The NMA models discussed above are hierarchical models that can be described by three stages: the first stage is the observational model p(y | α) with respect to the observed data y where α = (a, db, γ, ξ) includes all model parameters. Here a = (a1 j, . . . , aS j)denotes the vector of all baseline treatment effects in the model for trial-arm level data and is not needed for a model based on summary level data. The random effects vectorγ contains all trial-specific random effects γi. Likewise,ξ contains all incoherence random effectsξjkof the model based on summary level data orξjkl in the case of trial-arm level data. The second stage p(α | θ) is a latent Gaussian Markov random field (GMRF) as described in chapter 2 of Rue and Held (2005). The GMRF is controlled by hyperparameters, here θ = (τ2, κ2), which build the third stage p(θ).
The INLA approach proposed by Rue et al. (2009) generates accurate approximations to the marginal posterior distributions of the latent Gaussian modelα by applying a Laplace approximation (Tierney and Kadane, 1986) to the posterior distribution of θ and a second Laplace approximation to the posterior of the components ofα for selected values of the hyperparameters. The approximation of the marginal posterior distributions to the model parametersα is obtained by numerical integration over the hyperparameters. INLA were shown to deliver accurate approximations with reduced com- putational costs compared to MCMC in a variety of examples. Fong et al. (2010) demonstrated the applicability of INLA to generalized linear-mixed models. See Schr ¨odle et al. (2011) for applications to spatio-temporal models and Paul et al. (2010) for a bivariate meta-analysis of diagnostic tests with INLA. The available software package for INLA embraces a wide range of models that is progressively expanded as for example by measurement error models described by Muff et al. (2015).
The software r-inla implements the INLA approach, is available on http://www.r-inla.org/ and includes an R-interface. In the following Sections 3.1 and 3.2, we discuss the implementa- tion of NMA in r-inla for summary and a trial-arm level data models. The node-split approach with r-inla is introduced in Section 3.3. Each feature discussed in Subsections 3.1 to 3.3 is com- plemented by an application in Section 4. We compare the results obtained with the INLA ap- proach with MCMC, where we rely on the R-package gemtc (van Valkenhoef and Kuiper, 2014) using JAGS (Plummer, 2003) a generic MCMC sampler using BUGS-code. However, the gemtc- package can not fit models with both heterogeneity and incoherence random effects, as shown in Table 1 that gives an overview of which models can be estimated with the utilized software pack- ages. The NMA model with incoherence, for summary level data could be estimated by extending the gemtc-code and for trial-arm level data the BUGS-code provided by Lu and Ades (2006) on http://www.bristol.ac.uk/social-community-medicine/projects/mpes/code/ was adapted to fit into JAGS.
Table 1 Used software packages for NMA: ‘x’ can be estimated, ‘na’ not available.
r-inla gemtc JAGS
NMA models for summary level data:
random effects (heterogeneity+ inconsistency) x na x
NMA models for trial-arm level data:
fixed effects x x x
Biometrical Journal 57 (2015) 6 1043
Source code for MCMC sampling and the r-inla implementation and datasets together with an accompanying R-package to reproduce the results is available as Supporting Information on the journal’s web page.
3.1. Summary level data
The NMA model for summary level data described in Eq. (2) is a linear mixed model. Linear-mixed models with different random effects can be implemented in r-inla (Fong et al., 2010) and Schmidt and Nehmiz (2014) used INLA to estimate such a NMA model with hazard ratios as effect measures and compared the results to the ones by MCMC.
However, there are two peculiarities in the linear-mixed model (2) that need special attention: weighting the outcome with the inverse of the squared standard error σ2
i jk and inference for the functional contrasts df. As in a standard meta-analysis, we want to useσi jk2 as inverse weight. This can be done in r-inla by using the scale argument in the function call. As discussed in Sections 2.2 and 2.1, the functional contrasts df can be described as linear combination of dband we want to get the marginal posterior distributions of df. The r-inla software package allows to compute the marginal posterior distribution of any linear combination of the latent field (see Section 4.4 in Martins et al., 2013) and thus the full marginal distributions for df can be obtained. If one is interested in the functional contrasts one has to define these linear combinations in advance and hand them over in the lincomb argument of the r-inla function call. Further implementation details for these two peculiarities are described in the Supplementary Material.
3.2 Trial-arm level data
The NMA model for trial-arm level data described by (3) to (5) is a binary regression model with logit link function. The distribution of the response must be specified in r-inla by the family argument and the total number of study participants needs to be specified by the argument Ntrials. Functional contrasts can be obtained in r-inla by building linear combinations in the same way as described in Section 3.1. In the case of a multiarm trial we want to correlate the random effects of the latent field for all arms of the same trial i, as discussed in Section 2.2. The use of such correlated multivariate random effects is possible in r-inla. Riebler et al. (2012) use INLA for correlated multivariate age- period-cohort models. Their model includes several countries and for each country a second order random walk over time is used. These random walks are correlated across countries. Here, we want to introduce an exchangeable correlation structure across trial-arms. Correlating the random effects for the outcomes of a three-arm trial means that we correlate two components of the latent field.
To implement the correlation across study arms in r-inla we need to rewrite the homogeneous random effects covariance Ti. Using the uniform correlation matrix Ci= (1 − ρ)Ii+ ρJi, where Iiis the identity matrix and Jiis a matrix of ones, both of dimension(Ki− 1), we can write Ti= Ci⊗ τ2 where⊗ is the Kronecker product. This describes the random effects correlation structure for the response vector yi of trial i. As described in Section 2.2, we need to fix the hyperparameter ρ to the initial value 1/2 but this initial value must be transformed to the internal scale used by r-inla (see Section 1.2 in Supplementary Material). The homogeneous correlation model described here is implemented in r-inla under the name model=‘exchangeable’.
To identify the corresponding random effects in each trial, we need to define a grouping vector gi that is needed to determine the structure of Ci. The first entry of giis NA as no random effect is present in the baseline model (3). The remaining entries are numbered from 1 to Ki− 1, representing the random effectsγi j1, . . . , γi j,(Ki−1)in model (4). For example, if trial i compares treatments 1, 2, and 3, we have a bivariate random effects vectorγi= (γi12, γi13)and the grouping vector is gi= (NA, 1, 2). See section 4.6 in Martins et al. (2013) and the Supplementary Material for further details about this r-inla feature.
1044 R. Sauter and L. Held: Network meta-analysis with INLA
3.3. Assessing inconsistency with node-splitting
A key element of every NMA is to assess the degree and source of inconsistency in the network. Evidence inconsistencies can be modeled by random effects for treatment pairs (Section 2.1) or for independent loops (Section 2.2). An alternative to the random effects approach is to compare only the direct evidence of a relative treatment effect with the indirect evidence for the same relative treatment effect. This procedure corresponds to a cross-validation of the network leaving out all observations covering direct evidence on a specific treatment pair, as discussed by Lumley (2002, Section 5.2) for summary level, and by Lu and Ades (2006, Section 5.3) for trial-arm level data.
A modification of cross-validation is node-splitting as suggested by Dias et al. (2010, Section 3.5). As in cross-validation, the data are split into two independent sources of information on direct and indirect evidence for a pair of treatments j and k, say. However, in the node-split approach the heterogeneity hyperparameterτ2is estimated based on all the data. Assessing inconsistency by node-
splitting implies that the model is fitted repeatedly for every directly observed comparison of treatments
j and k where also indirect evidence is available. For every model fit, we obtain a posterior distribution
for the hyperparameterτ2, for the baseline treatment effects a and for the direct and indirect relative
treatment effects ddir.
jk and dind.jk .
The node-split approach can be implemented in r-inla using a joint model with two separate likelihood functions. Specifically, suppose treatment j and k are directly compared in s trials Djk⊂ {1, . . . , S}. In trial i ∈ Djkthere are 2 responses, yi jand yik, which form the direct evidence and are incorporated as vector
Y( j,k)= (yi
1j, yi1k, yi2j, yi2k, . . . , yisj, yisk)
in the first likelihood. There are K=Si=1Ki treatment arms in total, so the indirect evidence is represented by the remaining K− 2s responses, incorporated as vector Y−( j,k), say, in the second likelihood. The unknown parameters a, db, and γ must be organized accordingly in the two likelihoods. Of note, r-inla allows that the two likelihoods share some common parameters, here the heterogeneity varianceτ2. We can therefore use r-inla to compute the posterior distribution of the difference of the
direct treatment effect ddir.
jk and the indirect treatment effect dind.jk from the first and second likelihood, respectively, as a linear combination.
Martino et al. (2011) describe a joint model where one likelihood defines data for survival times while the second likelihood defines a quantitative measure for a longitudinal profile of the same patient. They implement a joint model in r-inla with two likelihoods where a hyperparameter for the frailties in the survival likelihood and the random effects in the longitudinal likelihood is shared. See Martins et al. (2013, Section 4.1) for further details on how to use two likelihoods in r-inla.