Bayesian method with horseshoe prior for incorporating multiple historical control data in randomized trials

概要

筑 波 大 学
博 士（ 医 学 ）学 位 論 文

Bayesian method with horseshoe prior
for incorporating
multiple historical control data
in randomized trials
(ランダム化比較試験で
既存対照データを組み込むための
馬蹄事前分布を用いたベイズ推定法)

2022
筑波大学大学院博士課程人間総合科学研究科 

大東 智洋

Contents

Abbreviation

3

List of Tables

4

List of Figures

5

1 Background

7

1.1

Historical data in clinical trials . . . . . . . . . . . . . . . . . .

7

1.2

Historical control data by Bayesian approaches . . . . . . . . . . 11

2 Purpose

16

3 Methods

17

3.1

3.2

Existing methods . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1

Meta-analytic approach

3.1.2

Power prior . . . . . . . . . . . . . . . . . . . . . . . . . 20

Proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1

Horseshoe prior . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2

Proposed method . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3

Property of the proposed method . . . . . . . . . . . . . 27

4 Results
4.1

. . . . . . . . . . . . . . . . . . 17

33

Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.1

Binary endpoint . . . . . . . . . . . . . . . . . . . . . . . 33
1

4.1.2
4.2

Time-to-event endpoint . . . . . . . . . . . . . . . . . . . 37

Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1

Binary endpoint . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2

Time-to-event endpoint . . . . . . . . . . . . . . . . . . . 48

5 Discussion

55

6 Conclusion

59

Summary figure

60

References

61

Acknowledgments

68

Source

70

Appendix

71

List of Tables in Appendix

72

A Additional tables in the simulation study

77

A.1 Binary endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2 Time-to-event endpoint . . . . . . . . . . . . . . . . . . . . . . . 102

2

Abbreviation
CI

Credible interval

CP

Calibrated power

DMPP

Dependent modified power prior

EBPP

Empirical Bayesian power prior

EHSS

Effective historical sample size

EMA

European Medicines Agency

ESS

Effective sample size

EX

Full exchangeability meta-analytic combined method

EXNEX

Robustified meta-analytic combined method

FDA

Food and Drug Administration

HS

Proposed method using horseshoe prior

ICH

International Conference on Harmonization

MAC

Meta-analytic combined

MAP

Meta-analytic predictive

MCMC

Markov chain Monte Carlo

MPP

Modified power prior

MPSD

Mean posterior standard deviation

PFS

Progression-free survival

RCT

Randomized controlled trial

RDMPP

Robust dependent modified power prior

RMSD

Root mean square deviation

SD

Standard deviation
3

List of Tables
Table 1

Observed response rate of the azathioprine and placebo groups
from five trials. . . . . . . . . . . . . . . . . . . . . . . . . . 34

Table 2

Summary statistics of posterior distribution of the treatment
effect in terms of the response rate (%). . . . . . . . . . . . . 35

Table 3

Available information in Project Data Sphere on extensive
stage small cell lung cancer trials. . . . . . . . . . . . . . . . 37

Table 4

Baseline characteristics recorded in common for the current
and historical trials.

Table 5

. . . . . . . . . . . . . . . . . . . . . . 39

Summary statistics of posterior distributions of the unadjusted and adjusted hazard ratio of the current treatment
group to the current control group. . . . . . . . . . . . . . . 40

Table 6

Settings for the number of historical controls, the number
of participants, the allocation ratio, and the between trial
heterogeeity for each scenario in the simulation study with
binary endpoint. . . . . . . . . . . . . . . . . . . . . . . . . 44

Table 7

Settings for the number of historical controls, the number
of participants, the allocation ratio, and the between trial
heterogeeity for each scenario in the simulation study with
time-to-event endpoint. . . . . . . . . . . . . . . . . . . . . . 50

4

List of Figures
Figure 1

(A): Using historical data instead of a concurrent control
group. (B) Incorporating historical data into current trials
while assigning participants to the treatment and control
groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2

“Exchangeable” assumption relating to parameters of current and historical controls. . . . . . . . . . . . . . . . . . . 12

Figure 3

“Equal but discounted” assumption relating to parameters
of current and historical controls. . . . . . . . . . . . . . . . 13

Figure 4

“Potential biases” assumption relating to parameters of
current and historical controls. . . . . . . . . . . . . . . . . 14

Figure 5

Comparison of the horseshoe, Cauchy, and Laplace densities. 25

Figure 6

Posterior distributions of meff in the scenarios with 30 participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 7

Posterior distributions of meff in the scenarios with 90 participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 8

Posterior distributions of λh and τ in the scenarios with 30
participants. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 9

Posterior distributions of λh and τ in the scenarios with 90
participants. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5

Figure 10

Posterior distributions of the potential bias between the
log-odds of the current control and each historical control
using HS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 11

Kaplan–Meier plots for each group.

. . . . . . . . . . . . . 38

Figure 12

Posterior distributions of the potential bias between the
log-hazard ratios of the current treatment group to the current control group and those of the current treatment group
to each historical control group using HS with unadjusted
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 13

Posterior distributions of the potential bias between the
log-hazard ratios of the current treatment group to the
current control group and those of the current treatment
group to each historical control group using HS with adjusted analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 42

Figure 14

Type I error rate (%) of the treatment effect in the simulation study with a binary endpoint. . . . . . . . . . . . . . 45

Figure 15

Power (%) of the treatment effect in the simulation study
with a binary endpoint. . . . . . . . . . . . . . . . . . . . . 46

Figure 16

Type I error rate (%) of the treatment effect in the simulation study with a time-to-event endpoint. . . . . . . . . . 51

Figure 17

Power (%) of the treatment effect in the simulation study
with a time-to-event endpoint. . . . . . . . . . . . . . . . . 52

6

Chapter 1
Background
1.1

Historical data in clinical trials

A randomized controlled trial (RCT) with placebo or standard care is the gold
standard for determining the efficacy of a test treatment. RCTs contribute to
the generation of unbiased treatment effect estimates and control for type I
error in hypotheses testing. For medical product development in major disease
areas, it is relatively easy to conduct RCTs with appropriate sample sizes,
which regulatory agencies require. Appropriate RCTs play an important role in
evidence-based medicine. However, it is often difficult to conduct appropriate
RCTs involving participants with rare diseases or children.
The prevalence of rare diseases is defined as less than five per 10,000 persons (Orphan Medicinal Product Regulation, 2000), meaning the number of
patients is small. However, due to the presence of several rare diseases in the
world, many patients suffer from rare diseases (Unkel et al., 2016). There
are several obstacles to the development of drugs and medical devices for the
treatment of rare diseases. First, it is difficult to design and conduct appropriate clinical trials to investigate the efficacy of a test treatment because of

7

the small number of patients. Hence, in the development of rare diseases, it
is difficult to construct sufficient evidence from two independent RCTs, as is
the case with major diseases. Second, the development of rare diseases may be
hampered by insufficient knowledge of their clinical course. Third, the small
number of patients with rare diseases may diminish the commercial interest
toward further development. To solve these problems, the research on clinical
trial methodologies and statistical issues to evaluate novel therapies for rare
diseases has increased in recent years. Additionally, the European Medicines
Agency (EMA) published a guideline on clinical trials for small populations
(European Medicines Agency, 2006). In this guideline, the options of internal
controls or external controls, which may be historical, are presented as control
groups for clinical trials. In many cases, it is preferable to use internal control
groups; however, under certain circumstances, it is acceptable to use historical
controls. Historical control data include participants assigned to placebo or
the (current) standard of care in previous clinical trials. The U.S. Food and
Drug Administration (FDA) guidance suggests that the use of historical controls may be acceptable, in particular, in cases of serious rare diseases with
unmet medical needs (Food and Drug Administration, 2015). Therefore, regulatory agencies have recognized that historical controls can be used to solve
the problem of insufficient sample sizes for rare diseases.
Conducting clinical trials in children presents several challenges. The ethical issues that place children at risk in clinical trials and small sample sizes have
limited the development of pediatric medical products. To address these issues,
design and analysis methodologies unique to pediatric clinical trials are being
considered instead of applying the usual methodologies of adult clinical trials.
One way to increase the feasibility of pediatric clinical trials is to extrapolate
data from adult and other pediatric trials (Dunne et al., 2011; Gamalo-Siebers
8

et al., 2019). ...