\\ \end{pmatrix}$ | 67 | 509 | 350.6 | 322.4 | 400.2
Tri |  | 0.803 | 2 | 77 | 24.2 | 16.7 | 15.4 | 19.1
Po | 0.025 | 0.801 | 2 | $\begin{pmatrix}0&2.442\\\ 0&2.241\end{pmatrix}$ | $\begin{pmatrix}2.442&2.442\\\ 3.169&2.241\end{pmatrix}$ | 76 | 514 | 403.1 | 285.5 | 448.1
OBF |  | 0.805 | 2 | 70 | 24.5 | 19.2 | 13.6 | 21.3
Po | 0.025 | 0.804 | 2 | $\begin{pmatrix}0&2.440\\\ 0&2.440\end{pmatrix}$ | $\begin{pmatrix}2.440&2.440\\\ 2.440&2.440\end{pmatrix}$ | 76 | 540 | 419.9 | 291.7 | 429.0
Po |  | 0.805 | 2 | 78 | 25.7 | 20.0 | 13.8 | 20.4
Po | 0.025 | 0.804 | 2 | $\begin{pmatrix}0&2.440\\\ 0.834&2.358\end{pmatrix}$ | $\begin{pmatrix}2.440&2.440\\\ 2.501&2.358\end{pmatrix}$ | 76 | 540 | 374.2 | 284.3 | 423.6
Tri |  | 0.803 | 2 | 78 | 25.7 | 17.8 | 13.5 | 20.2
Tri | 0.025 | 0.806 | 2 | $\begin{pmatrix}0.834&2.360\\\ 0&2.241\end{pmatrix}$ | $\begin{pmatrix}2.503&2.360\\\ 3.170&2.241\end{pmatrix}$ | 77 | 515 | 381.9 | 287.8 | 425.9
OBF |  | 0.802 | 2 | 69 | 24.5 | 18.2 | 13.7 | 20.3
Tri | 0.025 | 0.801 | 2 | $\begin{pmatrix}0.834&2.358\\\ 0&2.440\end{pmatrix}$ | $\begin{pmatrix}2.501&2.358\\\ 2.440&2.440\end{pmatrix}$ | 76 | 536 | 394.5 | 292.2 | 403.9
Po |  | 0.800 | 2 | 77 | 25.5 | 18.8 | 13.9 | 19.2
Tri | 0.025 | 0.802 | 2 | $\begin{pmatrix}0.834&2.358\\\ 0.834&2.358\end{pmatrix}$ | $\begin{pmatrix}2.501&2.358\\\ 2.501&2.358\end{pmatrix}$ | 76 | 540 | 351.8 | 285.8 | 400.8
Tri |  | 0.804 | 2 | 78 | 25.7 | 16.8 | 13.6 | 19.1

Key: $SB_{k}$ is the stopping boundary shape for treatment $k$; Tri is the
triangular stopping boundary shape; OBF is the O’Brien and Fleming stopping
boundary shape; Po is the Pocock stopping boundary shape; NSk is the maximum
number of stages for active treatment $k$; $E(N|H_{0.G})$,
$E(N|\text{LFC}_{k})$, $E(T|H_{0.G})$, $E(T|\text{LFC}_{k})$ is the expected
sample size and trial duration under the null and under the LFC for treatment
$k$, respectively.

Table 3: The results of different stopping boundary shapes on the design
configuration for the example trial under Setting 2.

$\text{SB}_{1}$ | FWER | LFC1 | NS1 | $L$ | $U$ | $n_{1}$ | $\max(N)$ | $E(N|H_{G})$ | $E(N|\text{LFC}_{1})$ | $E(N|\text{LFC}_{2})$
---|---|---|---|---|---|---|---|---|---|---
$\text{SB}_{2}$ |  | LFC2 | NS2 |  |  | $n_{2}$ | $\max(T)$ | $E(T|H_{G})$ | $E(T|\text{LFC}_{1})$ | $E(T|\text{LFC}_{2})$
OBF | 0.025 | 0.807 | 3 | $\begin{pmatrix}0&0&2.239\\\ 0&2.231&-\\\ \end{pmatrix}$ | $\begin{pmatrix}3.878&2.742&2.239\\\ 3.154&2.231&-\\\ \end{pmatrix}$ | 41 | 440 | 334.3 | 333.6 | 367.0
OBF |  | 0.800 | 2 | 69 | 21.0 | 15.9 | 15.9 | 17.5
OBF | 0.025 | 0.801 | 3 | $\begin{pmatrix}0&0&2.239\\\ 0&2.433&-\\\ \end{pmatrix}$ | $\begin{pmatrix}3.878&2.742&2.239\\\ 2.433&2.433&-\\\ \end{pmatrix}$ | 39 | 460 | 349.3 | 346.5 | 340.6
Po |  | 0.802 | 2 | 76 | 21.9 | 16.6 | 16.5 | 16.2
OBF | 0.025 | 0.802 | 3 | $\begin{pmatrix}0&0&2.239\\\ 0.831&2.351&-\\\ \end{pmatrix}$ | $\begin{pmatrix}3.878&2.742&2.239\\\ 2.494&2.351&-\\\ \end{pmatrix}$ | 39 | 460 | 313.5 | 332.9 | 336.3
Tri |  | 0.801 | 2 | 76 | 21.9 | 14.9 | 15.9 | 16.0
Po | 0.025 | 0.803 | 3 | $\begin{pmatrix}0&0&2.547\\\ 0&2.236&-\end{pmatrix}$ | $\begin{pmatrix}2.547&2.547&2.547\\\ 3.163&2.236&-\end{pmatrix}$ | 48 | 476 | 358.8 | 298.4 | 391.2
OBF |  | 0.805 | 2 | 71 | 22.7 | 17.1 | 14.2 | 18.6
Po | 0.025 | 0.806 | 3 | $\begin{pmatrix}0&0&2.547\\\ 0&2.436&-\end{pmatrix}$ | $\begin{pmatrix}2.547&2.547&2.547\\\ 2.436&2.436&-\end{pmatrix}$ | 47 | 496 | 373.4 | 308.4 | 362.8
Po |  | 0.802 | 2 | 77 | 23.6 | 17.8 | 14.7 | 17.3
Po | 0.025 | 0.806 | 3 | $\begin{pmatrix}0&0&2.547\\\ 0.832&2.355&-\end{pmatrix}$ | $\begin{pmatrix}2.547&2.547&2.547\\\ 2.497&2.355&-\end{pmatrix}$ | 47 | 496 | 337.3 | 298.9 | 358.8
Tri |  | 0.801 | 2 | 77 | 23.6 | 16.1 | 14.2 | 17.1
Tri | 0.025 | 0.806 | 3 | $\begin{pmatrix}0&1.472&2.404\\\ 0&2.235&-\end{pmatrix}$ | $\begin{pmatrix}2.776&2.454&2.404\\\ 3.161&2.235&-\end{pmatrix}$ | 48 | 472 | 332.2 | 296.3 | 376.2
OBF |  | 0.801 | 2 | 70 | 22.5 | 15.8 | 14.1 | 17.9
Tri | 0.025 | 0.802 | 3 | $\begin{pmatrix}0&1.472&2.404\\\ 0&2.435&-\end{pmatrix}$ | $\begin{pmatrix}2.776&2.453&2.404\\\ 2.435&2.435&-\end{pmatrix}$ | 46 | 492 | 347.0 | 306.9 | 353.2
Po |  | 0.804 | 2 | 77 | 23.4 | 16.5 | 14.6 | 16.8
Tri | 0.025 | 0.802 | 3 | $\begin{pmatrix}0&1.472&2.404\\\ 0.832&2.353&-\end{pmatrix}$ | $\begin{pmatrix}2.776&2.453&2.404\\\ 2.496&2.353&-\end{pmatrix}$ | 46 | 492 | 303.3 | 296.6 | 347.8
Tri |  | 0.803 | 2 | 77 | 23.4 | 14.4 | 14.1 | 16.6

$SB_{k}$ is the stopping boundary shape for treatment $k$; Tri is the
triangular stopping boundary shape; OBF is the O’Brien and Fleming stopping
boundary shape; Po is the Pocock stopping boundary shape; NSk is the maximum
number of stages for active treatment $k$; $E(N|H_{0.G})$,
$E(N|\text{LFC}_{k})$, $E(T|H_{0.G})$, $E(T|\text{LFC}_{k})$ is the expected
sample size and trial duration under the null and under the LFC for treatment
$k$, respectively.

##  Web Appendix E Distribution of sample size

The distribution of the total sample size and the sample size of each
treatment when the triangular stopping boundaries are used for Setting 1 under
the global null is given in Web Figure 1 with the probability mass function
for the total sample size given in Equation (Web Appendix E.1). In this
example the triangular stopping boundaries for Setting 1 gives the
interquartile range of 308 to 384 and median of 308 under the global null for
the total sample size.

$\text{Distribution of the total sample size Setting
1}=\begin{cases}0.006,&\text{if}\ N=152\\\ 0.641,&\text{if}\ N=308\\\
0.161,&\text{if}\ N=384\\\ 0.156,&\text{if}\ N=464\\\ 0.035,&\text{if}\
N=540\\\ \end{cases}$ (Web Appendix E.1)

Figure 1: Cumulative distribution functions (CDF) of the number of treatments
needed for the trial in Setting 2 of the total sample size and for each arm
individually

The probability mass function for the total sample size for Setting 2 is given
in Equation (Web Appendix E.2)

$\text{Distribution of the total sample size Setting
2}=\begin{cases}0.003,&\text{if}\ N=92\\\ 0.402,&\text{if}\ N=246\\\
0.369,&\text{if}\ N=292\\\ 0.098,&\text{if}\ N=400\\\ 0.034,&\text{if}\
N=415\\\ 0.071,&\text{if}\ N=446\\\ 0.023,&\text{if}\ N=492\\\ \end{cases}$
(Web Appendix E.2)

##  Web Appendix F Robustness to the timing of the actual adding for Setting 1

The same 3 approaches, as given in Section 4, to adding the second treatment
earlier or later are studied here for Setting 1 as can be seen in Web Figure
2.

Figure 2: The effect of adding the treatment later or earlier than planned
using three different approaches for Setting 1 on power and FWER. With sub-
figure (a) showing the FWER under the global null, and in sub-figure (b)
showing the power under the LFC for Treatment 1 and the power under the LFC
for Treatment 2.

##  Web Appendix G The effect of a larger $\theta_{0}$ when using triangular
stopping boundaries

For this example the same variables are used as the ones discussed in Section
3 apart from $\theta_{0}=-\log(0.80)$. The main focus of this section is to
show how this larger $\theta_{0}$ can result in the third approach to adding
treatments earlier or later, as discussed in Section 4, does not control the
power of all the treatments when the treatment is added earlier. The stopping
boundaries, sample size and expected sample size for both settings are given
in Web Table 4. The effect of adding the treatment earlier or later than
planned using the three approaches discussed in Section 4 is given in Web
Figure 3 and Web Figure 4 for Settings 1 and 2, respectively. Both these
figures now show that the power under the LFC for Treatment 1 is no longer
controlled for any of the approaches when the second treatment is added
earlier.

Table 4: The results of the triangular stopping boundaries on the design
configuration for the example trial with $\theta_{0}=-\log(0.80)$ for both
settings.

Setting | $n_{1}$ | $n_{2}$ | $\max(N)$ | $E(N|H_{G})$ | $E(N|\text{LFC}_{1})$ | $E(N|\text{LFC}_{2})$
---|---|---|---|---|---|---
1 | 73 | 144 | 795 | 501.7 | 384.8 | 492.8
2 | 50 | 108 | 632 | 388.3 | 374.8 | 422.7

Key: $E(N|H_{0.G})$, $E(N|\text{LFC}_{k})$ is the expected sample size under
the null and under the LFC for treatment $k$, respectively.

Figure 3: The effect of adding the treatment later or earlier than planned
using three different approaches for Setting 1 on power and FWER when
$\theta_{0}=-\log(0.80)$. With sub-figure (a) showing the FWER under the
global null, and in sub-figure (b) showing the power under the LFC for
Treatment 1 and the power under the LFC for Treatment 2.

Figure 4: The effect of adding the treatment later or earlier than planned
using three different approaches for Setting 2 on power and FWER when
$\theta_{0}=-\log(0.80)$. With sub-figure (a) showing the FWER under the
global null, and in sub-figure (b) showing the power under the LFC for
Treatment 1 and the power under the LFC for Treatment 2.