# Design of platform trials with a change in the control treatment arm

Peter Greenstreet1,2,⋆, Thomas Jaki3,4, Alun Bedding5, Pavel Mozgunov3
1 Department of Mathematics and Statistics, Lancaster University, Lancaster,
UK
2 Exeter CTU, University of Exeter, Exeter, UK
3 MRC Biostatistics Unit, University of Cambridge, Cambridge, UK
4 University of Regensburg, Regensburg, Germany
5 Roche Products Ltd., Welwyn Garden City, UK
⋆<EMAIL_ADDRESS>
###### Abstract

Platform trials are a more efficient way of testing multiple treatments
compared to running separate trials. In this paper we consider platform trials
where, if a treatment is found to be superior to the control, it will become
the new standard of care (and the control in the platform). The remaining
treatments are then tested against this new control. In such a setting, one
can either keep the information on both the new standard of care and the other
active treatments before the control is changed or one could discard this
information when testing for benefit of the remaining treatments. We will show
analytically and numerically that retaining the information collected before
the change in control can be detrimental to the power of the study.
Specifically, we consider the overall power, the probability that the active
treatment with the greatest treatment effect is found during the trial. We
also consider the conditional power of the active treatments, the probability
a given treatment can be found superior against the current control. We prove
when, in a multi-arm multi-stage trial where no arms are added, retaining the
information is detrimental to both overall and conditional power of the
remaining treatments. This loss of power is studied for a motivating example.
We then discuss the effect on platform trials in which arms are added later.
On the basis of these observations we discuss different aspects to consider
when deciding whether to run a continuous platform trial or whether one may be
better running a new trial.

##  1 Introduction

Clinical trials take many years and are very costly to run (Mullard,, 2018)
which has therefore lead to multiple developments in methodology on how to
efficiently design them (Pallmann et al.,, 2018). One of these developments
has been the idea of platform trials in which multiple treatments are tested
against a common control group (Urach and Posch,, 2016; Royston et al.,, 2003;
Wason and Jaki,, 2012; Bennett and Mander,, 2020). Platform trials can be
advantageous due to having a shared trial infrastructure and shared control
groups (Burnett et al.,, 2020). The interest in these types of trials has
increased since the beginning of the COVID-19 pandemic (Lee et al.,, 2021;
Stallard et al.,, 2020), as platform trials can result in therapies being
identified faster while reducing cost and time (Cohen et al.,, 2015).

One additional ability one may want from a platform trial’s design is to be
able to change the control group to a beneficial new treatment found within
the trial. A change of control has happened in multiple platform trial such as
STAMPEDE (Sydes et al.,, 2012) and RECOVERY (Horby et al.,, 2021). When
changing the control group one may think of using all the data collected to
calculate the future test statistics. There is little work currently
investigating whether keeping the data collected prior to the change of the
control group treatment is the most efficient approach. In this paper we will
consider two settings: (i) We keep all the concurrent data from before the
change in control; (ii) We do not keep any of the data prior to the control
changing. Concurrent data means only participants recruited to the current
control arm at the same time as the active arm of interest are used in the
comparisons. For work on non-concurrent controls see recent work by Lee and
Wason, (2020); Marschner and Schou, (2022); Saville et al., (2022); Wang et
al., (2022).

This work will focus on multi-arm multi-stage trials (MAMS) in which
additional treatments can be planned to be added at multiple points during the
trial. Multi-arm trials allow for multiple treatments to be compared at once
against a common control treatment. Multi-stage trials have interim analyses
which allow for ineffective treatments to be dropped for futility (or lack of
benefit) earlier. As a result interim analyses can improve a trials operating
characteristics (Pocock,, 1977; Todd et al.,, 2001). They also can allow
treatments to stop early if a superior treatment is found, however in the case
studied here the first time this happens this superior treatment will become
the new control.

We will focus our investigation on two types of power: (i) Conditional power
of a treatment - the probability a given treatment can be found superior
against the current control. (ii) Overall power of the trial - the probability
that the active treatment with the greatest treatment effect is found during
the trial.

In Section 2 we introduce the notation, the null hypotheses of interest and
discuss type I error. Section 3 studies the conditional power for a general
design, where treatments can be added at different points in a preplanned
manner. Then Section 3 gives theorems to when keeping the old data is
guaranteed to be detrimental in MAMS trials where all the treatments begin at
the same time. In Section 4 we give the formulation for the overall power
along with its definition and give theorems to when keeping the old data is
guaranteed to be detrimental. A motivating example is then studied in Section
5 for both the case when arms begin the trial at the same time and also when
one starts later. Finally we discuss the considerations one needs to make when
deciding whether to use the pre-change data or not.

##  2 Notation and type I error control

Consider a clinical trial with up to $K$ experimental arms that will be tested
against one common control arm. The primary outcome on each patient is
independent and normally distributed with known variance $\sigma^{2}$. Each
active treatment is tested at $J$ analyses with $J-1$ interim analyses. Let
$n_{k,j}$ denote the number of patients recruited to treatment $k$ by the end
of its $j^{\text{th}}$ stage assuming that recruitment of this arm had begun
at the start. For this paper the focus will be on equal sample size and
allocation ratio for each treatments and equally spaced interim analyses for
all treatments, as this ensures equal pairwise error for each treatment
without needing to have multiple boundary shapes (Greenstreet et al.,, 2021).
Therefore the number of patients recruited between interim analyses is equal
i.e. $n_{k,j}-n_{k,j^{\star}}=n_{k^{\star},j}-n_{k^{\star},j^{\star}}$ for all
$k,k^{\star}=0,\ldots,K$ and $j,j^{\star}=0,\ldots,J$. Let $n_{k,0}$ define
the number of patients already recruited to an active treatment that started
the trial before treatment $k$ enters the trial. We have
$k_{n_{k^{\prime},j^{\prime}}}^{\prime}$ denoting the current control
treatment at point $n_{k^{\prime},j^{\prime}}$, where $j^{\prime}$ is the
stage for treatment $k^{\prime}$ where it became the control, with
$k^{\prime}=0,\ldots,K$ and $j^{\prime}=0,\ldots,J$. Therefore
$n_{k^{\prime},j^{\prime}}$ denotes the number of patients recruited prior to
treatment $k^{\prime}$ becoming control at its $j^{\prime}$ ${}^{\text{th}}$
stage. For simplicity we drop the subscript from
$k_{n_{k^{\prime},j^{\prime}}}^{\prime}$ as the focus of this work will be on
only changing the control group once, with, $k^{\prime}=0$ at the beginning of
the trial.

The null hypotheses of interest are
$H_{k^{\prime}1}:\mu_{1}\leq\mu_{k^{\prime}},H_{k^{\prime}2}:\mu_{2}\leq\mu_{k^{\prime}},...,H_{k^{\prime}K}:\mu_{K}\leq\mu_{k^{\prime}},$
where $\mu_{1},\ldots,\mu_{K}$ are the mean responses on the $K$ experimental
treatments and $\mu_{k^{\prime}}$ is the mean response of the current control
group with $\mu_{0}$ being the mean response of the initial control. Each of
the $K$ hypotheses is potentially tested at a series of analyses indexed by
$j=\ddot{j}_{k,k^{\prime},j^{\prime}}+1,\ldots,J$ where
$\ddot{j}_{k,k^{\prime},j^{\prime}}$ is the last stage for $k$ before
$k^{\prime}$ became the control. When one is in the case that all the
treatments begin at once then $\ddot{j}_{k,k^{\prime},j^{\prime}}=j^{\prime}$
as each interim for each treatment happens at the same time. However if
treatments are added at different points this may not be the case. For example
in a 3 arm trial if treatment 1 becomes the control at its first stage and
treatment 2 is not added till after treatment 1’s first analysis then
$\ddot{j}_{2,1,1}=0$. At analysis $j$ for treatment $k$, to test
$H_{k^{\prime}k}$ it is assumed that responses, $X_{k,i}$ and
$X_{k^{\prime},i^{\prime}}$, from patients $i=n_{k,0},\ldots,n_{k,j}$ and
$i^{\prime}=n_{k^{\prime},0},\ldots,n_{k,j}$ are observed respectively. These
hypotheses are tested at given analysis $j$ using the test statistic:

$\displaystyle Z_{k,k^{\prime},j}=$
$\displaystyle\frac{\sum^{n_{k,j}}_{i=\max(n_{k,0},n_{k^{\prime},0})+1}X_{k,i}-\sum^{n_{k,j}}_{i=\max(n_{k,0},n_{k^{\prime},0})+1}X_{k^{\prime},i}}{\sigma\sqrt{2(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0}))}}.$

In order to ensure only concurrent controls are used we have
$\max(n_{k,0},n_{k^{\prime},0})$. If only the data post the change in the
control is used the test statistics are:

$Z^{\star}_{k,k^{\prime},j,j^{\prime}}=\frac{\sum^{n_{k,j}}_{i=\max(n_{k,0},n_{k^{\prime},0},n_{k^{\prime},j^{\prime}})+1}X_{k,i}-\sum^{n_{k,j}}_{i=\max(n_{k,0},n_{k^{\prime},0},n_{k^{\prime},j^{\prime}})+1}X_{k^{\prime},i}}{\sigma\sqrt{2(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0},n_{k^{\prime},j^{\prime}}))}},$

where $\max(n_{k,0},n_{k^{\prime},0},n_{k^{\prime},j^{\prime}})$ includes the
point in which the control changes and only if $n_{k,0},n_{k^{\prime},0}\leq
n_{k^{\prime},j^{\prime}}$ will $Z^{\star}_{k,k^{\prime},j,j^{\prime}}\neq
Z_{k,k^{\prime},j}$. These test statistics are used to test $H_{k^{\prime}k}$.
Upper and lower stopping boundaries, $U=(u_{1},\ldots,u_{J})$ and
$L=(l_{1},\ldots,l_{J})$, are used for the decision-making as follows. If
$Z_{k,k^{\prime},j}>u_{j}$ then $H_{k^{\prime}k}$ is rejected and the
conclusion that treatment $k$ is superior to the current control is made. If
$Z_{k,k^{\prime},j}<l_{j}$ 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 if $l_{j}\leq Z_{k,k^{\prime},j}\leq u_{j}$. If
the post change data is only used the same rules apply now replacing
$Z_{k,k^{\prime},j}$ with $Z^{\star}_{k,k^{\prime},j,j^{\prime}}$. If multiple
treatments exceed their given upper boundary at the same time point, for
example treatments $k_{1}$ and $k_{2}$ where $k_{1},k_{2}=1,\ldots K$, then
one finds $Z_{k_{1},k_{2},j}$ and if this is negative then one takes forward
treatment $k_{2}$ as the new control treatment and otherwise $k_{1}$ is taken.

These upper and lower stopping boundaries are group-sequential bounds which
are pre-defined in order to control the original type I error control aimed
for in the original trial. Therefore for example they could be aiming to
control the pairwise error rate (PWER) (Wason et al.,, 2014; Choodari-Oskooei
et al.,, 2020), the family wise error rate (FWER) (Burnett et al.,, 2020;
Magirr et al.,, 2012; Greenstreet et al.,, 2021) or the false discovery rate
(FDR) (Robertson et al.,, 2022; Cui et al.,, 2023). Typically when continuing
to use the same boundary as already pre-defined there is no longer a guarantee
this will control the type I error of interest after the change. This is
because the original bounds were not designed for this.

##  3 Conditional power

The conditional power for a given treatment $k^{\star}$ is the probability
that given treatment $k^{\prime}$ is the new standard of care after its
$j^{\prime\text{th}}$ stage that treatment $k^{\star}$ is found superior to
the new control $k^{\prime}$, when tested for treatment $k^{\star}$ remaining
analyses. The conditional part of conditional power can be split into 3
events. Event 1 ($E^{1}_{k^{\star},k^{\prime},j^{\prime}}$) is the event that
treatment $k^{\prime}$ becomes the control at its $j^{\prime\text{th}}$ stage.
Event 2 ($E^{2}_{k^{\star},k^{\prime},j^{\prime}}$) is that treatment
$k^{\star}$ is still in the trial when treatment $k^{\prime}$ becomes the
control. Event 3 ($E^{3}_{k^{\star},k^{\prime},j^{\prime}}$) is that none of
the other $k$ treatments become the control. The detailed formulations for
$E^{1}_{k^{\star},k^{\prime},j^{\prime}}$,
$E^{2}_{k^{\star},k^{\prime},j^{\prime}}$,
$E^{3}_{k^{\star},k^{\prime},j^{\prime}}$ are given in the Appendix A. The
conditional power is therefore defined as:

######  Definition 3.1.

The conditional power for treatment $k^{*}$ for given $k^{\prime}$ and
$j^{\prime}$ is

$P(\text{reject
}H_{k^{\prime}k^{\star}}|E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}}).$

From Definition 3.1 the conditional power is the probability we reject
$H_{k^{\prime}k^{\star}}$ given that at $j^{\prime}$ stage for treatment
$k^{\prime}$ it became the control and treatment $k^{\star}$ is still being
tested. Figure 1 shows the difference in the data included in the calculation
of the conditional power based on the motivating example discussed in Section
5. The motivating example has 2 stages and 4 arms to begin the trial. In this
figure it is assumed that treatment 3 has stopped for futility after the first
stage and treatment 1 has become the new control at this stage. Therefore the
conditional power of interest is that treatment 2 is found superior to
treatment 1, the new control. The area highlighted in blue represents the data
used if all the data is retained. This therefore covers the whole length of
the trial. Whereas the area in pink is if only the data post the change in
control is used therefore only covers the second stage of the trial.

Figure 1: Illustration of the difference in area of data used for conditional
power when comparing using all the data to only the data post change. The area
highlighted in blue represents the data used if all the data is retained. The
area in pink represents the data if only the data post the change in control
is used.

In order to equate the conditional power one can use the conditional
probability definition to remove the need to calculate any highly truncated
normal distributions. The conditional power is:

$\displaystyle\begin{cases}0&\text{ if }n_{k^{\star},J}\leq
n_{k^{\prime},j^{\prime}}\\\
\frac{P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{4}_{k^{\star},k^{\prime},j^{\prime}})}{P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}})}&\text{ if
}n_{k^{\star},J}>n_{k^{\prime},j^{\prime}}\end{cases}.$

where $E^{4}_{k^{\prime},k^{\star},j^{\prime}}$ is the event that we
$\text{reject }H_{k^{\prime}k^{\star}}$ within the rest of the trial. The
formulations for $E^{1}_{k^{\star},k^{\prime},j^{\prime}}$,
$E^{2}_{k^{\star},k^{\prime},j^{\prime}}$,
$E^{3}_{k^{\star},k^{\prime},j^{\prime}}$ and
$E^{4}_{k^{\star},k^{\prime},j^{\prime}}$ are given in the Appendix A. This
can be calculated using multivariate normal distributions as discussed for the
motivating example in Section 5, using the mean of each test statistic
$Z_{k,k^{\prime},j}$,

$\displaystyle\frac{(\mu_{k}-\mu_{k^{\prime}})\sqrt{(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0}))}}{\sigma\sqrt{2}},$

and the correlation matrix, $\Sigma$. The correlation matrix can be split into
multiple values,
$\psi_{(k_{1},k^{\prime}_{1},j_{1}),(k_{2},k^{\prime}_{2},j_{2})}$, that
depend on the correlation between $Z_{k_{1},k^{\prime}_{1},j_{1}}$ and
$Z_{k_{2},k^{\prime}_{2},j_{2}}$, and
$\psi_{(k_{1},k^{\prime}_{1},j_{1}),(k_{2},k^{\prime}_{2},j_{2})}$ equals,

$\begin{cases}0&\text{for }k_{1}\neq
k_{2},k^{\prime}_{2}\;\&\;k^{\prime}_{1}\neq k_{2},k^{\prime}_{2}\\\
\frac{\max(0,n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0},n_{k^{\prime}_{2},0}))}{2\sqrt{(n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0}))(n_{k_{1},j_{2}}-\max(n_{k_{1},0},n_{k^{\prime}_{2},0}))}}&\text{for
}k_{1}=k_{2}\;\&\;k^{\prime}_{1}\neq k^{\prime}_{2}\;\&\;n_{k_{1},j_{1}}\leq
n_{k_{2},j_{2}}\\\
-\frac{\max(0,n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k_{2},0},n_{k^{\prime}_{1},0}))}{2\sqrt{(n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0}))(n_{k_{2},j_{2}}-\max(n_{k_{2},0},n_{k_{1},0}))}}&\text{for
}k_{1}=k^{\prime}_{2}\;\&\;k^{\prime}_{1}\neq k_{2}\;\&\;n_{k_{1},j_{1}}\leq
n_{k_{2},j_{2}}\\\
\frac{\max(0,n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k_{2},0},n_{k^{\prime}_{1},0}))}{2\sqrt{(n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0}))(n_{k_{2},j_{2}}-\max(n_{k_{2},0},n_{k^{\prime}_{1},0}))}}&\text{for
}k_{1}\neq k_{2}\;\&\;k^{\prime}_{1}=k^{\prime}_{2}\;\&\;n_{k_{1},j_{1}}\leq
n_{k_{2},j_{2}}\\\
\sqrt{\frac{n_{k_{1},j_{1}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0})}{n_{k_{1},j_{2}}-\max(n_{k_{1},0},n_{k^{\prime}_{1},0})}}&\text{for
}k_{1}=k_{2}\;\&\;k^{\prime}_{1}=k^{\prime}_{2}\;\&\;n_{k_{1},j_{1}}\leq
n_{k_{2},j_{2}}.\end{cases}$

It is worth noting that one does not need to consider the case that
$k_{2}=k^{\prime}_{1}$ when $n_{k_{1},j_{1}}<n_{k_{2},j_{2}}$ as it is not
possible for a treatment to go from being a control back to being an active
treatment.

In the case of only considering the data post changing the control, the test
statistics before the change are now independent of the test statistics post
the change. Therefore one only needs the event that we $\text{reject
}H_{k^{\prime}k}$ within the rest of the trial. For the case where only the
post change data is used we define this as $E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}}$. If one is in the case where treatment
$k^{\star}$ joins the trial after treatment $k^{\prime}$ becomes the control
then $E^{4}_{k^{\star},k^{\prime},j^{\prime}}=E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}}$ as there is no data pre the change that
is shared. The formulations for $E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}}$ is given in the Appendix A. The
conditional power in this case is:

$\displaystyle\begin{cases}0&\text{ if }n_{k,J}\leq
n_{k^{\prime},j^{\prime}}\\\ P(E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}})&\text{ if
}n_{k,J}>n_{k^{\prime},j^{\prime}}\end{cases}.$ (3.1)

Once again this can be calculated using multivariate normal distributions as
discussed for the motivating example in Section 5 using the mean of each test
statistic $Z_{k,k^{\prime},j}$,

$\displaystyle\frac{(\mu_{k}-\mu_{k^{\prime}})\sqrt{(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0},n_{k^{\prime},j^{\prime}}))}}{\sigma\sqrt{2}};$

and the correlation matrix which can be split into multiple
$\psi_{i,i^{\star}}$ that depend on the correlation between
$Z^{\star}_{k,k^{\prime},j_{1},j^{\prime}}$ and
$Z^{\star}_{k,k^{\prime},j_{2},j^{\prime}}$, and equals,

$\psi_{i,i^{\star}}=\begin{cases}\sqrt{\frac{n_{k,j_{1}}-\max(n_{k,0},n_{k^{\prime},j^{\prime}})}{n_{k,j_{2}}-\max(n_{k,0},n_{k^{\prime},j^{\prime}})}}&\text{for
}j_{1}\leq j_{2}.\end{cases}$

When one is in the case that all the treatments begin at once this simplifies
the equations as now $n_{k,0}=0$ for all $k=0,\ldots K$. Additionally as shown
below one can now prove when it is guaranteed that there is no benefit to
retaining the information pre change in control treatment when considering
conditional power and using the predefined boundaries. These can not be proven
for when treatments are added later however as shown in Section 5 there can
still be a negative effect from keeping the data pre the change in the control
treatment.

###  3.1 When all the treatment start at the beginning of the trial

When all treatments begin at the same time, it can be proven that for many
cases there can never be benefit to retaining the information pre change in
control treatment when considering conditional power and using the predefined
boundaries. The first theorem (Theorem 3.1) states that if there is only one
stage left and the upper boundary is positive, then keeping the historic data
is detrimental to the conditional power.

######  Theorem 3.1.

If a treatment $k^{\prime}$ becomes the control group treatment at stage $J-1$
($E^{1}_{k^{\star},k^{\prime},J-1}\cap E^{3}_{k^{\prime},k^{\prime},J-1}$) and
$u_{J}\geq 0$ then the conditional power for treatment $k^{\star}$ when
retaining the data before the control changed is less than or equal to the
conditional power for treatment $k^{\star}$ when not retaining the pre-change
data.

The proof of Theorem 3.1 is given in Appendix B. This theorem can be further
extended. First in Theorem 3.2 which states that if there are multiple stages
of the trial left and both the upper and lower boundaries are greater than or
equal to 0 then retaining the post change data is detrimental to the
conditional power. The second extension is Theorem 3.3 which states that if
there are multiple stages of the trial left and the upper boundaries are
positive and there is no lower boundaries then retaining the post change data
is detrimental to the conditional power.

######  Theorem 3.2.

If a treatment $k^{\prime}$ becomes the new control group treatment at stage
$j^{\prime}-1$ ($E^{1}_{k^{\star},k^{\prime},j^{\prime}-1}\cap
E^{3}_{k^{\prime},k^{\prime},j^{\prime}-1}$) and $u_{j}\geq 0$ and $l_{j}\geq
0$ for all $j=(j^{\prime}+1)\ldots J$ then the conditional power for treatment
$k^{\star}$ when retaining the data before the control changed is less than or
equal to the conditional power for treatment $k^{\star}$ when not retaining
the pre-change data.

######  Theorem 3.3.

If a treatment $k^{\prime}$ becomes the new control group treatment at stage
$j^{\prime}-1$ ($E^{1}_{k^{\star},k^{\prime},j^{\prime}-1}\cap
E^{3}_{k^{\prime},k^{\prime},j^{\prime}-1}$) and $u_{j}\geq 0$ and there are
no lower boundaries for all $j=(j^{\prime}+1)\ldots J$ then the conditional
power for treatment $k^{\star}$ when retaining the data before the control
changed is less than or equal to the conditional power for treatment
$k^{\star}$ when not retaining the pre-change data.

The proof for Theorem 3.2 is given in the Appendix B. The proof for Theorem
3.3, which is similar to the proof of Theorem 3.2, is given in the
Supplementary Materials Section 1. Furthermore as we will show in the
Supplementary Materials Section 6 even if $l_{j}<0$ for some
$j=(j^{\prime}+1)\ldots J$ then one will find that retaining the old
information is likely detrimental for the conditional power. However in
Supplementary Materials Section 7 it is shown that there are cases when
$l_{j}<0$ where keeping the old data can be beneficial for conditional power.

##  4 Overall power

Overall power of a treatment gives the probability that during the trial the
active treatment with the greatest positive treatment effect is either taken
forward as the new control or is declared superior compared to a new control,
if the control has already changed. Therefore overall power can be thought of
as two main parts: either the correct treatment becomes the new control first,
or another treatment becomes the new control and subsequently this treatment
is found to be better than this new control. This gives the overall power
definition as:

######  Definition 4.1.

The overall power for the treatment $k^{\star}$ which has the greatest
treatment effect, $\mu_{k^{\star}}\geq\mu_{k}\forall k=1,\ldots,K,$ equals

$P(\bigcup^{J}_{j^{\star}=1}[E^{1}_{k^{\star},k^{\star},j^{\star}}\cap
E^{3}_{k^{\star},k^{\star},j^{\star}}]\cup\bigcup_{k^{\prime}\in\\{1,\ldots,K\\}/k^{\star}}\bigcup^{J}_{j^{\prime}=1}[E^{1}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{2}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{3}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{4}_{k^{\prime},k^{\star},j^{\prime}}]).$

Due to the multiple disjoint sets within Definition 4.1, the overall power can
be split into multiple, easy to compute, parts. The first of these is the
probability that at each interim $j^{\star}$, treatment $k^{\star}$ becomes
the control ($\Xi_{k^{\star},j^{\star}}$) and this equals:

$\Xi_{k^{\star},j^{\star}}=P(E^{1}_{k^{\star},k^{\star},j^{\star}}\cap
E^{3}_{k^{\star},k^{\star},j^{\star}}).$ (4.1)

The probability another treatment becomes the new control and then this
treatment is found to be better then the new control
($\Omega_{k^{\star},k^{\prime},j^{\prime}}$) can be split into every possible
$k^{\prime}$ and $j^{\prime}$.

$\Omega_{k^{\star},k^{\prime},j^{\prime}}=\begin{cases}0&\text{ if
}n_{k,J}\leq n_{k^{\prime},j^{\prime}}\\\
P(E^{1}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{2}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{3}_{k^{\prime},k^{\star},j^{\prime}}\cap
E^{4}_{k^{\prime},k^{\star},j^{\prime}})&\text{ if
}n_{k,J}>n_{k^{\prime},j^{\prime}}\end{cases}.$ (4.2)

Combining Equation (4.1) and Equation (4.2) the overall power is:

$\sum^{J}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,\ldots,K\\}/k^{\star}}\sum^{J}_{j^{\prime}=1}\Omega_{k^{\prime},k^{\star},j^{\prime}}.$

When we consider only using the data post change in control the probability
another treatment becomes the new control and then this treatment is found to
be better then the new control
($\Omega^{\star}_{k^{\star},k^{\prime},j^{\prime}}$) becomes:

$\Omega^{\star}_{k^{\star},k^{\prime},j^{\prime}}=\begin{cases}0&\text{ if
}n_{k,J}\leq n_{k^{\prime},j^{\prime}}\\\
P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}})P(E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}})&\text{ if
}n_{k,J}<n_{k^{\prime},j^{\prime}}\end{cases}.$

This is due to the independence of event 4 with the rest of the events.
Therefore the overall power is:

$\sum^{J}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,\ldots,K\\}/k^{\star}}\sum^{J}_{j^{\prime}=1}\Omega^{\star}_{k^{\star},k^{\prime},j^{\prime}}.$

As shown below one can now prove when it is guaranteed that there is no
benefit to retaining the information pre change in control treatment when
considering overall power and using the predefined boundaries. These can not
be proven for when treatments are added later, however as shown in Section 5
there can still be a negative effect from keeping the data pre the change in
the control treatment.

###  4.1 When all the treatment start at the beginning of the trial

In the scenario where all the treatments begin at once, from Theorem 3.2 and
Theorem 3.3 for the conditional power, one can prove similar results for the
overall power.

######  Theorem 4.1.

If $u_{j}\geq 0$ and $l_{j}\geq 0$ for all $j=1,\ldots,J$ then the overall
power when retaining the data before the control changed is less than or equal
to the overall power when not retaining the pre-change data.

######  Theorem 4.2.

If $u_{j}\geq 0$ and there are no lower boundaries for all $j=1,\ldots,J$ then
the overall power when retaining the data before the control changed is less
than or equal to the overall power when not retaining the pre-change data.

The proof for Theorem 4.1 and Theorem 4.2 is given in Appendix C. Furthermore
as is shown in Supplementary Materials Section 6 even if $l_{j}<0$ for any
$j=1,\ldots,J$ then there are cases that retaining information pre the change
in the control group is detrimental for the overall power. This is shown in
the example in the Supplementary Materials Section 6 as the difference in
conditional power between keeping and discarding the pre change data is
negative, therefore, so will the overall power.

##  5 Motivating trial example

We consider the motivating trial of TAILoR (Pushpakom et al.,, 2020). The
TAILoR trial was a 4 arm trial which studied the effect of different doses of
a treatment on HIV. The study had 1 interim analysis. We are going to use the
operating characteristics from this study to see the effects on overall and
conditional power if the control was changed mid trial if a treatment was
found superior. In the original design the family wise error rate (FWER)
(Pushpakom et al.,, 2015) was controlled at 5% one sided for a normal
continuous endpoint and there was a planned 90% power. The trial was planned
to have equal allocation across stages. In addition the clinically relevant
effect of $\theta_{1}=0.545$ and uninteresting effect $\theta_{0}=0.178$
assuming the variance $\sigma^{2}=1$ was used.

Triangular stopping boundaries will be used (Whitehead,, 1997) as recommended
in Wason and Jaki, (2012). The stopping boundaries when all the treatments
start at once will be calculated using the approach given in Magirr et al.,
(2012) to control FWER for the design before the change in control. In
addition we will consider the design if one of the treatments was added at the
end of the first stage. Therefore the stopping boundaries will be found using
the approach given in Greenstreet et al., (2021) to control FWER for the
design before the change in control. The calculations of the power will be
done using Greenstreet et al., (2023) in-order to control the pairwise power
for each treatment. This is chosen as it is similar to that used in the
original trial but is designed for trials which continues after a treatment is
taken forward. 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); the upper and lower
boundaries when all the treatments start at once where found using MAMS (Jaki
et al.,, 2019) and the code was parallelised using packages doParallel (Daniel
et al., 2022a, ) and foreach (Daniel et al., 2022b, ). The code is available
at https://github.com/pgreenstreet/change_control_platform.

This section will be split into two parts: the first case (Case 1) will look
at the case when all treatment start at the beginning; the second (Case 2)
will look at the case where one of the treatments is added a stage later. Case
1 and Case 2 are depicted in Figure 2(a) and Figure 2(b) respectively.

(a)

(b)

Figure 2: Illustration of the motivating trial. Figure 2(a) illustrates when
all treatments start at the beginning and Figure 2(b) illustrates when one
treatment starts at the end of the first stage.

###  5.1 Case 1: All treatments start at the same time

Using the approach by Magirr et al., (2012) the triangular upper and lower
stopping boundaries are found to be

$U=\begin{pmatrix}2.330&2.197\\\ 2.330&2.197\\\
2.330&2.197\end{pmatrix},\;\;\;L=\begin{pmatrix}0.777&2.197\\\ 0.777&2.197\\\
0.777&2.197\end{pmatrix}.$

Using Greenstreet et al., (2023) the maximum sample size is 344 based on 43
patients per arm per stage to ensure pairwise power of 90%. Due to each
treatment getting the same number of treatments per stage we define
$n=n_{k,1}$ for all $k=0,1,2,3$ therefore the number of patients recruited at
the second stage for a treatment which runs for both stages is $n+n=2n$. This
subsection begins by discussing the formulation of the equations for both
conditional power and overall power. After this we then study the results of
the conditional power and overall power under different treatment effects and
compare the effect of using all available data compared to only using the data
post change.

####  5.1.1 Conditional power

There is only one place in the trial where the conditional power is not zero
as there is only one interim analysis in the study and all the treatments
begin at the same point. This happens when a treatment becomes the new control
at the first stage. We define the treatment of interest as $k^{\star}$, define
the new control as $k^{\prime}$ and define the other treatment in the study as
$k_{1}$. The conditional power for treatment $k^{\star}$ when treatment
$k^{\prime}$ becomes the new control at stage 1 is:

$\frac{P(E^{1}_{k^{\star},k^{\prime},1}\cap E^{2}_{k^{\star},k^{\prime},1}\cap
E^{3}_{k^{\star},k^{\prime},1}\cap
E^{4}_{k^{\star},k^{\prime},1})}{P(E^{1}_{k^{\star},k^{\prime},1}\cap
E^{2}_{k^{\star},k^{\prime},1}\cap E^{3}_{k^{\star},k^{\prime},1})}.$ (5.1)

When calculating Equation (5.1) one can take advantage of the fact that all
the treatment start at the same time. Therefore,
$Z_{k,k^{\prime},j^{\prime}}<0\cup Z_{k,0,j^{\prime}}<u_{j^{\prime}}$ can be
simplified to $Z_{k,k^{\prime},j^{\prime}}<0$. This is because testing
$Z_{k,0,j^{\prime}}>u_{j^{\prime}}$ and $Z_{k,k^{\prime},j^{\prime}}<0$ is
equivalent to testing
$Z_{k,k^{\prime},j^{\prime}}=Z_{k,0,j^{\prime}}-Z_{k^{\prime},0,j^{\prime}}<0$
and $Z_{k^{\prime},0,j^{\prime}}>u_{j^{\prime}}$ for treatment $k^{\prime}$ to
be taken forward when all the treatments start at the same point. When we only
retain the new information the conditional power is

$P(E^{\star 4}_{k^{\star},k^{\prime},1}).$ (5.2)

In the Supplementary Materials Section 2 the formulations used to calculate
Equation (5.2) and Equation (5.1) are given.

####  5.1.2 Overall power

To calculate overall power in addition to the calculations above one needs the
probability the treatment of interest $k^{\star}$ becomes the control at stage
1 or stage 2 of the trial. The two other arms in this case are defined as
$k_{1}$ and $k_{2}$. Due to all the arms starting at the same point this
simplifies the calculation of both $\Xi_{k,1}$ and $\Xi_{k,2}$. The complete
formulation for $\Xi_{k,1}$ and $\Xi_{k,2}$ is given in the Supplementary
Materials Section 3. Using the calculations for the conditional power one can
find both $\Omega_{k^{\star},k^{\prime},1}$ and
$\Omega^{\star}_{k^{\star},k^{\prime},1}$. The overall power for treatment
$k^{\star}$ when the old information is retained is

$\sum^{2}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,2,3\\}/k^{\star}}\Omega_{k^{\star},k^{\prime},1}.$

When only new data is used the overall power is

$\sum^{2}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,2,3\\}/k^{\star}}\Omega^{\star}_{k^{\star},k^{\prime},1}.$

####  5.1.3 Results

In Figure 3 the difference between conditional power when retaining all the
old data and not retaining the data can be seen. The conditional power for
treatment 2 when treatment 1 is the new control after the first stage is
studied. The y-axis gives the treatment effect of treatment 2 compared to the
original control treatment. The x-axis gives effect of treatment 1 compared to
the original control treatment. The colour as given on the scale, to the right
of the figure, defines the difference in conditional power between retaining
the information pre the change and not. The effect of different values of
$\mu_{3}$ is very small. As it has very little effect on the probability that
treatment 2 is found superior to treatment 1 in the final stage. Therefore we
will focus on the results for when $\mu_{3}-\mu_{0}=0$. However in the
Supplementary Materials the effect of $\mu_{3}-\mu_{0}$ having an
uninteresting treatment effect is shown.

As shown in the Supplementary Materials Section 4, for the conditional power
when all the data is retained, if the difference between $\mu_{2}$ and
$\mu_{1}$ is greater than 1, the probability that treatment 2 is found
superior to the treatment 1 is at least 88.9%. When $\mu_{2}$ is less then
$\mu_{1}$ the probability of incorrectly taking treatment 2 forward is at
worst 0.01%. This is in comparison to when only the data post the change is
retained. Now when the difference between $\mu_{2}$ and $\mu_{1}$ is greater
than 0.760 we have conditional power of above 90%. When $\mu_{2}$ is less than
$\mu_{1}$ the probability of incorrectly taking treatment 2 forward is at
worst 1.15%. The figures illustrating the conditional power for these can be
seen in the Supplementary Materials Section 4.

The difference between conditional power when retaining all the old data and
not retaining the data can be seen in Figure 3 when $\mu_{3}-\mu_{0}=0$. As
can be seen in Figure 3 when $\mu_{2}-\mu_{1}$ is around 0.5 then the loss in
conditional power is maximised. This can be greater than 50%. However as this
difference becomes a lot more extreme the loss becomes close to 0. This is
because at this point either approach has almost an 100% chance of finding
treatment 2 superior to treatment 1. In the Supplementary Materials Section 5
we study the effect on conditional power of different possible values of
$Z_{(1,0),1}$ and $Z_{(2,0),1}$ for one of these points, $\mu_{1}=-0.25$ and
$\mu_{2}=0.75$. Here we can ignore the value of $Z_{(3,0),1}$ as this does not
influence the probabilities as shown in the proof to Theorem 1 in Appendix B.
It is shown here that even in this case where there is on average very little
benefit in only retaining the new information there are potential values of
$Z_{(1,0),1}$ and $Z_{(2,0),1}$ where there is large benefit in only using the
new data. However the probability of these $Z$ values happening is very small
for the given $\mu_{1}$ and $\mu_{2}$. When $\mu_{2}-\mu_{1}<0$ the difference
in conditional power is small. This is because for both approaches the
probability that treatment 2 is found superior to treatment 1 when in fact it
is not is small.

Figure 3: The difference in conditional power between keeping the data pre
change and not, for treatment 2 given that treatment 1 has gone forward at the
first stage.

The overall power is very similar between retaining the data or not as shown
in Figure 4. Once again the focus being $\mu_{3}-\mu_{0}$ equal to zero.
However in the Supplementary Materials Section 4 the results are also shown
when $\mu_{3}-\mu_{0}=0.178$ and $\mu_{3}-\mu_{0}=-0.178$. This is along with
the figures illustrating the overall power for when the data is retained or
not. The y-axis gives the treatment effect of treatment 2 compared to the
original control and the x-axis gives effect of treatment 1 compared to the
original control. The colour as given on the scale to the right defines the
difference in overall power at that given point.

The maximum difference in overall power is 1.7%. This is compared to a maximum
change of 52.6% for the difference in conditional power. This is because when
calculating the overall power most of the time the correct treatment will be
taken forward compared to the original control instead of one of the other
treatments, however, when studying the conditional power this is not
considered. Therefore when calculating the overall power the probability of a
mistake is taken into account. This effect can be seen in Figure 5. This
figure gives the probability of the treatment which does not have the greatest
treatment effect becoming the control at the first stage. This shows that in
many of the areas, where the difference in conditional power was at its
greatest, it is unlikely that treatment 1 or 3 would have been taken forward
instead of treatment 2.

Figure 4: The difference in overall power between keeping the data pre change
and not. Figure 5: The probability of the treatment which does not have the
greatest treatment effect becoming the control at the first stage.

###  5.2 Case 2: A treatment added later

This subsection will study the case where one of the treatments is added a
stage later as shown in Figure 2(b). Using the approach by Greenstreet et al.,
(2021) the triangular stopping boundaries are found to be

$U=\begin{pmatrix}2.358&2.223\\\ 2.358&2.223\\\
2.358&2.223\end{pmatrix},\;\;\;L=\begin{pmatrix}0.786&2.223\\\ 0.786&2.223\\\
0.786&2.223\end{pmatrix}.$

Based on 43 patients per arm per stage the maximum sample size is now 387 in
order to control the pairwise power at 90% (Greenstreet et al.,, 2023). This
addition accounts for the patients which would need to be added for the later
treatment as seen in Figure 2(b). As each treatment gets the same number of
treatments per stage we define $n=n_{k,1}-n_{k,0}=n_{k,2}-n_{k,1}$ for all
$k=0,1,2,3$.

We begin this subsection by discussing the equations for both conditional
power and overall power. For both of these we will split the calculations into
two. The first is for the treatments who begin the trial. The second is for
the treatment which joins after 1 stage. For brevity only an explanation of
the calculation required is given below, however, in the Supplementary
Materials Section 9 the equations required are given. After this we then study
the results of the conditional power and overall power under different
treatment effects and compare the effect of using all available data compared
to only using the data post change.

####  5.2.1 Conditional power

The only non zero conditional power for the 2 treatments that start the trial
are if a treatment becomes the control at the first stage. Therefore the
conditional power when old data is retained is very similar to the one given
in Subsection 5.1.1, however now the 3rd treatment is no longer considered, as
it is unable to become the control at stage 1 as it is yet to be tested. When
considering the conditional power when only new data is retained then Equation
(5.2) is used as once again the conditional power is independent of the other
continuing treatments.

The conditional power for treatment 3 is non zero in two cases. The first is
when the new control has been declared at the first stage of the trial - so at
the point treatment 3 begins. In this case the conditional power is the same
if the old data is retained or not. This is because only concurrent controls
are used therefore there is no difference between the two as no old data is
available. One can therefore take advantage of the fact that event
$E^{4}_{k^{\star},k^{\prime},1}$ is independent of
$E^{1}_{k^{\star},k^{\prime},1}$, $E^{2}_{k^{\star},k^{\prime},1}$ and
$E^{3}_{k^{\star},k^{\prime},1}$. The conditional power given that treatment 1
or 2 becomes the control at its second stage is more numerically complex.
However one can still use the fact that treatment 1 and 2 start at the same
time therefore, $Z_{k_{1},k^{\prime},j^{\prime}}<0\cup
Z_{k_{1},0,j^{\prime}}<u_{j^{\prime}}$ can be simplified to
$Z_{k_{1},k^{\prime},j^{\prime}}<0$ in this case. The complete equations for
these can be seen in the Supplementary Materials Section 9.

####  5.2.2 Overall power

Calculating $\Xi_{k,1}$ and $\Xi_{k,2}$ for the treatments that start at the
beginning of the trial is very similar to the calculations given in Subsection
5.1.2. However for $\Xi_{k,1}$ one only needs to consider the other treatment
that began the trial at the beginning. Similar for $\Xi_{k,2}$ one only needs
to consider the first stage for treatment 3. Therefore the overall power for
treatment $k^{\star}$ given it is treatment 1 or 2 is

$\sum^{2}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\Omega_{k^{\star},k^{\prime}=\\{1,2\\}/k^{\star},1}.$

If the third treatment is the superior treatment then one needs to calculate
$\Xi_{k,1}$ and $\Xi_{k,2}$ accounting for the fact this treatment has been
added at a later stage. One needs to include, all the possible outcomes for
the other treatments that ensures that treatment 3 is taken forward first. As
a result this requires 9 integrals and 4 integrals for $\Xi_{k,1}$ and
$\Xi_{k,2}$, respectively. The reason for the reduction in integrals for the
second is if treatment 3 becomes the new control at it second stage then this
guaranties that the other treatments have gone below their lower boundaries at
some point, where as this is not the case for $\Xi_{k,1}$. This can be seen
clearly in the equations given in the Supplementary Materials Section 9. The
overall power for treatment $k^{\star}$ given it is treatment 3 is

$\sum^{2}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,2\\}}\sum^{2}_{j^{\prime}=1}\Omega_{k^{\star},k^{\prime},j^{\prime}}.$

####  5.2.3 Results

In this section the main focus will be on the conditional and overall power of
treatment 3. This is because as shown in the Supplementary Materials Section
10 the results for the conditional and overall power for the earlier
treatments are almost identical to the case when all the treatments start at
the same time as seen in Section 5.1. For the conditional power we are going
to therefore focus on the case that treatment 1 becomes the new control at its
second stage and treatment 2 has treatment effect equal to that of the
original control. This is the focus as if treatment 1 becomes the control at
its first stage there is no difference between the conditional power for
treatment 3 if old data is retained or not. This is as the same data will be
used in both cases, as only concurrent data is used. The difference in
conditional power can be seen in Figure 6.

As can be seen here once again there is no benefit found in keeping the
historic data. This is because, as detailed in the proof to Theorem 3.1, it is
still unlikely that Equation (B.1) is true given that treatment $k^{\prime}$
became the control. However it is worth noting that the difference in power is
less than it was for the case when all the treatments started at once with the
maximum difference being 39.8%.

(a)

(b)

Figure 6: The difference in conditional power for treatment 3 given that
treatment 1 has gone forward at the second stage (Figure 6(a)) and the overall
power (Figure 6(b)) when treatment 3 starts after the first stage.

When we consider the difference in overall power we once again see that
keeping the old data is detrimental as shown in Figure 6. However as seen in
the case when all treatments start at the same time the effect of keeping the
old data is a lot less for the overall power compared to the conditional
power. Figure 6 is no longer symmetric as the treatment effect of treatment 3
has no effect on the difference for treatment 1. This is because at the first
opportunity treatment 3 could be taken forward, treatment 1 will be at its
final analyses as discussed in Section 5.2.1.

Through this section it has been shown that for an example, even for the later
arm, there is no benefit in keeping the data. However this is not always true.
For example when considering the case when there is only one analysis for each
treatment there is more likely to be benefit from keeping the old data for the
later treatment as can be seen in the Supplementary Materials Section 11.

##  6 Discussion

In this paper we have studied the effect of keeping or discarding the data
post a change in control in a platform study, with the focus being on the
power of the study. This work has shown that in many cases one is likely to be
better off not retaining the data, when using the same stopping boundaries.
Therefore in this case it would be potentially beneficial to start a new
trial. This would also give time for decisions with regards to which
treatments should be compared to the new control. There are likely more
benefits in starting a new trial including being able to adjust the research
question, the target population, and the treatment dose as well as many more.
However these would involve creating a new protocol and setting up a new trial
which may be more administrative and logistical work compared to continuing
the trial.

In Section 3 and Section 4 it was shown for many cases when all arm start at
the same time one can prove that the overall and conditional power will be
lower by retaining the old data. However even in scenarios where one can not
prove that the power will be lost by retaining the old data it has been shown
in the Supplementary Materials Section 6 that one is still likely better of
not retaining the previous data. This section looks at the effect of using the
symmetric boundary shape of O’Brien and Fleming (O’Brien and Fleming,, 1979)
for a 3 stage example using the same operating characteristics as given in the
motivating example of TAILor (Pushpakom et al.,, 2015). However in the
Supplementary Materials Section 7 there is an example when all arm start at
ones where keeping the old data can be beneficial. In Section 5 it was shown
that the loss in conditional power can be very large when old data is
retained. It was shown that for overall power the loss is less than that for
conditional power, but there is still a loss in overall power.

This work has shown that when adding additional treatments later that
depending on the boundary shapes there may not be benefit in keeping the
historic data for the later treatments as well as the ones which start the
trial. However in this case it is not ensured that keeping the old data will
be detrimental as is shown in the Supplementary Materials Section 11 for an
example where each treatment gets one analysis.

Equal allocation ratio between arms has been used in this work. This has been
done for simplicity, so the pairwise error is equal for every treatment before
the change in control, one could consider extending this work to consider
different allocation ratios for each treatment and one could also extend this
work to consider changing the control and a change in allocation ratio.
However one needs to be aware of the effect of time trends as discussed in
Greenstreet et al., (2021); Roig et al., (2023) and one may wish to use a
modelling approach (Lee and Wason,, 2020).

Furthermore in this work we have looked at a ideal example where the trial has
equal allocation as planned. However in reality the probability of having
equal allocation is very slim depending on the treatment allocation method.
Therefore we have also considered the effect of using simple random
allocation. This therefore means criteria of Theorem 3.2 or Theorem 3.3 are no
longer met. We have investigated this for three cases. We studied the number
of times out of 100,000,000 simulations that keeping the data has resulted in
the treatment of interest being taken forward when this would not have been
the case using only the new data. This probability is still very small
(0.0006% in the example studied) relative to the probability that discarding
the old data has resulted in the treatment of interest being taken forward
when this would not have been the case using all the data. This can be seen in
the Supplementary Materials Section 8.

Overall this paper has highlighted the importance to consider what to do if
you change control during a platform trial to one which has been found
superior to the control. Therefore one should consider if they should continue
the current trial or stop and start a new trial with the new control.

##  Appendix A Formulation of the events for calculating the conditional power
and overall

The event $E^{1}_{k^{\star},k^{\prime},j^{\prime}}$ which is the event that
treatment $k^{\prime}$ becomes the control at it’s $j^{\prime\text{th}}$ stage
equals,

$\displaystyle
E^{1}_{k^{\star},k^{\prime},j^{\prime}}=\bigcap^{j^{\prime}-1}_{i=1}(l_{i}\leq
Z_{k^{\prime},0,i}\leq u_{i})\cap Z_{k^{\prime},0,j^{\prime}}\geq
u_{j^{\prime}}.$

The event $E^{2}_{k^{\star},k^{\prime},j^{\prime}}$ which is that treatment
$k^{\star}$ is still in the trial when treatment $k^{\prime}$ becomes the
control equals,

$\displaystyle
E^{2}_{k^{\star},k^{\prime},j^{\prime}}=(n_{k^{\star},1}>n_{k^{\prime},j^{\prime}})\cup(n_{k^{\star},1}\leq
n_{k^{\prime},j^{\prime}})\cap\bigg{\\{}\bigg{[}(n_{k^{\star},{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}<n_{k^{\prime},j^{\prime}})\cap\bigcap^{{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}_{i=1}(l_{i}\leq
Z_{k^{\star},0,i}\leq u_{i})\bigg{]}$
$\displaystyle\cup\bigg{[}(n_{k^{\star},{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}=n_{k^{\prime},j^{\prime}})\bigcap^{{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}-1}_{i=1}(l_{i}\leq
Z_{k^{\prime},0,i}\leq
u_{i})\cap(l_{{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}\leq
Z_{k^{\star},0,{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}})$
$\displaystyle\cap[(Z_{k^{\star},0,{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}\leq
u_{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}})\cup(Z_{k^{\star},k,{\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}}}\leq
0)]$

The event $E^{3}_{k^{\star},k^{\prime},j^{\prime}}$ which is that none of the
other $k$ treatments become the control is

$\displaystyle E^{3}_{k^{\star},k^{\prime},j^{\prime}}$
$\displaystyle=\bigcap_{k\in(1\ldots
K)/k^{\star},k^{\prime}}\Bigg{(}(n_{k,1}>n_{k^{\prime},j^{\prime}})\cup(n_{k,1}\leq
n_{k^{\prime},j^{\prime}})\cap\bigg{\\{}\bigg{[}\bigcup^{\ddot{j}_{k,k^{\prime},j^{\prime}}-1}_{i=1}\bigcap^{i-1}_{i^{\star}=1}(l_{i^{\star}}\leq
Z_{k,0,i^{\star}}\leq u_{i^{\star}})$ $\displaystyle\cap(Z_{k,0,i}\leq
l_{i})\bigg{]}\cup\bigg{(}\bigg{[}(n_{k,{\ddot{j}_{k,k^{\prime},j^{\prime}}}}<n_{k^{\prime},j^{\prime}})\cap\bigcap^{{\ddot{j}_{k,k^{\prime},j^{\prime}}}-1}_{i=1}(l_{i}\leq
Z_{k,0,i}\leq u_{i})$
$\displaystyle\cap(Z_{k,0,{\ddot{j}_{k,k^{\prime},j^{\prime}}}}\leq
u_{\ddot{j}_{k,k^{\prime},j^{\prime}}})\bigg{]}\cup\bigg{[}(n_{k,{\ddot{j}_{k,k^{\prime},j^{\prime}}}}=n_{k^{\prime},j^{\prime}})\cap\bigcap^{{\ddot{j}_{k,k^{\prime},j^{\prime}}}-1}_{i=1}(l_{i}\leq
Z_{k,0,i}\leq u_{i})$
$\displaystyle\cap[(Z_{k,0,{\ddot{j}_{k,k^{\prime},j^{\prime}}}}\leq
u_{\ddot{j}_{k,k^{\prime},j^{\prime}}})\cup(Z_{k,k^{\prime},{\ddot{j}_{k,k^{\prime},j^{\prime}}}}<0)]\bigg{]}\bigg{)}\bigg{\\}}\Bigg{)}.$

The event $E^{4}_{k^{\prime},k^{\star},j^{\prime}}$ which is the event that we
$\text{reject }H_{k^{\prime}k}$ within the rest of the trial equals,

$\displaystyle
E^{4}_{k^{\star},k^{\prime},j^{\prime}}=\bigcup^{J}_{i=\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}+1}\bigcap^{i-1}_{i^{\star}=\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}+1}(l_{i^{\star}}\leq
Z_{k^{\star},k^{\prime},i^{\star}}\leq
u_{i^{\star}})\cap(u_{i}<Z_{k^{\star},k^{\prime},i}).$

The event $E^{\star 4}_{k^{\star},k^{\prime},j^{\prime}}$ which is the event
that we $\text{reject }H_{k^{\prime}k}$ within the rest of the trial when not
retaining the information post the change in control treatment equals,

$\displaystyle E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}}=\bigcup^{J}_{i=\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}+1}\bigcap^{i-1}_{i^{\star}=\ddot{j}_{k^{\star},k^{\prime},j^{\prime}}+1}(l_{i^{\star}}\leq
Z^{\star}_{k^{\star},k^{\prime},i^{\star},j^{\prime}}\leq
u_{i^{\star}})\cap(u_{i}<Z^{\star}_{k^{\star},k^{\prime},i,j^{\prime}}).$

##  Appendix B Proof of Theorem 1 and Theorem 2

The proof of Theorem 3.1 is:

###### Proof.

Define $\hat{Z}_{k,k^{\prime},j^{\prime}}$, where
$\hat{Z}_{k,k^{\prime},j^{\prime}}$ equals $Z_{k,k^{\prime},j}$ at
$n_{k^{\prime},j^{\prime}}$, so:

$\displaystyle\hat{Z}_{k,k^{\prime},j^{\prime}}=$
$\displaystyle\frac{\sum^{n_{k^{\prime},j^{\prime}}}_{i=\max(n_{k,0},n_{k^{\prime},0})+1}X_{k,i}-\sum^{n_{k^{\prime},j^{\prime}}}_{i=\max(n_{k,0},n_{k^{\prime},0})+1}X_{k^{\prime},i}}{\sigma\sqrt{2(n_{k^{\prime},j^{\prime}}-\max(n_{k,0},n_{k^{\prime},0}))}}.$

Therefore,

$\displaystyle
Z_{k,k^{\prime},j}=\frac{\hat{Z}_{k,k^{\prime},j^{\prime}}\sqrt{n_{k^{\prime},j^{\prime}}-\max(n_{k,0},n_{k^{\prime},0})}+Z^{\star}_{k,k^{\prime},j,j^{\prime}}\sqrt{n_{k,j}-n_{k^{\prime},j^{\prime}})}}{\sqrt{(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0}))}}.$

The same boundaries $U$ and $L$ ,as predefined for the trial, are used so if
the old data is kept one can rearrange $Z_{k,k^{\prime},j}>u_{j}$ to be:

$\displaystyle
Z^{\star}_{k,k^{\prime},j,j^{\prime}}>\frac{u_{j}\sqrt{(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0}))}-\hat{Z}_{k,k^{\prime},j^{\prime}}\sqrt{n_{k^{\prime},j^{\prime}}-\max(n_{k,0},n_{k^{\prime},0})}}{\sqrt{n_{k,j}-n_{k^{\prime},j^{\prime}}}},$

compared to $Z^{\star}_{k,k^{\prime},j,j^{\prime}}>u_{j}$ for only new data.
There is only increased chance of going above $u_{j}$ when keeping the
historic data if:

$\displaystyle\hat{Z}_{k,k^{\prime},j^{\prime}}>\frac{u_{j}[\sqrt{(n_{k,j}-\max(n_{k,0},n_{k^{\prime},0}))}-\sqrt{n_{k,j}-n_{k^{\prime},j^{\prime}}}]}{\sqrt{n_{k^{\prime},j^{\prime}}-\max(n_{k,0},n_{k^{\prime},0})}}.$
(B.1)

For an increase chance of rejecting the null hypothesis $H_{k,k^{\prime}}$ at
the next stage if pre change data is kept compared to discard it one requires
$\hat{Z}_{k,k^{\prime},j^{\prime}}$ to be positive if $u_{j}$ is positive.
Using Equation (B.1) if all treatment are added at the same point it is worth
keeping the historic data if:

$\displaystyle\hat{Z}_{k,k^{\prime},j^{\prime}}>\frac{u_{j}[\sqrt{n_{k,j}}-\sqrt{n_{k,j}-n_{k^{\prime},j^{\prime}}}]}{\sqrt{n_{k^{\prime},j^{\prime}}}}\geq
0.$ (B.2)

However $\hat{Z}_{k,k^{\prime},j^{\prime}}<0$ as treatment $k^{\prime}$ is the
new control not treatment $k^{*}$. ∎

The proof of Theorem 3.2 is:

###### Proof.

Define the following:

$\displaystyle B_{k^{\star},j}(\delta_{1,j},\delta_{2,j})=$
$\displaystyle[(\delta_{2,j}l_{j}+\delta_{1,j})<Z^{\star}_{k^{\star},k^{\prime},j}<(\delta_{2,j}u_{j}+\delta_{1,j})]$
$\displaystyle C_{k^{\star},j}(\delta_{1,j},\delta_{2,j})=$
$\displaystyle[(\delta_{2,j}u_{j}+\delta_{1,j})<Z^{\star}_{k^{\star},k^{\prime},j}].$

From Definition 3.1 the conditional power equals:

$R(\delta_{1,j},\delta_{2,j})=\bigcup^{J}_{j=j^{\prime}+1}\bigg{[}\bigcap^{j-1}_{i=j^{\prime}+1}(B_{k^{\star},i}(\delta_{1,i},\delta_{2,i}))\cap
C_{k^{\star},j}(\delta_{1,j},\delta_{2,j})\bigg{]}.$

When no data is taken $\delta_{1,j}=0$ and $\delta_{2,j}=1$. However when old
data is taken forward
$\delta_{1,j}=\frac{-\hat{Z}_{k,k^{\prime},j^{\prime}}\sqrt{n_{k^{\prime},j^{\prime}}}}{\sqrt{n_{k,j}-n_{k,j^{\prime}}}}$
and $\delta_{2,j}=\frac{\sqrt{n_{k,j}}}{\sqrt{n_{k,j}-n_{k,j^{\prime}}}}$.
Therefore when old data is retained $\delta_{1,j}\geq 0$ and $\delta_{2,j}\geq
1$ as $\hat{Z}_{k,k^{\prime},j^{\prime}}<0$.

Then under the assumption $u_{j}\geq 0$ and $l_{j}\geq 0$ for all
$j=(j^{\prime}+1)\ldots J$. For any $\epsilon_{1,j}\geq 0$ and
$\epsilon_{2,j}\geq 0$ let

$w=(Z^{\star}_{k^{\star},k^{\prime},j^{\prime}+1},\ldots,Z^{\star}_{k^{\star},k^{\prime},J})\in\bigcup^{J}_{j=j^{\prime}+1}\bigg{[}\bigcap^{j-1}_{i=j^{\prime}+1}(B_{k^{\star},i}(\delta_{1,i}+\epsilon_{1,i},\delta_{2,i}+\epsilon_{2,i}))\cap
C_{k^{\star},j}(\delta_{1,j}+\epsilon_{1,j},\delta_{2,j}+\epsilon_{2,j})\bigg{]},$

for some $q\in\\{j^{\prime}+1,\ldots,J\\}$ for which
$Z^{\star}_{k^{\star},k^{\prime},q}\in
C_{k^{\star},q}(\delta_{1,q}+\epsilon_{1,q},\delta_{2,q}+\epsilon_{2,q})$ and
$Z^{\star}_{k^{\star},k^{\prime},h}\in
B_{k^{\star},h}(\delta_{1,h}+\epsilon_{1,h},\delta_{2,h}+\epsilon_{2,h})$ for
$h=j^{\prime}+1,\ldots q-1$. $Z^{\star}_{k^{\star},k^{\prime},q}\in
C_{k^{\star},q}(\delta_{1,q}+\epsilon_{1,q},\delta_{2,q}+\epsilon_{2,q})$
implies that $Z^{\star}_{k^{\star},k^{\prime},q}\in
C_{k^{\star},q}(\delta_{1,q},\delta_{2,q})$. Furthermore
$Z^{\star}_{k^{\star},k^{\prime},q}\in
B_{k^{\star},q}(\delta_{1,q}+\epsilon_{1,q},\delta_{2,q}+\epsilon_{2,q})$
implies that $Z^{\star}_{k^{\star},k^{\prime},q}\in
B_{k^{\star},q}(\delta_{1,q},\delta_{2,q})\cup
C_{k^{\star},q}(\delta_{1,q},\delta_{2,q})$ for some $h=j^{\prime}+1,\ldots
q-1$. Therefore,

$w=(Z^{\star}_{k^{\star},k^{\prime},j^{\prime}+1},\ldots,Z^{\star}_{k^{\star},k^{\prime},J})\in\bigcup^{J}_{j=j^{\prime}+1}\bigg{[}\bigcap^{j-1}_{i=j^{\prime}+1}(B_{k^{\star},i}(\delta_{1,i},\delta_{2,i}))\cap
C_{k^{\star},j}(\delta_{1,j},\delta_{2,j})\bigg{]}.$

As a result $P(R(0,1))\geq
P(R(\frac{-\hat{Z}_{k,k^{\prime},j^{\prime}}\sqrt{n_{k^{\prime},j^{\prime}}}}{\sqrt{n_{k,j}-n_{k,j^{\prime}}}},\frac{\sqrt{n_{k,j}}}{\sqrt{n_{k,j}-n_{k,j^{\prime}}}}))$.
∎

##  Appendix C Proof of Theorem 4 and Theorem 5

The proof of both Theorem 4.1 and Theorem 4.2 is:

###### Proof.

Let the treatment with the greatest positive treatment effect be treatment
$k^{\star}$. Then one can write the conditional power of treatment k if no pre
change data is kept as:

$\displaystyle\frac{P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}})P(E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}})}{P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}})}.$

From Theorem 3.2 when $u_{j}\geq 0$ and $l_{j}\geq 0$ for all $j=1\ldots J$ is
true we know for a given $k^{\prime}$ and $j^{\prime}$

$\displaystyle P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{4}_{k^{\star},k^{\prime},j^{\prime}})\leq
P(E^{1}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{2}_{k^{\star},k^{\prime},j^{\prime}}\cap
E^{3}_{k^{\star},k^{\prime},j^{\prime}})P(E^{\star
4}_{k^{\star},k^{\prime},j^{\prime}}).$

Additionally this is known for when $u_{j}\geq 0$ and there are no lower
boundaries for all $j=1\ldots J$ from Theorem 3.3. This is true for every
$k^{\prime}\in\\{1,\ldots,K\\}/k^{*}$ and $j^{\prime}\in 1,\ldots,J$, so

$\sum^{J}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,\ldots,K\\}/k^{\star}}\sum^{J}_{j^{\prime}=1}\Omega_{k^{\star},k^{\prime},j^{\prime}}\leq\sum^{J}_{j^{\star}=1}\Xi_{k^{\star},j^{\star}}+\sum_{k^{\prime}\in\\{1,\ldots,K\\}/k^{\star}}\sum^{J}_{j^{\prime}=1}\Omega^{\star}_{k^{\star},k^{\prime},j^{\prime}}.$

∎

## SOFTWARE

Software in the form of R code can be found at:
https://github.com/pgreenstreet/change_control_platform.

## Competing interests

The authors declare no potential conflict of interests. Alun Bedding is a
shareholder of Roche Products Ltd.

## Acknowledgments

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). TJ and PM also received funding from UK Medical Research Council (MC
UU 00002/14 and MC UU 00002/19, respectively). This paper is based on work
completed while PG was part of the EPSRC funded STOR-i centre for doctoral
training (EP/S022252/1). For the purpose of open access, the author has
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

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