Network meta-analysis with integrated nested Laplace approximations

Network meta-analysis with integrated nestedLaplace approximations

Burak Kursad Gunhan

Supervised by Prof. Dr. Leonhard Held and Rafael SauterMaster exam

Zurich, 01 March 2016

Meta-analysis Network meta-analysis Conclusions References

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

Burak Kursad Gunhan 2/ 30


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



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



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



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



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).



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



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



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



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



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



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



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

]



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.



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.



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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4



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



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

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0 5

d14

MCMC

INLA

0.0

0.5

1.0

1.5

0 1 2

τ2

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0 1 2

κ2

Marginal posterior densityestimates of db, τ2 and κ2

Results show very goodagreementEvidence for severeheterogeneity, no evidencefor substantial inconsistency



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



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



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



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 )



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



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.



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.



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.



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)



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”.



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


Health & Medicine

Network meta-analysis with integrated nested Laplace approximations