One MCMC run with 100’000 iterations and a burnin of 200’000 samples was used to get the posterior distributions for all models presented in Section 4.3 in the main text. The rather large number of MCMC iterations is the same as indicated by Dias et al. (2010) motivated by the node- splitting. They mention that most of the models reach convergence for much less iterations while some of the node-split models need so many sampling iterations to satisfy the applied diagnostic convergence criteria. For MCMC sampling we rely on the R-package gemtc (van Valkenhoef and Kuiper, 2014) using JAGS (Plummer, 2003).
The thrombolytic treatment dataset compares 9 different treatments (1: streptokinase , 2: t- PA, 3: accelerated t-PA, 4: streptokinase and t-PA, 5: reteplase, 6: tenecteplase, 7: PTCA, 8:
Biometrical Journal 52 (2004) 61 11
urokinase, 9: anistreptilase) and reports the number of deaths in 30 or 35 days and number of pa- tients in each treatment arm for 50 different trials. There are two three arm trials, one comparing treatments 1, 3 and 4 and one comparing treatments 1, 2 and 9. The network provides direct evi- dence for 16 different pairwise treatment comparisons (see also Figure 1). The pairwise treatment comparison for which a node-split is possible, meaning that independent direct and indirect evi- dence is available in the network, was assessed by using the function mtc.nodesplit.comparisons available in the gemtc package. As treatment 6 was only compared in one trial with treatment 3 there is no indirect evidence and thus there is no node-split possible for d36. There is also
no node-split for treatment 1 and 4 as there is only one other three arm trial which compares treatment 4 with treatment 3. As we assume no inconsistency within a multi-arm trial we have no other independent source of indirect evidence for d14. There remain 14 possible node-splits for
the direct relative treatment comparisons. The 14 possible node-splits between treatments t1 and t2are contained in the data frame nodecomp_thrombdat produced by the R-code above:
nodecomp_thrombdat t1 t2 1 1 2 2 1 3 3 1 5 4 1 7 5 1 8 6 1 9 7 2 7 8 2 8 9 2 9 10 3 4 11 3 5 12 3 7 13 3 8 14 3 9
We use the same measure of inconsistency as proposed by Dias et al. (2010). They define the measure for the degree of inconsistency (ddiff.
jk ) as the difference of the log-odds ratios based on
direct and indirect evidence, i.e. ddiff.
jk = djkdir.− djkind.. We thus compute the posterior distribution
of ddiff.
jk . The result for the inconsistency estimates are shown in Table 2 for the fixed effects model
and the model with random effects for heterogeneity obtained by MCMC and INLA. The results are consistent with the ones discussed in table 2 in Dias et al. (2010).
The largest inconsistency in the random and fixed effect model is found if the node for treatment 3 and 9 is split. The estimate for the marginal posterior mean of the relative treatment d39based
on the direct evidence is 1.36 for the random effect model by INLA. The corresponding estimate based on indirect evidence is 0.16 while the analysis based on the full data under the consistency assumption gives a marginal posterior mean estimate ˆd39= 0.30 with INLA. A cross-validation
approach which does not use the complete data to estimate the baseline treatment effect and heterogeneity hyperparameter yields an inconsistency estimate for the node d39 equal to 1.28
with a standard error equal to 3.65. The difference to the inconsistency estimate based on node splitting is with 1.21 not very large. The estimate for the standard error of the inconsistency is with node splitting only equal to 0.43 as reported in Table 2. The large difference in the uncertainty
12 Rafael Sauter and Leonhard Held: Supplementary Material for network meta-analysis with INLA
Fixed effects Random effects
MCMC INLA MCMC INLA
Mean Stdev. Mean Stdev. Mean Stdev. Mean Stdev.
ddiff. 12 -0.342 0.258 -0.186 0.234 -0.342 0.258 -0.242 0.277 ddiff. 13 0.088 0.105 0.090 0.104 0.088 0.105 0.251 0.247 ddiff. 15 0.115 0.120 0.116 0.121 0.115 0.120 0.397 0.337 ddiff. 17 -0.273 0.219 -0.269 0.220 -0.273 0.219 -0.227 0.246 ddiff. 18 -0.203 0.574 -0.184 0.559 -0.203 0.574 -0.126 0.585 ddiff. 19 -0.453 0.255 -0.406 0.252 -0.453 0.255 -0.394 0.281 ddiff. 27 -0.078 0.430 -0.049 0.422 -0.078 0.430 0.017 0.450 ddiff. 28 -0.156 0.453 -0.143 0.446 -0.156 0.453 -0.126 0.474 ddiff. 29 -0.419 0.246 -0.426 0.245 -0.419 0.246 -0.510 0.290 ddiff. 34 -0.588 0.706 -0.649 0.668 -0.588 0.706 -0.868 0.726 ddiff. 35 -0.116 0.121 -0.116 0.121 -0.116 0.121 -0.397 0.337 ddiff. 37 0.263 0.211 0.258 0.210 0.263 0.211 0.213 0.240 ddiff. 38 0.284 0.463 0.267 0.453 0.284 0.463 0.216 0.484 ddiff. 39 1.233 0.418 1.196 0.409 1.233 0.418 1.209 0.425
Table 2 Inconsistency mean and standard deviation for all node-splits of the thrombolytic treatment network for the fixed effects and the random effects model obtained by MCMC and by INLA.
direct comparison of d39 only based on two observations. In the case of the fixed effect model
the difference between cross-validation and node-splitting would be smaller as the heterogeneity parameter τ2is equal to zero. The node-split for the moderately large network of the thrombolytic
treatment network with 14 node-splits and heterogeneity random effects took 192.9 minutes with 80’000 MCMC iterations which is 827 seconds per node-split or model. INLA was about 276 times faster and only used 42 seconds to complete the 14 node-splits which is 3.0 seconds per node-split. This difference in computation time is directly scalable with an increasing number of node-splits in a network.
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