Biomarker-Guided Adaptive Enrichment Design with Threshold Detection for
Clinical Trials with Time-to-Event Outcome

Kaiyuan Hua1, Hwanhee Hong1, Xiaofei Wang1
1 Department of Biostatistics and Bioinformatics, Duke University School of
Medicine, Durham, North Carolina, 27705, USA

Abstract: Biomarker-guided designs are increasingly used to evaluate
personalized treatments based on patients’ biomarker status in Phase II and
III clinical trials. With adaptive enrichment, these designs can improve the
efficiency of evaluating the treatment effect in biomarker-positive patients
by increasing their proportion in the randomized trial. While time-to-event
outcomes are often used as the primary endpoint to measure treatment effects
for a new therapy in severe diseases like cancer and cardiovascular diseases,
there is limited research on biomarker-guided adaptive enrichment trials in
this context. Such trials almost always adopt hazard ratio methods for
statistical measurement of treatment effects. In contrast, restricted mean
survival time (RMST) has gained popularity for analyzing time-to-event
outcomes because it offers more straightforward interpretations of treatment
effects and does not require the proportional hazard assumption. This paper
proposes a two-stage biomarker-guided adaptive RMST design with threshold
detection and patient enrichment. We develop sophisticated methods for
identifying the optimal biomarker threshold, treatment effect estimators in
the biomarker-positive subgroup, and approaches for type I error rate, power
analysis, and sample size calculation. We present a numerical example of re-
designing an oncology trial. An extensive simulation study is conducted to
evaluate the performance of the proposed design.

Key words: Biomarker adaptive design; Continuous biomarker; Patient
enrichment; Restricted mean survival time; Threshold detection.

##  1 Introduction

Due to advancements in targeted therapies and precision medicine over the last
two decades, there is a growing trend in the use of biomarker-guided designs
for assessing personalized treatments in Phase II and Phase III clinical
trials, especially in the field of oncology [1, 2, 3]. Predictive biomarkers
are often used in biomarker-guided trials to develop classifiers to identify
appropriate patients as either excellent or poor candidates for clinical
decisions to optimize therapy selections [4, 5]. Recently, designs with
adaptive enrichment have become increasingly attractive for biomarker-guided
therapies as they provide additional flexibility during the trial [6, 2]. In
this paper, we consider the setting in which a single continuous biomarker is
available at baseline. In general, a fixed threshold of the biomarker is pre-
specified to dichotomize patients into “biomarker-positive” and “biomarker-
negative” subgroups, and it is assumed that the biomarker-positive patients
will benefit more from the new therapy over the active control. The biomarker-
guided adaptive enrichment designs are typically conducted in a two-stage
manner [7, 8]. In the first stage, they start with randomizing patients from
the whole population. At the end of this stage, the accumulating data are used
to decide whether to restrict accrual in the biomarker-positive subgroup in
the second stage. Such adaptive enrichment could avoid further including
patients who do not benefit from an intervention and may experience some
negative effects [9, 10, 11]. Existing literature demonstrates that such
designs improve the efficiency of clinical trials by reducing the number of
patients needed for randomization, particularly when the therapy of interest
is known to be effective only in biomarker-positive patients and the biomarker
assessment is highly accurate [12].

However, the optimal cutoff for classifying the patients into positive and
negative subgroups is often not available and challenging to be pre-specified
at the design stage of a trial [6]. To address this challenge, more flexible
adaptive enrichment designs have been proposed by researchers [6, 13, 14].
While the trial is ongoing, the observed data is used to identify the subgroup
of patients most likely to benefit from a treatment (i.e., biomarker-positive
subgroup) based on the biomarker values and certain utility functions. These
designs allow simultaneously identifying the optimal biomarker threshold,
defining the biomarker-positive subgroup, and estimating and testing the
treatment effect in this subgroup. These approaches provide much efficiency of
the enrichment strategy without the need to select a subset in advance, and
the meaningful treatment effect estimates without being diluted by biomarker-
negative patients who receive no or fewer benefits [15].

Moreover, limited biomarker-adaptive enrichment designs have been proposed for
time-to-event outcomes [16, 17, 18]. These designs were developed based on
hazard ratio (HR), which is commonly estimated by the Cox PH model [19] or
requires testing treatment effect with a log-rank test [20]. A fundamental
assumption for Cox regression models is the PH assumption, which assumes that
the HR between two groups does not vary over time. However, non-constant HRs
have been recently observed in many oncology trials due to delayed treatment
effects or other biomedical reasons [21, 22, 23]. In such scenarios, the PH
assumption becomes problematic, rendering HR a potentially misleading and
inappropriate summary of treatment effects [24]. Violations of the PH
assumption may also considerably impact the statistical power of the log-rank
test to detect the treatment difference [25]. To address this issue, various
alternative tests have been proposed [26], and we emphasize a compelling
alternative method based on Restricted Mean Survival Time (RMST) [27, 28, 29]
in this paper.

RMST summarizes the survival time up to a clinically relevant and usually pre-
specified truncation time $t^{*}$. It is defined as the mean of the truncated
event time $Y=\text{min}(T,t^{*})$, and can be estimated by the area under the
survival curve $S(t)=P[T>t]$ from $t=0$ to $t=t^{*}$ [30]:

$\mu(t^{*})=E(Y)=E[T\wedge t^{*}]=\int_{0}^{t^{*}}S(t)dt.$ (1.1)

The difference or ratio of RMST between two treatment groups measures the
relative treatment effect concerning a gain or loss of event-free survival
time up to $t^{*}$ [31]. Estimating RMSTs does not necessitate model
assumptions, and their interpretations are more straightforward than HRs
across any distribution of time-to-event outcomes [32]. Leveraging these
advantages, many RMST-based clinical trial design methods have recently been
proposed [33, 34, 35, 36]. However, limited attention has been given to the
biomarker-adaptive enrichment design. We address this methodological and
practical gap by proposing a two-stage biomarker-guided adaptive RMST design
with threshold detection and patient enrichment.

Our proposed design is motivated by a biomarker-guided randomized Phase III
oncology trial, JAVELIN Lung 200 [23]. This trial compared the efficacy and
safety of avelumab, an anti-PD-L1 immune checkpoint inhibitor antibody, with
docetaxel, a standard care for patients with non-small-cell lung cancer
(NSCLC). PD-L1 expression on tumor cells is a well-studied biomarker in NSCLC
[37], and previous studies have indicated a positive association between high
PD-L1 expression levels and progression-free survival (PFS) and overall
survival (OS) after treatment with PD-L1 inhibitors for NSCLC [38]. In JAVELIN
Lung 200, avelumab was not found to have lower OS compared to docetaxel within
“PD-L1-positive” patients, defined as patients with PD-L1 expression $\geq
1\%$, but exploratory analyses revealed that avelumab was associated with
improved OS and PFS compared to docetaxel in patients with higher PD-L1
expression at $\geq 50\%$ and $\geq 80\%$ cutoffs. However, these exploratory
analyses were post hoc, and the biomarker thresholds were pre-specified and
thus possibly not optimal. Moreover, the PH assumption was violated for both
OS and PFS (Supplemental Figures 1 and 3 in [23]), so the characterization of
the treatment effect based on the HR and its confidence interval estimated
from the Cox PH model was not valid, and the hypothesis testing based on the
log-rank test was less powerful. Consequently, there is a need for an RMST-
based design to identify an optimal threshold and define the biomarker-
positive subgroup while the trial is ongoing. Enrichment is enabled to
increase the accrual of patients from the selected subgroup and enhance the
design’s efficiency and power.

##  2 Design

###  2.1 Notations and Assumptions

We consider a two-stage enrichment design where patients are recruited and
randomized between experimental and control treatments. Let $Z$ be the
treatment indicator, where $Z=1$ for the experimental and $Z=0$ for the
control. Let $X\in[0,1]$ represent a single continuous biomarker of interest
and is assumed to follow a standard uniform distribution, i.e.,
$X\sim\text{Unif}[0,1]$. If $X$ is a continuous marker with other
distributions, the proposed design is still applicable when the values of $X$
can be scaled to $[0,1]$ through percentile transformations.

Following the potential outcomes framework [39], let $T(1)$ and $T(0)$ be the
potential time-to-event under treatment and control, respectively. By assuming
the consistency of potential outcomes, the time-to-event is
$T=T(1)Z+T(0)(1-Z)$. Let $C$ be the censoring time, and we assume
conditionally independent censoring given $Z$ and $X$, written as
$C\perp\\!\\!\\!\\!\perp T|Z,X$. In the presence of right censoring, we
observe $U=\text{min}(T,C)$ and the censoring indicator $\delta=I[T\leq C]$,
where $I[\cdot]$ is the indicator function.

We assume piecewise exponential hazard, $h_{z}(t|X)$, for patients with $Z=z$:

$\displaystyle h_{z}(t|X)$ $\displaystyle=$
$\displaystyle\lambda_{z}(t)\times\text{exp}\\{\gamma_{z}X\\},\mbox{ for
}z=0,1$ (2.1)

where $\lambda_{z}(t)$ is the baseline hazard and $\gamma_{z}$ is the log-HR
representing a constant association between biomarker and hazard. Consider
$J_{z}$ time intervals for each treatment with change points
$0=\tau_{z,0}<\tau_{z,1}<\cdots<\tau_{z,J_{z}}=\infty$. We assume
$\lambda_{z}(t)$ is constant within each interval, so that
$\lambda_{z}(t)=\lambda_{z,j},\mbox{ for }t\in[\tau_{z,j-1},\tau_{z,j})$. Note
that our proposed design is versatile and applicable to various hazard
functions. The hazard function in Equation (2.1) is employed here for
illustrative purposes, and alternative hazard models can be considered based
on the requirement of a specific design.

###  2.2 Optimal Biomarker Threshold

Several methods have been proposed for selecting the biomarker threshold and
identifying biomarker-positive subgroups [40, 41, 14]. For example, the
subgroup can be selected by either maximizing the product of its sample size
and effect size [41] or maximizing the test statistic of the treatment effect
difference between two subgroups [40]. However, defining the best subgroup
using these methods involves a trade-off between subgroup and treatment effect
size [14]. For instance, selecting a smaller subgroup with a higher average
treatment effect may not be the best choice. Furthermore, these approaches
typically necessitate pre-specifying a set of candidate thresholds or
employing a greedy search to identify the optimal one.

To address these issues, we adopt the methods introduced by [14] to identify
the optimal threshold by the intersection point of the biomarker response
curves of two treatment groups (i.e., the RMST curves as functions of
biomarker value). This method parametrically models the biomarker response
curves without mandating the pre-specification or greedy search for
thresholds. Another advantage of using the intersection point is that it
ensures the benefit of the experimental treatment for the biomarker-positive
subgroup including all patients with positive treatment effects.

Here, the biomarker-positive subgroup comprises patients whose biomarker
values are within the range where the conditional RMST differences between the
experimental and control are all positive, given $X$. Conversely, the
biomarker-negative subgroup comprises the range of $X$ where the conditional
RMST differences are all negative. We refer to the threshold between
biomarker-positive and negative subgroups as the biomarker cutpoint, denoted
as $c_{t^{*}}$ concerning the truncation time $t^{*}$. Without loss of
generality, we assume $X$ is positively associated with the RMST difference
between treatments. As such, the biomarker-positive and negative subgroups are
defined as $X\in(c_{t^{*}},1]$ and $X\in[0,c_{t^{*}})$, respectively.

We denote $\mu_{z}(t^{*}|X)$ as the conditional RMST function for
$z\in\\{0,1\\}$, which is written as:

$\mu_{z}(t^{*}|X)=\int_{0}^{t^{*}}S_{z}(t|X)dt=\int_{0}^{t^{*}}\text{exp}\left\\{-\int_{0}^{t}h_{z}(u|X)du\right\\}dt,$

where $S_{z}(t|X)$ is the conditional survival function. The biomarker
cutpoint, $X=c_{t^{*}}$, can be identified by solving the following equation
concerning $X$:

$\mu_{1}(t^{*}|X)-\mu_{0}(t^{*}|X)=0.$ (2.2)

Note that $c_{t^{*}}$ may fall outside the range of $[0,1]$. In such
instances, the biomarker cutpoint will be truncated at 0 or 1, which is
formally written as $c_{t^{*}}=\text{min}\\{\text{max}\\{c_{t^{*}},0\\},1\\}$.

###  2.3 Two-Stage Adaptive Enrichment Design

Figure 1 visualizes the diagram of the proposed design. Stage I is conducted
from the calendar time 0 to $t_{1}$, and accrues $n_{1}$ patients per
treatment arm regardless of the biomarker value. At the end of Stage I (at
$t_{1}$), a biomarker cutpoint, denoted as $\hat{c}_{0}$, is estimated
concerning the RMST difference at a pre-specified truncation time $t^{*}$
related to a specific research question. Based on the result, two enrichment
criteria can be introduced for Stage II: 1) accrue patients regardless of the
biomarker value (i.e., no enrichment) in Stage II if $\hat{c}_{0}$ is not
identified (e.g., all biomarker values are positive or negative to patients);
2) enrich and restrict the accrual in Stage II to patients in the biomarker-
positive subgroup $(\hat{c}_{0},1]$. Note that the trial will not be
terminated in the first condition as the data are not mature in the early
stage, and the treatment effect is not tested.

Figure 1: Diagram of the two-stage adaptive and enrichment design.

In Stage II, $n_{2}$ patients per treatment arm are accrued, and the
enrollment continues from $t_{1}$ to $t_{2}$, resulting in a total of
$n=2n_{1}+2n_{2}$. All patients are continuously followed up until $t_{3}$,
and we assume $t_{3}>t^{*}$. After completing the follow-up, we estimate a
biomarker cutpoint, denoted as $\hat{c}_{t^{*}}$, and assume a total of
$n_{+}$ patients are included from both stages in the estimated biomarker-
positive subgroup $(\hat{c}_{t^{*}},1]$. Additionally, we test whether there
exists a statistically significant positive relationship between the biomarker
and treatment effect. This hypothesis test, denoted as Hypothesis Zero, can be
written as follows:

  * •

Null hypothesis $H_{00}$: There is no positive biomarker-by-treatment
interaction.

  * •

Alternative hypothesis $H_{a0}$: There is a positive biomarker-by-treatment
interaction.

Based on the testing result, two mutually exclusive hypotheses for conditional
treatment effects (i.e., RMST difference) are considered. First, if $H_{00}$
is rejected, we estimate and test the RMST difference within
$(\hat{c}_{t^{*}},1]$. This hypothesis test, denoted as Hypothesis One, is
written as:

  * •

Null hypothesis $H_{01}$: The RMST difference in the biomarker-positive
subgroup $=0$.

  * •

Alternative hypothesis $H_{a1}$: The RMST difference in the biomarker-positive
subgroup $>0$.

Second, if $H_{00}$ is not rejected, we estimate and test the RMST difference
in the overall population. This hypothesis test, denoted as Hypothesis Two, is
written as:

  * •

Null hypothesis $H_{02}$: The RMST difference in the overall population $=0$.

  * •

Alternative hypothesis $H_{a2}$: The RMST difference in the overall population
$>0$.

In Sections 3 to 5, we will further describe the methods for identifying
biomarker cutpoint, estimating treatment effects, and testing the
aforementioned hypotheses.

##  3 Biomarker Cutpoint Estimation

We explore two methods for identifying the biomarker cutpoint: 1) the
prediction method used in Stage I and 2) the RMST regression method used in
Stage II.

###  3.1 Prediction Method

The prediction method involves two key steps. First, it fits a hazard
function, $h_{z}(t|X)$, for each treatment based on the observed data. Second,
it integrates the fitted hazard functions into Equation (2.2) and numerically
estimates the biomarker cutpoint by solving this equation. This approach
necessitates certain parametric assumptions on the survival curves or hazard
functions within each treatment group. In this context, we focus on the
piecewise exponential models, specified in Equation (2.1).

For the $i^{th}$ patient and $j^{th}$ time interval, the hazard function can
be written as follows:

$\displaystyle\text{log}(\lambda_{z_{i},ij})=\text{log}(\lambda_{z_{i},j})+\gamma_{z_{i}}X_{i},\mbox{
for }t\in[\tau_{z_{i},j-1},\tau_{z_{i},j}),$ (3.1)

where $z_{i}\in\\{0,1\\}$ is the treatment indicator and $\lambda_{z_{i},j}$
is the baseline hazard in the $j^{th}$ interval. We use a Poisson log-linear
regression method [42] to fit this model and estimate the parameters
$\hat{\lambda}_{z,j}$ and $\hat{\gamma}_{z}$. The details are presented in Web
Appendix A.

The strength of the prediction method lies in its ability to predict a
biomarker cutpoint concerning a future truncation time $t^{*}$, which is
particularly valuable in survival adaptive design where an early enrichment is
sought to accrue more patients benefiting from the new therapy. In the context
of our design, this approach is particularly advantageous in Stage I of our
proposed design when the longest follow-up time $t_{1}$ is shorter than
$t^{*}$. However, one limitation is its dependence on a parametric assumption
on the survival curves. Moreover, under piecewise exponential models, it
assumes no change point in the hazard functions between $t_{1}$ and $t^{*}$.
As such, when a sufficient number of patients are being followed up beyond
$t^{*}$, an alternative method (e.g., RMST regression method) can be used to
estimate the biomarker cutpoint without the parametric model assumptions.

###  3.2 RMST Regression Method

The RMST regression method adapts the inverse probability of censoring
weighted (IPCW) RMST regression [43] to estimate the biomarker cutpoint. We
define the RMST up to $t^{*}$ given the biomarker and treatment as
$\mu(t^{*}|X,Z)=E(Y|X,Z)$. The RMST regression model is written as follows:

$\displaystyle g(\mu(t^{*}|X,Z))=\beta_{0}+\beta_{1}Z+\beta_{2}X+\beta_{3}ZX,$
(3.2)

where $g(\cdot)$ is the link function and identity link is used throughout
this paper. The model parameters are denoted as
$\hat{\mbox{\boldmath$\beta$}}=[\hat{\beta}_{0},\hat{\beta}_{1},\hat{\beta}_{2},\hat{\beta}_{3}]^{T}$.
The conditional RMSTs given $X$ under the experimental and control treatments
are
$\hat{\mu}(t^{*}|X,1)=\hat{\beta}_{0}+\hat{\beta}_{1}+\hat{\beta}_{2}X+\hat{\beta}_{3}X$
and $\hat{\mu}(t^{*}|X,0)=\hat{\beta}_{0}+\hat{\beta}_{2}X$, respectively.
Consequently, the conditional RMST difference is
$\hat{\mu}(t^{*}|X,1)-\hat{\mu}(t^{*}|X,0)=\hat{\beta}_{1}+\hat{\beta}_{3}X$.
Under the assumption that the treatment effect is positively related to the
biomarker value, i.e., $\beta_{3}>0$, the biomarker cutpoint is
$\hat{c}_{t^{*}}=-\hat{\beta}_{1}/\hat{\beta}_{3}$ with the biomarker-positive
subgroup defined as $X\in[-\hat{\beta}_{1}/\hat{\beta}_{3},1]$.

The advantage of this approach is that it does not require parametric
assumptions on the hazard functions. However, it is essential to ensure that
the truncation time $t^{*}$ does not surpass the maximum of the observed event
time (i.e., $U=\text{min}(T,C)$) within each treatment group. At the end of
Stage II of the proposed design, we utilize this method to estimate the
biomarker cutpoint as the final analysis time $t_{3}>t^{*}$.

##  4 Treatment Effect Estimators

Let $c_{t^{*}}$ denote the true biomarker cutpoint, the treatment effect
estimand, denoted as $\Delta_{t^{*}}^{(P)}$, is defined by the marginal RMST
difference between the experimental and control treatments among the true
biomarker-positive subgroup $(c_{t^{*}},1]$. It can be formally written as:

$\displaystyle\Delta_{t^{*}}^{(P)}=\int_{c_{t^{*}}}^{1}\left\\{\mu_{1}(t^{*}|x)-\mu_{0}(t^{*}|x)\right\\}dF_{+}(x),$
(4.1)

where $\mu_{1}(t^{*}|x)$ and $\mu_{0}(t^{*}|x)$ are the conditional RMST for
the experiment and control treatments, respectively, and $F_{+}(x)$ is the
cumulative distribution function of $X$ within the true biomarker-positive
subgroup. Note that $\Delta_{t^{*}}^{(P)}=\Delta_{t^{*}}^{(O)}$ when
$c_{t^{*}}=0$ in Equation (4.1).

###  4.1 Naive Unadjusted Estimator

The Naive unadjusted method employs the RMST definition in Equation (1.1) to
estimate $\Delta_{t^{*}}$, written as follows:

$\displaystyle\hat{\Delta}_{1}=\int_{0}^{t^{*}}\left\\{\hat{S}_{1}(t)-\hat{S}_{0}(t)\right\\}dt,$
(4.2)

where $\hat{S}_{z}(t)$ is the standard Kaplan-Meier survival curve for
Treatment $z$. The estimator $\hat{\Delta}_{1}$ is consistent with
$\Delta_{t^{*}}^{(P)}$ when the estimated biomarker cutpoint $\hat{c}_{t^{*}}$
is unbiased to $c_{t^{*}}$ and the biomarker distribution of $n_{+}$ patients
aligns with $F_{+}(x)$. We discuss its large sample properties in Web Appendix
B.1.

However, since the proposed design uses a different biomarker cutpoint
$\hat{c}_{0}$ estimated at the end of Stage I for enrichment, the distribution
of the included biomarker-positive patients may not align with the
distribution for $X\in(\hat{c}_{t^{*}},1]$. Specifically, when
$\hat{c}_{0}\leq\hat{c}_{t^{*}}$, the biomarker value of the enriched patients
in Stage II would cover the range of $X\in(\hat{c}_{t^{*}},1]$, while if
$\hat{c}_{0}>\hat{c}_{t^{*}}$, the biomarker values of the enriched patients
represent only a subset of $(\hat{c}_{t^{*}},1]$. In such a situation, the
distribution of those patients’ biomarker would be truncated to
$X\in(\hat{c}_{0},1]$, leading to an overestimation of $\hat{\Delta}_{1}$
under the positive assumption of biomarker-by-treatment interaction. As such,
a weighting approach is needed to balance the distribution of the included
$n_{+}$ biomarker-positive patients against the true distribution of
$F_{+}(x)$.

###  4.2 Calibration Weighted Estimators

We use a calibration weighting (CW) method to balance the biomarker
distribution. This method has been applied in causal inference for
generalizing treatment effects from randomized trials to the target population
[44, 45, 46]. It allows covariate distributions of the samples to empirically
match a specific target population without fitting parametric models for the
weight functions. In our proposed design, we leverage the CW method to
generalize the RMST difference from the included positive patients to the true
biomarker-positive subgroup with distribution $F_{+}(x)$. The calculation of
the calibration weights $\hat{p}_{i}$ is introduced in Web Appendix B.2.

The CW-adjusted estimators of the RMST difference have been proposed by [47].
Based on their approach, we propose four unbiased CW estimators of
$\Delta_{t^{*}}$ in our two-stage adaptive enrichment design: CW Kaplan-Meier
estimator ($\hat{\Delta}_{2}$), CW G-Formula Estimator ($\hat{\Delta}_{3}$),
CW Hajek Estimator ($\hat{\Delta}_{4}$), and CW Augmented Estimator
($\hat{\Delta}_{5}$). The details and large sample properties of each
estimator are provided in Web Appendix B.3.

##  5 Type I Error Rate, Power, and Sample Size

###  5.1 Type I Error Rate and Critical Values

We focus on the approaches that control the type I error rate under the global
null setting such that $h_{0}(t|X)=h_{1}(t|X)$ for any $X\in[0,1]$. As
outlined in Section 2.3, our proposed design incorporates one null hypothesis
$H_{00}$ for testing positive biomarker-by-treatment interaction and two null
hypotheses $H_{01}$ and $H_{02}$ for testing conditional treatment effects. We
discuss the type I error rate controls on these two aspects.

First, we assign $\alpha_{0}$ as the significant level for testing $H_{00}$.
Based on the RMST regression (Equation (3.2)) fitted at the end of Stage II,
we evaluate significant positive biomarker-by-treatment interaction by testing
if $\beta_{3}>0$. As such, the Hypothesis Zero can be rewritten as: $H_{00}$:
$\beta_{3}=0$ vs. $H_{a0}$: $\beta_{3}>0$. The test statistic is defined as
$Z_{\beta_{3}}=\sqrt{n}\hat{\beta}_{3}/\hat{\sigma}_{\beta_{3}}$, where
$\hat{\sigma}_{\beta_{3}}/\sqrt{n}$ is the standard error of
$\hat{\beta}_{3}$. Under $H_{00}$, $Z_{\beta_{3}}\sim N(0,1)$. To control the
type I error rate at $\alpha_{0}$, the critical value for rejecting $H_{00}$
is $q_{0}=\Phi^{-1}(1-\alpha_{0})$, where $\Phi^{-1}(\cdot)$ is the inverse
cumulative distribution function of standard normal distribution.

Second, we define the family-wise type I error rate $\alpha$ for testing the
treatment effect as the probability of rejecting $H_{01}$ or $H_{02}$ under
the global null. We allocate $\alpha_{1}$ and $\alpha_{2}$ to test $H_{01}$
and $H_{02}$, respectively. Since $H_{01}$ and $H_{02}$ are mutually
exclusive, we have $\alpha=\alpha_{1}+\alpha_{2}$. We denote $Z_{l}^{(P)}$ and
$Z_{l}^{(O)}$ as the test statistics when evaluating the treatment effect in
the biomarker-positive and overall patients, respectively. According to the
five treatment effect estimators proposed in Section 4, the test statistics
are defined as
$Z_{l}^{(P)}=\sqrt{n_{+}}\hat{\Delta}_{l}^{(P)}/\hat{\sigma}_{l}^{(P)}$ and
$Z_{l}^{(O)}=\sqrt{n}\hat{\Delta}_{l}^{(O)}/\hat{\sigma}_{l}^{(O)}$, for
$l=1,...,5$. As such, $\alpha_{1}$ and $\alpha_{2}$ are calculated as follows:

$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle P(H_{01}\text{ is
rejected }|H_{00}\text{ is rejected})\times P(H_{00}\text{ is rejected
}|\text{ global null})$ (5.1) $\displaystyle=$ $\displaystyle
P(Z_{l}^{(P)}>q|H_{00}\text{ is rejected})\times P(Z_{\beta_{3}}>q_{0}|\text{
global null})$ $\displaystyle=$ $\displaystyle\pi_{1}\times\alpha_{0},$
$\displaystyle\alpha_{2}$ $\displaystyle=$ $\displaystyle P(H_{02}\text{ is
rejected }|H_{00}\text{ is not rejected})\times P(H_{00}\text{ is not
rejected}|\text{ global null})$ (5.2) $\displaystyle=$ $\displaystyle
P(Z_{l}^{(O)}>q|H_{00}\text{ is not rejected})\times P(Z_{\beta_{3}}\leq
q_{0}|\text{ global null})$ $\displaystyle=$
$\displaystyle\pi_{2}\times(1-\alpha_{0}).$

Here, $q$ is the critical value for testing the conditional treatment effects
in both biomarker-positive and overall patients. We assume a significance
level of $\tilde{\alpha}$ for testing the conditional treatment effects, such
that $q=\Phi^{-1}(1-\tilde{\alpha})$. We let $\pi_{1}$ be the conditional
error rate for $H_{01}$ when $H_{00}$ is rejected and $\pi_{2}$ be the
conditional error rate for $H_{02}$ when $H_{00}$ is not rejected. When
$H_{00}$ is not rejected, $Z_{l}^{(O)}\sim N(0,1)$, thus we have
$\pi_{2}=\tilde{\alpha}$. When $H_{00}$ is rejected, a biomarker is estimated
from the prediction method as introduced in Section 3.1, and we calculate
$\pi_{1}$ through the Monte Carlo method. Taking JAVELIN Lung 200 as an
example, $\pi_{1}$ can be approximated by bootstrapping the data from patients
with docetaxel (the standard care) for both treatment groups. As a result, the
family-wise type I error rate can be controlled at $\alpha$ by adjusting the
values of $q$ and $q_{0}$.

###  5.2 Global Power

To calculate the power of our design, we consider a specific alternative
setting with hazard functions $h_{1}(t|X)$ and $h_{0}(t|X)$. We denote
$H_{a1}$ as an alternative setting such that a positive biomarker-by-treatment
interaction exists and the RMST difference in the positive group is greater
than 0. Once the critical values $q$ and $q_{0}$ are determined in Equations
(5.1) and (5.2), the global power can be calculated as:

Global Power $\displaystyle=$ $\displaystyle P(Z_{l}^{(P)}>q|H_{00}\text{ is
rejected},H_{a1})\times P(Z_{\beta_{3}}>q_{0}|H_{a1})$ $\displaystyle+$
$\displaystyle P(Z_{l}^{(O)}>q|H_{00}\text{ is not rejected},H_{a1})\times
P(Z_{\beta_{3}}\leq q_{0}|H_{a1}).$

###  5.3 Sample Size Calculation

Given the global power and the critical values $q$ and $q_{0}$, we calculate
the sample size using Equation (5.2) and the asymptotic distribution of the
proposed estimators (see Web Appendix B). Let $n_{+}^{*}$ and $n^{*}$
represent the required sample size in the biomarker-positive subgroup and the
overall patients, respectively. Note that $n_{+}^{*}=f(n^{*},c_{t^{*}})$ is a
function of $n^{*}$ and $c_{t^{*}}$ based on the enrichment strategy and
accrual rate in the proposed design. For example, in the all-comer design with
no enrichment, $f(n^{*},c_{t^{*}})=n^{*}(1-c_{t^{*}})$.

Under $H_{a1}$, we define the true RMST difference in the biomarker-positive
subgroup and the overall patients as $\Delta^{(P)}_{t^{*}}$ and
$\Delta^{(O)}_{t^{*}}$, respectively. Then we have:

$\displaystyle\frac{\hat{\Delta}_{l}^{(P)}-\Delta^{(P)}_{t^{*}}}{\hat{\sigma}_{l}^{(P)}/\sqrt{n_{+}^{*}}}\sim
N(0,1)\text{ and
}\frac{\hat{\Delta}_{l}^{(O)}-\Delta^{(O)}_{t^{*}}}{\hat{\sigma}_{l}^{(O)}/\sqrt{n^{*}}}\sim
N(0,1),\text{ for }l=1,\dots,5.$

To test the positive biomarker-by-treatment interaction, we have
$\frac{\hat{\beta}_{3}-\beta_{3}}{\hat{\sigma}_{\beta_{3}}/\sqrt{n^{*}}}\sim
N(0,1)$ under $H_{a1}$. Therefore,
$P(Z_{\beta_{3}}>q_{0}|H_{a1})=1-\Phi(q_{0}-\frac{\beta_{3}}{\hat{\sigma}_{\beta_{3}}/\sqrt{n^{*}}})$.
Let $\eta=P(Z_{\beta_{3}}>q_{0}|H_{a1})$, the Equation (5.2) can be written
as:

$\displaystyle\text{Global
Power}=\eta\left[1-\Phi(q-\frac{\Delta^{(P)}_{t^{*}}}{\hat{\sigma}_{l}^{(P)}/\sqrt{f(n^{*},c_{t^{*}})}})\right]+(1-\eta)\left[1-\Phi(q-\frac{\Delta^{(O)}_{t^{*}}}{\hat{\sigma}_{l}^{(O)}/\sqrt{n^{*}})}\right]$
(5.4)

By solving this equation, we can calculate the required sample size to achieve
the specific global power in our proposed design. The values of
$\Delta^{(P)}_{t^{*}}$ and $\Delta^{(O)}_{t^{*}}$ can be theoretically derived
from the hazard functions $h_{1}(t|X)$ and $h_{0}(t|X)$. Additionally, both
$\sigma_{l}^{(P)}$ and $\sigma_{l}^{(O)}$ can be estimated by Monte Carlo
method as described in [36], details are provided in Web Appendix C.

##  6 A Numerical Example

In this section, we illustrate how a biomarker-guided adaptive trial with
threshold detection and enrichment can be designed using our proposed
methodology by redesigning the JAVELIN Lung 200 study. In the new design, we
choose PFS as the endpoint of interest and use the RMST difference to quantify
the treatment effects, which is justified by notable PH violations observed in
the effect of avelumab. Necessary design parameters and assumptions are made
in the new design to mimic the results reported in [23]. We denote $Z=1$ to
represent avelumab and $Z=0$ for docetaxel, the randomization ratio is one-to-
one. The hazard functions for each treatment are assumed as follows:

$\displaystyle h_{0}(t|X)$ $\displaystyle=$ $\displaystyle 2.5\text{log}(2),$
$\displaystyle h_{1}(t|X)$ $\displaystyle=$
$\displaystyle\begin{cases}6\text{log}(2)\times\text{exp}\\{-0.8X\\}&\text{for
$t\leq 1/6$}\\\ 2\text{log}(2)\times\text{exp}\\{-0.8X\\}&\text{for
$t>1/6$}.\end{cases}$ (6.1)

Here, time is measured in years. We designate the PD-L1 expression on tumor
cells as the biomarker $X$. Similar to in JAVELIN Lung 200, we focus on the
patients with PD-L1 expression $\geq 1\%$ in this design. As such, we assume
$X\sim\text{Unif}(0.01,1)$. As docetaxel is not a biomarker-related therapy,
the hazard function for $Z=0$ is supposed to follow an exponential model and
is unrelated to the biomarker. The hazard function for avelumab is assumed to
be piecewise exponential, and according to Supplemental Figure 3 in [23], we
posit a single change point at 2 months (i.e., $t=1/6$).

We assume patients are uniformly enrolled from time 0 to $t_{1}=0.5$ year in
Stage I. In Stage II, patients are uniformly enrolled from $t_{1}$ to
$t_{2}=1$ year, with restriction in the biomarker-positive subgroup. We assume
equivalent accrual rates across both stages. All patients will be followed up
until $t_{3}=2.5$ years. We assume the lost to follow-up time $L_{i}$ follows
an exponential distribution with a rate $-\text{log}(1-0.05)/2$.

Treatment effect is measured as the RMST difference between avelumab and
docetaxel up to $t^{*}=1.5$ years. Based on Equations (2.2) and (6), the
optimal biomarker cutpoint is $c_{t^{*}}=29.6\%$. Consequently, the marginal
RMST difference among the biomarker-positive subgroup $(c_{t^{*}},1]$ and
overall patients are $\Delta_{t^{*}}^{(P)}=0.137$ and
$\Delta_{t^{*}}^{(O)}=0.082$ years, respectively.

We use Equation (5.4) to calculate the global power across a range of sample
sizes $n^{*}$ under each proposed estimator. Given the equivalent accrual time
and rates in two stages,
$n_{+}^{*}=\frac{n^{*}}{2}(1+\frac{1-c_{t^{*}}}{0.99})$. We plan to control
the family-wise type I error rate at 2.5%, and we allocate $\alpha_{0}=2.5\%$.
The critical values are determined as $q_{0}=\Phi^{-1}(0.975)$ and
$q=\Phi^{-1}(0.977)$ using a Monte Carlo method. The details can be found in
Web Appendix C.

Figure 2 presents the curves of global power across a range of total sample
sizes from the proposed enrichment design using the proposed estimators.
Additionally, we compare the result from an all-comer design with no
enrichment using the Naive unadjusted estimator. Notably, different estimators
employed in the enrichment design yield very similar results. Compared to the
all-comer design, the enrichment design requires fewer sample sizes to achieve
the same global power. For example, to attain 90% of the global power, the
total sample size required in the all-comer design is $n=940$, while in the
enrichment design with the Augmented estimator, it reduces to $n=845$.
Furthermore, within the enrichment design, the G-Formula estimator demands the
smallest sample size to achieve the same global power, while the Hajek
estimator requires the largest sample size.

Figure 2: Curves of the global power by total sample size required in the all-
comer design using naive unadjusted estimator and enrichment design using
Kaplan-Meier (KM), G-Formula (GF), Hajek (HJ), and Augmented (AG) estimators.

##  7 Simulation Studies

We conduct simulation studies to assess the properties of the proposed
biomarker cutpoint estimating methods and treatment effect estimators within
the framework of our proposed two-stage adaptive enrichment design.
Additionally, we seek to evaluate the efficacy of the design and compare it
with the all-comer design with no enrichment. Table 1 describes four design
scenarios in the simulation. Details of the data-generating mechanism,
methods, and performance measures to assess operating characteristics are
presented in Web Appendix D.

Table 1: Design scenarios in the simulation study. Enrichment designs are conducted using different settings in the prediction method to identify $\hat{c}_{0}$ at the end of Stage I. Different settings are regarding whether the number of change points in the piecewise exponential hazard functions and their locations are known or not. |  | Change Points
---|---|---
Design Scenario | Description | Number | Locations
1 | All-comer design without | - | -
| enrichment |  |
2 | Enrichment design using | Known | Known
| Prediction Method A |  |
3 | Enrichment design using | Known | Unknown
| Prediction Method B |  |
4 | Enrichment design using | Unknown | Unknown
| Prediction Method C |  |

###  7.1 Results for Biomarker Cutpoint Estimation

Table 2 summarizes the estimated biomarker cutpoints, $\hat{c}_{0}$ and
$\hat{c}_{t^{*}}$, in Stages I and II under each design scenario. In Stage I,
the Prediction Method A yields the most accurate prediction (bias = 0.001)
since the number of change points and their locations in the piecewise
exponential models are known. However, under Prediction Methods B and C with
less known information on the change points, the bias of $\hat{c}_{0}$
increases. In Stage II, the RMST regression method yields unbiased estimates
of $\hat{c}_{t^{*}}$ in all design scenarios regardless of the choice of the
prediction method in Stage I.

Table 2: Estimated biomarker cutpoints, $\hat{c}_{0}$ and $\hat{c}_{t^{*}}$, in Stages I and II in the simulation study under four design scenarios. $\hat{c}_{0}$ is estimated by the prediction method and $\hat{c}_{t^{*}}$ is estimated by the RMST regression method. S.D.: standard deviation. | Stage I | Stage II
---|---|---
Design Scenario | $\hat{c}_{0}$ |  Bias of $\hat{c}_{0}$ |  S.D. of $\hat{c}_{0}$ | $\hat{c}_{t^{*}}$ |  Bias of $\hat{c}_{t^{*}}$ |  S.D. of $\hat{c}_{t^{*}}$
1 | - | - | - | 0.522 | 0.003 | 0.062
2 | 0.520 | 0.001 | 0.130 | 0.519 | 0.000 | 0.067
3 | 0.506 | -0.013 | 0.132 | 0.520 | 0.001 | 0.066
4 | 0.509 | -0.010 | 0.163 | 0.519 | 0.000 | 0.067

###  7.2 Results for Treatment Effect Estimation

Table 3 displays the estimated RMST differences in the biomarker-positive
subgroup and their coverage probabilities under four design scenarios using
the estimated biomarker cutpoint $\hat{c}_{t^{*}}$. In the all-comer design,
all five estimators are unbiased with coverage probabilities over 98%. In the
enrichment designs, different prediction methods applied in Stage I do not
alter the treatment effect estimations, as Design Scenarios 2-4 give similar
results. Overall, the Naive unadjusted estimator overestimates the treatment
effect, but all CW-adjusted estimators show unbiased results because the
calibration weighting method balances the distribution of the included
biomarker-positive patients to the true distribution of $F_{+}(x)$. The
associated coverage probabilities are around 98%. We also evaluate the
estimated marginal RMST difference using the true biomarker cutpoint
$c_{t^{*}}$, which helps verify the correctness of the proposed variance of
each estimator. The results are presented in eTable 1 of the Supplementary
Materials.

Table 3: Estimated marginal RMST differences in the estimated biomarker-positive subgroup under four simulation design scenarios, using the estimated biomarker cutpoint $\hat{c}_{t^{*}}$. RMSTD: marginal RMST difference in the estimated biomarker-positive subgroup, C.P.: coverage probability. $\hat{\Delta}_{1}$: Naive unadjusted estimator; $\hat{\Delta}_{2}$: CW Kaplan-Meier estimator; $\hat{\Delta}_{3}$: CW G-formula estimator; $\hat{\Delta}_{4}$: CW Hajek estimator; $\hat{\Delta}_{5}$: CW Augmented estimator. Design Scenario | Estimator | Est. RMSTD | Bias of RMSTD | C.P. of RMSTD
---|---|---|---|---
1 | $\hat{\Delta}_{1}$ | 0.135 | 0.001 | 98.5%
(All-Comer) | $\hat{\Delta}_{2}$ | 0.134 | 0.000 | 98.4%
| $\hat{\Delta}_{3}$ | 0.134 | 0.000 | 99.0%
| $\hat{\Delta}_{4}$ | 0.134 | 0.000 | 98.8%
| $\hat{\Delta}_{5}$ | 0.135 | 0.001 | 98.8%
2 | $\hat{\Delta}_{1}$ | 0.144 | 0.010 | 96.5%
(Enrichment) | $\hat{\Delta}_{2}$ | 0.134 | 0.000 | 97.8%
| $\hat{\Delta}_{3}$ | 0.135 | 0.001 | 98.5%
| $\hat{\Delta}_{4}$ | 0.135 | 0.001 | 98.2%
| $\hat{\Delta}_{5}$ | 0.135 | 0.001 | 98.1%
3 | $\hat{\Delta}_{1}$ | 0.143 | 0.009 | 96.8%
(Enrichment) | $\hat{\Delta}_{2}$ | 0.135 | 0.001 | 97.8%
| $\hat{\Delta}_{3}$ | 0.135 | 0.001 | 98.5%
| $\hat{\Delta}_{4}$ | 0.135 | 0.001 | 98.1%
| $\hat{\Delta}_{5}$ | 0.135 | 0.001 | 98.1%
4 | $\hat{\Delta}_{1}$ | 0.145 | 0.010 | 95.4%
(Enrichment) | $\hat{\Delta}_{2}$ | 0.135 | 0.001 | 97.5%
| $\hat{\Delta}_{3}$ | 0.135 | 0.001 | 98.2%
| $\hat{\Delta}_{4}$ | 0.135 | 0.001 | 97.8%
| $\hat{\Delta}_{5}$ | 0.135 | 0.001 | 97.8%

###  7.3 Results for Operating Characteristics

Table 4 summarizes the results from the power analysis. We first compare the
average numbers of included negative patients in each design scenario. The
all-comer design includes 1048 (51.9%) true negative patients on average.
However, the enrichment designs include significantly fewer negative patients.
When less prior information on piecewise exponential hazard functions is used
in the prediction method in Stage I, the average number of true negative
patients increases from 628 (Design Scenario 2) to 653 (Design Scenario 4).

Subsequently, we compare the global power across five estimators. Within each
design scenario, using the CW G-formula estimator, $\hat{\Delta}_{3}$, yields
the largest global power due to its smallest variance, as outlined in eTable
1. Notably, the high global powers from the Naive unadjusted estimator,
$\hat{\Delta}_{1}$, in the enrichment design stem from its overestimation of
the RMST difference (see Table 3). The global powers from the other three CW-
adjusted estimators are similar but lower than those from the CW G-formula
estimator due to their larger variance. When considering the same estimator,
the global power in the all-comer design is smaller than in the enrichment
designs. Within the enrichment designs, the global powers are similar between
using the first and second prediction methods (Design Scenario 2 and 3), and
they are slightly higher than using the third prediction method (Design
Scenario 4).

Table 4: Results from power analysis. Design | Avg. Number of | Power of Using
---|---|---
Scenario | True Negative Patients | $\hat{\Delta}_{1}$ | $\hat{\Delta}_{2}$ | $\hat{\Delta}_{3}$ | $\hat{\Delta}_{4}$ | $\hat{\Delta}_{5}$
1 | 1048 | 83.7% | 80.0% | 89.8% | 77.4% | 77.7%
2 | 628 | 95.8% | 90.2% | 96.7% | 88.9% | 88.8%
3 | 640 | 95.6% | 90.6% | 96.5% | 89.1% | 89.1%
4 | 653 | 95.1% | 88.5% | 96.3% | 86.8% | 86.8%

In eTables 2 and 3 of the Supplementary Materials, we present the type I error
rates for testing Hypothesis Zero and the family-wise type I error rate across
various combinations of nominal significance levels of $\tilde{\alpha}$ and
$\alpha_{0}$ for each estimator under different design scenarios. Our findings
demonstrate that the proposed design and estimators effectively retain a well-
controlled family-wise type I error rate.

##  8 Discussion

We presented a two-stage adaptive RMST design incorporating biomarker
threshold detection and patient enrichment. Our approach involves two methods
to identify the biomarker cutpoint and five estimators of the RMST difference
in the biomarker-positive subgroup. Furthermore, we defined the family-wise
type I error rate and global power in the context of our design and proposed a
method for controlling the family-wise type I error rate under the global null
by numerically determining the critical values for test statistics and a
complementary method for calculating the sample size. We re-designed the
JAVELIN Lung 200 using our design scheme and illustrated the sample size
calculation process. Our findings demonstrated that the proposed enrichment
design offers substantial reductions in the required sample size compared to
the all-comer design while maintaining equivalent global power. Another
important strength is its independence from needing a pre-specification of the
biomarker-positive threshold, particularly for a continuous biomarker.
Therefore, adopting the new design to oncology trials could avoid the
potential failures of betting on an inaccurate biomarker cutpoint at the time
of trial design, which is especially crucial when investigators lack
sufficient data. Our flexible design allows adaptively identifying the
threshold while simultaneously determining the biomarker-positive subgroup and
estimating and testing the treatment effect in the “selected” positive group.
Furthermore, this RMST design applies to designs with time-to-event data when
non-proportional hazards are expected.

Our theoretical work and extensive simulation study demonstrated that the
proposed design and related methods addressed three key questions with
satisfaction. First, in Stage I, the prediction methods can effectively
identify the biomarker cutpoint for enrichment, and the accuracy increases as
more prior information on the piecewise hazard functions becomes available. In
Stage II, the RMST regression method provides an unbiased estimation of the
biomarker cutpoint. Second, the calibration weighting method successfully
balances the distribution of the included biomarker-positive patients with the
true distribution of positive biomarkers. Consequently, the CW-adjusted
estimators are all unbiased in the enrichment designs. The CW G-formula
estimator demonstrates the smallest variance with the largest statistical
power among all estimators. Third, compared to the all-comer design with an
equivalent overall sample size, our proposed enrichment design features
significantly more biomarker-positive patients on average and achieves higher
global power while retaining a well-controlled family-wise type I error rate.

When outlining the proposed design for time-to-event outcomes, we primarily
focused on utilizing the piecewise exponential model for hazard functions.
This model is flexible as we allow an arbitrary number of change points,
enabling us to approximate a wide range of event distributions. Notably, our
proposed RMST regression method for estimating the biomarker cutpoint and the
associated estimators for RMST difference does not necessitate the piecewise
exponential hazard assumption. Additionally, it is crucial to underscore that
our proposed design applies to other types of hazard functions, provided there
is sufficient prior information on the parametric assumptions involved.

We focused on a practical scenario in the clinical trial with time-to-event
endpoints, where the enrichment is implemented early to reduce the inclusion
of biomarker-negative patients. The treatment effects are not tested at the
end of Stage I as the follow-up time in this stage may not be sufficiently
long. However, an additional statistical test can be introduced at the end of
Stage I if the follow-up time exceeds the truncation time $t^{*}$ or if the
RMST difference test uses a different truncation time earlier than $t^{*}$.
Furthermore, the proposed design could be extended with a more complicated
group-sequential RMST design [35, 36]. This extension allows for incorporating
$k\geq 1$ additional interim analysis in Stage II. At each interim analysis,
enrichment can be simultaneously conducted with decisions on early termination
for futility or efficacy. While our proposed design primarily considers a
single truncation time, it is essential to note that challenges may arise when
multiple truncation times for RMSTs are used to define treatment effects. For
example, the multiplicity testing issues due to different treatment effect
estimands, and the difficulties when identifying the biomarker cutpoint for
enrichment under various truncation times. These topics remain for further
research. Furthermore, future research can focus on evaluating the proposed
design’s performance regarding follow-up time and enrollment ratio between
stages, exploring optimal or minimax designs, and extending from a single
biomarker to multiple biomarkers for advanced methods in variable and cutpoint
selection.

## Acknowledgements

Dr. Hong was partially supported by the National Institute of Mental Health
(R01 MH126856) and the Patient-Centered Outcomes Research Institute
(ME-2020C3-21145). Dr. Wang was partially supported by the NCI (P01 CA142538)
and the NIA (R01 AG066883).

## Code

The relevant R code for the methodology, simulation study, and numerical
example is available on is available on Github
(https://github.com/kimihua1995/RMST_DESIGN).

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