A propensity-score integrated approach to Bayesian dynamic power prior borrowing Jixian Wanga Hongtao Zhangb Ram Tiwaric

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A propensity-score integrated approach to

Bayesian dynamic power prior borrowing

Jixian Wanga∗

, Hongtao Zhangb, Ram Tiwaric

aBristol Myers Squibb, Boudry, Switzerland;

bMerck & Co., Inc., North Wales, Pennsylvania, USA

cBristol Myers Squibb, Berkeley Heights, New Jersey, USA

October 5, 2022

Abstract

Use of historical control data to augment a small internal control arm

in a randomized control trial (RCT) can lead to signiﬁcant improvement

of the eﬃciency of the trial. It introduces the risk of potential bias, since

the historical control population is often rather diﬀerent from the RCT.

Power prior approaches have been introduced to discount the historical

data to mitigate the impact of the population diﬀerence. However, even

with a Bayesian dynamic borrowing which can discount the historical

data based on the outcome similarity of the two populations, a consid-

erable population diﬀerence may still lead to a moderate bias. Hence,

a robust adjustment for the population diﬀerence using approaches such

as the inverse probability weighting or matching, can make the borrow-

ing more eﬃcient and robust. In this paper, we propose a novel approach

∗CONTACT Jixian Wang: jixian.wang@bms.com

arXiv:2210.01562v1 [stat.ME] 4 Oct 2022

integrating propensity score for the covariate adjustment and Bayesian dy-

namic borrowing using power prior. The proposed approach uses Bayesian

bootstrap in combination with the empirical Bayes method utilizing quasi-

likelihood for determining the power prior. The performance of our ap-

proach is examined by a simulation study. We apply the approach to two

Acute Myeloid Leukemia (AML) studies for illustration.

Key words: Bayesian bootstrap; Dynamic borrowing; Empirical

Bayesian; Power prior; Propensity score

1 Introduction

In situations when the control arm of a randomized clinical trial (RCT) is smaller

than the test arm in order to treat more patients with the test treatment, in

order that statistical inference for treatment comparison is not compromised

due to the small control arm, use of external data, which may be from another

clinical trial or real-world data (RWD), has become a valuable source for aug-

menting the internal control of the RCT. This approach is often referred to as

borrowing external controls. Regulatory guidance documents on using RWD,

to aid drug development, have been published (EMA, 2020; FDA 2018). How-

ever, the use of external data also introduces the risk of potential bias, as the

historical control population may be rather diﬀerent from the RCT.

There are several approaches to eliminate or reduce the bias due to popu-

lation diﬀerence. In particular, robust adjustment for the population diﬀerence

using propensity score (PS) based approaches such as the inverse probability

weighting or matching can make the borrowing more eﬃcient and robust to

model misspeciﬁcation to some extent (Rosenbaum & Rubin, 1983; Robins et

al., 1994). If we assume that the adjustment can eliminate the bias, one may

be tempted to use a single arm trial with adjusted external control completely.

However, some assumptions such as no unobserved confounders can not be veri-

ﬁed based on the data. Therefore, using adjusted external control has potential

risk of introducing confounding bias.

An RCT, with a small control arm, may provide an internal reference for

evaluating the diﬀerence between the internal and external control populations.

To deal with the population diﬀerence, power prior approaches (Ibrahim et al.,

2000, 2003; Hobbs et al., 2011, 2013; Neuenschwander et al. 2009) can be used

to discount the historical data to mitigate the impact of the bias. The amount

of borrowing can be either ﬁxed or determined by discounting the historical

data, based on the similarity of the outcomes of the two populations, known as

Bayesian dynamic borrowing. However, a considerable diﬀerence may still exist

and lead to a moderate bias. To mitigate the impact of population diﬀerence,

some recently developed approaches used PS matching or stratiﬁcation to reduce

the diﬀerence within matched pairs or strata, and then applied the power prior

within them (Wang et al., 2019a, 2019b; Sachdeva et al., 2021). An alternative

approach used a linear outcome model assuming exchangeability after covariate

adjustment (Kotalik et al., 2021).

As a further development based on the above-mentioned work, we propose

a novel approach integrating PS based approaches for the covariate adjustment

and Bayesian dynamic borrowing using the power prior (Ibrahim et al., 2000,

2003; Hobbs et al., 2011, 2013; Chen et al., 2011; Gravestock et al., 2017,

2018; Wang et al., 2019a, 2019b). The proposed approach combines the ad-

vantages of propensity score based approaches for adjusting confounding bias

without specifying the outcome model, and the power prior that down-weights

the information from the historical data if, after adjustment, it is still consid-

erably diﬀerent from the internal control. Our approach is an approximate full

Bayesian that takes the uncertainty of model ﬁtting and weighting into account.

The major challenge is that the PS based methods are frequentist approaches

with minimum model speciﬁcation, while the power prior methods are built in

the Bayesian framework. Our approach is partially built on the work of approx-

imate PS-based Bayesian approaches (Graham et al., 2016; Saarela et al., 2016;

Capistrano et al., 2019), in which the inverse probability weighting (IPW) and

doubly robust (DR) estimation approaches were put in the Bayesian framework

and the posterior distribution was approximated by Bayesian bootstrap (BB)

(Rubin, 1981). Our approach is considerably simpler than the outcome based

full Bayesian approach using MCMC such as Kotalik et al. (2021).

2 A review of relevant approaches

2.1 Borrowing historical controls to augment an internal

control arm

First, we state our approach of borrowing historical controls to augment an in-

ternal control arm formally. Let Di= (yi,Xi, Hi), i = 1, ..., n, be the outcome,

covariates, and an indicator of being in the historical control for the ith subject

in the combined of the internal trial population and the historical control pop-

ulation. The sample sizes and the means of the internal and historical control

populations are: nh=Pn

i=1 Hi, n0=n−nh, ¯y0=Pn

i=1(1 −Hi)yi/n0and

¯yh=Pn

i=1 Hiyi/nh, respectively. We also denote the whole dataset as D=

(D1, ..., Dn), and those of the internal and external controls as D0=D|Hi= 0

and Dh=D|Hi= 1, respectively. Although our ﬁnal goal is to evaluate the

treatment eﬀect of the treatment applied in the treated arm in the trial popula-

tion, the key issue we concentrate on is the evaluation of treatment eﬀect under

the control: µ=E(yi|Hi= 0), borrowing historical control data with confound-

ing adjustment. Here, yicould be either a continuous or binary variable, in the

latter case, µis the rate or proportion of the outcome. Due to population dif-

ference between the trial and historical control, E(¯yh) = E(yi|Hi= 1) is likely

diﬀerent from µ, hence adjustment for population diﬀerence is often necessary.

2.2 Propensity score based adjustment

A commonly used approach for population adjustment is based on PS deﬁned

as the probability of belonging to the historical control, given the covariates Xi:

ei=P(Hi= 1|Xi) (1)

which is often modelled by a logistic regression as P(Hi|Xi, γ) with parameters

γ. Under some technical conditions, yi⊥Hi|ei, hence we can use the inverse

probability waiting (IPW) estimator

ˆµipw = (

i=1

Hiwi)−1

i=1

Hiwiyi(2)

where wi= (1 −ei)/ei, to estimate µ. Using the property yi⊥Hi|ei, it is

straightforward to show that E(ˆµipw) = µ. Therefore, when the PS model is

correctly speciﬁed, one can combine ˆµipw with the internal control mean ¯y0

for more accurate estimation of µ. This IPW estimator (2) is slightly diﬀerent

from the standard IPW estimator for the average treatment eﬀect (ATE) for

the whole population, since we aim at estimating the control eﬀect in the trial

population.

2.3 Propensity score based Bayesian methods

Although PS based approaches were proposed from frequentist aspect, eﬀort has

been made to use PS in the Bayesian framework to provide a robust Bayesian

approach for population adjustment (Zigler, 2016; Zigler et al., 2013, 2014).

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Apropensity-scoreintegratedapproachtoBayesiandynamicpowerpriorborrowingJixianWanga*,HongtaoZhangb,RamTiwaricaBristolMyersSquibb,Boudry,Switzerland;bMerck&Co.,Inc.,NorthWales,Pennsylvania,USAcBristolMyersSquibb,BerkeleyHeights,NewJersey,USAOctober5,2022AbstractUseofhistoricalcontroldatatoaugmentasmal...

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A propensity-score integrated approach to Bayesian dynamic power prior borrowing Jixian Wanga Hongtao Zhangb Ram Tiwaric

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