# Deep Learning Based RIS Channel Extrapolation with Element-grouping

Shunbo Zhang, Shun Zhang, _Senior Member, IEEE_ , Feifei Gao, _Fellow, IEEE_ ,
Jianpeng Ma, _Member, IEEE_ , Octavia A. Dobre, _Fellow, IEEE_ S. Zhang, S.
Zhang and J. Ma are with the State Key Laboratory of Integrated Services
Networks, Xidian University, Xi’an 710071, P. R. China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>jpmaxdu@gmail.com). F. Gao is with Institute for Artificial Intelligence,
Tsinghua University (THUAI), State Key Lab of Intelligent Technologies and
Systems, Tsinghua University, Beijing National Research Center for Information
Science and Technology (BNRist), Department of Automation,Tsinghua University
Beijing, P.R. China (email: feifeigao@ieee.org). O. A. Dobre is with Faculty
of Engineering and Applied Science, Memorial University, St. John’s NL
AIC-5S7, Canada (e-mail: odobre@mun.ca).

###### Abstract

Reconfigurable intelligent surface (RIS) is considered as a revolutionary
technology for future wireless communication networks. In this letter, we
consider the acquisition of the cascaded channels, which is a challenging task
due to the massive number of passive RIS elements. To reduce the pilot
overhead, we adopt the element-grouping strategy, where each element in one
group shares the same reflection coefficient and is assumed to have the same
channel condition. We analyze the channel interference caused by the element-
grouping strategy and further design two deep learning based networks. The
first one aims to refine the partial channels by eliminating the interference,
while the second one tries to extrapolate the full channels from the refined
partial channels. We cascade the two networks and jointly train them.
Simulation results show that the proposed scheme provides significant gain
compared to the conventional element-grouping method without interference
elimination.

###### Index Terms:

Deep learning, channel extrapolation, reconfigurable intelligent surface,
element-grouping.

##  I Introduction

The increasingly demanding objectives for the sixth generation (6G) wireless
communication networks have spurred recent research activities on novel
wireless hardware architectures [1]. As an artificial planar structure with
integrated electronic circuits, the reconfigurable intelligent surface (RIS)
can be programmed to manipulate an incoming electromagnetic field in a wide
variety of functionalities [2]. The amplitude and phase information at each
RIS element can be adjusted independently, so as to digitally manipulate the
electromagnetic waves and realize signal propagation direction regulation and
in-phase superposition in the three-dimensional space, which will improve the
quality of received signals and the performance of wireless communication [3].
Hence, incorporating RISs in wireless networks has been advocated as a
revolutionary means to transform any naturally passive wireless communication
environment, including the set of objects between a transmitter and a
receiver, to an active one, and has become a hot research topic [1, 2, 3, 4].

To achieve good beamforming performance, RIS usually contains a large number
of elements. However, the more elements are used, the more unknown parameters
exist, which inevitably increases the overhead and complexity of channel
estimation. To tackle this problem, Yang _et al._ proposed an element-grouping
strategy [5]. In the element-grouping strategy, every RIS element in each
group is in the open state, shares the same reflection coefficient and is
assumed to have the same channel state information, which greatly reduces the
pilot overhead and enhances the system’s spectrum efficiency [6]. In [7],
Zheng _et al._ implemented two efficient channel estimation schemes with the
element-grouping strategy. However, the channels related with different RIS
elements cannot be ensured to be identical, and only partial channel
information can be obtained within the element-grouping scheme.

Fortunately, for a given RIS structure, there exists deterministic mapping
between the partial and the full channel information. As mentioned in [8], we
can exploit the universal approximation capability of neural networks (NNs),
adopt deep leaning (DL) to characterize the above mapping, and extrapolate the
full RIS channel information from the partial one. In fact, DL has been
utilized for channel estimation over RIS-assisted systems. In [9], Khan _et
al._ applied DL methods to estimate channels and phase angles in RIS-assisted
wireless communication systems. The trained DeepRIS model is deployed to
estimate channels and phase angles from the received symbols. In [10], Elbir
_et al._ proposed a federated learning (FL) framework for channel estimation
and designed a convolutional NN (CNN) trained on the local datasets of the
users without sending them to the base station (BS). With respect to the
element-grouping scheme, different channels related with distinguished RIS
elements of each group would interfere with each other. The larger the
difference among the channels related with different RIS elements in a group
is, the stronger the interference is. This observation should be carefully
considered within the design of the DL scheme.

In this letter, we consider the acquisition of the cascaded channels in a RIS-
assisted communication system and adopt the element-grouping strategy to
reduce the pilot overhead. Specifically, we investigate the impact of the
channel interference among distinguished RIS elements. We first utilize the
pilot orthogonality to eliminate the interference among different element
groups. After that, we construct a DL-based network to eliminate the
interference within each element group and acquire the refined partial
cascaded channels. With these partial channels, we further adopt an NN to
extrapolate the full cascaded channels. Lastly, we design a joint network
training scheme to optimize the two networks.

_Notations:_ We use lowercase (uppercase) boldface to denote vector (matrix).
$(\cdot)^{T}$, $(\cdot)^{H}$, $\|\cdot\|_{F}$ and $\mathbb{E}\\{\cdot\\}$
represent the transpose, Hermitian, Frobenius norm and expectation,
respectively. $\otimes$ is the Kronecker product operator and $|\mathcal{A}|$
is the number of elements in set $\mathcal{A}$.
$v\sim\mathcal{CN}(0,\sigma^{2})$ means that $v$ follows the complex Gaussian
distribution with zero-mean and variance $\sigma^{2}$. $\mathbf{I}_{N}$ and
$\mathbf{1}_{N}$ represent a $N\times N$ identity matrix and a $N\times 1$
all-one vector, respectively. $\text{diag}\\{\mathbf{x}\\}$ is a diagonal
matrix whose diagonal elements are formed with the elements of $\mathbf{x}$.

##  II System Model

Figure 1: The RIS-assisted uplink communication system.

As shown in Fig. 1, we consider an RIS-assisted communication system where a
single antenna user tries to communicate with the access point (AP) via the
RIS reflection. The AP is equipped with $M$ antennas in the form of a uniform
linear array (ULA) and RIS consists of $N_{v}\times N_{h}=N$ passive
reflection elements in the form of a uniform planar array (UPA), where $N_{h}$
and $N_{v}$ are the number of elements on the vertical and horizontal
dimensions, respectively. Moreover, RIS is connected to a dedicated controller
for dynamically adjusting the individual reflection of each element. Without
loss of generality, in this letter we focus on the uplink communication. Since
the direct link between the user and AP tends to be blocked by possible
obstacles like buildings and human bodies, we only consider the RIS-assisted
link and ignore the direct link. The $M\times N$ uplink channel matrix
$\mathbf{G}=[\mathbf{g}_{1},\mathbf{g}_{2},\ldots,\mathbf{g}_{N}]$ from RIS to
AP is expressed as

$\displaystyle\mathbf{G}=\sum_{p=1}^{P_{g}}\alpha_{g,p}\mathbf{a}_{A}(\theta_{g,p})\mathbf{a}_{R}^{H}(\vartheta_{g,p},\varphi_{g,p}),$
(1)

where $\mathbf{g}_{n}$ denotes the channel between the $n$-th RIS element and
AP, $P_{g}$ is the number of the scattering path from the RIS to AP,
$\alpha_{g,p}$ is the equivalent complex channel gain of the $p$-th path, and
$\mathbf{a}_{A}(\theta_{g,p})\\!\in\\!\mathbb{C}^{M\times 1}$ and
$\mathbf{a}_{R}(\vartheta_{g,p},\varphi_{g,p})\\!\in\\!\mathbb{C}^{N\times 1}$
respectively denote the spatial steering vector of AP and RIS, with
$\theta_{g,p}$, $\vartheta_{g,p}$ and $\varphi_{g,p}$ as the angle of arrival
(AoA) at AP, the azimuth and the elevation angle of departure (AoD) at the
RIS, separately. $\mathbf{a}_{A}(\theta_{g,p})\\!=\\![1,e^{\jmath\frac{2\pi
d}{\lambda}\sin\theta_{g,p}},\cdots,e^{\jmath\frac{2\pi
d}{\lambda}(M-1)\sin\theta_{g,p}}]^{T}$, where $d$ is the antenna spacing and
$\lambda$ denotes the carrier wavelength.
$\mathbf{a}_{R}(\vartheta_{g,p},\varphi_{g,p})=\mathbf{a}_{el}(\varphi_{g,p})\otimes\mathbf{a}_{az}(\vartheta_{g,p},\varphi_{g,p})$,
where $\mathbf{a}_{el}(\varphi_{g,p})=[1,e^{\jmath\frac{2\pi
d}{\lambda}\cos\varphi_{g,p}},\cdots,e^{\jmath\frac{2\pi
d}{\lambda}(N_{v}-1)\cos\varphi_{g,p}}]^{T}$ is the steering vector in the
vertical direction and
$\mathbf{a}_{az}(\vartheta_{g,p},\varphi_{g,p})=[1,e^{\jmath\frac{2\pi
d}{\lambda}\sin\varphi_{g,p}\cos\vartheta_{g,p}},\cdots,e^{\jmath\frac{2\pi
d}{\lambda}(N_{h}-1)\sin\varphi_{g,p}\cos\vartheta_{g,p}}]^{T}$ is the
steering vector in the horizontal direction. The $N\times 1$ uplink channel
$\mathbf{h}=[h_{1},h_{2},\cdots,h_{N}]^{T}$ from user to RIS is

$\displaystyle\mathbf{h}=\sum_{p=1}^{P_{h}}\alpha_{h,p}\mathbf{a}_{R}(\vartheta_{h,p},\varphi_{h,p}),$
(2)

where $h_{n}$ is the channel between the $n$-th RIS element and the user,
$P_{h}$ is the number of the scattering path from the user to the RIS,
$\alpha_{h,p}$ is the equivalent complex channel gain of the $p$-th path, and
$\mathbf{a}_{R}(\vartheta_{h,p},\varphi_{h,p})\in\mathbb{C}^{N\times 1}$ has a
similar definition as $\mathbf{a}_{R}(\vartheta_{g,p},\varphi_{g,p})$.
$\vartheta_{h,p}$ and $\varphi_{h,p}$ represent the azimuth and the elevation
AoA for RIS, respectively.

To estimate the uplink channels, the user transmits a pilot symbol $s_{t}$ to
AP in the $t$-th time slot and RIS dynamically changes the phase shift of each
element in different time slots. The $M\times 1$ received signal is expressed
as

$\displaystyle\mathbf{y}_{t}=\sqrt{P}\mathbf{G}\bm{\Phi}_{t}\mathbf{h}s_{t}+\mathbf{n}_{t},$
(3)

where $P$ is the user transmit power,
$\bm{\Phi}_{t}=\text{diag}\\{e^{\jmath\theta_{1,t}},e^{\jmath\theta_{2,t}},\cdots,e^{\jmath\theta_{N,t}}\\}\in\mathbb{C}^{N\times
N}$ denotes the reflection pattern of the RIS in the $t$-th time slot,
$\theta_{n,t}\in[-\pi,\pi)$ is the phase shift, and
$\mathbf{n}_{t}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{M})$ represents the
$M\times 1$ additive white Gaussian noise (AWGN) vector. The signal-to-noise
(SNR) is defined as $\text{SNR}=P/\sigma^{2}$. Correspondingly, the received
signal (3) can be rewritten as

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=\sqrt{P}\sum_{n=1}^{N}\mathbf{g}_{n}h_{n}s_{t}e^{\jmath\theta_{n,t}}+\mathbf{n}_{t}=\sqrt{P}\sum_{n=1}^{N}\mathbf{a}_{n}x_{n,t}+\mathbf{n}_{t}=\sqrt{P}\mathbf{A}\mathbf{x}_{t}+\mathbf{n}_{t},$
(4)

where
$\mathbf{A}=[\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{N}]=[\mathbf{g}_{1}h_{1},\mathbf{g}_{2}h_{2},\cdots,\mathbf{g}_{N}h_{N}]\in\mathbb{C}^{M\times
N}$ is the cascaded channel matrix and
$\mathbf{x}_{t}=[x_{1,t},x_{2,t},\cdots,x_{N,t}]^{T}=[s_{t}e^{\jmath\theta_{1,t}},s_{t}e^{\jmath\theta_{2,t}},\cdots,s_{t}e^{\jmath\theta_{N,t}}]^{T}\in\mathbb{C}^{N\times
1}$ is the equivalent pilot signal in the $t$-th time slot. In this letter, we
focus on the acquisition of the cascaded channel matrix $\mathbf{A}$.

##  III DL-based Channel Extrapolation Scheme

###  III-A Element-grouping Strategy

Theoretically, we can design an $N\times T$ channel training matrix
$\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{T}]$ that
satisfies $\mathbf{X}\mathbf{X}^{H}=\mathbf{I}_{N}$ to directly estimate
$\mathbf{A}$ with the linear estimator, where $T$ is the time duration of the
pilot signal. Within the Bayesian framework [11], $T\geq N$ must be satisfied
to effectively recover $\mathbf{A}$, which will cause large pilot overhead. To
overcome this bottleneck, we adopt the element-grouping strategy on RIS and
divide $N$ RIS elements into $\widetilde{N}$ groups, as depicted in Fig. 1. In
each group, $K=N/\widetilde{N}$ adjacent elements share a common phase shift
and are assumed to possess the same channel condition. The equivalent pilot
signal in the $t$-th time slot can be expressed as
$\mathbf{x}_{t}=\tilde{\mathbf{x}}_{t}\otimes\mathbf{1}_{K}$, where
$\tilde{\mathbf{x}}_{t}=[\tilde{x}_{1,t},\tilde{x}_{2,t},\cdots,\tilde{x}_{\widetilde{N},t}]^{T}=[s_{t}e^{\jmath\tilde{\theta}_{1,t}},s_{t}e^{\jmath\tilde{\theta}_{2,t}},\cdots,s_{t}e^{\jmath\tilde{\theta}_{\widetilde{N},t}}]^{T}$
and $\tilde{\theta}_{n,t}$ is the common phase shift of the $n$-th element
group. The grouped cascaded channel matrix can be denoted as
$\bar{\mathbf{A}}=\widetilde{\mathbf{A}}\otimes\mathbf{1}_{K}^{T}$, where
$\widetilde{\mathbf{A}}=[\tilde{\mathbf{a}}_{1},\tilde{\mathbf{a}}_{2},\cdots,\tilde{\mathbf{a}}_{\widetilde{N}}]\in\mathbb{C}^{M\times\widetilde{N}}$
and the $n$-th column in $\widetilde{\mathbf{A}}$ is equal to the
$((n-1)K+1)$-th column in $\mathbf{A}$. Then, to estimate $\bar{\mathbf{A}}$,
the minimum time duration of the pilot signal can be reduced from $N$ to
$\widetilde{N}$.

After some tedious operations, we can rewrite (4) as

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=K\sqrt{P}\sum_{n=1}^{\widetilde{N}}\widetilde{\mathbf{a}}_{n}\widetilde{x}_{n,t}+\sqrt{P}\sum_{n=1}^{\widetilde{N}}\left(\mathbf{a}_{(n-1)K+2}-\widetilde{\mathbf{a}}_{n}\right)\widetilde{x}_{n,t}$
$\displaystyle+\cdots+\sqrt{P}\sum_{n=1}^{\widetilde{N}}\left(\mathbf{a}_{(n-1)K+K}-\widetilde{\mathbf{a}}_{n}\right)\widetilde{x}_{n,t}+\mathbf{n}_{t}$
$\displaystyle=K\sqrt{P}\widetilde{\mathbf{A}}\widetilde{\mathbf{x}}_{t}+\sqrt{P}\underbrace{(\mathbf{A}-\bar{\mathbf{A}})}_{\mathbf{V}}\mathbf{x}_{t}+\mathbf{n}_{t},$
(5)

where $\mathbf{V}=\mathbf{A}-\bar{\mathbf{A}}$ represents the channel
interference matrix.

As shown in (III-A), the information of the entire cascaded channel
$\mathbf{A}$ is incorporated in $\widetilde{\mathbf{A}}$ and $\mathbf{V}$,
which are coupled with each other. The interference term $\mathbf{V}$ contains
both the inter-group and intra-group interferences. Let
$\widetilde{\mathbf{X}}=[\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2},\cdots,\tilde{\mathbf{x}}_{T}]$
be the pilot matrix. By leveraging the orthogonality of
$\widetilde{\mathbf{X}}$, i.e.,
$\widetilde{\mathbf{X}}\widetilde{\mathbf{X}}^{H}=\mathbf{I}_{\widetilde{N}}$,
the inter-group interference can be eliminated. However, the intra-group
interference cannot be ignored and would severely impact the accuracy of the
channel acquisition. In this letter, we propose two DL-based networks and
jointly train them to eliminate the interference and extrapolate the cascaded
channel.

###  III-B DL-based Interference Elimination and Channel Extrapolation

Figure 2: The flowchart and architecture of the designed network.

The flowchart and the network architectures are depicted in Fig. 2. In this
letter, both the interference elimination and the channel extrapolation are
similar to the super-resolution in the field of image processing [12], which
try to enhance the visual quality and recover the complete information from
the partial information.

With the perfect knowledge about $\mathbf{V}$ and
$\mathbf{N}=[\mathbf{n}_{1},\mathbf{n}_{2},\cdots,\mathbf{n}_{T}]$ ,
$\widetilde{\mathbf{A}}$ can be written as

$\displaystyle\widetilde{\mathbf{A}}=\frac{1}{K\sqrt{P}}(\mathbf{Y}\widetilde{\mathbf{X}}^{H}-\sqrt{P}\mathbf{V}(\mathbf{I}_{\widetilde{N}}\otimes\mathbf{1}_{K})-\mathbf{N}\widetilde{\mathbf{X}}^{H}),$
(6)

where $\mathbf{Y}=[\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{T}]$ and
$\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{T}]=\widetilde{\mathbf{X}}\otimes\mathbf{1}_{K}$.
In practice, due to the unknown interference term $\mathbf{V}$ and noise term
$\mathbf{N}$ in (6), $\widetilde{\mathbf{A}}$ can not be obtained accurately.
Inspired by the outstanding learning ability of NNs, we propose a DL-based
Interference Elimination Network (IENet) to refine the least square (LS)
estimation of $\widetilde{\mathbf{A}}$, i.e.,
$\widetilde{\mathbf{A}}^{(0)}=\mathbf{Y}\widetilde{\mathbf{X}}^{H}/(K\sqrt{P})$,
through weakening the effect of both $\mathbf{V}$ and $\mathbf{N}$ as

$\displaystyle\widehat{\widetilde{\mathbf{A}}}=f(\widetilde{\mathbf{A}}^{(0)};\bm{\Theta}),$
(7)

where $f(\cdot)$ denotes the refinement function learned by IENet,
$\widehat{\widetilde{\mathbf{A}}}$ is the output that represents the refined
version of $\widetilde{\mathbf{A}}$ and $\bm{\Theta}$ contains the tunable
parameters of IENet.

As shown in Fig. 2, since the DL model can only deal with real values, we add
a pre-processing module to the head of IENet to first separate the real and
imaginary parts of $\widetilde{\mathbf{A}}^{(0)}$ and stack them up on a new
dimension. Thus, the input dimension of IENet is transformed from
$M\times\widetilde{N}$ to $2\times M\times\widetilde{N}$. We construct IENet
with 3 convolutional layers (“Conv_I1”-“Conv_I3” in Fig. 2) and $N_{IR}$
cascaded residual blocks (RBs) that have the same structure. Each RB contains
3 identical dense blocks (DBs); each DB is composed of 5 convolutional layers
(“Conv_D1”-“Conv_D5” in Fig. 2) and the _Leaky ReLU_ 111Leaky ReLU is an
improvement of ReLU; when its input $x\geq 0$, its output is $y=x$, and when
$x<0$, its output is $y=\alpha x$, where $\alpha\in(0,1)$ (LReLU) is adopted
as the activation function for the first 4 layers in one DB. Conv_I1 applies
32 kernels with size 5$\times$5 to extract the features of the input with a
bigger receptive field. All the subsequent convolutional layers in IENet apply
32 kernels with size 3$\times$3 except Conv_I3, which applies 2 kernels with
size 3$\times$3 to output the refined $\widehat{\widetilde{\mathbf{A}}}$.
Inspired by the residual network (ResNet)[13], we add many shortcuts in the
designed networks to avoid the _gradient vanishing_ , _gradient exploding_ and
_network degradation_ in training a deep NN. For example, within both RB and
DB, the result of the _residual_ mapping, i.e., the result learned by the
stacked nonlinear transformations, is scaled down by multiplying a constant
$\beta$ and then added to the result of the _identity_ mapping, i.e., the
input of RB/DB, for a stable network training. To further enhance the network
capacity, within one DB, the feature maps of all previous layers are
empirically concatenated as the input of each subsequent convolutional layer,
which is referred to as the dense connection.

Since the reflection elements of RIS are placed close to each other, the
channels at different elements are highly correlated and it is possible to
infer $\mathbf{A}$ from $\widehat{\widetilde{\mathbf{A}}}$. Therefore, we
construct a DL-based Channel Extrapolation Network (CENet) to extrapolate the
full cascaded channel $\mathbf{A}$ as

$\displaystyle\widehat{\mathbf{A}}=g(\widehat{\widetilde{\mathbf{A}}};\bm{\Xi}),$
(8)

where $g(\cdot)$ denotes the mapping function learned by CENet,
$\widehat{\mathbf{A}}$ is the output that represents the extrapolated version
of $\mathbf{A}$, and $\bm{\Xi}$ contains the tunable parameters of CENet.

TABLE I: Structures of IENet and CENet. |  Layer name |  Output size |  Kernel size |  Activation function
---|---|---|---|---
IENet | Conv_I1 | $32\times M\times\widetilde{N}$ | $5\times 5$ | -
Conv_D1-Conv_D4 | $32\times M\times\widetilde{N}$ | $3\times 3$ | LReLU
Conv_D5 | $32\times M\times\widetilde{N}$ | $3\times 3$ | -
Conv_I2 | $32\times M\times\widetilde{N}$ | $3\times 3$ | -
| Conv_I3 | $2\times M\times\widetilde{N}$ | $3\times 3$ | -
CENet | Conv_C1 | $32\times M\times N$ | $5\times 5$ | -
Conv_D1-Conv_D4 | $32\times M\times N$ | $3\times 3$ | LReLU
Conv_D5 | $32\times M\times N$ | $3\times 3$ | -
Conv_C2 | $2\times M\times N$ | $3\times 3$ | -

As shown in Fig. 2, to keep the dimensions of CENet’s output consistent with
that of $\mathbf{A}$, we expand the last dimension of
$\widehat{\widetilde{\mathbf{A}}}$ from $\widetilde{N}$ to $N$ via an
operation similar to the generation of $\bar{\mathbf{A}}$ in Section III-A. We
utilize 2 convolutional layers (“Conv_C1” and “Conv_C2” in Fig. 2) and
$N_{CR}$ RBs to construct CENet. Similar to Conv_I1 and Conv_I3 in IENet,
Conv_C1 applies 32 kernels with size 5$\times$5 to extract the features of the
input, and Conv_C2 applies 2 kernels with size 3$\times$3 to match the
dimension of the output. Since the structure of the RBs in CENet is the same
as that in IENet, we omit the corresponding description due to space
limitation. Note that $\mathbf{A}=\mathbf{V}+\bar{\mathbf{A}}$; we follow the
structure of ResNet and adopt the residual mapping of CENet to learn
$\mathbf{V}$, which is then added to the identical mapping of CENet to output
the extrapolation result. Finally, we combine the real and imaginary parts of
the resultant in the post-processing module and obtain $\widehat{\mathbf{A}}$.
The structures of IENet and CENet are listed in TABLE I.

###  III-C Training Procedure

Instead of separately training IENet and CENet, we train them simultaneously.
Before training the two networks, we first determine the grouping factor $K$
and then send the pilot signal with length $T$ at the user side. To generate
the dataset, we change the location of the user and collect all the received
signal $\mathbf{Y}$ into a set $\mathcal{Y}$ at the AP side. With the known
pilot matrix $\widetilde{\mathbf{X}}$ and $\mathcal{Y}$, we generate the
network training dataset as $\mathcal{D}$, where $|\mathcal{D}|=N_{tr}$ is the
number of training samples. One sample in $\mathcal{D}$ contains three
matrices and is denoted as
$(\widetilde{\mathbf{A}}^{(0)},\widetilde{\mathbf{A}},\mathbf{A})$, where
$\widetilde{\mathbf{A}}^{(0)}$ is the initial input, and
$\widetilde{\mathbf{A}}$ and $\mathbf{A}$ are the labels of IENet and CENet,
respectively. More details about the dataset generation are presented in
Section IV.

During the training, the tunable parameters $\bm{\Theta}$ and $\bm{\Xi}$ are
obtained by minimizing the mean square errors (MSEs) between the outputs of
the two networks and their corresponding labels. The loss of IENet is written
as

$\displaystyle\mathcal{L}_{I}=\frac{1}{M_{tr}M\widetilde{N}}\sum_{\mu=1}^{M_{tr}}\left\|[\widetilde{\mathbf{A}}]_{\mu}-[\widehat{\widetilde{\mathbf{A}}}]_{\mu}\right\|_{F}^{2},$
(9)

where $[\cdot]_{\mu}$ denotes the $\mu$-th sample in the mini-batch and
$M_{tr}$ is the batch size for training.

The loss of CENet is written as

$\displaystyle\mathcal{L}_{C}=\frac{1}{M_{tr}MN}\sum_{\mu=1}^{M_{tr}}\left\|[\mathbf{A}]_{\mu}-[\widehat{\mathbf{A}}]_{\mu}\right\|_{F}^{2}.$
(10)

The above two losses are added together as
$\mathcal{L}=\mathcal{L}_{C}+\rho\mathcal{L}_{I}$ to jointly update the
parameters of IENet and CENet during training, where $\rho$ is the penalty
multiplier that evaluates the importance of different losses. For the networks
training, the adaptive moment estimation (Adam) [14] algorithm is adopted to
achieve the optimal $\bm{\Theta}$ and $\bm{\Xi}$.

##  IV Numerical Results

To evaluate the performance of the designed networks, we resort to the indoor
scenario ‘I1’ of the DeepMIMO dataset [15], which is widely used in DL
applications for massive MIMO systems. The BS 10 in the ‘I1’ scenario is
adopted as the RIS and is set as a UPA with $8\times 8$ ($N=64$) reflection
elements. The $300$-th user in the $500$-th row is assigned as AP with $M=16$
antennas in the form of a ULA and the users are located in the region from the
1-st row to the 400-th row. We set the carrier frequency as $f_{c}=2.4$ GHz.
For both the RIS and AP, the antenna spacing $d$ is set to $\lambda/2$, with
$\lambda$ as the carrier wavelength. The number of the scattering paths for
both $\mathbf{h}$ and $\mathbf{G}$ is 5. To generate the channel dataset, we
first obtain $\mathbf{h}$, $\mathbf{G}$ and $\mathbf{A}$ with the location of
users and AP. Next, we determine the group size $K$ and pilot length $T$ to
generate $\widetilde{\mathbf{X}}$ and $\mathbf{Y}$. With $\mathbf{A}$,
$\widetilde{\mathbf{X}}$ and $\mathbf{Y}$, we further generate the input-
labels tuple
$(\widetilde{\mathbf{A}}^{(0)},\widetilde{\mathbf{A}},\mathbf{A})$ as a sample
in the dataset. Since each row in the user’s region contains 201 users, the
total number of the cascaded channel samples is $80400$.

TABLE II: Performance Comparison of Different Structures under Different Group
Sizes.

$K$ | Structure
---|---
| IENet+CENet
---
| IENet+CENet
---
no dense connection in DBs
| Cascaded
---
CNN
2 | -31.8776 dB | -30.3112 dB | -25.5706 dB
4 | -29.9801 dB | -25.5156 dB | -18.1910 dB
8 | -26.3107 dB | -23.5888 dB | -17.0309 dB
16 | -22.4233 dB | -19.4551 dB | -14.3472 dB

We employ 90$\%$ of the dataset for training and the rest for validation.
Specifically, to achieve a trade-off between the network performance and the
network training overhead, we adopt $N_{IR}=2$ for IENet and $N_{CR}=4$ for
CENet. For both IENet and CENet, $\beta$ is empirically set as $0.2$. The
total iterations for training is $N_{iter}=300000$ and $\rho=0.1$. We set the
initial learning rate $\eta$ as $1e-3$ and halve it every 50000 iterations.
The Adam optimizer is used for network training with batch size $M_{tr}=16$.
To evaluate the channel extrapolation performance, we use the normalized MSE
(NMSE)
$\mathbb{E}\left\\{\frac{\|\mathbf{A}-\widehat{\mathbf{A}}\|_{F}^{2}}{\|\mathbf{A}\|_{F}^{2}}\right\\}$
as the metric for the designed scheme.

In TABLE II, we compare the extrapolation performance of three different
network structures, i.e., the structure designed in this letter, the structure
without the dense connections in DBs and the cascaded CNN. We set the SNR as
20 dB. For fairness, the three structures have the same number of layers and
the same kernel size at each layer. It can be seen that the dense connections
provide performance gains compared to the structure without the dense
connections. Moreover, both structures with or without the dense connections
are superior to the cascaded CNN, which illustrates the effectiveness of the
designed structures.

Figure 3: NMSE of channel extrapolation vs. network training iteration for
different $K$ values.

Fig. 3 depicts the channel extrapolation performance of the designed jointly
training scheme on the validation set versus the network training iterations.
The SNR is set as 20 dB and 4 element group sizes, i.e., $K=2,4,8,16$, are
investigated. It can be seen that the NMSE decreases with the training
iterations and almost achieves convergence after 200000 iterations.
Furthermore, Fig. 3 shows that the smaller the group size is, the better the
extrapolation performance is; this can be explained, as with a smaller group
size, more partial channel information can be obtained, which reduces the
difficulty of channel extrapolation.

Figure 4: NMSE of channel extrapolation vs. $K$ for different extrapolation
schemes.

Fig. 4 shows the channel extrapolation performance versus $K$ with SNR=10 dB
and 20 dB. We compare the performance of three different schemes.
Specifically, curves marked with “LS” represent the performance that directly
expands the LS estimation, i.e.,
$\widetilde{\mathbf{A}}^{(0)}\otimes\mathbf{1}_{K}^{T}$, as the extrapolated
channel, curves marked with “CENet” represent the performance that adopts
$\widetilde{\mathbf{A}}^{(0)}\otimes\mathbf{1}_{K}^{T}$ as the input of CENet,
and curves marked with “IENet+CENet” represent the performance of the designed
jointly training scheme. It can be seen that due to the intra-group
interference, the performance of LS estimation is poor. Both “CENet” and
“IENet+CENet” can provide huge gains compared to LS estimation, which shows
the powerful extrapolation ability of CENet. Moreover, due to the refinement
of the partial channels, “IENet+CENet” outperforms “CENet”, which verifies the
effectiveness of IENet.

Figure 5: NMSE of channel extrapolation vs. SNR for different $K$ values.

In Fig. 5, we investigate the channel extrapolation performance of the
designed jointly training scheme versus SNR with different $K$ values. As the
SNR increases, the NMSE decreases. The NMSE with smaller $K$ is always lower
than that with larger $K$, which confirms the explanation provided for the
result in Fig. 3.

##  V Conclusion

In this letter, we have analysed the channel interference caused by the
element-grouping strategy and designed two DL-based networks, i.e., IENet and
CENet, to obtain the cascaded channels of RIS. The former aims to obtain the
refined partial channel information from the coarse LS estimation, while the
latter tries to extrapolate the full channel information from the refined
partial one. To achieve enhanced performance, we have cascaded the two
networks and jointly trained them. Simulation results have illustrated that
the proposed scheme provides significant gain compared to the conventional
element-grouping method without interference elimination.

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# Combining Real-World and Randomized Control Trial Data Using Data-Adaptive
Weighting via the On-Trial Score

Joanna Harton1, Brian Segal2, Ronac Mamtani3,
Nandita Mitra1∗, and Rebecca A. Hubbard1∗

11footnotetext: Department of Biostatistics, Epidemiology, and Informatics,
University of Pennsylvania, Philadelphia, PA, USA22footnotetext: Indigo Ag,
Boston, MA, USA22footnotetext: Penn Medicine, Philadelphia, PA,
USA11footnotetext: Co-senior authors††footnotetext: Corresponding author:
Joanna Harton, Department of Biostatistics, Epidemiology, and Informatics, 423
Guardian Drive, Philadelphia, PA 19104, USA. Email:
<EMAIL_ADDRESS>
## Abstract

Clinical trials with a hybrid control arm (a control arm constructed from a
combination of randomized patients and real-world data on patients receiving
usual care in standard clinical practice) have the potential to decrease the
cost of randomized trials while increasing the proportion of trial patients
given access to novel therapeutics. However, due to stringent trial inclusion
criteria and differences in care and data quality between trials and community
practice, trial patients may have systematically different outcomes compared
to their real-world counterparts. We propose a new method for analyses of
trials with a hybrid control arm that efficiently controls bias and type I
error. Under our proposed approach, selected real-world patients are weighted
by a function of the “on-trial score,” which reflects their similarity to
trial patients. In contrast to previously developed hybrid control designs
that assign the same weight to all real-world patients, our approach upweights
of real-world patients who more closely resemble randomized control patients
while dissimilar patients are discounted. Estimates of the treatment effect
are obtained via Cox proportional hazards models. We compare our approach to
existing approaches via simulations and apply these methods to a study using
electronic health record data. Our proposed method is able to control type I
error, minimize bias, and decrease variance when compared to using only trial
data in nearly all scenarios examined. Therefore, our new approach can be used
when conducting clinical trials by augmenting the standard-of-care arm with
weighted patients from the EHR to increase power without inducing bias.

##  1 Introduction

Randomized clinical trials are the gold-standard for testing a new treatment
though there can be some potential disadvantages to running a traditional
clinical trial. Clinical trials for rare diseases can take a long time to
accrue patients due to the rarity of the disease, which makes the clinical
trial and drug approval process take longer. Additionally, if a prior Phase II
trial has shown superiority of the new treatment over the standard treatment,
it may not be ethical to randomize patients in a 1:1 ratio, rather it may be
preferred to use a 2:1 or 3:1 ratio favoring the intervention over the
standard-of-care. This can result in lower power and the inability to detect
an effect even if one truly exists.

It is therefore appealing to consider combining clinical trial data on
patients receiving a novel treatment with data on patients receiving the
control therapy derived from electronic health records (EHR). [1] The appeal
of including external patients receiving the standard-of-care is that it
reduces or even completely eliminates the need to randomize patients to a
control arm in the current trial. [2] In a trial with an external control arm,
all data on the standard-of-care is derived from EHR, while, in a hybrid
control arm external patients receiving the standard-of-care from an EHR are
combined with randomized trial patients in the control arm. [3]

Electronic health records (EHR) contain a vast amount of data that can be
relatively easily leveraged for research. These data are by nature
observational, and, as such, there are a few key features of EHR data that are
worth noting. First, EHR were developed for clinical care and billing
purposes. Therefore, some of the information that researchers may be
interested in may not be collected or may be contained in narrative text
notes, which are difficult to analyze. Furthermore, the healthcare system in
the United States, as well as in many other countries, is fragmented such that
a patient’s medical records may be spread across the databases of multiple
healthcare systems, which can result in an incomplete or inadequate picture of
a patient’s health when relying on data from an individual EHR database.

Though conducting research with EHR data has distinct challenges, approaches
have been developed to make beneficial use of this data source.

In 1976, Pocock proposed a method to combine randomized patients receiving the
standard-of-care from historical clinical trials with intervention arm
patients from a new trial to address the fact that many studies of the day
examining the efficacy of a new treatment did not contain a randomized control
arm, which made it difficult to draw causal conclusions. [4] The approach
represents a hybrid control arm combining patients randomized to the trial
control arm with patients from the control arm of a historical trial. Pocock
proposed six criteria for evaluating what constitutes an acceptable historical
control arm as well as how much weight to assign to historical control
patients relative to randomized control patients. [4]

Pocock’s six criteria can also be applied to the use of EHR data to construct
the hybrid control arm of a trial with varying success. These six proposed
guidelines along with considerations for application to an EHR-derived hybrid
control arm are presented below:

  1. 1.

_“Such a group must have received a precisely defined standard treatment which
must be the same as the treatment for the randomized controls.”_ Patients
derived from EHR data can be selected such that they are receiving the same
primary treatment as the randomized trial control patients. However,
supportive care and the care environment may differ between the trial and
routine clinical practice.

  2. 2.

_“The group must have been part of a recent clinical study which contained the
same requirements for patient eligibility.”_ By definition the EHR patients
are not part of a recent clinical trial but efforts should be made to identify
an EHR-derived cohort using eligibility criteria as similar to the trial as
possible. Due to limitations of EHR data capture, it may be difficult to apply
all clinical trial inclusion/exclusion criteria. [5]

  3. 3.

_“The methods of treatment evaluation must be the same.”_ This criterion may
or may not be met, depending on the outcome and method of outcome
ascertainment used in the trial. While it is imperative for the outcome
measure to be the same, outcome ascertainment may differ between the trial and
EHR data capture. For example, even an outcome such as death can have
sensitivity between $30\%$ and $90\%$, meaning that there are deaths that are
not recorded in the EHR. [6]

  4. 4.

_“The distributions of important patient characteristics in the group should
be comparable with those in the new trial.”_ Patients receiving the control
treatment in routine care may differ in many ways from patients participating
in a trial. The requirement of the same distribution of patient
characteristics in the EHR patient pool as the trial patients is possible but
highly unlikely to be exact; similarity, however, should be striven for.

  5. 5.

_“The previous study must have been performed in the same organization with
largely the same clinical investigators.”_ This criterion is not able to be
met by definition.

  6. 6.

_“There must be no other indications leading one to expect differing results
between the randomized and historical controls.”_ This is unlikely to be met
completely due to many differences between real-world and clinical trial care.
However, if the EHR data is contemporaneous with the current trial it is
reasonable to assume that care received by the EHR patients is similar to the
care received by the trial control patients.

Overall, if an EHR cohort can be constructed such that the patients are
contemporaneous with the clinical trial, the same inclusion/exclusion criteria
are applied to ensure comparable patient populations are under study, EHR
control patients receive the same treatment as the clinical trial control arm
patients, and the outcome and method of outcome ascertainment are as similar
between EHR data capture and the trial as possible, it may be appropriate to
use EHR data as part of a hybrid control arm.

There are several approaches to using external control patients when
estimating treatment effects, including Pocock’s approach which was among the
first. Although these methods were developed in the context of historical
controls, where control patients are drawn from a previously conducted
clinical trial, they can also be applied to the case where control patients
are drawn from an EHR database. This does not affect the implementation of the
methods, though interpretation must be made carefully. We assume throughout
this paper that only patients in the trial receive the intervention therapy.
Pocock’s method assumes that the parameter of interest is the difference
between the mean outcome in the intervention arm patients and the mean outcome
in the standard-of-care arm patients in the clinical trial. Pocock’s method
also assumes that the true mean value for the trial standard-of-care patients
follows a normal distribution centered at a weighted sum of the sample mean of
the trial standard-of-care patients and the sample mean of the external
standard-of-care patients, where weights are selected based on the extent to
which external standard-of-care patients are believed to be representative of
trial standard-of-care patients and with a standard deviation also dependent
upon these factors. [4]

Chen, et al. introduced a Bayesian approach to estimating the effect of a
treatment when using hybrid control arms that relies on power priors. [7, 8]
This approach incorporates external standard-of-care arm data with the current
trial data by taking only a fraction of the information from each external
standard-of-care patient. [7, 8] The power prior can estimate many different
estimands, including differences in means or proportions, hazard ratios, or
odds ratios. In this method, the pool of external standard-of-care patients is
weighted as a whole and the external standard-of-care patients are assigned
anywhere from $0\%$ to $100\%$ of the weight that a current trial participant,
whether intervention or standard-of-care, receives in the final model. [7, 8]
When $\alpha$, the weight assigned to external standard-of-care patients, is
$0$, the power prior approach is equivalent to using no data from the external
standard-of-care patients, and when $\alpha$ is $1$, the power prior approach
is the same as fully pooling the external standard-of-care patients with the
current trial data. [7, 8] This method may be more interpretable than Pocock’s
method as the amount of information incorporated is quantified directly
through $\alpha$ rather than through a variance parameter. [7, 8] Similar to
Pocock’s method, the amount of information incorporated from the external
standard-of-care patients must be prespecified by the researcher and
sensitivity analyses are recommended to determine the robustness of results to
choice of $\alpha$. [7, 8, 4]

Duan and Ye in 2008 [9] and Neuenschwander, et al. in 2009 [10] concurrently
developed the normalized power prior approach, which extended the power prior
model to estimate $\alpha$ from the data rather than using a prespecified
$\alpha$. [9, 10] The normalized power prior approach allows for more weight
to be allocated to the external standard-of-care patients when the external
standard-of-care patients are similar in terms of outcome to the current trial
patients and less weight when they are dissimilar. [9, 10]

These earlier methods focused on weighting external standard-of-care patients
as a group, whether the weight is pre-specified or data-driven. More recent
extensions have considered individual-weighting of external standard-of-care
patients based on the similarity of each individual to patients in the trial
standard-of-care arm. One such adaptation of the power prior approach proposes
dividing the external standard-of-care patients into subgroups based on their
similarity to the trial patients and assigning a weight to each subgroup. [11]
Another recently proposed method uses a modification of the propensity score,
called the on-trial score, to create matches between the external standard-of-
care patients and the trial standard-of-care patients in order to create a
hybrid standard-of-care arm consisting of patients most similar to those in
the clinical trial intervention arm. [12]

In this paper, we propose a new data-adaptive weighting method that addresses
the limitation of assigning a single weight to the entire group of external
standard-of-care patients by assigning weights to each individual in the
external standard-of-care arm based on similarity to trial patients using the
on-trial score. The use of individualized weights helps to account for the
fact that patients included in EHR databases may be more heterogeneous than
patients included in clinical trials. The proposed approach incorporates more
information from standard-of-care patients who are more similar to trial
participants than those who are not.

The structure of the paper is as follows: in section 2 we define notation,
outline the existing approaches used for hybrid standard-of-care arms and
introduce our proposed method for combining trial standard-of-care patients
with external standard-of-care patients. In section 3, simulations are
presented to assess the relative performance of our new method compared to
existing methods. Section 4 applies all methods discussed to a clinical study
for patients with metastatic castration-resistant prostate cancer comparing
the standard-of-care treatment of prednisone with the new treatment of
abiraterone acetate in conjunction with prednisone with with external
standard-of-care patients from a pseudo EHR, and section 5 provides a summary
and discussion.

##  2 Methods

###  2.1 Notation

We assume the existence of a trial of size $N_{Trial}$ and an external data
source, e.g. an EHR database consisting of patients meeting comparable
inclusion/exclusion criteria and receiving the same treatment as trial
standard-of-care arm patients, of size $N_{External}$. Let
$N=N_{Trial}+N_{External}$. Let each patient in each database have information
on a set of $k$ covariates,
$\mathbf{X}_{Nxk}=\\{\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{k}\\}$.
Additionally, let $\mathbf{R}$ be an indicator such that $\mathbf{R}_{i}$
takes the value of 1 if the $i^{th}$ patient is in the external data source
and 0 if the $i^{th}$ patient is enrolled in the trial. Similarly, let
$\mathbf{T}$ be a treatment indicator such that $\mathbf{T}_{i}$ takes a value
of 0 if the $i^{th}$ patient receives the standard-of-care and 1 if the
$i^{th}$ patient receives the intervention. In our numerical experiments
below, we a have a time to event outcome variable,
$\mathbf{Y}=min(\mathbf{F},\mathbf{C})$, where $\mathbf{F}$ is the time of the
event of interest (failure time), $\mathbf{C}$ is the censoring time. Also,
let $\mathbf{S}$ be a status indicator where $\mathbf{S}_{i}$ takes a value of
1 if $\mathbf{T}_{i}<\mathbf{C}_{i}$ and 0 otherwise. Extensions to outcomes
of other variable types follow directly from the likelihood-based formulation
below. We denote the data available for the set of external standard-of-care
patients as
$\mathbf{D_{0}}=\\{\mathbf{X},\mathbf{S},\mathbf{Y}|\mathbf{R}_{i}=1\\}$, the
set of randomized standard-of-care patients as
$\mathbf{D_{C}}=\\{\mathbf{X},\mathbf{S},\mathbf{Y}|\mathbf{R}_{i}=0,\mathbf{T}_{i}=0\\}$,
the set of randomized intervention arm patients as
$\mathbf{D_{T}}=\\{\mathbf{X},\mathbf{S},\mathbf{Y}|\mathbf{R}_{i}=0,\mathbf{T}_{i}=1\\}$,
and the set of all trial patients as
$\mathbf{D}=(\mathbf{D_{T}},\mathbf{D_{C}})$. We also let $\theta$ denote a
target parameter of interest that represents treatment efficacy which could be
parameterized as a difference in mean event times or hazard ratios comparing
intervention arm and standard-of-care patients.

Below we summarize existing and proposed approaches to incorporating data from
external standard-of-care patients into an analysis of treatment efficacy.

###  2.2 Existing Approaches to Hybrid Control Trials

####  2.2.1 Naïve Approaches

Ignoring the external data and using only the current trial data serves as a
positive control with regards to the minimum bias that can be attained when
estimating the parameter of interest. In this method, only the data from the
patients enrolled in the trial are analyzed and the patients from the external
data source are left out.

Fully pooling the external data with the trial data serves as a negative
control with regards to the amount of bias that is likely to occur when
estimating the treatment effect. In this case, the external patients are given
the same weight as the trial patients and all external patients are included
in the analysis.

####  2.2.2 Bayesian Approaches

The power prior (PP) approach combines the external patients with the trial
patients such that each external patient has a weight less than 1. [8] The
power prior approach assigns the same weight, $\alpha$, such that
$0<\alpha<1$, to all patients in the external data source and the value of
$\alpha$ is prespecified by the researcher. The power prior approach proposes
the following prior distribution for $\theta$:
$\pi(\theta|{\mathbf{D_{0}}},\alpha)\propto\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)$
which yields the following posterior distribution for $\theta$:
$\pi(\theta|{\mathbf{D_{0}}},\mathbf{D},\alpha)\propto\mathscr{L}(\theta|\mathbf{D})[\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)]$.

The normalized power prior approach (NPP) is similar to the power prior
approach except that $\alpha$ is estimated from the data rather than being
pre-specified by the researcher. [9] The normalized power prior approach
specifies a conditional prior for $\theta$ given $\alpha$ and a marginal
distribution for $\alpha$. The normalized power prior approach has the form:

$\pi(\theta,\alpha|{\mathbf{D_{0}}})\propto\frac{\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)}{\int_{\Theta}\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)d\theta}\pi(\alpha),$

which results in the following posterior:

$\pi(\theta,\alpha|{\mathbf{D_{0}}},\mathbf{D})\propto\mathscr{L}(\theta|\mathbf{D})\pi(\theta,\alpha|{\mathbf{D_{0}}})\propto\frac{\mathscr{L}(\theta|\mathbf{D})\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)\pi(\alpha)}{\int_{\Theta}\mathscr{L}(\theta|{\mathbf{D_{0}}})^{\alpha}\pi(\theta)d\theta}.$

If $\pi(\alpha)$ is proper then the normalized power prior will also be
proper.

####  2.2.3 Lin’s Method

Lin’s method uses an on-trial score, similar to a propensity score, where the
outcome of interest is inclusion in the trial to construct a matched set of
external standard-of-care patients and weight their likelihood contribution.
[12] The on-trial score is estimated as the probability that a patient is in
the clinical trial given their baseline covariates using a logistic regression
model. Next, optimal pair matching is performed using the on-trial score so
that each trial patient receiving the intervention is matched with an external
standard-of-care patient. The selected external standard-of-care patients form
a pool from which $N_{T}-N_{C}$ patients are randomly drawn so that the
augmented trial has a 1:1 ratio of treated to standard-of-care patients. In
the outcome model the external standard-of-care patients are weighted by their
on-trial scores while the trial patients are given a weight of one. [12]

###  2.3 Proposed Approach: Data-Adaptive Weighting

In our proposed approach, we let the on-trial score be defined as the
probability that the $i^{th}$ patient is included in the trial given the
observed baseline covariates, $P(R_{i}=0|\mathbf{X}_{i})=e(\mathbf{X}_{i})$.
To maximize the similarity between external and randomized standard-of-care
patients, we then limit the set of external standard-of-care patients to the
subset with the highest on-trial scores, such that the number of external
standard-of-care patients selected results in a hybrid control arm of the same
size as the intervention arm. Let $D^{*}_{0}$ represent data for the subset of
$D_{0}$ with the $N_{D_{T}}-N_{D_{C}}$ largest on-trial scores. The on-trial
scores are then transformed to obtain values for $\hat{\gamma}_{i}$ such that
$\hat{\gamma}_{i}=e(\mathbf{X}_{i})/(1-e(\mathbf{X}_{i}))$ and standardized
such that
$\hat{\gamma}_{i}^{*}=\hat{\gamma}_{i}N_{\mathbf{D^{*}_{0}}}/\sum_{i=1}^{N_{\mathbf{D^{*}_{0}}}}\hat{\gamma}_{i}$.
The inverse odds weight is used as we are interested in the average treatment
effect on the treated, or in this case, the average treatment effect for those
on-trial. This weighting method assigns all trial patients their full weight
and only up- or down-weights the selected external standard-of-care patients.

Estimation for data-adaptive weighting then uses a prior for $\theta$ of the
form:

$\pi(\theta|\bm{\hat{\gamma}^{*}},{\mathbf{D^{*}_{0}}})\propto\prod_{i=1}^{N_{\mathbf{D^{*}_{0}}}}\big{[}\mathscr{L}(\theta|{{\mathbf{D^{*}_{0}}}_{i}})^{\hat{\gamma}^{*}_{i}}\big{]}\pi(\theta),$
(1)

which gives the following posterior:

$\pi(\theta|\bm{\hat{\gamma}^{*}},{\mathbf{D^{*}_{0}}},\mathbf{D})\propto\prod_{j=1}^{N_{\mathbf{D}}}\big{[}\mathscr{L}(\theta|\mathbf{D}_{j})\big{]}\prod_{i=1}^{N_{\mathbf{D^{*}_{0}}}}\big{[}\mathscr{L}(\theta|{{\mathbf{D^{*}_{0}}}_{i}})^{\hat{\gamma}^{*}_{i}}\big{]}\pi(\theta)$
(2)

We note that all patients are used in the estimation of the on-trial score as,
assuming trial patients are randomly assigned to intervention arm, the
distribution of baseline covariates is the same for standard-of-care arm and
intervention arm patients. Here the on-trial score is estimated via a logistic
regression, rather than being jointly estimated with $\theta$. However, the
on-trial score may be estimated using more flexible modeling such as a random
forest or ensemble machine learning if desired.

###  2.4 Estimation

The Bayesian estimation approach for the models presented above, under certain
conditions, can be approximated using a frequentist analog. For example, in
the case of the DAW method, when a non-informative prior is used for $\theta$,
the prior for DAW,
$\pi(\bm{\theta}|\mathbf{D_{0}},\bm{\hat{\gamma}^{*}})\propto\big{[}\mathscr{L}(\bm{\theta}|\mathbf{D_{0}})^{\hat{\gamma}_{i}^{*}}\big{]}\pi(\bm{\theta})$,
is equivalent to
$\pi(\bm{\theta}|\mathbf{D_{0}},\bm{\hat{\gamma}^{*}})\propto\mathscr{L}(\bm{\theta}|\mathbf{D_{0}})^{\hat{\gamma}_{i}^{*}}$.
In this case the posterior mode corresponds to the maximum likelihood
estimator for a weighted parametric survival model. In this case, estimates
from a weighted Cox proportional hazards model with weights of 1 for trial
patients and $\bm{\hat{\gamma}^{*}}$ for selected external standard-of-care
patients will provide similar estimates to the Bayesian estimates with flat
priors. This approach may be preferable to a fully Bayesian estimation
approach because of its relative computational efficiency and insensitivity to
prior specification. In numerical experiments below, we investigate
performance of estimation using this weighted Cox approach for all methods
described above.

Our proposed method, data-adaptive weighting (DAW), builds on the power prior
approach and Lin’s approach. While the power prior approach and normalized
power prior approach both use the same $\alpha$ value for all patients, DAW
uses individual weights for the external patients depending on their
similarity to the trial patients. Lin’s approach uses individual weights for
each of the external patients. However, the selected external patients are
based upon matching on the on-trial score and selected subjects are directly
weighted by the on-trial score.

##  3 Simulation Study

We conducted a simulation study to investigate the bias, efficiency, effective
sample size, and type I error of the existing methods outlined above and the
data-adaptive weighting method. Data were simulated with the objective of
generating simulated data resembling a real-world study using trial data and
external standard-of-care data from an EHR database.

Data were simulated for four covariates ($\bm{X}$), a real-world indicator
($\bm{R}$), a treatment indicator ($\bm{T}$), a failure time ($\bm{T}$), and a
censoring time ($\bm{C}$). We simulated data for trials of two different sizes
($100,1000$), each with two different randomization ratios of the number of
patients in the intervention arm to the number of patients in the standard-of-
care arm (2:1 and 3:1). The number of external standard-of-care patients
available was equal to the number of trial patients. These values were
selected to mirror real-world scenarios for unbalanced clinical trials and
provide enough potential EHR patients to distinguish between the performance
of the various methods. The hazard ratio for failure for patients in the
intervention arm versus randomized standard-of-care patients (treatment
effect) was allowed to take values of $0.5,0.75,0.875,$ and $1$. Two different
strengths of confounding of the relationship between baseline covariates and
failure were also explored: mild confounding and strong confounding. We note
that these covariates confound the relationship between enrollment in the
trial and the outcome because the baseline covariate distribution differs
between trial and external standard-of-care patients. Analyses limited to the
trial population are unconfounded because there is no relationship between
trial arm and baseline covariates due to randomization. Censoring rate was
held constant across all simulation scenarios. External standard-of-care
patients had censoring times arising from an exponential distribution with
rate 0.4 and trial patients had censoring times arising from an exponential
distribution with rate 0.1.

Relationships among the simulated variables and the complete set of
distributions and parameter values used to simulate data, along with examples
of EHR-derived covariates used to motivate the simulation study are provided
in Table 1.

Table 1: Data generation scheme and parameter values used for simulation study. Variable | Trial Distribution | External Distribution | Analogous Variable
---|---|---|---
$X_{1}$ | $\text{Bernoulli}(0.5)$ | $\text{Bernoulli}(0.55)$ | Gender
$X_{2}$ | $\text{Bernoulli}(0.6)$ | $\text{Bernoulli}(0.4)$ | College Degree
$X_{3}$ | $\text{Normal}(60,5)-60$ | $\text{Normal}(60,10)-60$ | HDL Cholesterol
$X_{4}$ | $\text{Normal}(21,2)-21$ | $\text{Normal}(23,2)-21$ | BMI
T |  $\text{Bernoulli}(t)$, $t=\\{0.67,0.75\\}$ | 0 | Treatment Indicator
$Y_{failure}$ |  | $\text{Exp}[log(\eta)T+log(\bm{\beta})\bm{X}]$
---
Failure Time
| $\eta=$
---
$\bm{\beta}=$
.
| $\\{0.5,0.75,0.875,1\\}$
---
$\\{(1.25,0.67,0.98,1.06),$
$(2.25,0.4,0.93,1.21)\\}$
$Y_{censor}$ | $\text{Exp}(0.1)$ | $\text{Exp}(0.4)$ | Censoring Time

In numerical examples, the on-trial score was estimated using logistic
regression including all baseline covariates as predictors. Weighted Cox
proportional hazards models with a treatment indicator as the sole covariate
were fit as the outcome model to estimate the treatment effect.

Each simulation scenario was repeated 1,000 times. We estimated bias relative
to the true marginal treatment effect, empirical variance, 95% confidence
interval coverage probabilities, power (for scenarios with non-null treatment
effects), and type I error (for scenarios with a null treatment effect). Power
and type I error are estimated for hypothesis testing using a significance
threshold of 0.05.

We first examined the performance of alternative methods as we varied the
ratio of patients in the intervention arm to standard-of-care patients in the
trial. In these analyses neither the proportion of trial patients in the
intervention arm nor the number of EHR patients available as a function of the
number of trial patients affected the pattern of the results. Therefore, only
a 2:1 intervention arm to standard-of-care arm ratio with the same number of
EHR patients available as trial patients are presented here. Sample Kaplan-
Meier curves for each treatment hazard ratio and confounding level combination
show the difference in survival over time across the three groups of patients
(Supplemental Figure S1).

###  3.1 Simulation Results

Figure 1: Bias for methods when applied to a trial with a 2:1 randomization
ratio. PP = power prior, NPP = normalized power prior, DAW = data-adaptive
weighting. Figure 2: Variance for methods when applied to a trial with a 2:1
randomization ratio. PP = power prior, NPP = normalized power prior, DAW =
data-adaptive weighting. Figure 3: Power for methods when applied to a trial
with a 2:1 randomization ratio. $80\%$ power is marked with a dotted line. PP
= power prior, NPP = normalized power prior, DAW = data-adaptive weighting.
Figure 4: Coverage of the true treatment effect hazard ratio when a 95$\%$
confidence interval is used for methods when applied to a trial with a 2:1
randomization ratio. 95$\%$ coverage is marked with a dotted line. PP = power
prior, NPP = normalized power prior, DAW = data-adaptive weighting. Figure 5:
Zoomed-in view of coverage of the true treatment effect hazard ratio when a
95$\%$ confidence interval is used for methods when applied to a trial with a
2:1 randomization ratio. 95$\%$ coverage is marked with a dotted line. PP =
power prior, NPP = normalized power prior, DAW = data-adaptive weighting.

The results of the simulation study show that fully pooling the data from the
available EHR patients with the trial data results in large biases under all
conditions examined (Figure 1). As expected, using only the trial patient data
results in negligible bias. The power prior method was also substantially
biased across all three $\alpha$ values explored, with $\alpha=0.25$ having
the least bias and $\alpha=0.75$ having the most bias (Figure 1). The
normalized power prior was biased when the trial size was 100 but displayed
minimal bias when the trial size was 1,000. This is explained by the fact that
$\hat{\alpha}$ was estimated to be approximately 0.22-0.36 for the trial size
of 100 and 0.01-0.03 for the trial size of 1,000 (Table 4). Therefore,
although the normalized power prior approach performed well in terms of bias
this is because little information from the EHR patients was incorporated.
Both Lin’s method and DAW displayed low bias across all scenarios examined.
DAW had consistently lower bias than Lin’s method (Figure 1).

The variances of the estimates reflect the extent to which data from external
standard-of-care subjects was incorporated. The trial only approach had the
largest variance and full pooling of all EHR patients with the trial patients
had the smallest variance (Figure 2). The power prior approach had variance
between the trial only and full pooling methods, with variance inversely
proportional to $\alpha$. The normalized power prior had substantially smaller
variance than the trial only approach when the trial size was 100 due to the
larger $\hat{\alpha}s$ (Figure 2, Table 4). DAW had smaller variance than
Lin’s method under all conditions examined. This is due to the larger
effective sample size for any given scenario (Figure 2, Table 3).

DAW was able to achieve the targeted 1:1 intervention to standard-of-care
ratio, while Lin’s method resulted in substantially smaller effective sample
sizes. DAW includes about twice as many external patients as Lin’s method does
in the scenarios examined (Table 3).

The full pooling and power prior methods had high power due to their large
effective sample sizes and biased treatment effect estimated which were biased
away from the null; however, had the covariate effects been in the other
direction the bias would have been towards the null and these methods would
have had lower power (Figures 3, 1). The normalized power prior approach had
higher power relative to the trial only method when the trial size was 100 but
not when the trial size is 1,000. Lin’s method and DAW had higher power than
the trial only approach and were quite similar to one another when the trial
size was 100; Lin’s method had slightly higher power than DAW when the trial
size was 1,000 (Figure 3).

Clearly, unless the strength of confounding and the trial size are small, full
pooling of all EHR patients or using the power prior with one of the $\alpha$
values examined provides very poor coverage of the true HR (Figure 4). NPP
only provides nominal coverage when the trial size is 1,000. Both Lin’s method
and DAW had nominal coverage in all scenarios except under strong confounding
when the trial size was 1,000; in that case Lin’s method had approximately
$93-94\%$ coverage and DAW had $94\%$ coverage except when the marginal
treatment ratio is nearly 1, in which case the coverage dropped to $9\%$
(Figures 4, 5).

Type I error was controlled at the 5% level when only the trial data were
analyzed, and was poorly controlled under the full pooling and power prior
methods (Table 2). As expected, NPP controlled type I error when
$\hat{\alpha}$ was small (i.e., sample size of 1,000) and when there was mild
confounding (Tables 2, 4). Both Lin’s method and DAW controlled type I error
under all scenarios examined except when there was strong confounding and a
trial size of 1,000; type I error was slightly inflated in this case to around
$6\%$ (Table 2).

Table 2: Type I error for trials with a 2:1 randomization ratio. PP = power
prior, NPP = normalized power prior, DAW = data-adaptive weighting.

| Mild Confounding | Strong Confounding
---|---|---
| Trial = 100 | Trial = 1,000 | Trial = 100 | Trial = 1,000
Trial Only | 0.05 | 0.051 | 0.052 | 0.046
Full Pooling | 0.126 | 0.716 | 0.356 | 0.999
PP, $\bm{\alpha=0.25}$ | 0.053 | 0.177 | 0.098 | 0.619
PP, $\bm{\alpha=0.5}$ | 0.069 | 0.403 | 0.192 | 0.953
PP, $\bm{\alpha=0.75}$ | 0.097 | 0.583 | 0.287 | 0.996
NPP | 0.061 | 0.061 | 0.117 | 0.047
Lin | 0.049 | 0.046 | 0.044 | 0.060
DAW | 0.052 | 0.048 | 0.050 | 0.059

Table 3: Effective sample size for trials with a 2:1 randomization ratio. PP =
power prior, NPP = normalized power prior, DAW = data-adaptive weighting

| Mild Confounding | Strong Confounding
---|---|---
| Trial = 100 | Trial = 1,000 | Trial = 100 | Trial = 1,000
Trial Only | 100 | 1000 | 100 | 1000
Full Pooling | 200 | 2000 | 200 | 2000
PP, $\bm{\alpha=0.25}$ | 125 | 1250 | 125 | 1250
PP, $\bm{\alpha=0.5}$ | 150 | 1500 | 150 | 1500
PP, $\bm{\alpha=0.75}$ | 175 | 1750 | 175 | 1750
NPP | 135 | 1033 | 123 | 1013
Lin | 116 | 1166 | 116 | 1166
DAW | 134 | 1340 | 134 | 1340

Table 4: Mean $\alpha$ (alpha) values for normalized power prior approach for
trials with a 2:1 randomization ratio. PP = power prior, NPP = normalized
power prior, DAW = data-adaptive weighting

| Mild Confounding | Strong Confounding
---|---|---
| Trial = 100 | Trial = 1,000 | Trial = 100 | Trial = 1,000
HR: 0.5 | 0.36 | 0.03 | 0.22 | 0.01
HR: 0.75 | 0.36 | 0.03 | 0.23 | 0.01
HR: 0.875 | 0.36 | 0.03 | 0.24 | 0.01
HR: 1 | 0.35 | 0.03 | 0.23 | 0.01

Note: Hazard ratios listed are the conditional treatment hazard ratios as
opposed to the marginal treatment hazard ratios.

##  4 Case Study: Metastatic Castration-Resistant Prostate Cancer

Table 5: Characteristics of patients in the Janssen MCRPC cohort by data
source

|  | Clinical Trial
---|---|---
|  | Pseudo
---
EHR
| Standard-of-Care
---
Arm
| Intervention
---
Arm
| N = 1000 | N = 394 | N = 791
Age, median (IQR) | 69 (63 - 76) | 69 (63 - 75) | 69 (64 - 75)
ECOG PS, N (%) |  |  |
0 | 244 (24.4) | 140 (35.5) | 262 (33.1)
1 | 572 (57.2) | 209 (53.0) | 447 (56.5)
2 | 184 (18.4) | 45 (11.4) | 82 (10.4)
Gleason Score, N (%) |  |  |
1 | 66 (6.6) | 0 (0.0) | 1 (0.1)
2 | 11 (1.1) | 15 (3.8) | 31 (3.9)
3 | 12 (1.2) | 1 (0.3) | 2 (0.3)
4 | 4 (0.4) | 1 (0.3) | 3 (0.4)
5 | 19 (1.9) | 3 (0.8) | 4 (0.5)
6 | 78 (7.8) | 2 (0.5) | 24 (3.0)
7 | 304 (30.4) | 32 (8.1) | 76 (9.6)
8 | 152 (15.2) | 151 (38.3) | 286 (36.2)
9 | 354 (35.4) | 76 (19.3) | 159 (20.1)
10 | 0 (0.0) | 113 (28.7) | 205 (25.9)
PSA, median (IQR) | 399 (98 - 914) | 139 (41 - 412) | 120 (37 - 354)
LDH, median (IQR) | 293 (225 - 397) | 235 (188 - 321) | 222 (187 - 308)
ALP, median (IQR) | 246 (109 - 447) | 126 (83 - 268) | 125 (79 - 254)
Hb, median (IQR) | 11.1 (10.1 - 12.5) | 12.0 (10.8 - 12.8) | 11.9 (0.9 - 12.9)
Testosterone, median (IQR) | 11.5 (5.6 - 20.0) | 12.0 (5.7 - 20.0) | 13.0 (5.8 - 20.2)

The objective of this analysis was to compare the performance of alternative
methods described above in a real-world context in which data were available
from a clinical trial and a pseudo EHR dataset, which was constructed from the
standard-of-care arm from the clinical trial. We compared the effect of
abiraterone acetate (an androgen synthesis inhibitor) plus prednisone compared
to prednisone alone on overall survival in patients with metastatic
castration-resistant prostate cancer progressing after chemotherapy.
Metastatic castration-resistant prostate cancer (MCRPC) is a type of advanced
prostate cancer that no longer completely responds to treatments that lower
testosterone. [13] The study sample included patients with MCRPC from a phase
3 randomized double-blind clinical trial (NCT00638690) conducted by Janssen
Research $\&$ Development, L.L.C.. Complete details of trial eligibility and
treatment protocols have been previously published. [14]

The clinical trial population included 1,185 patients with MCRPC progressing
after taxane chemotherapy. [15] Patients were randomized in a 2:1 ratio to
treatment with abiraterone acetate plus prednisone or treatment with
prednisone alone. Patients were enrolled from 2008 to 2009 and followed for
five years, or until death. Treatment arm was classified based on the arm to
which a patient was randomized, regardless of whether they crossed over to
open-label abiraterone acetate at any point (Table 5).

The pseudo EHR dataset was constructed by sampling patients from the standard-
of-care arm in the clinical trial such that the baseline covariate
distribution of the resultant sample differed between the EHR and clinical
trial (Table 5, Supplemental Figure S2). We assume that all patients had the
same set of covariates associated with poor performance for patients with
MCRPC recorded at the baseline encounter. [16, 17, 18, 19, 20, 21]
Specifically, all standard-of-care arm patients were sampled with replacement
to create a population of size 10,000 from which we could draw our pseudo EHR
sample. Next, each patient was assigned a probability of sampling according to
a non-linear function of the baseline covariates. By constructing this
sampling probability using a non-linear functional form, the estimated on-
trial score in the DAW approach will be mis-specified. This reflects the real-
world scenario where we are unlikely to be able to correctly specify this
model. A psuedo EHR sample of size 1,000 was then drawn. The sampling
probabilities were generated such that patients were more likely to be
included in the psuedo EHR if they: were younger, had a higher ECOG score, had
a higher Gleason score, had a higher lab value for PSA, LDH, Hb, and ALP, or
had a lower testosterone value. The logistic sigmoid function was used to
relate the ECOG and Gleason scores to the sampling probability in order to
separate out those with high versus low scores rather than having alinear
additive effect as the score increased. The square root of the testosterone
lab value was used to shrink the effect of a high testosterone value on being
included in the sample.

Due to missingness in some variables, multiple imputation via predictive mean
matching was used with 5 imputations. The median of the imputed covariates was
calculated across imputations and included in the on-trial score, which is
valid in the case where the covariates do not inform treatment assignment such
as in a clinical trial. [22] Post-imputation covariate distributions
stratified by data source were similar to pre-imputation covariate
distributions.

###  4.1 Case Study Results

In order to appropriately interpret the results of the case study we first
must evaluate each of Pocock’s six criteria for the external standard-of-care
data. [4] In this case meeting most of the criteria was trivial due to the
fact that the pseudo EHR was created from the standard-of-care arm of the
clinical trial, except for $\\#6$, since the pseudo EHR had patients selected
in a biased fashion such that their covariate distribution was slightly
different and their outcomes were somewhat worse than the clinical trial.

The case study of MCRPC patients had a high rate of death. Of the 791
intervention arm patients in the trial there were 645 deaths, of the 394
patients receiving the standard-of-care in the trial there were 331 deaths,
and of the 1000 patients in the pseudo EHR there were 937 deaths. The median
survival time was 15.6 months (95$\%$ CI: 14.7-16.8) for the intervention arm
in the trial , 11.2 months (95$\%$ CI: 10.4-13.3) for the standard-of-care arm
of the trial, and 8.0 months (95$\%$ CI: 7.9-8.6) for the pseudo EHR. It is
clear that the patients in the pseudo EHR had inferior survival relative to
both arms of the clinical trial. This is likely to be true in reality as
patients often receive more supportive care in a clinical trial than in
regular clinical practice and tend to have different covariate distributions
due to restrictive inclusion/exclusion criteria.

The hazard ratio for death for patients on abiraterone acetate plus prednisone
as compared to patients on prednisone alone was 0.86 (95$\%$ CI: 0.75-0.98)
using only patients enrolled in the clinical trial. When all pseudo EHR
patients were added to the analysis population, the hazard ratio for death was
0.60 (95$\%$ CI: 0.55-0.66) (Figure 7). The hazard ratio for death as
estimated by the power prior method with the three different $\alpha$ values
were between the trial-only and full pooling methods, as expected. The
normalized power prior method estimated $\alpha=0.004$ and therefore was
virtually identical to the trial only method as it effectively borrowed
information from only four registry patients. Lin’s method returned results
somewhat similar to the trial only method, with an estimated hazard ratio of
0.76 (95$\%$ CI: 0.68-0.86), adding just over 200 patients to the analysis
(Figure 7). The estimated hazard ratio using DAW was 0.85 (95$\%$ CI:
0.76-0.94), which is almost identical to that obtained with the trial only
data and also had a smaller confidence interval due to the fact that 397
patients were added so that the augmented trial had a 1:1 randomization ratio
(Figure 7). With the exception of NPP, which only added 4 patients, DAW
returned results most similar to the trial-only result while achieving
improved efficiency (Figure 7).

Figure 6: Kaplan-Meier curves for overall survival patients in the clinical
trial and the pseudo EHR. Figure 7: Hazard ratio for death and 95$\%$
confidence intervals (CI) for patients on abiraterone acetate plus prednisone
compared to patients on prednisone alone. FP = full pooling, PP = power prior,
NPP = normalized power prior ($\hat{\alpha}=0.004$), DAW = data-adaptive
weighting. Vertical line denotes hazard ratio estimated with only the trial
data and the grey band denotes the 95$\%$ confidence interval when only the
trial data is used. ESS = estimated sample size.

##  5 Discussion

Data-adaptive weighting allows for the external patients who are most similar
to the clinical trial to be selected and weighted such that the augmented
trial has a 1:1 randomization ratio, which results in minimal bias and tighter
confidence intervals as compared to using only the trial data. We compared the
performance of alternative methods for constructing and analyzing a hybrid
control arm in terms of bias, variance, power, confidence interval coverage,
and type I error across various scenarios for trial size, trial randomization
ratio, strength of covariate effects, and treatment effect.

Based on the results from our simulation study it is clear that fully pooling
external patients with trial patients has the potential to produce highly
biased results, have poor coverage, and substantially inflate type I error
rates. Similarly, the power prior at all three alpha values examined exhibited
poor performance, although results were attenuated towards the trial only
analyses. The normalized power prior either exhibited moderate bias and poor
coverage and type I error rate or had little bias but failed to incorporate
much information from the EHR database. The case study examined here shows how
the methods perform on real data when Pocock’s criteria are met when creating
a hybrid control trial. While observed covariates can be accounted for using
on-trial scores as in Lin’s method and DAW, the other criteria are extremely
important to avoid confounding by unobserved characteristics of patients or
their care environment, and creation of a hybrid control arm is not
recommended if they are not met.

Lin’s method has reduced bias and variance compared to the more traditional
methods, though DAW was able to achieve lower bias and variance than Lin’s
method. Both methods performed very similarly with regards to confidence
interval coverage and type I error rates. While this may initially cause one
to conclude that Lin’s method and DAW are both good methods to use for hybrid
control arms, there are several points to be made regarding Lin’s method.
First, Lin’s method becomes more difficult to implement with larger trial
and/or external data source sizes as optimal matching can be cumbersome or
even impossible with large samples. Second, while the number of EHR patients
selected by Lin’s method nominally produces a 1:1 ratio, weighting by the on-
trial score results in an effective sample size that has fewer standard-of-
care than intervention arm patients, reducing efficiency of this approach.
Finally, it is unclear what estimand Lin’s method estimates as the external
patients are weighted by their on-trial score and the trial patients are given
a weight of 1. These issues are addressed by the DAW as the IOW are scaled to
ensure that the 1:1 ratio is preserved and the use of IOW allows one to
estimate the estimand of interest: the average treatment effect for those on-
trial.

While our simulation study evaluated a large combination of possible
characteristics of a trial and real-world data source, there are additional
factors that could have an effect on the results that were not examined,
including differential covariate effects on the outcome between EHR and trial
patients and potential differential error in outcome ascertainment between EHR
and trial patients. Furthermore, due to the computational demands of Bayesian
estimation for these methods, we have evaluated performance of a frequentist
analogue, which targets a different estimand and may produce different results
from a Bayesian implementation, particularly for small sample sizes where the
Bayesian central limit theorem has little effect. One must also consider the
direction of the bias induced by covariates that differ between the trial and
real-world populations in order to determine the effect of the differential
covariate distributions on power and type I error.

Based on the results of these simulations and the real-world data example,
when working with hybrid-control arm data we recommend using the DAW method in
order to minimize bias and variance while maximizing coverage and properly
controlling the type I error rate. Additionally, DAW estimates the average
treatment effect for those on-trial, which is the estimand of interest.

##  6 Acknowledgements

This study, carried out under YODA Project $\\#2020-4453$, used data obtained
from the Yale University Open Data Access Project, which has an agreement with
JANSSEN RESEARCH $\&$ DEVELOPMENT, L.L.C.. The interpretation and reporting of
research using this data are solely the responsibility of the authors and does
not necessarily represent the official views of the Yale University Open Data
Access Project or JANSSEN RESEARCH $\&$ DEVELOPMENT, L.L.C..

##  7 Data Availability

The data that support the findings of this study are available from JANSSEN
RESEARCH $\&$ DEVELOPMENT, L.L.C. via the Yale University Open Data Access
Project. Restrictions apply to the availability of these data, which were used
under license for this study. Data are available at https://yoda.yale.edu/
with the permission of the Yale University Open Data Access Project.

##  8 Funding

Research reported in this publication was supported in part by NIH grant
R21CA227613. The content is solely the responsibility of the authors and does
not necessarily represent the official views of the National Institutes of
Health.

##  9 Declaration of conflicting interests

The author(s) declared no other potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.

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##  10 Supplement

Figure S1: Kaplan-Meier curves for sample simulated datasets for each of four
conditional treatment hazard ratios and two strengths of confounding. Figure
S2: Covariate distributions in the clinical trial and pseudo EHR. Figure S3:
On-trial score distributions for patients in the clinical trial and the pseudo
EHR. Vertical lines denote mean on-trial score in each group.