# A Seamless Phase II/III Design with Dose Optimization for Oncology Drug
Development

Yuhan Li  Department of Statistics, University of Illinois Urbana-Champaign
Yiding Zhang Correspondence to<EMAIL_ADDRESS>Biostatistics and
Programming, Sanofi  Gu Mi  Biostatistics and Programming, Sanofi  Ji Lin
Biostatistics and Programming, Sanofi

###### Abstract

The US FDA’s Project Optimus initiative that emphasizes dose optimization
prior to marketing approval represents a pivotal shift in oncology drug
development. It has a ripple effect for rethinking what changes may be made to
conventional pivotal trial designs to incorporate a dose optimization
component. Aligned with this initiative, we propose a novel Seamless Phase
II/III Design with Dose Optimization (SDDO framework). The proposed design
starts with dose optimization in a randomized setting, leading to an interim
analysis focused on optimal dose selection, trial continuation decisions, and
sample size re-estimation (SSR). Based on the decision at interim analysis,
patient enrollment continues for both the selected dose arm and control arm,
and the significance of treatment effects will be determined at final
analysis. The SDDO framework offers increased flexibility and cost-efficiency
through sample size adjustment, while stringently controlling the Type I
error. This proposed design also facilitates both Accelerated Approval (AA)
and regular approval in a “one-trial” approach. Extensive simulation studies
confirm that our design reliably identifies the optimal dosage and makes
preferable decisions with a reduced sample size while retaining statistical
power.

##  1 Introduction

Oncology drug development relies heavily on clinical trials to assess the
safety and efficacy of novel therapies for cancer patients. However,
conventional approaches such as separated Phase II and Phase III studies may
not be the optimal option given the ever-changing competitive and treatment
landscapes, the ever-increasing development costs, and the high attrition
rates across different stages of development. According to Wong et al[1], the
probability of success rate in oncology is only 57.6% from Phase I to Phase
II, and 32.7% from Phase II to Phase III, both of which are the lowest
compared to other therapeutic areas. There is a justified urgency to adopt a
“quick-to-win, fast-to-fail” mindset and utilize more efficient designs for
oncology drug development to expedite patient access to potentially life-
saving treatments.

The European Medicine Agency (EMA) Reflection Paper [2] has a pioneering
effort in shaping the regulatory direction of adaptive designs. It has been
released during a period of extensive discussion on topics like methodology,
implementation, and regulatory validation of adaptive designs. Since then,
there has been a growing interest in adopting a more integrated and adaptive
approach known as “seamless design” in oncology clinical trials [3, 4, 5].
Seamless Phase II/III trials, in particular, aim to overcome the limitations
of traditional designs by seamlessly transitioning from exploratory to
confirmatory phases within a single trial framework. This approach offers
several potential benefits, including the concurrent recruitment of
participants for both Phase II and Phase III [5], and the flexibility to
adjust sample size based on interim analysis while allowing conventional test
procedure to be performed at the final analysis [6]. These desired features
have the potential to expedite the development process and save costs while
maintaining statistical rigor and facilitating efficient decision-making in
oncology drug development.

Most recently, with the initiation of Project Optimus by the Oncology Center
of Excellence (OCE) within the US Food and Drug Administration (FDA) in 2021
[7], the exploration of efficient dose-finding strategies has become a focal
point in oncology drug development. As described in the draft Guidance for
Industry on dose optimization [8] and several key publications on Project
Optimus [9, 10, 11], dose optimization is preferably addressed at earlier
stages and before marketing approval. The challenge of dose optimization,
which has been a subject of extensive investigation in clinical trials, has
been explored long before the initiation of Project Optimus. Various methods
have been proposed to balance the risk-benefit trade-offs, such as EffTox
[12], BOIN12 [13], and those based on backfill designs [14, 15]. Following the
introduction of Project Optimus, there has been a surge in innovative and
practical methodologies aiming to integrate dose optimization with seamless
design frameworks, such as the MATS design [16], the DROID design [17], the
controlled amplification approach [18], and the Bi3+3 design [19] for early-
phase studies. For late-phase development, integrating seamless Phase II/III
design with dose optimization has emerged as an intriguing area of research.
Several extensions of the original seamless design have already been proposed
[20, 21, 22, 23].

Existing seamless Phase II/III designs generally assess multiple doses through
randomized Phase II trials. Interim analysis results typically determine
whether to proceed with each arm into the Phase III part of the study.
Specifically, Jin and Zhang [21] proposed to evaluate each dose arm
independently, and one or multiple dose arms may advance to Phase III if they
individually demonstrate a significant treatment effect. In contrast, Zhang et
al. [20, 22] recommended selecting the optimal dose at interim analysis, and
then deciding whether to expand the optimal dose arm to Phase III. Jiang and
Yuan [23] extend this approach by considering designs that employ different
control arm enrollment strategies, aiming to address challenges in different
clinical settings. Type I error control for these methods is achieved by
selecting the appropriate cut-off values for test statistics when deciding
whether to advance to Phase III part of the study.

While existing seamless Phase II/III trial designs provide valuable and
practical options, several limitations exist. First, analytically these
approaches treat each dose level as an independent treatment, disregarding the
potential correlation of treatment effect among dose levels, which may be
readily available from Phase I dose-escalation studies (e.g., preliminary
depiction of the dose-response curve). Failure to utilize this valuable prior
information in a seamless design framework may incur a loss of efficiency.
Moreover, the existing framework does not include a sample size re-estimation
(SSR) procedure [24, 25, 26, 27]. Incorporating SSR adds adaptiveness to the
design of confirmatory trials and aligns with the recommendations in the FDA’s
Guidance for Industry on adaptive designs (see section V.B. of [28]).

To address the limitations of current methods, we introduce a novel “Seamless
Design with Dose Optimization” (hereinafter “SDDO”) framework, an adaptive
two-stage Phase II/III design that incorporates both dose optimization and
SSR. In Stage I, candidate dosages (treatment arms) and a control arm are
evaluated in a randomized setting primarily based on a short-term surrogate
endpoint, leveraging the Bayesian framework to facilitate information-
borrowing from prior knowledge. After Stage I, a formal interim analysis is
conducted with three objectives: (1) select the optimal dosage; (2) make
decisions regarding trial early termination, Phase II continuation, or Phase
III expansion; (3) SSR based on Predictive Probability of Success (PPoS) if
the decision is to expand to Phase III based on all observed data up till the
interim analysis. In Stage II, enrollment continues for the control arm and
the selected optimal dose arm. A final analysis is then conducted when the
adjusted event size is reached, combining data from both Stage I and Stage II,
and using traditional statistical tests to determine the significance of
treatment effect.

The remainder of this article is organized as follows. In Section 2, we first
provide an overview of the proposed SDDO framework, including the design
schema and model setup, clinical endpoints and decision-making procedure at
different stages, SSR with PPoS, and type I error control at the final
analysis. Scenarios for the comprehensive simulation studies are described and
simulation results are discussed in Section 3. We conclude the article in
Section 4 with some practical considerations on the SDDO framework and
potential future works.

##  2 Methodology

The primary objective for the development of the SDDO framework is to enable
judicious decision-making based on prior knowledge of the dose-response
relationship and accumulating data evidence collected by the time of the
interim analysis, while concurrently enabling more efficient sample size
utilization, expediting trial timeline and maintaining statistical rigor.
Figure 1 shows the design schema of the proposed SDDO framework.

Figure 1: Design schema of the proposed SDDO framework.

The trial begins by randomly assigning participants to either the control arm
or treatment of different candidate dosages deemed worthwhile for further
evaluations from early-phase studies. Interim analysis takes place when the
short-term surrogate endpoint reaches sufficient level of maturity (e.g., when
a pre-specified number of participants have been followed up for at least two
to three tumor assessments). Subsequently, by combining data obtained from the
surrogate endpoint and available (but pre-mature) data from the primary
endpoint (e.g., some time-to-event endpoints accepted for regular approval) at
interim analysis, we identify a single dose that demonstrates the most
significant treatment effect (which we call the “optimal dose”) among all
candidate doses evaluated. Based on this selected optimal dose, decisions
regarding trial continuation are made. Specifically, three scenarios may
occur:

  * •

If the optimal dose meets the “futility” criterion, the trial will be
terminated at the interim analysis, with “early futility” declared at the
study level.

  * •

If the optimal dose shows some treatment effects but falls short of being
favorable (i.e., it lies within the “promising zone” but outside the
“favorable zone” according to SSR terminology), the trial will proceed to
complete Phase II, focusing solely on the selected optimal dose and
maintaining the pre-determined sample size.

  * •

If the interim analysis yields a favorable result (i.e., within the “favorable
zone”), the trial will directly advance to phase III with the selected optimal
dose. Sample size re-estimation is conducted in this scenario to improve cost
efficiency and increase Phase III success, as more investment will be
committed compared to a (smaller) Phase II study.

The above decision criteria offer significant advantages and align closely
with the “quick to win, fast to fail” principle which is essential for both
ethical and economic considerations in clinical trial designs. The “quick to
win” objective is fulfilled when the interim analysis identifies the dose
level in the favorable zone, enabling a swift and seamless transition to Phase
III for favorable outcomes. This accelerates the development timeline and has
the potential to bring effective treatments to patients more quickly. The
“fast to fail” objective is achieved by allowing early termination of the
trial when even the “optimal dose” turns out to be nonpromising based on
interim analysis. This rapid identification of futility not only conserves
valuable resources but also minimizes patient exposure to ineffective or
potentially harmful treatments. Together, these features enhance cost-
efficiency and align with ethical considerations, making the SDDO framework
well-suited for modern clinical trials.

For many tumor types, one early surrogate endpoint commonly utilized is the
overall response rate (ORR), while the primary endpoint for Phase II studies
can be either ORR or progression-free survival (PFS) [29]. In Phase III
studies, the primary endpoint typically shifts to overall survival (OS) which
is considered as the gold standard in oncology [30]. We follow this convention
by using ORR as the endpoint for the interim analysis, PFS as the endpoint for
Phase II, and OS as the endpoint for Phase III. It is important to note that
this selection is based on the prevailing choices in oncology clinical trials,
while the framework can be easily adjusted to accommodate the utilization of
other endpoints as appropriate. For the rest of this paper, we assume equal
randomization ratio in the control arm and each treatment arm, and denote the
treatment candidate dose levels as $d_{1},d_{2},...,d_{I}$. Without loss of
generality, we assume ascending dosing strength as $d_{1}<d_{2}<...<d_{I}$. We
also denote the ORR for the $i$th dosage as $p_{i}$ for $i=1,2,...,I$ and the
ORR for the control arm as $p_{0}$. Similarly, the log hazard ratio for the
$i$th dosage of the treatment comparing with the control arm is denoted as
$\eta_{i}$ for PFS, and $\theta_{i}$ on OS for $i=1,2,...,I$.

###  2.1 Optimal Dose Selection

The first objective in conducting the interim analysis is to determine the
optimal dosage. To incorporate prior knowledge of the dose-response
relationship, we define the prior distribution of $p_{i}$ as a beta
distribution, denoted by $p_{i}\sim\text{Beta}(a(d_{i}),b(d_{i}))$, where
$a(d)$ and $b(d)$ are pre-determined functions of dose level $d$. In practice,
the choice of $a(d)$ and $b(d)$ can be flexible to reflect the prior knowledge
(or lack thereof) of the dose-response relationship. For instance, we could
specify $b(d)$ as a constant value, while $a(d)$ could be a sigmoid or
quadratic function of the dose level to characterize a monotonically
increasing or bell-shaped relationship, respectively. Figure 2 provides two
illustrations of prior distributions across various dose arms, with the
relationship implied by the relative orders of the color-coded curves. It is
evident that by appropriately specifying parameter values associated with the
beta distribution, we can effectively integrate prior knowledge about the
dose-response relationship within the SDDO framework.

Figure 2: Examples of prior distributions of $p_{i}$. In both figures, we
assume $\text{Dose 1}<\text{Dose 2}<\text{Dose 3}$, and $b(d)=20$ as a
constant. $a(d)$ follows a sigmoid function in the left panel and a quadratic
function in the right panel.

Suppose that by the time of the interim analysis, $n_{i}$ participants are
enrolled in the treatment arm at dose $d_{i}$, and $y_{i}$ of them are
responders. We assume that the number of responders $y_{i}$ follows a binomial
distribution $y_{i}|n_{i},p_{i}\sim\text{Bin}(n_{i},p_{i})$. Consequently, the
posterior distribution of $p_{i}$ can be derived as follows:

$\displaystyle\begin{split}f(p_{i}|n_{i},y_{i})&\propto
f(n_{i},y_{i}|p_{i})f(p_{i})\\\
&\propto\text{Beta}(a(d_{i})+y_{i},b(d_{i})+n_{i}-y_{i})\\\
&\approx\mathcal{N}\left(\frac{a(d_{i})+y_{i}}{a(d_{i})+b(d_{i})+n_{i}},\frac{(a(d_{i})+y_{i})(b(d_{i})+n_{i}-y_{i})}{(a(d_{i})+b(d_{i})+n_{i}+1)(a(d_{i})+b(d_{i})+n_{i})^{2}}\right).\end{split}$
(2.1)

It is worth noting that during the interim analysis, a certain number of OS
events may also be available. Although these observed events are likely to be
limited and may not provide mature information, they can still offer
additional evidence for decision-making during the interim analysis. Similar
to ORR, we assume that the prior distribution of the log hazard ratio follows
$\theta_{i}\sim\mathcal{N}(\mu(d),\frac{4}{\sigma(d)})$, where $\mu(d)$ and
$\sigma(d)$ are pre-defined functions of the dose level $d_{i}$ that reflect
prior beliefs regarding the relationship between dose levels and their
corresponding hazard ratios. Based on the data collected up until the interim
analysis, the data likelihood follows
$\text{data}|\theta_{i}\sim\mathcal{N}(\log(h_{i}),\frac{4}{M^{1}_{i}})$,
where $h_{i}$ represents the estimated hazard ratio based on the available
data, and $M^{1}_{i}$ denotes the total number of events in the control arm
and treatment arm of dose $d_{i}$ at interim analysis [31]. Consequently, the
posterior distribution at the interim analysis for the log hazard ratio of OS
is given by:

$\displaystyle\begin{split}f(\theta_{i}|\text{data})&\propto
f(\text{data}|\theta_{i})f(\theta_{i})\\\
&\propto\mathcal{N}\left(\frac{\log(h_{i})M^{1}_{i}+\sigma(d_{i})\mu(d_{i})}{M^{1}_{i}+\sigma(d_{i})},\frac{4}{M^{1}_{i}+\sigma(d_{i})}\right).\end{split}$
(2.2)

The correlation between ORR and log hazard ratio ($\log(\text{HR})$) has been
extensively studied in clinical trials [32, 33], and it is common practice to
specify a pre-determined correlation coefficient $\rho_{1}$ between ORR and HR
in a given clinical trial based on prior knowledge and external data [5].
Based on equations (2.1) and (2.2), the joint posterior distribution of
$(p_{i},\theta_{i})$ follows a bivariate normal distribution:

$\displaystyle\begin{bmatrix}p_{i}\\\
\theta_{i}\end{bmatrix}\sim\mathcal{N}\left(\begin{bmatrix}\mu_{p_{i}}\\\
\mu_{\theta_{i}}\end{bmatrix},\begin{bmatrix}\sigma_{p_{i}}^{2}&\rho_{1}\sigma_{p_{i}}\sigma_{\theta_{i}}\\\
\rho_{1}\sigma_{p_{i}}\sigma_{\theta_{i}}&\sigma_{\theta_{i}}^{2}\end{bmatrix}\right),$
(2.3)

where $\mu_{p_{i}}$, $\mu_{\theta_{i}}$, $\sigma_{p_{i}}$, and
$\sigma_{\theta_{i}}$ are obtained by substituting the corresponding posterior
means and standard deviations from equations (2.1) and (2.2).

The joint posterior distribution (2.3) incorporates both prior knowledge and
all available data at the interim analysis. The optimal dose level should be
thus determined corresponding to the posterior distribution that is farthest
away from the null distribution with mean
$[\mu_{p_{i}},\mu_{\theta_{i}}]=[p_{0},0]$ and estimated variance from
distribution (2.3). Therefore, we define the optimal dosage at the interim
analysis as:

$i^{*}=\operatorname*{arg\,max}_{i\in\\{1,2,...,I\\}}\mathbb{E}[p_{i}|\theta_{i}=\mu_{\theta_{i}}],$
(2.4)

and the optimal dosage is then $d_{i^{*}}$.

It’s important to note that the selection of optimal dose is primarily driven
by the surrogate endpoint (i.e., ORR), as defined in equation (2.4). This
choice is influenced by the fact that only ORR has matured sufficiently for
interim-stage decision-making due to the relatively short follow-up time.
However, by conditioning on the posterior mean of log hazard ratio of OS, we
effectively integrate supplementary information regarding the Phase III
primary endpoint into the dose selection process. In particular, when ORRs
among the treatment arms are comparable based on this criterion at the interim
analysis, preference is given to dose levels with lower HRs that could
potentially be indicative of stronger OS benefits at the final analysis.

###  2.2 Decision-Making at Interim Analysis

Once the optimal dose is selected based on the Bayesian framework introduced
in the previous section, the next step is to determine the appropriate course
of action: whether to terminate the trial at the interim analysis, continue
with the originally planned Phase II trial, or directly expand to a Phase III
trial with SSR.

Figure 3: Decision-making procedure for the proposed SDDO design.

As depicted in Figure 3, the decision at the interim analysis is made by
evaluating futility and significance based on the joint posterior distribution
of $(p_{i^{*}},\theta_{i^{*}})$. Specifically, the selected optimal dose would
be deemed futile if the probability satisfies:

$P(p_{i^{*}}-p_{0}\geq\tau_{0}|\theta_{i^{*}})\leq s_{0}.$ (2.5)

If the futility criterion (2.5) holds, the trial would be terminated: it
suggests that, although the dose itself “outperforms” other candidate doses,
it fails to demonstrate a desired treatment effect over the control arm so
that any further development is not justified. However, if the optimal dose
passes the futility criterion, we proceed to assess its significance by
checking:

$P(p_{i^{*}}-p_{0}\geq\tau_{1}|\theta_{i^{*}})\geq s_{1}.$ (2.6)

If the significance criterion (2.6) holds, indicating strong evidence of a
positive treatment effect at the interim analysis, the trial would directly
expand to Phase III with SSR to be discussed in detail in Section 2.3. This
approach saves time and sample size, as the participants already enrolled in
the treatment and control arms can be further followed and included as part of
the Phase III portion of the trial.

It should be highlighted that if the interim results fall into the favorable
zone, the option of seeking Accelerated Approval (AA) from the FDA becomes
viable. This can be pursued concurrently with the ongoing patient enrollment
process. Notably, the FDA’s Guidance for Industry on AA endorses the “one-
trial” approach as an efficient means of securing both accelerated and regular
approvals within the same study [34, 35, 36], while the proposed SDDO
framework aligns well with this “one-trial” approach. Specifically, meeting
criterion (2.6) in conjunction with a high PPoS to be discussed in Section 2.3
below would demonstrate compelling evidence of treatment benefits under the
selected optimal dose $d_{i^{*}}$, and thus warrant an application for AA.
Meanwhile, patient enrollment continues in this framework, with continued data
collection on the primary endpoint of OS to support the pursuit of a regular
approval. Our proposed SDDO framework fits into this “one-trial” approach and
has the potential to deliver novel agents to patients more efficiently.

Note that $s_{0},s_{1},\tau_{0}$, and $\tau_{1}$ are four crucial tuning
parameters that need to be pre-determined within the SDDO framework. These
parameters should be carefully selected to align with the trial’s specific
requirements and concerns [37], thereby maximizing the benefits of the SDDO
framework. As a practical approach, we suggest setting $s_{0}=s_{1}=0.9$ to
ensure high confidence in decision-making. The values of $\tau_{0}$ and
$\tau_{1}$ should be chosen with the aim of controlling the rate of false
decisions effectively. Our approach is driven by two key considerations.
Firstly, to ensure high confidence in crucial decision-making, such as
deciding whether to terminate a trial or to progress to Phase III—decisions
that are high-stakes and have far-reaching consequences—we recommend setting
$s_{0}$ and $s_{1}$ at higher values. Secondly, considering the limited data
available at interim analysis, which may obscure the true treatment effect, we
propose the selection of $\tau_{0}$ and $\tau_{1}$ using grid search and
simulation techniques. This approach can accommodate the inherent data
variability encountered at this early stage of the trial.

From Figure 3, given fixed values of $s_{0}$ and $s_{1}$, $\tau_{0}$ controls
whether the trial should be terminated at the interim stage, while $\tau_{1}$
determines if the trial should be directly expanded to Phase III or continue
with Phase II. Therefore, we propose to select $\tau_{0}$ by controlling the
probability of falsely terminating the trial early when at least one candidate
dosage shows clinically meaningful difference in ORR (i.e., a false negative
decision). It is important to manage this risk of false negative decision-
making, as such errors could result in missed opportunities in drug
development that are of market approval potential. Meanwhile, for the value of
$\tau_{1}$, we aim to control the rate of falsely expanding to Phase III when
none of the dosage is effective, as such false positive decisions could incur
substantial waste of resources and investments for Phase III study.

Analytically finding the value of $\tau_{0}$ and $\tau_{1}$ to achieve
acceptable levels of false negative and false positive rates is
computationally challenging, due to the $\max$ operator involved in (2.4) for
optimal dosage selection. Thus, we propose to identify the values of
$\tau_{0}$ and $\tau_{1}$ by grid search between $[-1,1]$. The results via
grid search are illustrated in Figure 4. Notably, $\tau_{0}$ tends to remain
consistent across varying numbers of candidate dosages. This stability can be
attributed to the ORR difference of 0.15 being significant enough to
distinguish the optimal dose arm from the control arm, and the influence of
additional candidate dosages on the value of $\tau_{0}$ is minimal. In
contrast, the value of $\tau_{1}$ would increase as the number of dosages
increases. This finding implies the necessity to apply a more stringent
multiple testing correction procedure as the number of dosages increases, for
the purpose of controlling false positives.

###  2.3 Sample Size Re-estimation (SSR) with PPoS

When the decision at the interim analysis is to directly expand to Phase III,
we propose to conduct sample size re-estimation (SSR) to alleviate the risk of
an underpowered test at final analysis and ensure cost and time efficiency
[24, 27]. Following our proposed Bayesian framework, we calculate the
Predictive Probability of Success (PPoS) at the interim analysis [38, 39], and
set the new sample size as the minimum sample size required to achieve the
desired PPoS.

As per the notations introduced in Section 2.1, the prior distribution of the
OS log hazard ratio for the optimal dosage $d_{i^{*}}$ is assumed to be
$\theta_{i^{*}}\sim\mathcal{N}(\mu(d_{i^{*}}),\frac{4}{\sigma(d_{i^{*}})})$.
The conditional distribution of $\theta_{i^{*}}(t)$ and $\theta_{i^{*}}(1-t)$
given $\theta_{i^{*}}$ follows
$\hat{\theta}_{i^{*}}(t)|\theta_{i^{*}}\sim\mathcal{N}(\theta_{i^{*}},\frac{4}{M_{i^{*}}^{1}})$
and
$\hat{\theta}_{i^{*}}(1-t)|\theta_{i^{*}}\sim\mathcal{N}(\theta_{i^{*}},\frac{4}{\widetilde{M}^{2}})$,
respectively, where $t=\frac{M^{1}_{i^{*}}}{M^{1}_{i^{*}}+\widetilde{M}^{2}}$
denotes the proportion of OS events observed at the interim analysis,
$\hat{\theta}_{i^{*}}(t)$ and $\hat{\theta}_{i^{*}}(1-t)$ refer to the
estimated log-hazard ratios based on data collected prior to and after interim
analysis, respectively, and $\widetilde{M}^{2}$ refers to the total number of
events on the control arm and treatment arm $d_{i^{*}}$ after interim
analysis. The posterior distribution of $\theta_{i^{*}}$ after interim
analysis follows
$\theta_{i^{*}}|\hat{\theta}_{i^{*}}(t)\sim\mathcal{N}\left(\frac{\log(h_{i^{*}})M^{1}_{i^{*}}+\sigma(d_{i^{*}})\mu(d_{i^{*}})}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})},\frac{4}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})}\right)$.

To derive the predictive distribution of
$\hat{\theta}_{i^{*}}(1-t)|\theta_{i^{*}}(t)$, we have:

$\displaystyle\begin{split}E\Big{(}\hat{\theta}_{i^{*}}(1-t)\mid\hat{\theta}_{i^{*}}(t)\Big{)}&=E\Big{(}E(\hat{\theta}_{i^{*}}(1-t)\mid\hat{\theta}_{i^{*}}(t),\theta_{i^{*}})\Big{)}\\\
&=E\Big{(}\theta_{i^{*}}\mid\hat{\theta}_{i^{*}}(t)\Big{)}\\\
&=\frac{\log(h_{i^{*}})M^{1}_{i^{*}}+\sigma(d_{i^{*}})\mu(d_{i^{*}})}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})},\\\
\end{split}$ (2.7)
$\displaystyle\begin{split}\operatorname{Var}\Big{(}\hat{\theta}_{i^{*}}(1-t)\mid\hat{\theta}_{i^{*}}(t)\Big{)}&=E\Big{(}\operatorname{Var}(\hat{\theta}_{i^{*}}(1-t)\mid\theta_{i^{*}},\hat{\theta}_{i^{*}}(t))\Big{)}+\operatorname{Var}\Big{(}E(\hat{\theta}_{i^{*}}(1-t)\mid\theta_{i^{*}},\hat{\theta}_{i^{*}}(t))\Big{)}\\\
&=\frac{4}{\widetilde{M}^{2}}+\operatorname{Var}\Big{(}\theta_{i^{*}}\mid\hat{\theta}_{i^{*}}(t)\Big{)}=\frac{4}{\widetilde{M}^{2}}+\frac{4}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})}.\end{split}$
(2.8)

Based on (2.7) and (2.8), the predictive distribution is then:

$\hat{\theta}_{i^{*}}(1-t)|\hat{\theta}_{i^{*}}(t)\sim\mathcal{N}\Big{(}\frac{\log(h_{i^{*}})M^{1}_{i^{*}}+\sigma(d_{i^{*}})\mu(d_{i^{*}})}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})},\frac{4}{\widetilde{M}^{2}}+\frac{4}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})}\Big{)}.$
(2.9)

At final analysis (i.e., $t=1$), the trial is considered a success if the
standardized test statistic
$Z(t)=Z(1)=\frac{\hat{\theta}_{i^{*}}(1)}{se(\hat{\theta}_{i^{*}}(1))}\leq
z_{\alpha}$, where $\alpha$ is the desired level of Type I error rate, and
then $\text{PPoS}=P(Z(1)\leq z_{\alpha}|\hat{\theta}_{i^{*}}(t))$. From [40],
we have $Z(1)=\sqrt{t}Z(t)+\sqrt{1-t}Z(1-t)$. Therefore,

$\displaystyle\begin{split}\text{PPoS}(\widetilde{M^{2}})&=P(Z(1)\leq
z_{\alpha}|\hat{\theta}_{i^{*}}(t))\\\ &=P(\sqrt{t}Z(t)+\sqrt{1-t}Z(1-t)\leq
z_{\alpha}|\hat{\theta}_{i^{*}}(t))\\\
&=P\Big{(}Z(1-t)\leq\frac{z_{\alpha}-\sqrt{t}Z(t)}{\sqrt{1-t}}\Big{|}\hat{\theta}_{i^{*}}(t)\Big{)}\\\
&=P\Big{(}Z(1-t)\sqrt{\frac{4}{\widetilde{M}^{2}}}\leq\frac{z_{\alpha}-\sqrt{t}Z(t)}{\sqrt{1-t}}\sqrt{\frac{4}{\widetilde{M}^{2}}}\Big{|}\hat{\theta}_{i^{*}}(t)\Big{)}\\\
&=P\Big{(}\hat{\theta}_{i^{*}}(1-t)\leq\frac{z_{\alpha}-\sqrt{t}Z(t)}{\sqrt{1-t}}\sqrt{\frac{4}{\widetilde{M}^{2}}}\Big{|}\hat{\theta}_{i^{*}}(t)\Big{)}\\\
&=\Phi\left(\frac{\frac{2\big{(}z_{\alpha}-\sqrt{t}Z(t)\big{)}}{\sqrt{(1-t)\widetilde{M}^{2}}}-\frac{\log(h_{i^{*}})M^{1}_{i^{*}}+\sigma(d_{i^{*}})\mu(d_{i^{*}})}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})}}{\sqrt{\frac{4}{\widetilde{M}^{2}}+\frac{4}{M^{1}_{i^{*}}+\sigma(d_{i^{*}})}}}\right).\end{split}$
(2.10)

The analytical solution of PPoS is given by equation (2.10). It is important
to note that all terms in equation (2.10) are fixed at interim analysis,
except for $\widetilde{M^{2}}$ and $t$. Consequently, PPoS can be considered
as a function of $\widetilde{M^{2}}$. To determine the adjusted sample size
for the Phase III study, we define:

$\widetilde{M^{2}}^{*}=\operatorname*{arg\,min}_{\widetilde{M^{2}}}\\{\text{PPoS}(\widetilde{M^{2}})\geq
1-\beta\\},$ (2.11)

where $1-\beta$ is the desired statistical power.

Solving equation (2.11) explicitly is computationally challenging due to the
complex analytical form in equation (2.10) and the non-monotonic nature of
PPoS with respect to $\widetilde{M^{2}}$ [41]. Furthermore, in clinical
trials, event sizes are generally only expected to increase [28], and
typically there is an upper limit on the number of subjects to be enrolled,
denoted as $M^{2}_{\text{max}}$, to ensure the budget is under control [21].
Consequently, the adjusted sample size denoted as $\widehat{M^{2}}^{*}$ should
fall within the range $\widehat{M^{2}}^{*}\in[M^{2},M^{2}_{\text{max}}]$,
where $M^{2}$ denotes the pre-determined Phase III event size (without SSR).

Taking these practical considerations into account, we propose a numerical
approach to solve for $\widehat{M^{2}}^{*}$ as follows. First, we check if
$\text{PPoS}(M^{2})\geq 1-\beta$. If this condition is satisfied, the event
size remains the same as pre-determined $M^{2}$, and no sample size re-
estimation is needed. However, if $\text{PPoS}(M^{2})<1-\beta$, we perform a
grid search for $\widehat{M^{2}}^{*}$ within the range
$[M^{2},M^{2}_{\text{max}}]$ to find the smallest value that achieves the
desired PPoS. If such a value exists, we take it as the adjusted event size
$\widehat{M^{2}}^{*}$; if no value within the range satisfies the desired
PPoS, we set $\widehat{M^{2}}^{*}=M^{2}_{\text{max}}$. The adjusted sample
size obtained from the approach above could be summarized as follows:

$\displaystyle\widehat{M^{2}}^{*}=\begin{cases}\operatorname*{arg\,min}\limits_{\widetilde{M^{2}}\in[M^{2},M^{2}_{\text{max}}]}\text{PPoS}(\widetilde{M^{2}})\geq
1-\beta,&\text{if
}\max\limits_{\widetilde{M^{2}}\in[M^{2},M^{2}_{\text{max}}]}\text{PPoS}(\widetilde{M^{2}})\geq
1-\beta,\\\ M^{2}_{\text{max}},&\text{otherwise}.\end{cases}$

By adopting this numerical approach and confining our grid search within the
range $[M^{2},M^{2}_{\text{max}}]$, we significantly reduce the computational
resources required relative to solving equation (2.11) directly.

###  2.4 Final Analysis and Type I Error Control

Previous sections provide a comprehensive discussion of the objectives and
analyses conducted at the interim stage for the SDDO framework, which can be
summarized into three components: (1) optimal dose selection, (2) expansion
decision, and the possibility of requesting an accelerated approval, and (3)
SSR with PPoS. Based on the decision made at the interim analysis, we proceed
to enroll participants with the adjusted sample size for the final analysis.
Specifically,

  * •

If the decision at the interim analysis is to declare futility, the trial will
be terminated.

  * •

If the decision is to continue with Phase II, the trial will continue patient
enrollment in both the control and optimal dose arms until reaching the pre-
determined total event size for Phase II study. Once the pre-planned PFS event
size is met, a log-rank test will be performed using the combined PFS data
from before and after the interim analysis to assess the significance of
treatment effects at the conclusion of the Phase II study.

  * •

In the case of expanding to Phase III, enrollment also continues with the
control arm and the optimal dose arm until the adjusted event size
$M^{1}_{i^{*}}+\widehat{M^{2}}^{*}$ for OS is achieved. Subsequently, a log-
rank test is conducted based on all the available OS data from the beginning
of trial to assess the significance of treatment effects.

Controlling Type I error at final analysis is a critical aspect in seamless
designs [5, 22]. In our proposed SDDO framework, there are two principal
sources that may contribute to the inflation of Type I error:

  1. 1.

Multiple Testing Issue: The process of selecting the optimal dose from
multiple candidate dosages during interim analysis introduces the issue of
multiple testing.

  2. 2.

Interim Analysis and Phase III expansion: Inclusion of data from the primary
endpoint for the Phase III study in the decision-making process at the interim
analysis stage can also incur Type I error inflation at the final analysis.

In the proposed SDDO framework, the two sources of Type I error inflation are
addressed through the careful selection of $\tau_{0}$ (for futility) and
$\tau_{1}$ (for significance). Specifically, the choice of $\tau_{0}$ results
in the early termination of a certain proportion of trials during the interim
analysis, thereby precluding their advancement to the final analysis stage.
This proportion of terminated trials would never have the opportunity to
detect any potential positive treatment effects in the final analysis.
Consequently, $\tau_{0}$ serves dual purposes: beyond controlling the
probability of erroneously terminating the trial, it also functions as a
“correction mechanism” to the multiple testing issue. This addresses the first
source of overall Type I error inflation.

In parallel, the selection of $\tau_{1}$ also serves dual purposes: it not
only restricts the probability of incorrect expansion to Phase III when none
of the treatment arms has demonstrated efficacy but also manages the
probability of directly expanding to Phase III. By constraining the
probability of advancing to Phase III, $\tau_{1}$ effectively controls the
second source of Type I error inflation.

While theoretical guarantees for Type I error control are challenging to
achieve when Bayesian methods are applied, we provide a theoretical proof for
Type I error control, assuming uniform priors, in Appendix Section B.
Extensive simulations, to be elaborated in Section 3, affirm that the proposed
SDDO framework succeeds in controlling Type I error rates while preserving
statistical powers.

##  3 Simulations

To evaluate our proposed SDDO framework, we conduct extensive simulation
studies to assess its operating characteristics in a wide spectrum of
realistic and representative scenarios. In all simulation settings, we
initiate the Phase II study with three candidate dosages and assume an equal
randomization ratio between each treatment arm and the control arm. An interim
analysis is performed after 60 participants in each arm have reached mature
ORR, and at least 30 OS events have been observed in the control arm. Unless
otherwise stated, we assume the correlation between the surrogate endpoint
(ORR) and the primary endpoint (OS) for the Phase III study (using
$\log(\text{HR}_{\text{OS}})$) to be $\rho_{1}=-0.5$.

The total event size for the Phase II study is set at 140, with the aim of
detecting a hazard ratio of $0.58$ with $90\%$ power and a one-sided Type I
error rate of $0.025$. For the pre-planned Phase III study, the event size is
set as $M^{2}=226$, and the maximum event size for Phase III is
$M^{2}_{\max}=507$. These settings are designed to detect hazard ratios of
$0.65$ and $0.75$, respectively, with $90\%$ power and a one-sided type I
error rate of $0.025$. For tuning parameters, we set $s_{0}=s_{1}=0.9$, and
select $\tau_{0}$ and $\tau_{1}$ by grid search to control the false positive
rate at 2.5% and false negative rate at 5% as described in Section 2.2. This
results in $\tau_{0}=-0.05$ and $\tau_{1}=0.10$.

In Table 1, 2, and 3, the first entries under ORR, $\text{HR}_{\text{PFS}}$,
and $\text{HR}_{\text{OS}}$ correspond to the control arm, while the
subsequent entries refer to three candidate dosages, assuming
$d_{1}<d_{2}<d_{3}$. The column labeled Optimal Dose Percentage demonstrates
the probability of selecting $d_{1},d_{2},$ or $d_{3}$ as the optimal dosage
during the interim analysis, respectively. Uninformative priors are assigned
to ORR and $\log(\text{HR}_{\text{OS}})$ across all simulation settings in
Table 1-3. Specifically, the prior distribution for ORR is modeled as
$\text{Beta}(2,2)$, and $\mathcal{N}(0,0.5)$ for $\log(\text{HR}_{\text{OS}})$
across all dosage levels. It should be noted that the reported expected event
sizes include only the events that occurred in the optimal dose arm selected
at the interim analysis and the control arm. Simulation results using
different prior distributions are presented in Tables 4 and 5.

| Simulation
---
Scenarios
| Optimal Dose
---
Percentage
|  | Interim
---
Decision
| Positive
---
Rate
| Expected
---
Event Size
| Expected
---
Study Duration
Global Null: ORR = [0.2,0.2,0.2,0.2] $\text{HR}_{\text{PFS}}$=[1,1,1,1] $\text{HR}_{\text{OS}}$=[1,1,1,1] | [33.32%,33.35%,33.33%] | Overall | – | 1.45% | 120 | 23.57
Terminate | 44.12% | 0 | 79 | 17.63
Phase II | 53.69% | 2.37% | 140 | 27.89
Phase III | 2.19% | 8.22% | 453 | 37.15
One Significant: ORR = [0.2,0.2,0.2,0.35] $\text{HR}_{\text{PFS}}$=[1,1,1,0.58] $\text{HR}_{\text{OS}}$=[1,1,1,0.7] | [1.11%,0.98%,97.91%] | Overall | – | 88.67% (88.65%) | 280 | 38.35
Terminate | 3.11% | 0 | 74 | 18.50
Phase II | 44.88% | 86.01% (85.98%) | 140 | 36.01
Phase III | 52.01% | 96.27% (96.25%) | 414 | 42.67
Two Significants: ORR = [0.2,0.2,0.35,0.35] $\text{HR}_{\text{PFS}}$=[1,1,0.58,0.58] $\text{HR}_{\text{OS}}$=[1,1,0.7,0.7] | [0.16%,50.88%,48.96%] | Overall | – | 93.98% (93.98%) | 327 | 40.67
Terminate | 0.66% | 0 | 74 | 19.27
Phase II | 29.45% | 89.24% (89.24%) | 140 | 36.30
Phase III | 69.89% | 96.85% (96.85%) | 408 | 42.72
Global Alternative: ORR = [0.2,0.35,0.35,0.35] $\text{HR}_{\text{PFS}}$=[1,0.58,0.58,0.58] $\text{HR}_{\text{OS}}$=[1,0.58,0.58,0.58] | [32.54%,34.03%,33.43%] | Overall | – | 98.61% (98.61%) | 302 | 42.81
Terminate | 0.05% | 0 | 68 | 19.73
Phase II | 11.94% | 89.86% (89.86%) | 140 | 36.40
Phase III | 88.01% | 99.85% (99.85%) | 324 | 43.69
Table 1: Operating characteristics of the SDDO framework based on 10,000
simulation runs under global null, only one significant dosage, two
significant dosages, and global alternative scenarios. “Positive Rate” column
shows overall positive effect rate, and in brackets, the rate of selecting the
optimal dose with a positive effect.

Table 1 presents the operating characteristics under various standard
scenarios. Under the global null hypothesis, where no candidate dosage
demonstrates treatment effect, the proposed SDDO framework successfully
controls the rate of false entry into Phase III trials at $2.19\%$. This is
achieved by selecting the appropriate value of $\tau_{1}$ through grid search.
Additionally, the overall Type I error rate is well-controlled at $1.45\%$. In
the second scenario with only one candidate dosage showing treatment effect,
the SDDO framework correctly identifies the optimal dose with a $97.91\%$
probability. Moreover, the chance of prematurely terminating the trial is
capped at $3.11\%$, thanks to the tuning parameter $\tau_{0}$. In the third
scenario when two dosages show treatment effect, they are almost equally
selected as the optimal dose, with similar operating characteristics as seen
in the ”one significant” scenario. Under the global alternative hypothesis
with all candidate dosages showing treatment effects, the proposed framework
exhibits strong statistical power, achieving a true positive rate of
$98.61\%$. Furthermore, the expected event size for Phase III can be
significantly reduced (i.e., 324 instead of 453, 414, and 408 in the other
three scenarios), as such a strong signal allows for effect detection with
fewer events. For the overall development timeline (study duration), in case
that none of the dosages is effective, the study on average can be completed
within two years (23.57 months), while in case that at least one dosage is
effective, the time from the first patient into the final Phase III OS readout
readily for regular approval is around 43 months (and an accelerated approval
request based on PPoS would likely be made at the earlier interim analysis
stage).

| Simulation
---
Scenarios
| Optimal Dose
---
Percentage
|  | Interim
---
Decision
| Positive
---
Rate
| Expected
---
Event Size
| Expected
---
Study Duration
Same ORR, Different HR: ORR = [0.2,0.35,0.35,0.35] $\text{HR}_{\text{PFS}}$=[1,0.8,0.7,0.6] $\text{HR}_{\text{OS}}$=[1,0.8,0.7,0.6] | [20.48%,32.25%,47.27%] | Overall | – | 86.96% (44.72%) | 332 | 41.02
Terminate | 0.22% | 0 | 73 | 19.53
Phase II | 20.01% | 61.37% (35.32%) | 140 | 33.83
Phase III | 79.77% | 93.62% (47.20%) | 380 | 32.89
Weak Correlation, $\rho_{1}=-0.1$: ORR = [0.2,0.35,0.35,0.35] $\text{HR}_{\text{PFS}}$=[1,0.8,0.7,0.6] $\text{HR}_{\text{OS}}$=[1,0.8,0.7,0.6] | [29.70%,33.65%,36.65%] | Overall | – | 71.86% (32.99%) | 274 | 38.02
Terminate | 1.43% | 0 | 69 | 17.47
Phase II | 49.72% | 56.21% (30.37%) | 140 | 33.46
Phase III | 48.85% | 89.89% (36.62%) | 416 | 43.27
Weak Effect: ORR = [0.2,0.25,0.27,0.3] $\text{HR}_{\text{PFS}}$=[1,0.95,0.9,0.85] $\text{HR}_{\text{OS}}$=[1,0.95,0.9,0.85] | [14.06%,29.20%,56.84%] | Overall | – | 19.13% (14.91%) | 221 | 31.69
Terminate | 7.44% | 0 | 78 | 18.49
Phase II | 66.03% | 11.83% (8.79%) | 140 | 29.73
Phase III | 26.53% | 42.67% (34.31%) | 462 | 40.26
Table 2: Operating characteristics of the SDDO framework based on 10,000
simulation runs under various scenarios assuming that a higher ORR can
translate into benefits in PFS and OS. “Positive Rate” column shows overall
positive effect rate, and in brackets, the rate of selecting the optimal dose
with a positive effect.

Table 2 presents simulation results under a range of realistic settings other
than the standard settings, assuming that a higher ORR can translate into
benefits in PFS and OS. In the first setting, we consider a scenario where all
dose levels yield comparable ORR but differ in PFS and OS benefits. By
conditioning the optimal dose selection at interim analysis on the OS data, we
achieve a higher probability (47.27%) of selecting the true optimal dose that
maximizes OS benefits while attaining a robust statistical power of 86.96%.
However, when only the correlation between the surrogate and primary endpoints
is changed to be lower, as illustrated in the second setting ($\rho_{1}=-0.1$
compared to $\rho_{1}=-0.5$ in other settings), both the likelihood of
identifying the true optimal dose (36.65% vs. 47.27%) and the overall
statistical power (71.86% vs. 86.96%) are diminished. This decline occurs
because a lower correlation $\rho_{1}$ results in a lower weighting of OS
benefits in the optimal dose selection, which further affects the overall
statistical power. This observation strongly suggests that choosing a
surrogate endpoint with a higher correlation to the primary endpoint can
enhance the performance of the proposed SDDO framework. Lastly, we examine
scenarios in which all dosages demonstrate a weak treatment effect. Our
results indicate that, while the SDDO framework could correctly select the
true optimal dose with a higher probability (56.84%), in most cases, it fails
to detect a significant treatment effect in the final analysis as expected
(only 11.83% and 42.67% for PFS and OS, respectively), and the false-positive
rate remains controlled at 19.13%.

| Simulation
---
Scenarios
| Optimal Dose
---
Percentage
|  | Interim
---
Decision
| Positive
---
Rate
| Expected
---
Event Size
| Expected
---
Study Duration
Strong PFS, Weak OS: ORR = [0.2,0.3,0.35,0.4] $\text{HR}_{\text{PFS}}$=[1,0.6,0.6,0.6] $\text{HR}_{\text{OS}}$=[1,0.95,0.9,0.85] | [5.80%,25.03%,69.17%] | Overall | – | 54.23% (39.15%) | 373 | 39.34
Terminate | 0.75% | 0 | 80 | 19.85
Phase II | 30.40% | 85.89% (52.23%) | 140 | 35.75
Phase III | 68.85% | 40.84% (33.80%) | 479 | 41.14
Weak PFS, Strong OS: ORR = [0.2,0.3,0.35,0.4] $\text{HR}_{\text{PFS}}$=[1,0.95,0.9,0.85] $\text{HR}_{\text{OS}}$=[1,0.75,0.7,0.65] | [5.44%,25.12%,69.44%] | Overall | – | 84.66% (61.00%) | 395 | 41.13
Terminate | 0.17% | 0 | 74 | 20.38
Phase II | 15.60% | 12.11% (9.03%) | 140 | 29.89
Phase III | 84.23% | 98.25% (70.75%) | 395 | 43.26
Strong ORR, Weak HR: ORR = [[0.2,0.3,0.35,0.4] $\text{HR}_{\text{PFS}}$=[1,1,0.95,0.9] $\text{HR}_{\text{OS}}$=[1,1,0.95,0.9] | [5.82%,25.12%,69.06%] | Overall | – | 15.62% (13.31%) | 366 | 36.41
Terminate | 0.92% | 0 | 81 | 19.60
Phase II | 34.10% | 7.68% (6.04%) | 140 | 29.06
Phase III | 64.98% | 20.01% (17.31%) | 489 | 40.50
Weak ORR, Strong HR: ORR = [0.2,0.22,0.25,0.28] $\text{HR}_{\text{PFS}}$=[1,0.7,0.65,0.6] $\text{HR}_{\text{OS}}$=[1,0.7,0.65,0.6] | [10.55%,29.23%,60.22%] | Overall | – | 83.19% (53.49%) | 212 | 37.39
Terminate | 3.36% | 0 | 78 | 18.69
Phase II | 56.59% | 76.75% (46.67%) | 140 | 34.99
Phase III | 40.05% | 99.27% (67.61%) | 326 | 42.35
Table 3: Operating characteristics of the SDDO framework based on 10,000
simulation runs under various scenarios when a higher ORR does not necessarily
translate into benefits in PFS or OS. “Positive Rate” column shows overall
positive effect rate, and in brackets, the rate of selecting the optimal dose
with a positive effect.

Lastly, we assess the performance of the proposed SDDO framework in scenarios
where a significant treatment effect in ORR does not invariably translate to
PFS or OS benefits. Such a discrepancy between the surrogate and primary
endpoints might elevate the likelihood of making sub-optimal decisions during
the interim analysis. Given that similar observations have emerged in certain
clinical trials, it is crucial to evaluate how well the SDDO framework can
manage these instances. The pertinent results under three scenarios are
discussed in Table 3. All scenarios are deliberately chosen to reflect some
associated real-world examples.

The first scenario mimics the evaluations of paclitaxel plus bevacizumab as
initial treatment for metastatic breast cancer where only PFS and ORR
demonstrated significant treatment effects in the E2100 trial [42] but
subsequent studies failed to show the investigational treatment’s OS
superiority over different chemotherapy regimens [43, 44]. Under this setting,
the proposed SDDO framework has demonstrated a commendable ability to select
the true optimal dose with a high probability, consistent with the results
presented in Tables 1 and 2. Nonetheless, due to the modest OS benefits in
this context, the positive rate of advancing to Phase III stands at only
40.84%. This outcome is attributable to the inconsistencies between the
surrogate and primary endpoints in predicting treatment benefits that may
result from multiple factors such as different cancer types and/or lines of
therapies [45]. Therefore, on the one hand, it underscores the importance of
meticulously choosing a surrogate endpoint that can reliably predict survival
benefits to optimize the performance of the SDDO framework; on the other hand,
in case when the surrogacy is poorly established, the SDDO framework is
advantageous over conventional designs for its capacity of quickly concluding
marginal OS benefits within the same study (in this case, it may be just
within months instead of multiple years of conducting separate confirmatory
trials, along with substantially reduced number of participants exposed to the
suboptimal treatments).

The second scenario mimics the evaluations of nivolumab as a treatment for
previously treated advanced renal-cell carcinoma where both ORR and OS
demonstrated significant treatment effects over everolimus but median PFS was
similar between the two groups [46]. Another example is sipuleucel-T as a
treatment for metastatic castration-resistant prostate cancer where time to
progression (TTP, a related endpoint to PFS) is similar but OS is positive
[47]. In comparison to the first scenario, the proposed SDDO framework
exhibits a higher likelihood of advancing to the Phase III study (84.23% vs.
68.85%). This is attributable to the integration of OS benefits during the
decision-making process (at the interim) concerning trial expansion.
Furthermore, the proposed SDDO framework yields a strong statistical power of
$84.66\%$ in detecting such OS benefits. This is particularly notable since it
doesn’t rely on any assumptions regarding the correlation between the
surrogate and primary endpoints for the Phase II study (in this case, an
inconsistent ORR and PFS). Such flexibility offers additional advantages for
employing the SDDO framework, as in such scenarios, traditional designs would
be terminated at the end of the Phase II study since no treatment effect is
observed in regard to PFS benefits, while other seamless designs require
assumptions on the correlation between the surrogate endpoint and the primary
endpoint for the Phase II study [6, 20, 22] which may incur a detrimental
effect on concluding the survival benefits in case of poor surrogacy.

The third and fourth scenarios in Table 3 evaluate the performance of the SDDO
framework when treatment effects measured between ORR and time-to-event
endpoints (PFS and OS) are inconsistent. Numerous Phase III failures exemplify
the third scenario as strong efficacy signals measured by early surrogate
endpoints do not necessarily translate into survival benefits in late-phase
development. For the fourth scenario, Kok et al. [48] revealed that relative
measures of tumor response are poor surrogate endpoints for OS for many immune
checkpoint inhibitors. Under these scenarios, the proposed framework is
capable of selecting the true optimal dosage (69.09% and 60.22%) by means of
strong ORR (scenario 3) or strong OS (scenario 4) at the interim. For the most
challenging and bewildering scenario 3, the SDDO framework managed to control
the overall positive rate at 15.62% with a high probability of concluding no
PFS or OS benefits at the end of the study. For scenario 4 typically seen in
immunotherapies, despite weak ORR signals in the interim, the SDDO framework
only prematurely terminated the study 3.36% of the time, while attaining high
probabilities of trial success by PFS or OS at the final analysis.

##  4 Discussion

The Oncology Center of Excellence (OCE) at FDA initiated a couple of Research
and Development Projects that “aim to advance the development and regulation
of medical products for patients with cancer” [49]. The proposed SDDO
framework provides possible solutions to two of the projects, namely Project
Optimus [7] and Project FrontRunner [50], which we will provide more context
below.

The Phase II part of the SDDO framework aims to determine the optimal dosage
by randomizing multiple treatment doses (and a control arm), which is in
accordance with the guiding principles of Project Optimus. One noteworthy
advantage of the SDDO framework is its capacity to formally incorporate prior
dose-response information obtained from Phase I studies (as illustrated in
Figure 2). This feature is very relevant and critical in contemporary oncology
development, which has evolved from the traditional chemotherapeutic agents to
a wide range of novel therapies such as molecular targeted agents,
immunotherapies, and cell therapies that potentially exhibit drastically
different dose-response relationships. Some of them may exhibit a “bell-
shaped” relationship, indicating “less (dose) is more (favorable benefit-risk
profile)” (e.g., the most recent example is tarlatamab, when the lower 10-mg
group showed better efficacy and safety profiles than the higher 100-mg group
[51]). The seamless incorporation of the dose optimization component with
simultaneous evaluations of proof-of-concept efficacy signals (or lack of) in
a randomized pivotal trial setting has made the SDDO framework a viable
development option.

The proposed SDDO design is also closely related to Project Frontrunner which
encourages evaluations earlier in the course of the disease (e.g., first and
second lines) and FDA’s accelerated approval pathway [34, 35]. By utilizing a
single randomized controlled trial (RCT) for both accelerated and regular
approvals instead of two trials (conventionally involving a single-arm trial
for AA and a separate trial for regular approval), the SDDO framework fits
into the “one-trial” approach [34, 52] with enhancements to achieve multiple
objectives, namely the optimal dose selection and potential AA at the interim
analysis via the use of (mature) early surrogate (e.g., ORR) and
(supplementary) long-term time-to-event endpoints (e.g., OS) within a Bayesian
framework; and potential regular approval at the final analysis by comparing
the selected optimized dosing arm with the concurrent control arm, with
possible SSR to increase the trial’s success rate. The many advantages brought
by the SDDO framework, such as shortening the overall study duration and
making the best use of patient resources, are expected to greatly increase the
efficiency of oncology development.

We propose ORR as the surrogate endpoint for the interim analysis (dose
optimization and preliminary efficacy evaluations) because it matures quicker
(e.g., two or three tumor assessments with median scan frequency of every 8
weeks [53]) than time-to-event endpoints such as PFS and OS, and had been
accepted in many cases for accelerated approvals depending on the context of
use [29]. Slight modifications of the SDDO framework would allow the use of
PFS for such decision-making (especially, when PFS is a better predictor of OS
than ORR). It is worth emphasizing that the choice of an appropriate surrogate
endpoint requires rigorous validations in terms of its clinical relevance to
the final primary endpoint, whether its improvement can be generally
“translated” into meaningful improvement as measured by the primary endpoint,
and consistency for the relationship between the surrogate and final endpoints
in the particular population of interest. We evaluated some challenging but
realistic scenarios when surrogacy is less established (see Table 3):
unsurprisingly, the performance of the SDDO framework is undermined relative
to those scenarios with good surrogacy. However, for conflicting cases between
ORR and OS (scenarios 3 and 4), the SDDO framework is still preferred over
traditional (two-trial) designs in that it can either “stop loss” much earlier
(scenario 3, when a promising ORR does not translate into survival benefits),
or give a “second chance” to those investigational agents which actual
survival benefits could not be predicted by an early surrogate endpoint
(scenario 4). In each case, a traditional design would likely require a
separate Phase III study to demonstrate the lack of survival benefits
(scenario 3), or would make an incorrect, pre-mature termination decision
(scenario 4).

The SDDO framework may encounter some operational and logistical challenges.
The adaptation decision is made when the surrogate endpoint matures with a
certain fraction of OS events observed in the control arm. Temporary pause of
study and potential delay of future site activations (for Phase III) may be
inevitable. However, inheriting the advantages of the adaptive 2-in-1 designs,
the SDDO framework is much more efficient than conventional designs that
conduct sequential Phase II and Phase III trials, in terms of overall trial
duration and sample size. In addition, unblinding data for adaptation
decision-making and the concurrent enrollment of “optimized” treatment arm and
control arm may require independent data monitoring committee or independent
statistical center. Lastly, robust infrastructure, sufficient resource
allocation, and efficient information circulation are needed to avoid
logistical-related problems for the supply of experimental agents and to
ensure quality and scientific integrity of the study, as well as validity of
the interpretation of final results [54]. Statisticians are expected to take a
leading role in such discussions with project teams and health regulators at
the trial planning stage, e.g. via a Type-B meeting, details of optimization
strategy and Phase II/III trial design should be aligned with health
regulators, to further reduce the operational hurdle[55].

The risk of inflating overall Type I error from the adaptation decision of
advancing the “promising” dose level at interim analysis can be well
controlled through careful choices of the tuning parameters $\tau_{0}$,
$\tau_{1}$, $s_{0}$, and $s_{1}$ (Appendix Section B). The adaptive decision-
making relies on the joint distribution of surrogate endpoint ORR and final
primary endpoint OS with priors associated with dose levels. Our proposed
priors can inclusively incorporate the preliminary efficacy information from
early-phase studies – even if such efficacy information is limited, it is
still clinically relevant to borrow information from early-phase to increase
the overall success rate of late-phase studies (Appendix Section C).
Extensions of adaptation decisions based on different surrogate and primary
endpoints, and priors that incorporate both safety and preliminary efficacy
information, are worth further explorations.

The SDDO framework includes multiple enhancements of the original 2-in-1
adaptive design which has been implemented and considered in oncology studies
over the years: the first published example is the INDUCE-3 study for head and
neck squamous cell carcinoma (Hensen et al., 2020[56]); in Li et al.
(2022)[57], the authors explored the option of a flexible 2-in-1 design for a
relapsed/refractory multiple myeloma study; in Cong and Rubin (2023)[58], the
authors mentioned a possible extension from 2-in-1 to a 3-in-1 design using
the SKYSCRAPER-06 study for first-line non-small cell lung cancer as an
example. All the aforementioned examples indicate that it is totally feasible
to consider such novel adaptive designs in real-world oncology trials.
Compared with conventional designs, we acknowledge that the proposed SDDO
framework is relatively complicated; however, we strongly believe in its
practical use as it is an efficient design option to address key questions
outlined in the clinical regulatory projects recently initiated by the FDA
Oncology Center of Excellence such as Project Optimus and Project FrontRunner.
Whenever needed, sponsors who plan to implement this novel approach can
utilize the Agency’s “Complex Innovative Trial Designs (CIDs)” program for
consultation (which includes “evaluating Bayesian approaches for the potential
to increase clinical trial efficiency”).

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##  Appendix A Tuning Parameter Selection

Figure 4 shows the selected value of $\tau_{0}$ and $\tau_{1}$ from grid
search, using the procedure described in Section 2.2. Notably, $\tau_{0}$
tends to remain consistent across varying numbers of candidate dosages. This
stability can be attributed to the ORR difference of 0.15 being significant
enough to distinguish the optimal dose arm from the control arm, and the
influence of additional candidate dosages on the value of $\tau_{0}$ is
minimal. In contrast, the value of $\tau_{1}$ would increase as the number of
dosages increases. This finding implies the necessity to apply a more
stringent multiple testing correction procedure as the number of dosages
increases, for the purpose of controlling false positives.

Figure 4: Mean and standard deviation of $\tau_{0}$ and $\tau_{1}$ with
respect to different numbers of dosages based on 1000 simulations. $\tau_{0}$
is selected to control the error rate of false negative decision at 5% with an
ORR difference of 0.15. $\tau_{1}$ is selected to control the error rate of
false positive decision at 2.5%. The prior distribution is set to be uniform
for all dosages.

##  Appendix B Type I Error Control

We prove that the proposed SDDO design could control the overall Type I error
in this section. Without the loss of generality, suppose the desired Type I
error is 0.025. To avoid the influence of prior distribution, we assume
uniform prior for both ORR and $\text{HR}_{\text{OS}}$.

Under the null hypothesis, suppose the probability of terminating the trials,
entering Phase II and Phase III study after interim analysis are
$a_{1},a_{2},a_{3}$ respectively ($a_{1}+a_{2}+a_{3}=1$), and the false
positive rate after entering Phase II and Phase III study are denoted as
$\alpha_{2},\alpha_{3}$. Therefore, the overall Type I error $\alpha$ could be
computed as $\alpha=a_{2}\alpha_{2}+a_{3}\alpha_{3}.$ Notice that $\tau_{1}$
is selected to control the error rate of entering Phase III to be 2.5% under
the null hypothesis, thus we could specify $a_{3}=0.025$. Additionally, since
Phase II study is performed following the pre-planned event size and is not
influenced by interim analysis, the Type I error would be controlled at the
targeted level, thus $\alpha_{2}=0.025$. Combining the above result, we have

$\displaystyle\alpha$ $\displaystyle=a_{2}\alpha_{2}+a_{3}\alpha_{3}$
$\displaystyle=0.025a_{2}+0.025\alpha_{3}.$

Therefore, we proceed to find the upper bound for $a_{2}$ and $\alpha_{3}$. We
first focus on the upper bound for $\alpha_{3}$. Based on the SDDO framework,
suppose we have $I$ candidate dosage at the beginning of Phase II study. we
denote $t$ as the proportion of event size available at interim analysis,
$Z(t),Z(1-t)$ refers to the standardized test statistics of the selected
optimal dose based on data before and after interim analysis, suppose the
clinical trials entering Phase III under null hypothesis we have

$\displaystyle\begin{split}\alpha_{3}&=P(Z(1)\leq Z_{\alpha})\\\
&=P\Big{(}\sqrt{t}Z(t)+\sqrt{1-t}Z(1-t)\leq Z_{\alpha}\Big{)}\\\ &\leq
P\Big{(}\sqrt{t}\min(Z_{1},Z_{2},...,Z_{I})+\sqrt{1-t}Z(1-t)\leq
Z_{\alpha}\Big{)}\quad(Z_{1},Z_{2},...Z_{I}\stackrel{{\scriptstyle
i.i.d}}{{\sim}}N(0,1),Z(1-t)\sim N(0,1))\\\
&=P\Big{(}\min(Z_{1},Z_{2},...,Z_{I})\leq\frac{Z_{\alpha}-\sqrt{1-t}Z(1-t)}{\sqrt{t}}\Big{)}\\\
&=\int^{\infty}_{-\infty}P\Big{(}\min(Z_{1},Z_{2},...,Z_{I})\leq\frac{Z_{\alpha}-\sqrt{1-t}z_{0}}{\sqrt{t}}\Big{|}Z(1-t)=z_{0}\Big{)}f_{Z(1-t)}(z_{0})dz_{0}\\\
&=\int^{\infty}_{-\infty}\Big{[}1-\Big{(}1-\Phi\big{(}\frac{Z_{\alpha}-\sqrt{1-t}z_{0}}{\sqrt{t}}\big{)}\Big{)}^{I}\Big{]}\phi(z_{0})dz_{0},\end{split}$
(B.1)

where $\Phi,\phi$ denote the cdf and pdf of standard normal distribution. Note
that Equation B.1 illustrates how the upper bound of $\alpha_{3}$ is
influenced by both $t$ and the number of candidate dosages, which need to be
determined prior to the start of the clinical study. Therefore, $\alpha_{3}$
can be more effectively controlled by appropriately specifying values for $t$
and $I$. Figure 5 demonstrates such relationship, clearly showing that

$\alpha_{3}\leq 0.125$ (B.2)

across all examined scenarios.

Figure 5: Upper bound of $\alpha_{3}$ as affected by varying the number of
candidate dosages and different values of $t$.

We then proceed to bound $a_{2}$. Recall that $a_{2}$ refers to the
probability of entering Phase II study after interim analysis, thus
$a_{2}=P\big{(}P(p_{i^{*}}-p_{0}\geq\tau_{0})>0.9\big{)}-a_{3}=P\big{(}P(p_{i^{*}}-p_{0}\geq\tau_{0})>0.9\big{)}-0.025$.
Notice that under null hypothesis $\theta_{i^{*}}=0$, thus it wouldn’t
influence the probability $P(p_{i^{*}}-p_{0}\geq\tau_{0})$, we omit the
conditional part in Equation (2.5) for simplicity.

Under the null hypothesis, at interim analysis of the proposed SDDO framework,
we have posterior distribution
$p_{0}\sim\mathcal{N}(\hat{\mu}_{0},\sigma^{2})$,
$p_{i^{*}}\sim\mathcal{N}(\hat{\mu}_{i^{*}},\sigma^{2})$, where
$\sigma=\sqrt{\frac{\hat{\mu}_{0}(1-\hat{\mu}_{0})}{n_{0}}}$,
$\hat{\mu}_{0}=\frac{y_{0}}{n_{0}},\hat{\mu}_{i^{*}}=\frac{y_{i^{*}}}{n_{0}}$,
$n_{0}$ is the number of participants enrolled in each dose arm and control
arm, and $y_{i}$ is the number of responders in each dose arm. Therefore,
$p_{i^{*}}-p_{0}\sim\mathcal{N}(\hat{\mu}_{i^{*}}-\hat{\mu}_{0},2\sigma^{2})$,
we then have

$\displaystyle P\big{(}P(p_{i^{*}}-p_{0}\geq\tau_{0})>0.9\big{)}$
$\displaystyle=P\Big{(}1-\Phi\Big{(}\frac{\tau_{0}-(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})}{\sqrt{2\sigma^{2}}}\Big{)}>0.9\Big{)}$
$\displaystyle=P\Big{(}\Phi\Big{(}\frac{\tau_{0}-(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})}{\sqrt{2\sigma^{2}}}\Big{)}<0.1\Big{)}$
$\displaystyle=P\Big{(}\frac{\tau_{0}-(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})}{\sqrt{2\sigma^{2}}}<-1.28\Big{)}\quad(\text{$\Phi^{-1}(0.1)=-1.28$})$
$\displaystyle=P\Big{(}\tau_{0}-(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})<-1.28\sqrt{2\sigma^{2}}\Big{)}$
$\displaystyle=P\Big{(}(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})>\tau_{0}+1.28\sqrt{2\sigma^{2}}\Big{)}.$

Notice that
$\hat{\mu}_{i^{*}}=\max(\hat{\mu}_{1},\hat{\mu}_{2},...,\hat{\mu}_{I})$ and
$\hat{\mu}_{0},\hat{\mu}_{1},\hat{\mu}_{2},...,\hat{\mu}_{I}\stackrel{{\scriptstyle
i.i.d}}{{\sim}}N(p_{0},\sigma^{2})$, therefore,

$\displaystyle\begin{split}P\big{(}P(p_{i^{*}}-p_{0}\geq\tau_{0})>0.9\big{)}&=P\Big{(}(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})>\tau_{0}+1.28\sqrt{2\sigma^{2}}\Big{)}\\\
&=1-P\Big{(}(\hat{\mu}_{i^{*}}-\hat{\mu}_{0})\leq\tau_{0}+1.28\sqrt{2\sigma^{2}}\Big{)}\\\
&=1-\int^{\infty}_{-\infty}P\Big{(}\hat{\mu}_{i^{*}}\leq\mu+\tau_{0}+1.28\sqrt{2\sigma^{2}}\Big{|}\hat{\mu}_{0}=\mu\Big{)}f_{\hat{\mu}_{0}}(\mu)d\mu\\\
&=1-\int^{\infty}_{-\infty}\Phi^{I}\Big{(}\frac{\mu+\tau_{0}+1.28\sqrt{2\sigma^{2}}}{\sigma}\Big{)}\phi\Big{(}\frac{\mu-
p_{0}}{\sigma}\Big{)}d\mu.\end{split}$

Therefore,

$\displaystyle\begin{split}a_{2}&=P\big{(}P(p_{i^{*}}-p_{0}\geq\tau_{0})>0.9\big{)}-0.025\\\
&=0.975-\int^{\infty}_{-\infty}\Phi^{I}\Big{(}\frac{\mu+\tau_{0}+1.28\sqrt{2\sigma^{2}}}{\sigma}\Big{)}\phi\Big{(}\frac{\mu-
p_{0}}{\sigma}\Big{)}d\mu.\end{split}$ (B.3)

Figure 6: Value of $a_{2}$ as affected by varying the number of candidate
dosages and different values of $p_{0}$ and $n_{0}$. The left panel assumes
$n_{0}=80$, and the right panel assumes $p_{0}=0.2$.

From Equation B, it is evident that the value of $a_{2}$ is influenced by
three factors: the number of candidate dosages $I$, the sample size prior to
interim analysis $n_{0}$, and the ORR under the null hypothesis $p_{0}$.
Figure 6 illustrates their relationship, which shows that

$a_{2}<0.8$ (B.4)

across all scenarios. Combining (B.2) and (B.4), the overall Type I error is
controlled by

$\displaystyle\alpha$ $\displaystyle=a_{2}\times 0.025+0.025\alpha_{3}$
$\displaystyle<0.025(0.125+0.8)<0.025.$

Therefore, the overall Type I error is controlled for the proposed SDDO
design.

##  Appendix C Additional Simulation Results

In this section, we evaluate the performance of the SDDO framework with
respect to various prior distributions assigned to ORR and
$\text{HR}_{\text{OS}}$. Tables 4 and 5 demonstrate the operating
characteristics under both uninformative and informative priors across
different scenarios. Specifically, in Table 4, we assume a bell-shaped dose-
response relationship, while in Table 5, a monotonic relationship is assumed.
For the informative prior, prior distributions that accurately reflect these
relationships were assigned to each dose level, while the uninformative prior
remains consistent across all dosage levels. Detailed illustrations of the
prior distributions are provided in Figures 7 and 8.

Figure 7: Prior distributions for each dose level on ORR and log hazard ratio
corresponding to the simulation scenario in Table 4. The uninformative prior
is designated as $\text{Beta}(2,2)$ for ORR, and $\mathcal{N}(0,0.5)$ for
$\log(\text{HR}_{\text{OS}})$ across all dosages. The informative prior for
ORR follows the distribution $\text{Beta}(\alpha(d),20)$, where $\alpha(d)$ is
a quadratic function of the dose level. Similarly, the informative prior for
$\log(\text{HR})$ is given by $\mathcal{N}(\mu(d),0.05)$, where $\mu(d)$ also
represents a quadratic function of the dose level.

Tables 4 and 5 clearly demonstrate that the proposed SDDO framework achieves a
higher probability of selecting the true optimal dosage when aided by an
informative prior derived from data from previous clinical studies, such as
Phase I dose-escalation trials. This improvement is attributed to the enhanced
decision-making certainty provided by the informative prior, which reinforces
the signal of treatment effects through the integration of prior knowledge
regarding the outcomes of interest. Additionally, as shown in Table 4, when
the treatment effect is significant, the probability of directly advancing to
a Phase III study is substantially higher with an informative prior (83.48% vs
70.43%), whereas this probability tends to decline when the treatment effect
is insignificant as shown in Table 5 (13.19% vs 26.53%). These observations
also suggest that informative priors could significantly improve the
probability of making a preferable choice regarding trial expansion, thereby
facilitating the “quick to win, fast to fail” principle in decision-making.

Figure 8: Prior distributions for each dose level on ORR and log hazard ratio
corresponding to the simulation scenario in Table 5. The uninformative prior
is designated as $\text{Beta}(2,2)$ for ORR, and $\mathcal{N}(0,0.5)$ for
$\log(\text{HR}_{\text{OS}})$ across all dosages. The informative prior for
ORR follows the distribution $\text{Beta}(\alpha(d),20)$, where $\alpha(d)$ is
a linear function of the dose level. Similarly, the informative prior for
$\log(\text{HR})$ is given by $\mathcal{N}(\mu(d),0.05)$, where $\mu(d)$ also
represents a linear function of the dose level.

Besides improving the probability of making optimal decisions regarding dose
selection and trial expansion, a smaller event size is required under both
scenarios with informative priors while achieving better accuracy at the final
analysis. When the treatment effect is significant as in Table 4, the
reduction in event size can be attributed to the informative prior
strengthening the original signal by accommodating additional information,
thus achieving a higher PPoS at the interim analysis for Phase III expansion
which could, in turn, contribute to a reduction in the required event size at
the final analysis. Meanwhile, when the treatment effect is insignificant as
in Table 5, such reduction is mainly due to the lower probability of entering
the Phase III study. These observations further indicate that the proposed
SDDO design could optimize the cost efficiency of the clinical study
significantly.

| Simulation
---
Scenarios
| Optimal Dose
---
Percentage
|  | Interim
---
Decision
| Positive
---
Rate
| Expected
---
Event Size
| Expected
---
Study Duration
Uninformative Prior: ORR = [0.2,0.32,0.35,0.3] $\text{HR}_{\text{PFS}}$=[1,0.7,0.65,0.75] $\text{HR}_{\text{OS}}$=[1,0.7,0.65,0.75] | [26.80%,58.63%,14.57%] | Overall | – | 86.17% (52.17%) | 313 | 40.01
Terminate | 0.56% | 0 | 72 | 18.84
Phase II | 29.01% | 61.43% (37.38%) | 140 | 33.72
Phase III | 70.43% | 97.04% (58.62%) | 386 | 42.76
Informative Prior: ORR = [0.2,0.32,0.35,0.3] $\text{HR}_{\text{PFS}}$=[1,0.7,0.65,0.75] $\text{HR}_{\text{OS}}$=[1,0.7,0.65,0.75] | [23.11%,68.98%,7.91%] | Overall | – | 89.49% (63.78%) | 253 | 39.28
Terminate | 0.01% | 0 | 75 | 24.60
Phase II | 16.51% | 62.83% (45.51%) | 140 | 33.86
Phase III | 83.48% | 94.76% (67.41%) | 276 | 40.36
Table 4: Operating characteristics of the SDDO framework based on 10,000 simulation runs under uninformative prior and informative prior. Dose response relationship follows a bell curve under this scenario, detailed illustrations of the prior distributions are provided in Figure 7. “Positive Rate” column shows overall positive effect rate, and in brackets, the rate of selecting the optimal dose with a positive effect. | Simulation
---
Scenarios
| Optimal Dose
---
Percentage
|  | Interim
---
Decision
| Positive
---
Rate
| Expected
---
Event Size
| Expected
---
Study Duration
Uninformative Prior: ORR = [0.2,0.25,0.27,0.3] $\text{HR}_{\text{PFS}}$=[1,0.95,0.9,0.85] $\text{HR}_{\text{OS}}$=[1,0.95,0.9,0.85] | [14.06%,29.10%,56.84%] | Overall | – | 19.13% (14.91%) | 221 | 31.69
Terminate | 7.44% | 0 | 78 | 18.49
Phase II | 66.03% | 11.83% (8.78%) | 140 | 29.73
Phase III | 26.53% | 42.67% (34.31%) | 462 | 40.26
Informative Prior: ORR = [0.2,0.25,0.27,0.3] $\text{HR}_{\text{PFS}}$=[1,0.95,0.9,0.85] $\text{HR}_{\text{OS}}$=[1,0.95,0.9,0.85] | [9.62%,25.70%,64.68%] | Overall | – | 18.13% (14.68%) | 185 | 30.98
Terminate | 2.54% | 0 | 80 | 20.01
Phase II | 84.27% | 13.13% (10.36%) | 140 | 29.86
Phase III | 13.19% | 53.52% (45.11%) | 495 | 40.19
Table 5: Operating characteristics of the SDDO framework based on 10,000
simulation runs under uninformative prior and informative prior. Treatment
effect is assumed to be insignificant for all dosages, detailed illustrations
of the prior distributions are provided in Figure 8. “Positive Rate” column
shows overall positive effect rate, and in brackets, the rate of selecting the
optimal dose with a positive effect.