Network meta-analysis using integrated nested Laplace approximations - PowerPoint PPT Presentation

Network meta-analysis using integrated nested Laplace approximations (INLA) Burak Kürsad Günhan 1 Tim Friede 1 Leonhard Held 2 1 Department of Medical Statistics, University Medical Center Göttingen, Göttingen, Germany 2 Epidemiology, Biostatistics and Prevention Institute, University of Zürich, Zürich, Switzerland Mainz, December 02, 2016 This project has received funding from the European Union’s Seventh Framework Programme for research, tech- nological development and demonstration under grant agreement number FP HEALTH 2013-602144.

Introduction Network meta-analysis Application Conclusions and outlook 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 criteria 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 Kürsad Günhan 2/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Conventional meta-analysis Only two treatments are compared 2 Trt 1 vs Trt 2 can be estimated ( d 1 , 2 ) Direct estimate Heterogeneity between trials 1 Pairwise meta-analysis Meta-regression Burak Kürsad Günhan 3/ 23

Introduction Network meta-analysis Application Conclusions and outlook References More than two treatments? Increasing number of 3 treatments 1 Solid lines indicate comparisons are available A generalization of pairwise meta-analysis Indirect estimate of 2 vs 3 2 d Ind 2 , 3 = d Dir 1 , 2 − d Dir 1 , 3 Burak Kürsad Günhan 4/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Terminology in NMA (Salanti, 2012) If both direct and indirect estimates are available for d 1 , 2 Consistency : No discrepancy between indirect and direct estimates d Dir 1 , 2 = d Ind 1 , 2 Consistency relation d Dir 1 , 2 = d Dir 1 , 3 − d Dir 2 , 3 Trials of different comparisons were undertaken in different periods Right-hand side parameters are basic parameters ( d b ) ⇒ Parametrization of the network Others are functional parameters ( d f ) Burak Kürsad Günhan 5/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Terminology in NMA From Graph theory: vertex, edge, cycle, spanning tree Design : set of treatments included in a trial; 1-2 design, 1-2-3 design 3 Example d b = { d 12 , d 13 , d 14 } (red 1 lines) ⇒ d f = d 24 = d 12 − d 14 4 Consistency relation ⇒ 3-cycle 2 Burak Kürsad Günhan 6/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Statistical models for NMA Arm-based instead of contrast-based models ⇒ Advantage: one-stage approach, exact likelihood Bayesian hierarchical models, more specifically generalized linear mixed models (GLMMs) Datasets with different endpoints (binomial, continuous, survival) can be modelled Basic model is same, but likelihood and link function can change Burak Kürsad Günhan 7/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Consistency models (Dias et al., 2011) For convenience, consider data with binomial endpoints In trial i ; j, k is treatment pair where j baseline treatment, k remaining treatment Number of events, y ik ∼ Bin ( π ik , n ik ) and y ij ∼ Bin ( π ij , n ij ) Logit link, model equations: logit ( π ij ) = µ i logit ( π ik ) = µ i + d jk + γ ijk where µ i nuisance parameter and d jk basic parameters Heterogeneity random effects: γ ijk ∼ N (0 , τ 2 ) Burak Kürsad Günhan 8/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Consistency models (Dias et al., 2011) (cont.) But, for a multi-arm trial: dependency within trial! Example: A three-arm trial i with the design 1-2-3 γ i = ( γ i 12 , γ i 13 ) T ∼ N 2 ( 0 , Σ γ ) A simple but a convenient structure is as follows (Higgins and Whitehead, 1996): � τ 2 τ 2 / 2 � Σ γ = τ 2 / 2 τ 2 Some comments Basic parameters can be any T − 1 treatment comparisons For continuous endpoints, normal likelihood and identity link Consistency is assumed in the network! Models are needed to account for inconsistency in the network Burak Kürsad Günhan 9/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Lu-Ades Model (Lu and Ades, 2006) Uses cycle-inconsistency approach Assumption: inconsistency only occurs from 3-cycles Basic parameters should form a spanning tree Cycle-specific inconsistency random effects: ω jkl ∼ N (0 , κ 2 ) Multi-arm trials are inherently consistent Number of inconsistency random effects: ICDF = # d f − S where S is the number of cycles only formed by a multi-arm trial Algorithm for ICDF (van Valkenhoef et al., 2012), but not efficient In the presence of multi-arm trials, results depend on treatment ordering! Burak Kürsad Günhan 10/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Jackson Model (Jackson et al., 2014) Uses design-inconsistency approach (Higgins et al., 2012) Design inconsistency : occurs between trials involving different designs 1,2,3 trials can be inconsistent with 1,2 trials Adding more inconsistency parameters to the model Inconsistency parameters as random effects logit ( π ik ) = a ij + d jk + γ ijk + ω D jk ω D = ( ω jk 1 , ω jk 2 , . . . ) ∼ N c ( 0 , Σ ω ) such that Σ ω has diagonal entries κ 2 and all others are κ 2 / 2 NMA-regression: incorporating trial-specific covariates to the model in order to explain sources of inconsistency Burak Kürsad Günhan 11/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Fully-Bayesian inference for NMA models Markov Chain Monte Carlo (MCMC) A simulation-based technique and the most popular Popular MCMC-tools: WinBUGS, JAGS or Stan Integrated Nested Laplace Approximations (INLA) An approximate Bayesian method (Rue et al., 2009) for latent Gaussian models (LGMs) Fast and accurate alternative to MCMC How INLA works (Rue et al., 2016)? Laplace approximations & numerical integration Implemented in R-INLA (http://www.r-inla.org/) Burak Kürsad Günhan 12/ 23

Introduction Network meta-analysis Application Conclusions and outlook References INLA for NMA models By Sauter and Held (2015), INLA can be used for many NMA models My goal: Extend INLA implementation to different NMA models (Jackson model, NMA-regression) and also automation How NMA models are LGMs? Three stages: Observational model: p ( y | α ) where α = ( µ , d b , x , γ , ω ) 1 Latent Gaussian field: p ( α | θ ) 2 Hyperparameters: θ = ( τ 2 , κ 2 ) 3 Burak Kürsad Günhan 13/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Smoking dataset (Hasselblad, 1998) Network Plot 24 trials investigating four 1 interventions to aid smoking cessation Coding; 1: no contact, 2: self-help, 3: individual 4 2 counseling and 4: group counseling Area of circle: participants; width of line: trials 3 8 designs, 1-3-4 and 2-3-4 three arm trials Burak Kürsad Günhan 14/ 23

Introduction Network meta-analysis Application Conclusions and outlook References MCMC vs INLA Consistency model d b = { d 12 , d 13 , d 14 } τ Priors: d 1 x ∼ N (0 , 1000) , d 14 τ ∼ U (0 , 5) , κ ∼ U (0 , 5) . d 13 MCMC using JAGS JAGS code (Jackson d 12 et al., 2014) 0.0 0.5 1.0 1.5 2.0 Convergence diagnostics MCMC 95% CI INLA 95% CI Burak Kürsad Günhan 15/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Jackson model 0.8 0.9 0.6 0.6 0.4 0.3 0.2 0.0 0.0 −4 0 4 8 −4 0 4 8 d 12 d 13 0.6 0.4 MCMC 0.2 INLA 0.0 −5 0 5 d 14 6 1.5 4 1.0 0.5 2 0.0 0 0 1 2 0 1 2 τ 2 κ 2 Burak Kürsad Günhan 16/ 23

Introduction Network meta-analysis Application Conclusions and outlook References Jackson vs Lu-Ades model using INLA ICDF κ τ 4 interventions, 4! = 24 Consistency 0 0.00 0.81 possibilities of coding Jackson 10 0.39 0.82 Lu-Ades model substantially Lu-ades depend on treatment 1234, 1243 3 0.52 0.84 ordering! 1324, 1423 3 0.60 0.83 Confirmation of Higgins 1342, 1432 3 0.55 0.84 et al. (2012) 2314, 3214 3 1.39 0.79 3412, 4213 3 1.40 0.79 Burak Kürsad Günhan 17/ 23

Introduction Network meta-analysis Application Conclusions and outlook References nmainla R package Installation via devtools (Wickham and Chang, 2016) R package devtools:: install_github ('gunhanb/nmainla') Data preparation SmokdatINLA <- create_INLA_dat (dat = Smokdat, # one-study-per-row dataset armVars = c ('treatment' = 't', 'responders' = 'r', 'sampleSize' = 'n'), nArmsVar = 'na', design = 'des') Fitting a Jackson model nma_inla (SmokdatINLA, likelihood = 'binomial', fixed.par = c (0, 1000), type = 'jackson', tau.prior = 'uniform', tau.par = c (0, 5), kappa.prior = 'uniform', kappa.par = c (0, 5)) Burak Kürsad Günhan 18/ 23