Bayesian multivariate meta-analysis for modelling surrogate endpoints in HTA

Bayesian multivariate meta-analysis
for modelling surrogate endpoints in HTA
Sylwia Bujkiewicz
University of Leicester
Acknowledgements: John Thompson, Keith Abrams, Sze Huey Tan
Richard Riley,
Funding: Medical Research Council, New Investigator Research Grant
York 10 March 2016
Sylwia Bujkiewicz (University of Leicester) 1 / 68

Outline
Introduction
Bayesian meta-analysis
Univariate meta-analysis
Multiple outcomes
Correlated eﬀects
Application in rheumatoid arthritis
Data
Results
Application in prostate cancer
Application to surrogate endpoints
Introduction
Use of bivariate meta-analysis
Methods
Illustrative example
Modelling scenarios
Discussion
References

Introduction
Introduction
Health Technology Assessment:
Evidence-based decision-making to make reimbursement decisions
requires careful synthesis of available evidence
relies heavily on meta-analysis of effectiveness of new interventions
evidence base typically obtained from the systematic literature review
of randomised controlled trials
inclusion of real world evidence (observational studies)
Challenges:
a lot of heterogeneity in reporting of clinical outcomes
variety of scales on which effectiveness can be measured
different time points at which different studies report their outcomes
different control arms
long follow up

Introduction
Multi-parameter evidence synthesis
Bayesian statistics provides ﬂexible framework for modelling complex data
structures
allows multiple parameters to be modelled simultaneously
Network meta-analysis (NMA) facilitates simultaneous comparison
of multiple treatments
multivariate meta-analysis (MVMA) allows to model jointly
treatment eﬀects on multiple correlated outcomes
Use in evidence based decision making
NMA is becoming a standard tool for synthesising evidence in HTA
MVMA has not been widely used despite of substantial
methodological developments
many advantages of this approach to evidence synthesis (potential for
reduced uncertainty, outcome reporting bias)

Introduction

Bayesian meta-analysis Univariate meta-analysis
Random-effects meta-analysis
In the presence of the between-study heterogeneity: a random effect
approach:
treatment effects Yi are assumed to estimate study-specific true
treatment effects µi
different in each study i
they follow a common distribution
Yi ∼ N(µi , σ2
i ),
µi ∼ N(β, τ2
),
τ2 – the between-studies variance (heterogeneity parameter)
In the Bayesian framework unknown parameters are given prior
distributions:
heterogeneity parameter, τ2 ∼ U[0, 10]
the pooled effect, β ∼ N(0.0, 103).

Bayesian meta-analysis Multiple outcomes
Use of bivariate MA: pooled eﬀects
study outcome 1 outcome 2
study 1
study 2
study 3
study 4
study 5
study 6
study 7
study 8
study 9
study 10
pooled ? ?

Bivariate random-effects meta-analysis
within–study variability
Y1i ∼ N(µ1i , σ2
1i ),
Y2i ∼ N(µ2i , σ2
2i ),
between–study variability
µ1i ∼ N(β1, τ2
1 ),
µ2i ∼ N(β2, τ2
2 ),
Hierarchical framework:
Y1i , Y2i – estimates of correlated effects µ1i , µ2i
σ2
1i , σ2
2i – within-study variances.
µ1i , µ2i – true effects
(β1, β2) – pooled estimates
τ2
1 , τ2
2 – between-study variances.

within–study model
Y1i
Y2i
∼ MVN
µ1i
µ2i
, Σi =
σ2
1i 0
0 σ2
2i
,
between–study model
µ1i
µ2i
∼ MVN
β1
β2
, T =
τ2
1 0
0 τ2
2
Σi – within-study covariance matrices of the estimates.
T – between-study covariance matrix.

Bayesian meta-analysis Correlated eﬀects
study 1
(Y11, Y21)

study 1
Y11
Y21
(Y11, Y21)

study 1 study 2 study N
. . .
(Y11, Y21) (Y12, Y22) (Y1N, Y2N)

Y1i
Y2i
∼ MVN
µ1i
µ2i
, Σi =
σ2
1i σ1i σ2i ρwi
σ1i σ2i ρwi σ2
2i
,
µ1i
µ2i
∼ MVN
β1
β2
, T =
τ2
1 τ1τ2ρb
τ1τ2ρb τ2
2
.

Trivariate random effects meta-analysis


Y1i
Y2i
Y3i

 ∼ N




µ1i
µ2i
µ3i

 , Σi

 , Σi =


σ2
1i σ1i σ2i ρ12
wi σ1i σ3i ρ13
wi
σ2i σ1i ρ12
wi σ2
2i σ2i σ3i ρ23
wi
σ3i σ1i ρ13
wi σ3i σ2i ρ23
wi σ2
3i




µ1i
µ2i
µ3i

 ∼ N




β1
β2
β3

 , T

 , T =


τ2
1 τ1τ2ρ12
b τ1τ3ρ13
b
τ2τ1ρ12
b τ2
2 τ2τ3ρ23
b
τ3τ1ρ13
b τ3τ2ρ23
b τ2
3


Y1i , Y2i and Y3i – estimates of correlated effects µ1i , µ2i and µ3i
µ1i , µ2i and µ3i – true effects
(β1, β2, β3) – pooled estimates

Application in rheumatoid arthritis Data

Example in rheumatoid arthritis
S. Lloyd, S. Bujkiewicz, A.J. Wailoo, A.J. Sutton & D. Scott,
The eﬀectiveness of anti-TNF-alpha therapies when used sequentially
in rheumatoid arthritis patients: a systematic review and
meta-analysis, Rheumatology (2010) 49(12), 2313-2321.
Standard instruments for measuring response to treatment in RA:
HAQ - Health Assessment Questionnaire
Disease Activity Score (DAS-28)
American College of Rheumatology (ACR) response criteria.
Multivariate meta-analysis developed to jointly model the eﬀect of
treatments on the multiple outcomes in evidence synthesis which aims
to estimate the change from baseline of the HAQ score.
The estimate of the HAQ is of a particular interest to clinicians and
decision-makers in health care as it is often used to estimate quality
of life of patients following treatments of RA.

Data
Study HAQ
Bennet 2005
Bingham 2009
Bombardieri 2007
Haroui 2004
Hyrich 2008
Iannone 2009
Navarro-Sarabia 2009
Van der Bijl 2008

Data
Study HAQ DAS28
Bennet 2005
Bingham 2009
Bombardieri 2007
Haroui 2004
Hyrich 2008
Iannone 2009
Van der Bijl 2008
Buch 2007
Cohen 2005
Di Poi 2007
Finckh 2007
Hjardem 2007
Laas 2008
Nikas 2006
Wick 2005

Data
Study HAQ DAS28 ACR20
Bennet 2005
Bingham 2009
Bombardieri 2007
Haroui 2004
Hyrich 2008
Iannone 2009
Van der Bijl 2008
Buch 2007
Cohen 2005
Di Poi 2007
Finckh 2007
Hjardem 2007
Laas 2008
Nikas 2006
Wick 2005
Karlsson 2008
Buch 2005
Van Vollenhoven 2003

External sources of evidence
Individual patients data (IPD)
The BROSG trial designed to assess the benefit of aggressive
DMARD treatment in patients with established RA [Symmons et al
(2006) Rheumatology 45, 558–565].
Two cohorts (1) symptomatic control of pain and stiffness in the
shared care setting, or (2) a more aggressive regime focussing on
control of symptoms and joint inflammation in the hospital setting.
External summary data
Meta-analysis of data from external studies which included those from
a technology appraisal of adalimumab, etanercept and infliximab in
treatment of rheumatoid arthritis in biologically naive patients, carried
out by [Chen et al (2006) HTA 10, (42), i-250], extended by more
recent data.

Example in rheumatoid arthritis: data sources
External
aggregate data
DAS, HAQ, ACR
Meta-analysis data
DAS, HAQ, ACR
Individual
patient data
das, haq, acr
Bayesian
meta-analysis
model
thebetween-studycorrelation
prioron
thewithin-studycorrelation
prioron

Application in rheumatoid arthritis Results
Example in rheumatoid arthritis: results
posterior mean [95% HPDI]
univariate analyses bivariate trivariate
HAQ, DAS-28
HAQ DAS-28 ACR20 HAQ & DAS-28 & ACR20
HAQ -0.25 -0.28 -0.28
[-0.43,-0.09] [-0.41,-0.14] [-0.43,-0.12]
DAS -1.57 -1.51 -1.52
[-1.84,-1.31] [-1.67,-1.35] [-1.71,-1.34]
ACR 0.62 0.61
[0.53,0.71] [0.52,0.71]
τH 0.21 0.21 0.22
[0.08,0.38] [0.10,0.35] [0.10,0.39]
τD 0.44 0.44 0.44
[0.25,0.67] [0.24,0.67] [0.25,0.67]
τA 0.52 0.53
[0.20,0.90] [0.21,0.91]
ρDH
b 0.89 0.83
[0.65,0.994] [0.45,0.99]
ρAH
b -0.14
[-0.63,0.49]

Application in rheumatoid arthritis Pooled eﬀects
Use of bivariate MA: pooled eﬀects
study 1
study 2
study 3
study 4
study 5
study 6
study 7
study 8
study 9
study 10
pooled ? ?

Application in rheumatoid arthritis Missing data
Use of bivariate MA: pooled eﬀects and missing data
study 1 NA
study 2
study 3 NA
study 4
study 5
study 6 NA
study 7
study 8 NA
study 9
study 10
pooled ? ?

Application in rheumatoid arthritis Missing data
Example in rheumatoid arthritis: results

Example in prostate cancer

Metastatic prostate cancer
Technology appraisal by the National Institute for Health and Care
Excellence
A systematic review and economic model of the clinical effectiveness
and cost-effectiveness of docetaxel in combination with prednisone or
prednisolone for the treatment of hormone-refractory metastatic
prostate cancer
No direct evidence of D+P or P compared to other established
chemotherapy regimens and best supportive care
Trials comparing mitoxantrone with other chemotherapies and
corticosteroids (best supportive care) identified
Indirect comparison meta-analysis to estimate overall survival
comparing D+P vs P
2-state Markov model developed for the cost-effectiveness analysis
Health Technology Assessment 2007; Vol. 11: No. 2

Example in prostate cancer: case study design
Trials Network for OS and PFS with a Two State Economic Markov Model in HTA Report
D+P P
M+P
OS
D+P P
M+P
PFS
OS
Stable
disease
Death
1 3
indirect
comparison
3
D+P – docetaxel in combination with either prednisone or prednisolone,
M+P – mitoxantrone plus prednisone

D+P P
M+P
OS
D+P P
M+P
PFS
OS
Stable
disease
Death
1 3
indirect
comparison
3
Proposed Bayesian model to connect the Trials Network for PFS leading to a Three State Model
D+P P
M+P
OS
D+P P
M+P
PFS
OS
PFS
Stable
disease
Death
Progression
1 3
indirect
comparison
3
indirect
comparison
predicted
eﬀect on PFS
Sze Huey Tan, PhD Thesis, Leicester 2015

Example in prostate cancer: data sources
OS HRs
from trials
in HTA Set
Indirect
comparison
OS HR
estimate
for D+P vs P
BRMA
PFS HRs from
trials in HTA Set
excluding TAX327
Indirect
comparison
PFS HR
estimate
for D+P vs P
PFS HR predicted
for TAX 327 trial
(D+P) vs (M+P)
Likelihood
Likelihood

OS HRs
from trials
in HTA Set
Indirect
comparison
OS HR
estimate
for D+P vs P
BRMA
PFS HRs from
trials in HTA Set
excluding TAX327
Indirect
comparison
PFS HR
estimate
for D+P vs P
PFS HR predicted
for TAX 327 trial
(D+P) vs (M+P)
Likelihood
Likelihood

OS HRs
from trials
in HTA Set
Indirect
comparison
OS HR
estimate
for D+P vs P
BRMA
PFS HRs from
trials in HTA Set
excluding TAX327
Indirect
comparison
PFS HR
estimate
for D+P vs P
PFS HR predicted
for TAX 327 trial
(D+P) vs (M+P)
Likelihood
Likelihood
Posterior
between-study
correlation
used as prior
distribution
BRMA
OS HRs
from EDS
PFS HRs
from EDS
Likelihood
Likelihood

D+P P
M+P
OS
D+P P
M+P
PFS
OS
Stable
disease
Death
1 3
indirect
comparison
3
Proposed Bayesian model to connect the Trials Network for PFS leading to a Three State Model
D+P P
M+P
OS
D+P P
M+P
PFS
OS
PFS
Stable
disease
Death
Progression
1 3
indirect
comparison
3
indirect
comparison
predicted
eﬀect on PFS
Sze Huey Tan, PhD Thesis, Leicester 2015

Application to surrogate endpoints Introduction
Surrogate endpoints
Treatment
Surrogate
endpoint
Final clinical
outcome
treatment effect
on surrogate endpoint
prediction
treatment effect
on final outcome
Surrogate outcome: A biomarker that is intended to substitute for a
clinical (final) outcome. A surrogate end point is expected to predict
clinical benefit
Biomarkers Definitions Working Group. Clin Pharmacol Ther 2001.

Surrogate endpoints: importance
Surrogate endpoints are of interest in drug development process if they
can be measured
less costly
less invasively
or require shorter follow-up time
compared to a target (final) clinical outcome.
They are increasingly important in health technology assessment
at the early stages of drug development
conditional licensing based on a biomarker
evidence on treatment effectiveness on a target outcome limited
evidence on treatment effectiveness on a target outcome limited

Surrogate endpoints: importance
Surrogate endpoints are of interest in drug development process if they
can be measured
less costly
less invasively
or require shorter follow-up time
compared to a target (ﬁnal) clinical outcome.
Validation on three levels
biological plausibility of association between outcomes
patient-level association between outcomes
study-level association
a surrogate endpoint is expected to predict clinical beneﬁt

Examples of potential surrogate endpoints
disease area surrogate endpoint ﬁnal clinical outcome
colorectal cancer1,2 progression-free survival overall survival
HIV infection3 CD4 count AIDS or death
gastric cancer4 event-free survival overall survival
multiple sclerosis5 relapse rate disability progression
1
Buyse M, Burzykowski T, Carroll K et al. Journal of Clinical Oncology 2007;
25:5218–5224.
2
Ciani et al, Journal of Clinical Epidemiology 2015, 68: 833–842.
3
Daniels MJ, Hughes MD. Statistics in Medicine 1997; 16:1965–1982.
4
Oba K, Paoletti X, Alberts S et al. Journal of the National Cancer Institute 2013;
105:1600–1607.
5
Sormani MP, Bonzano L, Roccatagliata L et al. Neurology 2010; 75:302–309.

De Gruttola et al, Controlled Clinical Trials 22:485502 (2001)

Application to surrogate endpoints Estimation of pooled eﬀects
Use of bivariate MA: pooled eﬀects and missing data
study 1 NA
study 2
study 3 NA
study 4
study 5
study 6 NA
study 7
study 8 NA
study 9
study 10
pooled ? ?

Application to surrogate endpoints Evaluating surrogate endpoints
Use of bivariate MA: validation of surrogate
endpoints (1)
surrogate ﬁnal
study outcome outcome
new study ?
historical study 1
historical study 2
historical study 3
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

endpoints (2)
surrogate ﬁnal
historical study 1 NA
historical study 2
historical study 3
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

endpoints (2)
surrogate ﬁnal
historical study 1
historical study 3
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

endpoints (2)
surrogate ﬁnal
historical study 1
historical study 2
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

endpoints (2)
surrogate ﬁnal
historical study 1
historical study 2
historical study 3
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

endpoints (3)
surrogate ﬁnal
new study NA
historical study 1
historical study 2
historical study 3
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

Use of mvmeta: validation of surrogate endpoints
(4)
surrogate surrogate ﬁnal
study outcome 1 outcome 2 outcome
new study NA
historical study 1
historical study 2
historical study 3
historical study 4
historical study 5
historical study 6
historical study 7
historical study 8
historical study 9
historical study 10

Application to surrogate endpoints Methods
Multivariate meta-analysis for surrogate endpoints

Application to surrogate endpoints Illustrative example
Illustrative example: MS
Disease area: multiple sclerosis
Target (ﬁnal) outcome: Disability progression rate ratio, the ratio
between the proportion of patients with a disability progression in the
experimental and the control arms.
Surrogate endpoint 1: MRI-eﬀect, The ratio between the average
number of MRI lesions in the experimental and the control arms.
Surrogate endpoint 2: Annualized relapse rate ratio, the ratio between
the relapse rate in the experimental and the control arms.
Sormani et al., Neurology 2011

Application to surrogate endpoints Illustrative example
Illustrative example: MS data

Application to surrogate endpoints Product normal formulation: model 1
Trivariate random effects meta-analysis 1


Y1i
Y2i
Y3i

 ∼ N




µ1i
µ2i
µ3i

 , Σi

 , Σi =


σ2
1i σ1i σ2i ρ12
wi σ1i σ3i ρ13
wi
σ2i σ1i ρ12
wi σ2
2i σ2i σ3i ρ23
wi
σ3i σ1i ρ13
wi σ3i σ2i ρ23
wi σ2
3i


F
S1 S2



µ1i ∼ N(η1, ψ2
1)
µ2i | µ1i ∼ N(η2i , ψ2
2)
η2i = λ20 + λ21µ1i
µ3i | µ1i , µ2i ∼ N(η3i , ψ2
3)
η3i = λ30 + λ31µ1i + λ32µ2i .
Criteria for surrogate markers (Daniels and Hughes, Statistics in Medicine 1997)
λ21 = 0 for the association between the treatment effects
ψ2
2 = 0 for perfect association
λ20 = 0 (no treatment effect on the surrogate endpoint gives no treatment effect on the target
outcome)



Y1i
Y2i
Y3i

 ∼ N




µ1i
µ2i
µ3i

 , Σi

 , Σi =


σ2
1i σ1i σ2i ρ12
wi σ1i σ3i ρ13
wi
σ2i σ1i ρ12
wi σ2
2i σ2i σ3i ρ23
wi
σ3i σ1i ρ13
wi σ3i σ2i ρ23
wi σ2
3i


F
S1 S2



µ1i ∼ N(η1, ψ2
1)
µ2i | µ1i ∼ N(η2i , ψ2
2)
η2i = λ20 + λ21µ1i
µ3i | µ1i , µ2i ∼ N(η3i , ψ2
3)
η3i = λ30 + λ31µ1i + λ32µ2i.
Criteria for surrogate markers
For multiple endpoints with N − 1 surrogate markers
λNX = 0, (X = 1, . . . , N − 1) for the association
ψ2
N = 0 for perfect association
λN0 = 0.



Y1i
Y2i
Y3i

 ∼ N




µ1i
µ2i
µ3i

 , Σi

 , Σi =


σ2
1i σ1i σ2i ρ12
wi σ1i σ3i ρ13
wi
σ2i σ1i ρ12
wi σ2
2i σ2i σ3i ρ23
wi
σ3i σ1i ρ13
wi σ3i σ2i ρ23
wi σ2
3i


assuming independence of µ1 and µ3 conditional on µ2:
FS1 S2



µ1i ∼ N(η1, ψ2
1)
µ2i | µ1i ∼ N(η2i , ψ2
2)
η2i = λ20 + λ21 µ1i
µ3i | µ2i ∼ N(η3i , ψ2
3)
η3i = λ30 + λ32 µ2i ,
Bujkiewicz et al., Statistics in Medicine 2016

Application to surrogate endpoints Comparison of results between models
Comparison of predictions from models (2D & 3D)
predicted
trivariate
study observed bivariate unstructured % red.
Paty (A) 1.00 0.86 (0.4, 1.74) 0.86 (0.43, 1.73) 0.67
Paty (B) 0.64 0.76 (0.37, 1.56) 0.77 (0.38, 1.54) 2.24
Simon 0.53 0.78 (0.36, 1.73) 0.80 (0.36, 1.77) 0.19
Milleﬁorini 0.13 0.62 (0.11, 3.46) 0.65 (0.12, 3.62) -0.29
Li (C) 0.73 0.79 (0.43, 1.45) 0.79 (0.45, 1.41) 4.46
Li (D) 0.63 0.79 (0.43, 1.44) 0.81 (0.44, 1.50) -1.07
Polman 0.50 0.59 (0.32, 1.11) 0.61 (0.35, 1.07) 9.67
Comi (E) 0.64 0.62 (0.35, 1.11) 0.62 (0.36, 1.08) 3.71
Comi (F) 0.69 0.64 (0.36, 1.13) 0.64 (0.38, 1.09) 5.77
Rudick 0.73 0.62 (0.37, 1.05) 0.61 (0.37, 1.01) 4.18
Sorensen 0.57 0.62 (0.24, 1.64) 0.64 (0.23, 1.82) -7.52
Clanet 1.00 0.89 (0.49, 1.63) 0.88 (0.49, 1.58) 3.21
Mikol 1.39 0.82 (0.44, 1.55) 0.83 (0.45, 1.53) 2.65
average % reduction in CrI 2.14%
increased precision when including the second surrogate endpoint

predicted
trivariate
study observed bivariate unstructured % red.
Paty (A) 1.00 0.86 (0.42, 1.74) 0.86 (0.43, 1.73) 0.67
Paty (B) 0.64 0.76 (0.37, 1.56) 0.77 (0.38, 1.54) 2.24
Simon 0.53 0.78 (0.36, 1.73) 0.80 (0.36, 1.77) 0.19
Milleﬁorini 0.13 0.62 (0.11, 3.46) 0.65 (0.12, 3.62) -0.29
Li (C) 0.73 0.79 (0.43, 1.45) 0.79 (0.45, 1.41) 4.46
Li (D) 0.63 0.79 (0.43, 1.44) 0.81 (0.44, 1.50) -1.07
Polman 0.50 0.59 (0.32, 1.11) 0.61 (0.35, 1.07) 9.67
Comi (E) 0.64 0.62 (0.35, 1.11) 0.62 (0.36, 1.08) 3.71
Comi (F) 0.69 0.64 (0.36, 1.13) 0.64 (0.38, 1.09) 5.77
Rudick 0.73 0.62 (0.37, 1.05) 0.61 (0.37, 1.01) 4.18
Sorensen 0.57 0.62 (0.24, 1.64) 0.64 (0.23, 1.82) -7.52
Clanet 1.00 0.89 (0.49, 1.63) 0.88 (0.49, 1.58) 3.21
Mikol 1.39 0.82 (0.44, 1.55) 0.83 (0.45, 1.53) 2.65

predicted
trivariate
study observed bivariate structured % red.
Paty (A) 1.00 0.86 (0.42, 1.74) 0.88 (0.44, 1.73) 3.16
Paty (B) 0.64 0.76 (0.37, 1.56) 0.77 (0.38, 1.54) 3.29
Simon 0.53 0.78 (0.36, 1.73) 0.79 (0.37, 1.72) 2.10
Milleﬁorini 0.13 0.62 (0.11, 3.46) 0.62 (0.11, 3.44) 0.32
Li (C) 0.73 0.79 (0.43, 1.45) 0.80 (0.45, 1.42) 5.19
Li (D) 0.63 0.79 (0.43, 1.44) 0.79 (0.45, 1.39) 7.23
Polman 0.50 0.59 (0.32, 1.11) 0.58 (0.33, 1.03) 8.62
Comi (E) 0.64 0.62 (0.35, 1.11) 0.62 (0.36, 1.07) 5.13
Comi (F) 0.69 0.64 (0.36, 1.13) 0.63 (0.37, 1.08) 5.93
Rudick 0.73 0.62 (0.37, 1.05) 0.61 (0.38, 0.98) 10.01
Sorensen 0.57 0.62 (0.24, 1.64) 0.63 (0.24, 1.66) 0.63
Clanet 1.00 0.89 (0.49, 1.63) 0.92 (0.53, 1.60) 8.30
Mikol 1.39 0.82 (0.44, 1.55) 0.85 (0.46, 1.57) 3.45

Application to surrogate endpoints Scenario 1
Scenarios for modelling of surrogate endpoints (1)
F
S1 S4
S2 S3
all outcomes correlated:
full unstructured covariance
10 correlations, 5 heterogeneity parameters

Application to surrogate endpoints Scenario 2
Scenarios for modelling of surrogate endpoints (2)
S1 S2 S3 S4 F
S1 S2 S3 S4
F

Discussion
Discussion: complexity
Difficulties in obtaining the within-study correlation between
treatment effects
Can be obtained from the individual patient data by bootstrapping
(Daniels and Hughes Stat Med 1997)
Can be obtained from the individual patient data by double bootstrap
(Bujkiewicz et al Stat Med 2013)
Obtained from reported correlations between other measures
(Wei and Higgins Stat Med 2013, Bujkiewicz et al Stat Med 2015)
Using a joint linear regression for multiple continuous outcomes
(Riley et al Res Synth Meth 2015)
Prior distribution for between-studies correlations
Can be counter-intuitive – U[−1, 1] not vague (Burke, Bujkiewicz and
Riley, Stat Meth Med Res 2016)
Can be difficult in higher dimensions
Cholesky or spherical decomposition (Wei and Higgins, Stat Med 2013)
Informative (empirical) prior on the whole correlation structure

Discussion

Discussion
Discussion: complexity
Difficulties in obtaining the within-study correlation between
treatment effects
Can be obtained from the individual patient data by bootstrapping
(Daniels and Hughes Stat Med 1997)
Can be obtained from the individual patient data by double bootstrap
Obtained from reported correlations between other measures
(Wei and Higgins Stat Med 2013, Bujkiewicz et al Stat Med 2015)
Using a joint linear regression for multiple continuous outcomes
(Riley et al Res Synth Meth 2015)
Prior distribution for between-studies correlations
Can be counter-intuitive – U[−1, 1] not vague (Burke, Bujkiewicz and
Riley, Stat Meth Med Res 2016)
Can be difficult in higher dimensions
Cholesky or spherical decomposition (Wei and Higgins, Stat Med 2013)
Informative (empirical) prior on the whole correlation structure

Discussion

Discussion
Discussion: advantages of multivariate meta-analysis
Employing this Bayesian approach to evidence synthesis can lead to a
signiﬁcant reduction in the uncertainty around the estimate of interest
We cannot predict the extent of the gain in the precision of the
estimate (or guarantee it will occur).
Multivariate approach can lead to a more appropriate estimate of the
clinical outcome in the presence of outcome reporting bias (Kirkham
JJ, Riley RD, Williamson PR, Stat Med 2012; 31:2179-95).
Can include external evidence (observational studies, expert opinions
etc ) in the form of prior distributions, for example on the correlation
(Bujkiewicz et al Stat Med 2013) or between-study heterogeneity
(Higgins and Whitehead Stat Med 1996).
Not more complex than network meta-analysis - methods can be used
interchangeably (Ian White, Stata Journal 2015).
Can be extended to multivariate network meta-analysis (Achana,
Cooper, Bujkiewicz et al, BMC Med Res Meth 2014, Efthimiou et al
Stat Med 2014).

Discussion

Discussion
Employing this Bayesian approach to evidence synthesis can lead to a
signiﬁcant reduction in the uncertainty around the estimate of interest
We cannot predict the extent of the gain in the precision of the
estimate (or guarantee it will occur).
Multivariate approach can lead to a more appropriate estimate of the
clinical outcome in the presence of outcome reporting bias (Kirkham
JJ, Riley RD, Williamson PR, Stat Med 2012; 31:2179-95).
Can include external evidence (observational studies, expert opinions
etc ) in the form of prior distributions, for example on the correlation
(Bujkiewicz et al Stat Med 2013) or between-study heterogeneity
(Higgins and Whitehead Stat Med 1996).

Discussion
Not more complex than network meta-analysis - methods can be used
interchangeably (Ian White, Stata Journal 2015).
Can be extended to multivariate network meta-analysis (Achana,
Cooper, Bujkiewicz et al, BMC Med Res Meth 2014, Efthimiou et al
Stat Med 2014).
Flexible tool for modelling of surrogate endpoints
Allows to take into account uncertainty around the treatment effect
on surrogate endpoint.
In the Bayesian framework can be parameterised in a convenient form
for making inference about association between effects on surrogate
and final endpoints.
Easily extended to adapt to complex data structures.

References
References
1. Bujkiewicz S et al. Multivariate meta-analysis of mixed outcomes: a Bayesian approach.
Statistics in Medicine, 2013; 32: 3926 - 3943.
2. Bujkiewicz S et al, Use of Bayesian multivariate meta-analysis to estimate HAQ for
mapping onto EQ-5D in rheumatoid arthritis, Value in Health, 2014; 17: 109-115.
3. Daniels MJ and Hughes MD. Meta-analysis for the evaluation of potential surrogate
markers. Statistics in Medicine 1997; 16:1965–1982.
4. Bujkiewicz S, Thompson JR, Spata E, Abrams KR. Uncertainty in the Bayesian
meta-analysis of normally distributed surrogate endpoints.
Statistical Methods in Medical Research; online ﬁrst 13 August 2015.
5. Bujkiewicz S, Thompson JR, Riley RD, Abrams KR, Bayesian meta-analytical methods to
incorporate multiple surrogate endpoints in drug development process,
Statistics in Medicine 2016; 35:1063-1089.
6. Wei Y,Higgins JPT. Bayesian multivariate meta-analysis with multiple outcomes.
Statistics in Medicine 2013; 32:291134.
7. Wei Y, Higgins JPT. Estimating within-study covariances in multivariate meta-analysis
with multiple outcomes. Statistics in Medicine 2013; 32: 1191-1205.
8. Burke DL, Bujkiewicz S, Riley RD, Bayesian bivariate meta-analysis of correlated eﬀects:
Impact of the prior distributions on the between-study correlation, borrowing of strength,
and joint inferences, Statistical Methods in Medical Research; in press.
9. Achana FA, Cooper NJ, Bujkiewicz S, et al. Network meta-analysis of multiple outcome
measures accounting for borrowing of information across outcomes. BMC Medical
Research Methodology 2014; 14:92.

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