Network meta-analysis with integrated nestedLaplace approximations
Burak Kursad Gunhan
Supervised by Prof. Dr. Leonhard Held and Rafael SauterMaster exam
Zurich, 01 March 2016
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Systematic review
Review of evidences from different studiesOn a specific question, methods to identify, select, appraiseand summarize similar but separate studiesStudy selection: inclusion and exclusion criterion
Meta-analysis (The analysis of analyses)
Quantitative part of systematic reviewSR may or may not include a meta-analysis!Using statistical methods to combine results from differentstudies
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TB dataset (Coldlitz et al., 1994)
13 vaccine controlledtrials of BCG forprevention of TBYear and Latitudevariables are givenMeasure of treatmenteffect: Log odds ratioObserved log oddsratios95 % Wald C.I.sArea of boxes: 1/σ2
i
Forest plot
log odds ratio
Tria
ls
1
2
3
4
5
6
7
8
9
10
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12
13
−2 −1 0 1 2
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Statistical methods for meta-analysis
1 Fixed effect modelAssumption: common true treatment effect
θ̂i ∼ N (θ, σ2i )
Inverse variance-weighted method (ωi = 1/σ2i )
θ̂IVW =∑ki=1 ωiθ̂i∑ki=1 ωi
and Var(θ̂IVW ) = 1∑ki=1 ωi
Between-trial variability?e. g. study populations
2 Random effects model: Accounting heterogeneity
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Different approaches for RE models
Likelihood approach, adapted from Lumley (2002)A linear mixed model containing components for samplingvariability and heterogeneity
θ̂i|θi ∼ N (θi, σ2i )
θi ∼ N (d+ γi, σ2i )
γi ∼ N (0, τ2) (1)
where d mean treatment effect and τ2 heterogeneity varianceMethod of moments (MOM), by DerSimonian and Laird(1986)
ωi = 1/(σ2i + τ2)
Available from metafor (Viechtbauer, 2010) R packageIf τ2 = 0, then fixed effect model
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Fully Bayes approach
The model formulation same as equation (1), but assigningprior distributions for d and τUsing uninformative priors: d ∼ N (0, 1000); τ ∼ U(0, 10)
Inference methodsMCMC: simulation-based technique, very popular
Implemented by using JAGS with R2jags R packageConvergence diagnostics checked!JAGS code is taken from Lunn et al. (2012)
INLA: An approximate Bayesian inference technique by Rueet al. (2009) with INLA R package
Shown to be suitable for meta-analysis inference by Sauter andHeld (2015)Main goal: INLA implementation of the models
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Two modelling approaches
Summary-level
Dataset: One-study-per-row structureZero entry problem?
Trial-arm levelDataset: One-arm-per-row structureUsing 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).
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Results of different models for TB dataset
Mean treatment effect
Mod
els
FE Summary(IVW)
FE Trial−arm(MCMC)
RE Summary(MOM)
RE Trial−arm(MCMC)
−1.0 −0.5 0.0 0.5
OtherINLA
Table: Heterogeneity variance
τ2
Trial-arm RE -INLA 0.50-MCMC 0.49
Summary RE -INLA 0.48-MOM 0.37
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Meta-regression
MotivationExplore and possibly explaining heterogeneityMainly, achieved by including the summary-level covariates tothe model
Statistical methodsRandom effects or fixed effect model using summary level ortrial-arm levelWeighted-least square technique (WLSQ), an extension ofMOM approach
Implemented in metafor (Viechtbauer, 2010)Fully-Bayes with INLA: summary level or trial-arm level
logit(πi2) = ai1 + d+ xiβ + γi
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Results of meta-regression for TB dataset
Table: WLSQ vs INLA
Mean 2.5 % 97.5 %
Lat. -0.03 -0.05 -0.01INLA -0.03 -0.05 -0.00Year 0.00 -0.02 0.03INLA 0.01 -0.03 0.04τ2 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)
obse
rved
log
odds
rat
ios
WLSQINLA
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The need for a broader approach
Consider three treatments (1, 2, 3)
3
1
2
Solid lines indicatecomparisons are availableBut, the estimate forcomparison Trt 2 vs Trt 3d23? Multi-arm trials?Indirect estimate of 2 vs 3
dInd23 = dDir12 − dDir13
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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
dInd23 = dDir23
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
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Terminology in NMA
Functional contrasts (df ): can be written as functions of dbthrough linear relationsDesign: set of treatments included in a trial; 1-2 design,1-2-3 design
1
3
2
4
Exampledb = {d12, d13, d14} (redlines)df = d24 = d12 − d14Consistency relation3-cycle
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The Lu-Ades model (Lu and Ades, 2006)
Trial-arm level approach, accounting for the multi-arm trialsTrial-specific heterogeneity random effects γiBut, 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 andWhitehead, 1996):
Σγ =[τ2 τ2/2τ2/2 τ2
]
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The Lu-Ades model (cont.)
Cycle-specific approachThe inconsistency random effects: ωjkl ∼ N (0, κ2)
Multi-arm trials are inherently consistentNumber of inconsistency random effects: ICDF = #df − S; Sis the number of cycles only formed by a multi-arm trial.No multi-arm trial: ICDF = #dfOtherwise, discount some 3-cycles!ICDF must be calculated by “hand”
If we assume κ2 = 0, the model reduces to the consistencymodel.
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The Jackson model (Jackson et al., 2014)
The design-by-treatment interaction model with randomeffects inconsistency parameters, Higgins et al. (2012) treatedthem as fixed effects.Advantage: average treatment comparison across designs canbe estimatedThe Jackson model using trial-arm level approachThis model differs from Lu-Ades model by introducingdesign-specific inconsistency random effects
logit(πik) = aij + djk + γijk + ωDjk (3)
ωD = (ωjk1 , ωjk2 , . . . ) ∼ Nc(0,Σω) such that Σω hasdiagonal entries κ2 and all others are κ2/2.
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Smoking dataset (Hasselblad, 1998)
24 trials investigating fourinterventions to aid smokingcessationCoding; 1: no contact, 2:self-help, 3: individualcounseling and 4: groupcounseling8 designs, 1-3-4 and 2-3-4three arm trialsArea of circle: participants;width of line: trials
Network Plot
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Results of NMA models for Smoking dataset
db = {d12, d13, d14}BUGS/JAGS codes aretaken from Jackson et al.(2014)nmainla:::creatINLAdatBlue points: post. medians,red lines: 95 % Cr.IINLA implementation ofJackson model is newκ ∼ U(0, 10)
Consistency model
Pa
ram
ete
rs
d12
d13
d14
Heter.Stdev.
−0.5 0.0 0.5 1.0 1.5 2.0
MCMCINLA
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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 densityestimates of db, τ2 and κ2
Results show very goodagreementEvidence for severeheterogeneity, no evidencefor substantial inconsistency
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Changing the coding of the interventions
4 interventions, 4! = 24different codingThe results of the fittedSmoking dataset withdifferent intervention codingvia INLALu-Ades model substantiallydepend on treatmentordering!But why?
ICDF κ τ
Consistency 0 0.00 0.81Jackson 10 0.49 0.82Lu-ades
1234, 1243 3 0.54 0.841324, 1423 3 0.62 0.831342, 1432 3 0.57 0.842314, 3214 3 2.01 0.793412, 4213 3 2.04 0.79
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2-4 treatment comparison
2
3
4
Design inconsistencybetween 2-4 (from two-armtrial) and 2-4 (frommulti-arm trial)However, some Lu-Adesmodels allows thisinconsistency in the network,whereas some other do notJackson model take intoaccount all possibleinconsistency in the network
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Jackson vs Lu-AdesWith the presence of multi-arm trials, Jackson model shouldbe preferredMoreover, Jackson model can be automatedNetwork with only two-arm trials, Lu-Ades may be preferred
Why INLA over MCMC for NMA
It is faster, not a simulation-based techniqueNo need to check any convergence diagnostics!
What we’ve learnedINLA implementation of pairwise meta-analysis models ordifferent NMA models is possible
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What we’ve contributed
INLA implementation of the Jackson model (including fork-arm trials)An R function meta.inla to fit various pairwise meta-analysismodels
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 INLAPosterior 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 )
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Future research
INLA implementation of network meta-regression withJackson modelAn R function, nma.inla to fit different NMA modelsFunction(s) for visualization (forest, bubble, network plots andmarginal posterior distributions)Including those functions to the nmainla R package orcreating a new package nmabayes and uploading to CRANTo make it more accessible for researchers
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References I
Acknowledgements
Prof. Dr. Leonhard HeldDr. Rafael Sauter
Coldlitz, G., Brewer, T., Berkey, C., Wilson, M., Burdick, E., Fineberg, H., andMosteller, F. (1994). Efficacy of bcg vaccine in the prevention oftuberculosis. 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. MedicalDecision Making, 18(1):37–43.
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References II
Higgins, J., Jackson, D., Barrett, J., Lu, G., Ades, A., and White, I. (2012).Consistency and inconsistency in network meta-analysis: concepts andmodels for multi-arm studies. Research Synthesis Methods, 3(2):98–110.
Higgins, J. and Whitehead, A. (1996). Borrowing strength from external trialsin a meta-analysis. Statistics in medicine, 15(24):2733–2749.
Jackson, D., Barrett, J. K., Rice, S., White, I. R., and Higgins, J. (2014). Adesign-by-treatment interaction model for network meta-analysis withrandom inconsistency effects. Statistics in medicine, 33(21):3639–3654.
Jackson, D., Boddington, P., and White, I. R. (2015). The design-by-treatmentinteraction model: a unifying framework for modelling loop inconsistency innetwork meta-analysis. Research synthesis methods, n/a–n/a.
Lu, G. and Ades, A. (2006). Assessing evidence inconsistency in mixedtreatment 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.
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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 inferencefor latent gaussian models by using integrated nested laplaceapproximations. Journal of the royal statistical society: Series b (statisticalmethodology), 71(2):319–392.
Sauter, R. and Held, L. (2015). Network meta-analysis with integrated nestedlaplace approximations. Biometrical Journal, 57(6):1038–1050.
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metaforpackage. Journal of Statistical Software, 36(3):1–48.
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Extra slides
The Jackson model using trial-arm level approachyij ∼ Bin(nij , πij) and yik ∼ Bin(πik, nik)
logit(πij) = aij
logit(πik) = aij + djk + γijk + ωDjk
where γi ∼ N (0,Σγ) and ωD ∼ N (0,Σω)
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The Lumley model (Lumley, 2002)
Summary level approach, only for networks with two arm trialsInconsistency random effects is added for each treatmentcomparison 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-Adesmodels with all different treatment orderings is thedesign-by-treatment interaction model”.
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The structure of covariance matrices
For heterogeneity
Σγ =[τ2 τ2/2τ2/2 τ2
]
The assumption: The homogeneity of between-studyvariations for every treatment comparisonFor inconsistency
Σω =[κ2 κ2/2κ2/2 κ2
]
The assumption: The homogeneity of inconsistency variationsfor every treatment comparison
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