Network meta-analysis with integrated nested Laplace approximations

Network meta-analysis with integrated nested
Laplace approximations
Burak Kursad Gunhan
Supervised by Prof. Dr. Leonhard Held and Rafael Sauter
Master exam
Zurich, 01 March 2016

Meta-analysis Network meta-analysis Conclusions References
Systematic review
Review of evidences from different studies
On a specific question, methods to identify, select, appraise
and summarize similar but separate studies
Study selection: inclusion and exclusion criterion
Meta-analysis (The analysis of analyses)
Quantitative part of systematic review
SR may or may not include a meta-analysis!
Using statistical methods to combine results from different
studies
Burak Kursad Gunhan 2/ 30

TB dataset (Coldlitz et al., 1994)
13 vaccine controlled
trials of BCG for
prevention of TB
Year and Latitude
variables are given
Measure of treatment
eﬀect: Log odds ratio
Observed log odds
ratios
95 % Wald C.I.s
Area of boxes: 1/σ2
i
Forest plot
log odds ratio
Trials
1
2
3
4
5
6
7
8
9
10
11
12
13
−2 −1 0 1 2

Statistical methods for meta-analysis
1 Fixed effect model
Assumption: common true treatment effect
ˆθi ∼ N(θ, σ2
i )
Inverse variance-weighted method (ωi = 1/σ2
i )
ˆθIV W =
k
i=1 ωi
ˆθi
k
i=1 ωi
and Var(ˆθIV W ) =
1
k
i=1 ωi
Between-trial variability?
e. g. study populations
2 Random effects model: Accounting heterogeneity

Different approaches for RE models
Likelihood approach, adapted from Lumley (2002)
A linear mixed model containing components for sampling
variability and heterogeneity
ˆθi|θi ∼ N(θi, σ2
i )
θi ∼ N(d + γi, σ2
i )
γi ∼ N(0, τ2
) (1)
where d mean treatment effect and τ2
heterogeneity variance
Method of moments (MOM), by DerSimonian and Laird
(1986)
ωi = 1/(σ2
i + τ2
)
Available from metafor (Viechtbauer, 2010) R package
If τ2 = 0, then fixed effect model

Fully Bayes approach
The model formulation same as equation (1), but assigning
prior distributions for d and τ
Using uninformative priors: d ∼ N(0, 1000); τ ∼ U(0, 10)
Inference methods
MCMC: simulation-based technique, very popular
Implemented by using JAGS with R2jags R package
Convergence diagnostics checked!
JAGS code is taken from Lunn et al. (2012)
INLA: An approximate Bayesian inference technique by Rue
et al. (2009) with INLA R package
Shown to be suitable for meta-analysis inference by Sauter and
Held (2015)
Main goal: INLA implementation of the models

Two modelling approaches
Summary-level
Dataset: One-study-per-row structure
Zero entry problem?
Trial-arm level
Dataset: One-arm-per-row structure
Using binomial structure of data directly: yi1 ∼ Bin(πi1, ni1)
and yi2 ∼ Bin(πi2, ni2)
logit(πi1) = ai1
logit(πi2) = ai1 + d + γi (2)
where γi ∼ N(0, τ2).

Results of diﬀerent models for TB dataset
Mean treatment effect
Models
FE Summary
(IVW)
FE Trial−arm
(MCMC)
RE Summary
(MOM)
RE Trial−arm
(MCMC)
−1.0 −0.5 0.0 0.5
Other
INLA
Table: Heterogeneity variance
τ2
Trial-arm RE -INLA 0.50
-MCMC 0.49
Summary RE -INLA 0.48
-MOM 0.37

Meta-regression
Motivation
Explore and possibly explaining heterogeneity
Mainly, achieved by including the summary-level covariates to
the model
Statistical methods
Random effects or fixed effect model using summary level or
trial-arm level
Weighted-least square technique (WLSQ), an extension of
MOM approach
Implemented in metafor (Viechtbauer, 2010)
Fully-Bayes with INLA: summary level or trial-arm level
logit(πi2) = ai1 + d + xiβ + γi

Results of meta-regression for TB dataset
Table: WLSQ vs INLA
Mean 2.5 % 97.5 %
Lat. -0.03 -0.05 -0.01
INLA -0.03 -0.05 -0.00
Year 0.00 -0.02 0.03
INLA 0.01 -0.03 0.04
τ2 0.07
INLA 0.12 0.01 0.76 −20 −10 0 10 20 30 40 50
−1.5−1.0−0.50.00.5
Bubble plot
Latitude (centered)
observedlogoddsratios
WLSQ
INLA

The need for a broader approach
Consider three treatments (1, 2, 3)
3
1
2
Solid lines indicate
comparisons are available
But, the estimate for
comparison Trt 2 vs Trt 3
d23? Multi-arm trials?
Indirect estimate of 2 vs 3
dInd
23 = dDir
12 − dDir
13

Terminology in NMA
From Graph theory: vertex, edge, cycle and spanning tree, i.e.
covering all vertices without any cycles
Consistency assumption
No discrepancy between indirect and direct estimates
dInd
23 = dDir
23
Need for statistical methods which account for inconsistency
The parametrization of the network
Determining the basic contrasts (db):
Treatment comparisons which deﬁne a spanning tree

Terminology in NMA
Functional contrasts (df ): can be written as functions of db
through linear relations
Design: set of treatments included in a trial; 1-2 design,
1-2-3 design
1
3
2
4
Example
db = {d12, d13, d14} (red
lines)
df = d24 = d12 − d14
Consistency relation
3-cycle

The Lu-Ades model (Lu and Ades, 2006)
Trial-arm level approach, accounting for the multi-arm trials
Trial-speciﬁc heterogeneity random eﬀects γi
But, for a multi-arm trial: dependency within trial!
Example: A three-arm trial i with the design 1-2-3
γi = (γi12, γi13)T
∼ Nc(0, Σγ)
A simple but a convenient structure is as follows (Higgins and
Whitehead, 1996):
Σγ =
τ2
τ2
/2
τ2
/2 τ2

The Lu-Ades model (cont.)
Cycle-specific approach
The inconsistency random effects: ωjkl ∼ N(0, κ2)
Multi-arm trials are inherently consistent
Number of inconsistency random effects: ICDF = #df − S; S
is the number of cycles only formed by a multi-arm trial.
No multi-arm trial: ICDF = #df
Otherwise, discount some 3-cycles!
ICDF must be calculated by “hand”
If we assume κ2 = 0, the model reduces to the consistency
model.

The Jackson model (Jackson et al., 2014)
The design-by-treatment interaction model with random
effects inconsistency parameters, Higgins et al. (2012) treated
them as fixed effects.
Advantage: average treatment comparison across designs can
be estimated
The Jackson model using trial-arm level approach
This model differs from Lu-Ades model by introducing
design-specific inconsistency random effects
logit(πik) = aij + djk + γijk + ωD
jk (3)
ωD = (ωjk1 , ωjk2 , . . . ) ∼ Nc(0, Σω) such that Σω has
diagonal entries κ2 and all others are κ2/2.

Smoking dataset (Hasselblad, 1998)
24 trials investigating four
interventions to aid smoking
cessation
Coding; 1: no contact, 2:
self-help, 3: individual
counseling and 4: group
counseling
8 designs, 1-3-4 and 2-3-4
three arm trials
Area of circle: participants;
width of line: trials
Network Plot
1
2
3
4

Results of NMA models for Smoking dataset
db = {d12, d13, d14}
BUGS/JAGS codes are
taken from Jackson et al.
(2014)
nmainla:::creatINLAdat
Blue points: post. medians,
red lines: 95 % Cr.I
INLA implementation of
Jackson model is new
κ ∼ U(0, 10)
Consistency model
Parameters
d12
d13
d14
Heter.
Stdev.
−0.5 0.0 0.5 1.0 1.5 2.0
MCMC
INLA

The Jackson model
0.0
0.2
0.4
0.6
−5 0 5
d12
0.00
0.25
0.50
0.75
1.00
−2.5 0.0 2.5 5.0 7.5
d13
0.0
0.2
0.4
0.6
0 5
d14
MCMC
INLA
0.0
0.5
1.0
1.5
0 1 2
τ2
0
2
4
6
0 1 2
κ2
Marginal posterior density
estimates of db, τ2 and κ2
Results show very good
agreement
Evidence for severe
heterogeneity, no evidence
for substantial inconsistency

Changing the coding of the interventions
4 interventions, 4! = 24
different coding
The results of the fitted
Smoking dataset with
different intervention coding
via INLA
Lu-Ades model substantially
depend on treatment
ordering!
But why?
ICDF κ τ
Consistency 0 0.00 0.81
Jackson 10 0.49 0.82
Lu-ades
1234, 1243 3 0.54 0.84
1324, 1423 3 0.62 0.83
1342, 1432 3 0.57 0.84
2314, 3214 3 2.01 0.79
3412, 4213 3 2.04 0.79

2-4 treatment comparison
2
3
4
Design inconsistency
between 2-4 (from two-arm
trial) and 2-4 (from
multi-arm trial)
However, some Lu-Ades
models allows this
inconsistency in the network,
whereas some other do not
Jackson model take into
account all possible
inconsistency in the network

Jackson vs Lu-Ades
With the presence of multi-arm trials, Jackson model should
be preferred
Moreover, Jackson model can be automated
Network with only two-arm trials, Lu-Ades may be preferred
Why INLA over MCMC for NMA
It is faster, not a simulation-based technique
No need to check any convergence diagnostics!
What we’ve learned
INLA implementation of pairwise meta-analysis models or
diﬀerent NMA models is possible

What we’ve contributed
INLA implementation of the Jackson model (including for
k-arm trials)
An R function meta.inla to ﬁt various pairwise meta-analysis
models
TB.datINLA <- creatINLAdat.dir(ntrt = TB$TRT, nctrl = TB$CON,
ptrt = TB$TRTTB, pctrl = TB$CONTB, cov1 = TB$Latitude,
cov2 = TB$Year)
inla.re.tb <- meta.inla(TB.datINLA, meanf = 0, varf = 1000,
mod = "RE", ul = 10, type = "trial-arm", mreg = FALSE)
print(inla.re.tb)
Call: meta.inla(datINLA = TB.datINLA, meanf = 0, varf = 1000, ul = 10,
mod = "RE", type = "trial-arm", mreg = FALSE)
Meta analysis using INLA
Posterior mean of treatment effect = -0.76 95% CrI ( -1.18, -0.35 )
Posterior mean of heterogeneity variance = 0.5 95% CrI ( 0.15, 1.29 )

Future research
INLA implementation of network meta-regression with
Jackson model
An R function, nma.inla to ﬁt diﬀerent NMA models
Function(s) for visualization (forest, bubble, network plots and
marginal posterior distributions)
Including those functions to the nmainla R package or
creating a new package nmabayes and uploading to CRAN
To make it more accessible for researchers

References I
Acknowledgements
Prof. Dr. Leonhard Held
Dr. Rafael Sauter
Coldlitz, G., Brewer, T., Berkey, C., Wilson, M., Burdick, E., Fineberg, H., and
Mosteller, F. (1994). Eﬃcacy of bcg vaccine in the prevention of
tuberculosis. J. Am. Med. Assoc, 271:698–702.
DerSimonian, R. and Laird, N. (1986). Meta-analysis in clinical trials.
Controlled clinical trials, 7(3):177–188.
Hasselblad, V. (1998). Meta-analysis of multitreatment studies. Medical
Decision Making, 18(1):37–43.

References II
Higgins, J., Jackson, D., Barrett, J., Lu, G., Ades, A., and White, I. (2012).
Consistency and inconsistency in network meta-analysis: concepts and
models for multi-arm studies. Research Synthesis Methods, 3(2):98–110.
Higgins, J. and Whitehead, A. (1996). Borrowing strength from external trials
in a meta-analysis. Statistics in medicine, 15(24):2733–2749.
Jackson, D., Barrett, J. K., Rice, S., White, I. R., and Higgins, J. (2014). A
design-by-treatment interaction model for network meta-analysis with
random inconsistency eﬀects. Statistics in medicine, 33(21):3639–3654.
Jackson, D., Boddington, P., and White, I. R. (2015). The design-by-treatment
interaction model: a unifying framework for modelling loop inconsistency in
network meta-analysis. Research synthesis methods, n/a–n/a.
Lu, G. and Ades, A. (2006). Assessing evidence inconsistency in mixed
treatment comparisons. Journal of the American Statistical Association,
101(474).
Lumley, T. (2002). Network meta-analysis for indirect treatment comparisons.
Statistics in medicine, 21(16):2313–2324.

References III
Lunn, D., Jackson, C., Best, N., Thomas, A., and Spiegelhalter, D. (2012).
The BUGS book: A practical introduction to Bayesian analysis. CRC press.
Rue, H., Martino, S., and Chopin, N. (2009). Approximate bayesian inference
for latent gaussian models by using integrated nested laplace
approximations. Journal of the royal statistical society: Series b (statistical
methodology), 71(2):319–392.
Sauter, R. and Held, L. (2015). Network meta-analysis with integrated nested
laplace approximations. Biometrical Journal, 57(6):1038–1050.
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor
package. Journal of Statistical Software, 36(3):1–48.

Extra slides
The Jackson model using trial-arm level approach
yij ∼ Bin(nij, πij) and yik ∼ Bin(πik, nik)
logit(πij) = aij
logit(πik) = aij + djk + γijk + ωD
jk
where γi ∼ N(0, Σγ) and ωD ∼ N(0, Σω)

The Lumley model (Lumley, 2002)
Summary level approach, only for networks with two arm trials
Inconsistency random eﬀects is added for each treatment
comparison and ωjk ∼ N(0, κ2)
Only for two arm trials!
Jackson et al. (2015)
It is proven that “The only model that contains all the Lu-Ades
models with all diﬀerent treatment orderings is the
design-by-treatment interaction model”.

The structure of covariance matrices
For heterogeneity
Σγ =
τ2 τ2/2
τ2/2 τ2
The assumption: The homogeneity of between-study
variations for every treatment comparison
For inconsistency
Σω =
κ2 κ2/2
κ2/2 κ2
The assumption: The homogeneity of inconsistency variations
for every treatment comparison

Network meta-analysis with integrated nested Laplace approximations

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