 Technical advance
 Open Access
 Published:
Hypothesis testing in Bayesian network metaanalysis
BMC Medical Research Methodology volume 18, Article number: 128 (2018)
Abstract
Background
Network metaanalysis is an extension of the classical pairwise metaanalysis and allows to compare multiple interventions based on both headtohead comparisons within trials and indirect comparisons across trials. Bayesian or frequentist models are applied to obtain effect estimates with credible or confidence intervals. Furthermore, pvalues or similar measures may be helpful for the comparison of the included arms but related methods are not yet addressed in the literature. In this article, we discuss how hypothesis testing can be done in a Bayesian network metaanalysis.
Methods
An index is presented and discussed in a Bayesian modeling framework. Simulation studies were performed to evaluate the characteristics of this index. The approach is illustrated by a real data example.
Results
The simulation studies revealed that the type I error rate is controlled. The approach can be applied in a superiority as well as in a noninferiority setting.
Conclusions
Test decisions can be based on the proposed index. The index may be a valuable complement to the commonly reported results of network metaanalyses. The method is easy to apply and of no (noticeable) additional computational cost.
Background
Network metaanalysis (NMA), as an extension of the classical pairwise metaanalysis, is gaining acceptance and popularity in medical research. The general idea is to include all evidence at hand about a specific research question in one single model. The classical pairwise metaanalysis is limited to twoarm comparisons of interventions that were directly compared in trials. An NMA can include any number of treatments as well as interventions that have not been investigated headtohead. Several approaches (frequentist and Bayesian) were introduced and extended during recent years. Thus, a framework of modeling techniques is available to implement an NMA in many different data situations. Efthimiou et al. and Dias et al. give very useful overview of recent developments [1, 2]. Alongside the benefits those procedures provide, many challenges arise when applying an NMA model. First, all the issues that are already known from pairwise metaanalysis, like heterogeneity, have to be addressed. In addition, new items, like inconsistency which denotes the problem of deviations between direct and indirect estimates, have to be taken into consideration (see, for example, Dias et al. [3]).
As a result of an NMA, point estimates with credible intervals of pairwise effects between treatment arms are obtained. In this article, we focus on the issue of testing for superiority or noninferiority between treatment arms in an NMA model. For Bayesian modelling, we present and discuss an index υ that can be used for hypothesis testing within the network. Similar ideas were presented in the article by Rücker and Schwarzer [4] in a frequentist framework. However, we focus on Bayesian modeling. Furthermore, while we apply the index for a test procedure, Rücker and Schwarzer use their approach to rank treatment arms.
General modeling in NMA
The concept of NMA in a Bayesian framework was introduced by Higgins and Whitehead [5]. Many extensions and discussions about the idea were published in recent years. Introductions and overviews can be found in the literature [1, 2, 6, 7]. Here, we only present the basic idea of the modeling procedure. For this, we assume throughout this paper that the outcome is binary (e.g., success / no success, or failure / no failure).
The following notation is used. N is the number of trials, K the number of arms, p_{ik} the success (or failure) probability, and N_{ik} the sample size of arm k in study i. In the setting of a binary outcome, we apply two different approaches: Either by use of the binomial distributions directly or by calculating the log odds ratios (OR) for each trial which are pooled in the model afterwards. In the former case, we use the logit function as link function and assume
which can be denoted as a fixedeffect model, where y_{ik} is the number of events, μ_{i} is the baseline value (and is seen as a nuisance parameter), \(d_{A_{i}k}\) is the log OR between arm k and arm A_{i} which is the baseline arm and has to be chosen for each trial. All arms are compared to this baseline treatment arm. These log ORs are of main interest in an NMA and are typically assumed to be approximately normally distributed. In a randomeffects model, the logits are modeled as
When the log ORs are used directly, the fixedeffect model is defined as
and a randomeffects model as
In this implementation, \(\phantom {\dot {i}\!}\psi _{{iA}_{i}k}\) is the log OR in trial i of treatment arm k compared to the baseline treatment arm A_{i}. The log OR together with its variance \(\phantom {\dot {i}\!}\text {var}(\psi _{{iA}_{i}k})\) have to be estimated using the data of study i. The estimation of \(\phantom {\dot {i}\!}\psi _{{iA}_{i}k}\) can be problematic when the number of events is rare (see [8–10], and the Cochrane Handbook, chapter 16.9.2 [11]). Thus, some care has to be taken when applying this approach. Further challenges and assumptions (as, for instance, the consistency assumption) but also extensions of these models are discussed and explained in the literature. Albeit there are important issues, we do not focus on them here.
Objective
In this paper, we want to introduce a simple method to obtain an index υ that can be interpreted similarly to a frequentist pvalue for an effect estimate within a Bayesian NMA. For this, we adapt an idea proposed by Kawasaki and Miyaoka [12, 13] where the authors introduce a similar index but to compare only two groups with respect to a binary outcome using Bayesian methods in a randomized trial. Our approach serves as a complement when presenting the results of an NMA reporting the effect estimates and the credible intervals. It can also be interpreted as the probability of superiority or noninferiority, respectively. Furthermore, the index might be useful to define boundaries when updating NMAs as proposed by Nikolakopoulou et al. [14] and may therefore be applied in sequential NMAs. In our simulation study and real data example, we discuss the characteristics of the proposed approach.
Methods
In this section, we present the definition of the index υ and how it can be used when comparing two treatment arms within an NMA model.
Definition of index υ
To explain our approach, we assume that there are three treatment arms compared (P: Placebo, S: standard treatment, and E: experimental treatment). Assuming that an event denotes a success, a log OR of d_{PE}>0 or d_{PS}>0 denotes a benefit of the experimental treatment or the standard treatment over placebo, respectively. To assess whether E is superior to S (by at least a certain (prespecified) relevant amount Δ≥0), we can estimate the probability
and base our decision on it. Under the consistency assumption, this equals to the definition
and therefore, this index υ can also be applied in any Bayesian (pairwise or network) metaanalysis.
Of course, Δ can be chosen to be negative as well leading to a noninferiority setting. Then, the probability of a treatment of being not less effective by more than a prespecified amount compared to another treatment arm is estimated. In the following, it will be shown how the estimation of this probability can be realized.
Estimation of υ
The log ORs are estimated via Bayesian methods. We assume that they are approximately normally distributed. As prior distributions, one can use (flat) normal distributions, resulting in a normal distribution as posterior. Let us assume that the posterior mean values of d_{PS} and d_{PE} are denoted by μ_{PS,post} and μ_{PE,post}, respectively. One can then define a Z statistic as
where
and
is the standard error of the difference of the log ORs. Thus, Z is asymptotically normally distributed as well.
Let Φ(·) denote the cumulative distribution function of the standard normal distribution. The probability of interest can then be approximated as
It has to be noted that this approach is based on the approximation of the distribution of the log ORs by the normal distribution and is, therefore, only an approximation of P(d_{PE}>d_{PS}+Δ).
An estimate of this probability is then
where \(\hat {d}_{PE}\) and \(\hat {d}_{PS}\) denote the estimates of the mean values of the posterior distribution of d_{PE} and d_{PS}, respectively. The estimated posterior variance is denoted by \(\widehat {\text {Var}}(d_{PE}  d_{PS}  \Delta) = \widehat {\text {Var}}(d_{PE}  d_{PS})\).
Estimation of this probability can be done within the MCMC approach in two different ways. The first approach is to estimate the (posterior) distributions of d_{PS}−d_{PE}−Δ directly. From this, we can estimate \(\hat {d}_{PS}  \hat {d}_{PE}  \Delta \) as well as the variance \(\widehat {\text {Var}}(d_{PE}  d_{PS}  \Delta)\). However, there is an even more intuitive way. In an MCMC estimation procedure, we store in every single iteration whether the parameter d_{PE} was larger than d_{PS}+Δ or not. After the MCMC estimation is finished, we evaluate the relative frequency of runs where d_{PE}>d_{PS}+Δ within the MCMC approach to estimate the probability \(\hat {P}(d_{PE} > d_{PS} + \Delta)\). An advantage of this approach is that it does not rely on the normal distribution and can therefore be applied in any NMA setting.
Use of υ for Bayesian hypothesis testing
The index υ can be used to estimate the probability of superiority or noninferiority between treatment arms with respect to the event probability. Therefore, it is a useful complement to the common results obtained in a NMA. Furthermore, this index can be used to make test decisions. Let us, again, assume that there are three treatment arms (P, S and E). Furthermore, we want to assess the following test problem:
with Δ∈R. We can now use the index υ to perform a Bayesian hypothesis test in an NMA. If the value of υ exceeds a prespecified value (for instance, 0.975, as an equivalent to a frequentist pvalue of 0.025 which is typically used in a onesided test procedure) we reject the nullhypothesis. Since the index υ is based on a Bayesian approach, it is unclear whether the test decisions coincide with the results of frequentist testing procedures. For this, a “probability matching prior” (PMP) has to be found as outlined, for example, in Datta and Sweeting [15]. We assume that the log ORs are normally distributed. It can be shown that in this case a uniform prior is a PMP [15]. In NMA, flat normal priors are commonly used which are very close to uniform priors if they are chosen sufficiently flat. However, since small deviations might still be present either because of the (flat) prior distribution or the approximation of the log OR via a normal distribution, we applied simulation studies to evaluate the characteristics of our approach.
Some technical issues
As already discussed in the “Background” section, there are two ways to define an NMA model with a binary outcome. Either using the number of observations and the number of events per treatment arm assuming a binomial distribution, or using the approximately normally distributed log ORs.
In the next section, results from simulation studies will be provided where both approaches are compared. Therein, the method where the binomial distribution is used, is called armbased approach. The method where ORs are modeled, is called contrastbased approach. The same distinction is done, for example, in the manual of the R package “netmeta” [16]. As a side note, the computation time of the contrastbased approach was substantially lower (in some situations about 40 times lower). Thus, from a computational point of view, this approach is much more efficient. From a technical point of view, the main difference between the armbased and the contrastbased approach is that an additional level in the hierarchy of the Bayesian model is used. In the armbased approach, a binomial distribution is estimated on the lower level, based on the number of successes (y_{ik}) and the number of observations (N_{ik}). On the upper level, the log ORs (\(d_{A_{i}k}\)) are estimated (model (1)). When using the trialspecific log ORs, there is only one level (model (2)).
Two different ways of estimating the probability P(d_{PE}>d_{PS}+Δ) have been presented above (note that this distinction is independent of the distinction between the contrastbased and the armbased approach). The first option is to estimate the (posterior) distribution of d_{PE}−d_{PS}−Δ and the second one is to estimate P(d_{PE}>d_{PS}+Δ) directly during the MCMC procedure. In all simulation studies, both approaches were used in parallel. It became clear that the differences between the results where negligibly small. Thus, only the results from the second approach are presented, since it is the simplest way to estimate the index υ.
Simulation study
Simulation studies were done to evaluate the testing approach. The main aim was to examine whether the approach maintains the type I error rate when used for hypothesis testing. For this, we have to define a cutoff value for a test decision. Analogously to a frequentist setting with a type I error rate of 0.025, we reject the null hypothesis H_{0}: d_{PE}≤d_{PS}+Δ if \(\hat {\upsilon } = \hat {P}(d_{PE} > d_{PS} + \Delta) \geq 0.975\).
A further issue was to examine the power of the approaches. Different settings regarding baseline risk, d_{PS}, d_{PE}, and Δ were used.
Binary data based on the assumption that the null hypothesis holds true were simulated and the rejection rate was estimated to examine the actual type I error rate. The boundary of the null hypothesis was considered, i.e., the data were simulated so that d_{PE}=d_{PS}+Δ holds true.
Three arms were compared (P: placebo; S: standard treatment; E: experimental treatment) in 16 studies, where four studies of each were simulated comparing P vs. S, P vs. E, and S vs. E, respectively, and another four studies were simulated including all three treatment arms. In each study, a sample size of 500 observations per treatment arm was used. We assume that the main interest was to compare the experimental treatment with the standard treatment. The success probabilities of the three arms were varied to examine the characteristics of our approach in different scenarios. The success probabilities of the placebo and the standard treatment arm were assumed to be equal which was done to simplify the simulation procedure; different values were chosen to evaluate different scenarios (p_{iP} = p_{iS} = 0.05, 0.1 or 0.2, i = 1, …,16). The success probability of the experimental arm was calculated such that d_{PE}=d_{PS}+Δ holds true. The values of Δ were chosen based on the ORs between the treatment arms. Eleven different ORs were used: log(1), log(1.05), log(1.1), log(1.2), log(1.5), log(2) (superiority), and log(1.05^{−1}), log(1.1^{−1}), log(1.2^{−1}), log(1.5^{−1}), log(2^{−1}) (noninferiority). The significance level was set to 0.025.
For each simulation scenarios, 50,000 iterations were used. Based on the results obtained in these scenarios, some further interesting data situations were examined. Firstly, a sample size of 1,000 observations per treatment arm with a success rate of 0.2 was used leading to a data situation where even approximate approaches should perform sufficiently well. Secondly, the sample size was lowered to 200 observations per treatment arm with a success rate of 0.1. The values for Δ were varied between log(0.9) and log(1.1) since the most often used values should be within this range. In a last scenario, extreme values of Δ were examined combined with a sample size of 400 observations per treatment arm using a success rate of 0.05.
We also evaluated our approach in situations where heterogeneity was present in the data. We used the same simulation settings as above (16 studies, 500 observations per arm). We did not vary Δ but set it to 0 thus considering a superiority setting. We simulated heterogeneity using the same values for τ^{2} as in Friede et al. [17]: 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, and 2. We, again, used three different baseline risk values: 0.05, 0.1, and 0.2. Randomeffects models were fitted and 10,000 iterations per scenario were performed.
In a last step, we lowered the sample size per arm and trial to 50 patients, used a baseline risk of 0.1, and applied the same values for τ^{2} as before. Again, with 10,000 replications per scenario, randomeffects models were fitted and evaluated.
Furthermore, the power of the testing approach was evaluated. Again, the main interest was to analyze the difference between the experimental and the standard treatment. The success rates in arm P and S were set to 0.1, assuming that d_{SE}=1.15, and the sample size was varied from 100 to 1,000 observations per treatment arm. Per scenario, 10,000 iterations were used.
In all simulation scenarios, the consistency as well as the similarity assumption was assumed to hold true. For parameter estimation, MCMC techniques were used. Two chains with a burnin of 20,000 followed by 40,000 runs with a thinning rate of 5 resulting in 8,000 samples per chain were generated to estimate the posterior distribution following Song et al. who used a similar setting [18]. The software R [19] in combination with JAGS (version 3.4.0 or higher, http://mcmcjags.sourceforge.net/) and the Rpackages rjags [20], doSNOW [21], foreach [22], coda [23], and iterators [24] were used to conduct the simulations. Since the computations were done on different systems and different work stations, different versions of the software packages were used. In the evaluation step, the package xtable [25] was used in addition.
Illustrative example
To further illustrate the approach, we analyzed a real data example that was already evaluated elsewhere [6, 26]. The data are provided by the Smoking Cessation Guideline Panel [27].
In the data set, 24 trials comparing four different treatments about smoking cessation are included (A: “no contact”, B: “selfhelp”, C: “individual counseling”, and D: “group counseling”). The number of cessations and the number of observations are presented in Table 1. In the following, it is tested whether the treatment effects of arm B, C, and D are different from that of treatment arm A using a fixedeffect model. Here, the following three test problems for superiority (i.e., Δ=0) are assessed (no adjustment for multiple testing is performed):
It should be mentioned that these hypotheses were not prespecified but the example is just presented to show the characteristics of our approach in a real data setting. Compared to the original data, the number of events was changed from 0 to 1 in two cases (study ID 9 and 20). This was done due to two reasons: If there are zero events in a treatment arm, an OR cannot be calculated. However, the contrastbased approach is based on ORs between treatment arms and thus the number of events had to be adjusted. As already mentioned above, the problem of rare events is common and discussed in the literature. In practice, a better choice may be to change the number of events from 0 to 0.5 and to add 0.5 to the number of observations [11]. However, the armbased approach is based on a binomial distribution which is a discrete distribution. Thus, only integers can be used as numbers of events. Since a comparison of both approaches should be provided, the number of events was thus changed to 1.
An MCMC approach was implemented to estimate the parameters with 500,000 iterations after a burnin of 100,000 iterations.
Results
Simulation study
In the following, we will present the simulation results. Due to convergence problems which resulted from zero counts, the results are sometimes based on slightly less than 50,000 or 10,000 runs, respectively. This is not mentioned in every single results description to improve readability.
Type I error rate: The main interest was whether the approach maintains the type I error rate. In Fig. 1, the results of the first part of the simulation studies are shown. The number of observations per treatment arm was kept fixed (at 500 per treatment arm) and the value of Δ was varied, where three different success rates for treatment arms P and S were assumed (0.05, 0.1 and 0.2). The type I error rate using the contrastbased approach is close to the nominal level if the success rates are 0.1 or 0.2 and Δ is between log(1.2^{−1}) and log(1.2) (Fig. 1). However, as soon as Δ is changed to more extreme values, it is slightly liberal in a noninferiority setting (exp(Δ)<1) and slightly conservative in a superiority setting (exp(Δ)≥1). This characteristics is even more pronounced when the success rate is set to 0.05. Furthermore, one can see that the type I error rate tends to be higher the higher the success rate is. In contrast, the actual level of the armbased approach is very close to the nominal one in most situations. Only if Δ and the success rate are relatively large, the type I error rates are slightly increased. If Δ is very small, the approach is slightly conservative. It is interesting to see that the lines in Fig. 1 cross. Thus, in some situations the armbased and in some other situations the contrastbased approach is more conservative or liberal, respectively.
In the setting with 1,000 observations per treatment arm and study and values for Δ very close to 0, both approaches lead to very similar results. Both nearly maintain the type I error rate. The situations with 200 observations per treatment arm and Δvalues varying between log(0.9) and log(1.1) might be more interesting, since these values are more common in practice. In all these scenarios, the armbased approach seems to perform slightly better than the contrastbased one, since it is less conservative but still maintains the type I error rate. Sometimes, the type I error rate was slightly above the nominal level. However, this exceedance can be regarded as negligible. In the last scenario, where extreme Δvalues were used, one can see that the contrastbased approach inflates the type I error rate in a noninferiority setting while it is very conservative in the superiority trials. In contrast, the armbased approach maintains the type I error rate in (even extreme) noninferiority scenarios but inflates the type I error rate in a superiority setting. Table 2 summarizes these results.
When introducing heterogeneity, we saw that the results for the two approaches (armbased and contrastbased) were more different. The armbased approach always maintains the type I error rate but becomes very conservative in case of strong heterogeneity (see Fig. 2). The contrastbased approach, however, leads to slightly increased type I error rates for higher values of heterogeneity. Lowering the sample size to 50 patients per study did not, in general, lead to inflated type I error rates when the armbased approach was used. Only in case of strong heterogeneity the type I error was slightly inflated, or the test behaved slightly too conservative in the situation of strong heterogeneity. In contrast, the effectbased approach led to an increased type I error rate in case of strong heterogeneity.
Power The investigations of the power showed that both approaches have a very similar performance. The armbased approach resulted in slightly higher power compared to the contrastbased one (see Fig. 3). The difference decreased with increasing sample size. This was to be expected since the type I error rates of the armbased approach were also slightly increased compared to the contrastbased one. However, one has to keep in mind that the armbased method did not maintain the significance level in some situations and thus has to be used with care.
Real data example
In Table 3, we provide the results for the data example. The estimated values for υ resulting from the armbased and the contrastbased approach are presented for each pair of hypotheses. We can see that the armbased approach always leads to a higher value of \(\hat {\upsilon }\) than the contrastbased approach. If the cutoff for a test decision of 0.975 is applied, the following test decisions result. The first null hypothesis H_{0,1} cannot be rejected for both approaches. This means that the group counseling and the individual counseling are not significantly different. The second null hypothesis (H_{0,2}) can be rejected according to both approaches that means that the individual counseling is significantly more effective than selfhelp. The third null hypothesis (H_{0,3}) can be rejected with the armbased but not when applying the contrastbased approach. Since we could see from our simulation study that the armbased approach leads to type I error rates that are very close to the nominal level, the armbased should be a proper choice. However, the safe (but maybe too conservative) option would be to apply the contrastbased approach and thus to maintain the null hypothesis in this case.
Discussion
In this article, a method for hypothesis testing in an Bayesian NMA is presented. For this, an index was introduced that describes the probability of superiority or noninferiority from a Bayesian perspective. We examined whether this index can also be used to make test decisions in a frequentistic sense. In a simulation study, two different approaches were compared, an armbased and a contrastbased one. When there was no heterogeneity present in the data and fixedeffects models were applied, the observed type I error rates were very close to the nominal significance level while the armbased approach led to slightly more favorable results in most situations. If the sample size is sufficiently high, both approaches maintain the type I error rate. If an extreme noninferiority margin is used, only the armbased approach led to valid results. An extremely large margin for relevant superiority, however, leads to an inflation of the type I error rate of the armbased approach, and the contrastbased approach is then the better choice. However, in most situations in practice the deviations from the nominal type I error rate observed in our simulation studies are negligible. We also investigated the situation where heterogeneity is present in the data and saw that this can have a stronger impact on the type I error rate. However, even when the sample size was lowered to 50 patients per arm and trial, the type I error was still very close to the nominal level and only deviated slightly from it in case of strong heterogeneity. It is worth mentioning that our concept is not identical to a Bayesian posterior predictive pvalue as described in Gelman et al. [28]. The index υ rather describes a Bayesian probability for superiority or noninferiority.
There are some limitations of our simulation study. Of course, there are by far more data situations as those considered. However, we covered a range of common situations in medical research. There is also a lot of discussion about inconsistency in NMA models in the literature (see, for example, Dias et al. [29], or Krahn et al. [30]). In our simulation scenarios, it was assumed that there is no inconsistency present in the data which is a limitation of our study. Consistency is an assumption typically made in a standard NMA model but might be problematic in practice. In recent publications, this issue was addressed and solutions were proposed by applying more complex models [31–35]. However, in this work we focused on the standard NMA model. Note that when examining the type I error rate, the null hypothesis is assumed to hold true. Thus, the success rates in all treatment arms are exactly the same by design (or the same plus a predefined Δ) and therefore there is no inconsistency per definition.
A test decision can also be based on the 95% credible intervals around the point estimate of the log OR. If Δ is not included, the null hypothesis can be rejected. We compared this approach to the methods suggested in this article. The type I error rate tended to be slightly increased if the test decision was based on the credible interval compared to the approach based on υ but overall the results were very similar. Thus, it is not a considerable improvement compared to a test decision based on the credible intervals but rather a complement on the existing methodology.
Conclusions
In conclusion, we proposed and discussed an index that can be used to test for superiority or noninferiority of a treatment arm compared to another one within a Bayesian NMA. The estimation is done during the NMA model estimation and does not result in any (noticeable) additional computational cost. At the same time, the implementation is very easy. Obviously, this approach can also be applied in a straightforward way in any other data situation than binary data, as continuous data or a survival time, and is therefore a flexible tool.
However, as already mentioned, we did not cover all possible scenarios in our simulation study and, therefore, the index has to be used and interpreted with care. For example, as shown by Friede et al. [17] coverage of the credibility intervals decreases (and the type I error rate increases) substantially in case of rare diseases (low number of events), small populations, and strong heterogeneity. We did not discuss these situations here but it is clear that the same results for the index υ would have been observed as well. This shows that it is easy to generate examples that lead to invalid results. The choice of a proper prior distribution affects the results as well, as also described by Friede et al. [17]. Therefore, an adequate assessment of the data situation at hand has to be done before applying the approach discussed here or, in general, any NMA approach. It is hardly possible to define an approach that is valid and optimal for any situation in practice and we emphasize the limitations of the approach described in this paper.
Abbreviations
 NMA:

Network metaanalysis
 OR:

Oddsratio
References
Efthimiou O, Debray T, Valkenhoef G, Trelle S, Panayidou K, Moons KG, Reitsma JB, Shang A, Salanti G, GetReal Methods Review Group. Getreal in network metaanalysis: a review of the methodology. Res Synth Methods. 2016; 7:236–63.
Dias S, Sutton AJ, Ades A, Welton NJ. Evidence synthesis for decision making 2: a generalized linear modeling framework for pairwise and network metaanalysis of randomized controlled trials. Med Dec Making. 2013; 33:607–17.
Dias S, Welton NJ, Sutton AJ, Caldwell DM, Lu G, Ades AE. Evidence synthesis for decision making 4: inconsistency in networks of evidence based on randomized controlled trials.Med Dec Making. 2013; 33:641–56.
Rücker G, Schwarzer G. Ranking treatments in frequentist network metaanalysis works without resampling methods. BMC Med Res Methodol. 2015; 15:58.
Higgins JPT, Whitehead A. Borrowing strength from external trials in a metaanalysis. Stat Med. 1996; 15:2733–49.
Lu G, Ades A. Assessing evidence inconsistency in mixed treatment comparisons. J Am Stat Assoc. 2006; 101:447–59.
Salanti G. Special issue on network metaanalysis. Res Synth Methods. 2012; 2:69–190.
Bradburn MJ, Deeks JJ, Berlin JA, Russell Localio A. Much ado about nothing: a comparison of the performance of metaanalytical methods with rare events. Stat Med. 2007; 26:53–77.
Lane PW. Metaanalysis of incidence of rare events. Stat Methods Med Res. 2013; 22:117–32.
Sweeting MJ, Sutton AJ, Lambert PC. What to add to nothing? Use and avoidance of continuity corrections in metaanalysis of sparse data. Stat Med. 2004; 23:1351–75.
Higgins J, Deeks J, Altman D. Chapter 16: Special topics in statistics In: Higgins S, Green S, editors. Cochrane Handbook for Systematic Reviews of Interventions Version 5.1.0 (updated March 2011). The Cochrane Collaboration, 2011: 2011. Available from http://handbook.cochrane.org/. Accessed 12 May 2016.
Kawasaki Y, Miyaoka E. A bayesian inference of p(π _{1}>π _{2}) for two proportions. J Biopharm Stat. 2012; 22:425–37.
Kawasaki Y, Miyaoka E. A bayesian noninferiority test for two independent binomial proportions.Pharm Stat. 2013; 12:201–6. https://doi.org/10.1002/pst.1571.
Nikolakopoulou A, Mavridis D, Egger M, Salanti G. Continuously updated network metaanalysis and statistical monitoring for timely decisionmaking. Stat Methods Med Res. 2018; 27:1312–30.
Datta GS, Sweeting TJ. Probability matching priors. Technical Report, Research Report No. 252. 2005. Department of Statistical Science, University College London.
Rücker G, Schwarzer G, Krahn U, König J. Netmeta: Network MetaAnalysis Using Frequentist Methods. 2015. R package version 0.80, http://CRAN.Rproject.org/package=netmeta. Accessed 25 Aug 2016.
Friede T, Röver C, S W, Neuenschwander B. Metaanalysis of few small studies in orphan diseases. Res Synth Meth. 2017; 8:79–91.
Song F, Clark A, Bachmann MO, Maas J. Simulation evaluation of statistical properties of methods for indirect and mixed treatment comparisons.BMC Med Res Methodol. 2012; 12:138.
R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2012. R Foundation for Statistical Computing. http://www.Rproject.org/. Accessed 25 Aug 2016.
Plummer M. Rjags: Bayesian Graphical Models Using MCMC. 2013. R package version 311, http://CRAN.Rproject.org/package=rjags. Accessed 25 Aug 2016.
Revolution Analytics, Weston S. doSNOW: Foreach Parallel Adaptor for the Snow Package. 2014. R package version 1.0.12, http://CRAN.Rproject.org/package=doSNOW.
Revolution Analytics. Foreach: Foreach Looping Construct for R. 2012. R package version 1.4.0, http://CRAN.Rproject.org/package=foreach. Accessed 25 Aug 2016.
Plummer M, Best N, Cowles K, Vines K. Coda: convergence diagnosis and output analysis for mcmc. R News. 2006; 6:7–11.
Revolution Analytics. Iterators: Iterator Construct for R. 2012. R package version 1.0.6, http://CRAN.Rproject.org/package=iterators. Accessed 25 Aug 2016.
Dahl DB. Xtable: Export Tables to LaTeX or HTML. 2014. R package version 1.74, http://CRAN.Rproject.org/package=xtable. Accessed 25 Aug 2016.
Hasselblad V.Metaanalysis of multitreatment studies. Med Dec Making. 1998; 18:37–43.
Smoking Cessation Guideline Panel. Smoking Cessation, Clinical Practice Guideline No. 18 (AHCPR Publication No. 960692). Rockville: MD: Agency for Health Care Policy and Research, U.S. Department of Health and Human Services; 1996.
Gelman A. Comment: Fuzzy and bayesian pvalues and uvalues. Statist Sci. 2005; 20:380–1.
Dias S, Welton NJ, Caldwell DM, Ades AE. Checking consistency in mixed treatment comparison metaanalysis. Stat Med. 2010; 29:932–44. https://doi.org/10.1002/sim.3767.
Krahn U, Binder H, König J. A graphical tool for locating inconsistency in network metaanalyses. BMC Med Res Methodol. 2013; 13:35.
Jackson D, Barrett J, Rice S, White I, Higgins J. A designbytreatment interaction model for network metaanalysis with random inconsistency effects. Stat Med. 2014; 33:3639–54.
Jackson D, Boddington P, White I. The designbytreatment interaction model: a unifying framework for modelling loop inconsistency in network metaanalysis. Res Synth Methods. 2016; 7:329–32.
Jackson D, Law M, Barrett J, Turner R, Higgins J, Salanti G, White I. Extending dersimonian and laird’s methodology to perform network metaanalyses with random inconsistency effects. Stat Med. 2016; 35:819–39.
Jackson D, Veroniki A, Law M, Tricco A, Baker R. Paulemandel estimators for network metaanalysis with random inconsistency effects. Res Synth Methods. 2017; 8:416–34.
Law M, Jackson D, Turner R, Rhodes K, W V. Two new methods to fit models for network metaanalysis with random inconsistency effectS. BMC Med Res Methodol. 2017; 16:87.
Acknowledgements
We acknowledge financial support by Deutsche Forschungsgemeinschaft and RuprechtKarlsUniversität Heidelberg within the funding programme Open Access Publishing. Furthermore, we would like to thank the editors and reviewers for their valuable proposals and comments that considerably improved our manuscript in different aspects.
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Open Access Publishing was funded by Deutsche Forschungsgemeinschaft and RuprechtKarlsUniversität Heidelberg within the funding programme Open Access Publishing.
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All data analyzed in our illustrative example are included in this published article.
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LU wrote the first draft of the manuscript, conducted the simulation study and applied the approach to the illustrative data example. KJ and MK contributed to the writing, and critically commented and revised the applied methods and the manuscript. All authors revised and approved the final version of the manuscript.
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Uhlmann, L., Jensen, K. & Kieser, M. Hypothesis testing in Bayesian network metaanalysis. BMC Med Res Methodol 18, 128 (2018). https://doi.org/10.1186/s128740180574y
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DOI: https://doi.org/10.1186/s128740180574y
Keywords
 Network metaanalysis
 Hypothesis testing
 Treatment comparison
 Superiority
 Noninferiority