A review and comparison of arm‐based versus contrast‐based network meta‐analysis for binary outcomes—Understanding their differences and limitations

Network meta‐analysis (NMA) is a statistical procedure to simultaneously compare multiple interventions. Despite the added complexity of performing an NMA compared with the traditional pairwise meta‐analysis, under proper assumptions the NMA can lead to more efficient estimates on the comparisons of interventions by combining and contrasting the direct and indirect evidence into a form of evidence that can be used to underpin treatment guidelines. Two broad classes of NMA methods are commonly used in practice: the contrast‐based (CB‐NMA) and the arm‐based (AB‐NMA) models. While CB‐NMA only focuses on the relative effects by assuming fixed intercepts, the AB‐NMA offers greater flexibility on the estimands, including both the absolute and relative effects by assuming random intercepts. A major criticism of the AB‐NMA, on which we aim to elaborate in this paper, is that it does not retain randomization within trials, which may introduce bias in the estimated relative effects in some scenarios. This criticism was drawn under the implicit assumption that a given relative effect is transportable, in which case the data generating mechanism favors the inference based on CB‐NMA, which models the relative effect. In this article, we aim to review, summarize, and elaborate on the underlying assumptions, similarities and differences, and also the advantages and disadvantages, between CB‐NMA and AB‐NMA methods. As indirect treatment comparison is susceptible to risk of bias no matter which approach is taken, it is important to consider both approaches in practice as complementary sensitivity analyses and to provide the totality of evidence from the data.

Comparative effectiveness research (CER) aims to inform health care decisions concerning the benefits and risks of different prevention strategies, diagnostic instruments and treatment options.The growth of interest in CER and evidence-based-medicine (EBM) has led to a dramatic increase in interest in systematic review and meta-analysis (SRMA) (Egger et al., 2001;Jackson et al., 2011;Sutton & Higgins, 2008).Meta-analysis (MA) uses statistical approaches to combine and contrast the results from multiple independent studies (Borenstein et al., 2009;Schmid et al., 2020).Systematic review and meta-analysis can detect certain biases such as publication bias and reporting bias, improve statistical efficiency (compared to individual studies), and identify patterns and sources of disagreement among study results (Lin et al., 2018;Lin & Chu, 2018a;Lin & Chu, 2018b).SRMAs have been applied to a wide range of scientific areas and are commonly considered to be at the top of evidence pyramid, and as a lens through which evidence is viewed (Murad et al., 2016).
Traditionally, a meta-analysis of randomized controlled trials compares only two treatments, typically an intervention versus a placebo.With an increasing number of interventions for a given condition, network meta-analysis (NMA) (also called mixed or multiple treatment comparisons meta-analysis) has been developed to simultaneously synthesize both direct comparisons of interventions within randomized controlled trials (RCTs) and indirect comparisons across trials (Dias et al., 2018).Considering the simplest setting of an NMA comparing three treatments A, B and C for a certain disease, the RCTs of A versus C provide direct evidence, while the RCTs of either A or C versus B provide indirect evidence for the comparison between A and C (Cipriani et al., 2013;Dias et al., 2018;Mills et al., 2012;Mills et al., 2013).
With appropriate assumptions (e.g., transitivity, evidence consistency), borrowing information from indirect evidence allows more precise estimates of treatment effects than can be obtained from pairwise meta-analysis focusing only on direct evidence (Lumley, 2002).Multiple guidelines and evaluation documents have been developed recently.For example, a modified, 32-item Preferred Reporting Items for Systematic reviews and Meta-Analyses (Page, McKenzie, et al., 2021;Page, Moher, et al., 2021) extension checklist (PRISMA-NMA) was developed to guide the reporting of NMAs (Hutton et al., 2015).The GRADE (grading of recommendations assessment, development, and evaluation) guidance has been proposed on how to draw conclusions from an NMA that includes randomized controlled trials addressing a single outcome (Brignardello-Petersen, Florez, et al., 2020;Brignardello-Petersen, Izcovich, et al., 2020).More recently, the Confidence in Network Meta-Analysis (CINeMA) framework has also been developed to evaluate the credibility of results from NMAs when multiple interventions are compared (Nikolakopoulou et al., 2020;Papakonstantinou et al., 2020).
For example, when the outcome is binary, the CB-NMA focuses on the weighted average of conditional (study-specific) relative effects such as odds ratio, while the AB-NMA can estimate both the treatment-specific event rates, and the marginal (population-averaged) effects and the weighted average of conditional (relative) effects without any additional external data.In the context of meta-analysis, conditional effects are calculated as the (weighted) average of study-specific effects, which measures the association between an outcome and an exposure/ treatment given the study levels in the model.Marginal effects are computed over the entire population, giving us the effect across the whole population represented in the meta-analysis.
Despite the advantage of flexibility for AB-NMA, CB-NMA models have been favored in recent applications.A major criticism of the AB-NMA, which we aim to elaborate in this paper, is that it does not retain randomization within trials, which may introduce bias in the estimated relative effects.This criticism has been made under the implicit assumption that the relative effect (i.e., odds ratio) is transportable (i.e., not dependent on the baseline risk), that is, when the data are generated favoring the CB-NMA model.
In this article, we aim to review, summarize, and elaborate on the similarities and differences, and also the advantages and disadvantages, between CB-NMA and AB-NMA methods.Specifically, we will investigate their relative performance under three simple assumptions using hypothetical binary data.The purpose is not to show that one model is better than another in practice, because the underlying data generating mechanism is unknown and undoubtedly determines the performance.As indirect treatment comparison is susceptible to risk of bias no matter which model is chosen, it is important to consider both approaches in practice as complementary sensitivity analyses and to provide the totality of evidence.

| OVERVIEW OF CONTRAST-BASED AND ARM-BASED NETWORK META-ANALYSIS MODELS
Let us consider an NMA of a binary outcome simultaneously comparing T interventions in I independently randomized clinical trials.A subset of the complete collection of T treatments is evaluated in trial i ¼ 1, 2, …, I ð Þ .Let t i and S i be the corresponding number and set of interventions that are evaluated in trial i.
gdenote the data available from the ith trial, where n it is the number of subjects and y it is the number of events for the tth treatment in the ith trial.
In this section, we will first provide the mathematical details of the two most commonly used NMAs: the contrast-based NMA (Lu & Ades, 2004;Lu & Ades, 2006;Lu & Ades, 2009) which models the relative effects, and the arm-based NMA which flexibly models both the absolute and relative effects (Hong et al., 2016;Lin et al., 2016;Lin, Zhang, et al., 2017;Zhang et al., 2014;Zhang et al., 2017).Second, we will describe the relationship between them and the estimands that can be computed by them.Last, we will summarize their differences, advantages, and disadvantages.Some details of the models described in this section are also available in White et al. (2019) and Karahalios et al. (2022).
At the first stage, both models assume that conditional on the probability of events P i ¼ p it f g, the elements y it of y i ¼ y it f g are independently binomially distributed with a likelihood function At the second stage, the contrast-based and arm-based NMAs differ in model specifications, which will be described in Sections 2.1 and 2.2.

| The contrast-based (CB) NMA approach
In the second stage, the classic heterogeneity CB-NMA uses the following random effects models, Corr Here, logit p ð Þ ¼ p 1Àp and b i is the study-specific "reference" treatment for the ith trial and White et al. (2019), we explicitly use b i instead of a commonly simplified notation b to replace b i in the above to emphasize that most NMAs do not have a common reference treatment.The study-specific log odds in the arm b i , μ ib i , is treated as a fixed effect.Here, the term "fixed effect" denotes a parameter that is estimated independently of the other parameters, whereas a "random effect" shares a distribution with other parameters and is drawn from a distribution.
The study-specific treatment contrast (i.e., the log odds ratio) of the treatment t relative to reference treatment b i for study i, δ ib i t , is assumed to be a random effect from a normal distribution, . The parameter d b i t is the average log odds ratio of the treatment t relative to treatment b i across trials.The consistency assumption d ht ¼ d kt À d kh means that, if any two treatments are compared indirectly through a third treatment, the result is consistent with the direct comparison.Here d ht measures the conditional treatment effect comparing treatment h versus treatment t (i.e., the conditional log odds ratio) because it is estimated conditional on study-specific baseline effects (i.e., the fixed intercepts).The hyper-parameters ρ b i ð Þ ht specify the correlations between two treatment contrasts within a trial.They are only needed when there are multiple arm trials on treatment triples (b i , h, t).
The models can be fitted in a Bayesian framework using non-informative or informative prior distributions for the parameters.Typically, non-informative prior distributions are chosen for μ ib i and d b i k , The specification of variance priors can be a difficult task for heterogeneous variance models due to variance constraints (Lu & Ades, 2009).However, when all variances are assumed to be equal, that is, σ 2 b i t ¼ σ 2 for any t T i , i ¼ 1, 2,…, I, we obtain a simplified homogeneity model with correlation parameters (Gelman et al., 2008;Ghosh et al., 2018) or an empirically-derived predictive prior distribution of heterogeneity parameter from Turner et al. (2012Turner et al. ( , 2015) ) can be chosen for σ 2 .The homogeneity model is most frequently used in practice due to its simplicity.When comparing CB-NMA with AB-NMA, we will focus on the homogeneity model.This is consistent with Model 1 White et al. (2019), in which Þand the conditional treatment effects d 1t compare each treatment t with the reference treatment 1 for t > 1, and d 11 ¼ 0.
It is worth mentioning that in a classic CB-NMA, the conditional treatment effect on the log odds ratio scale d 1t (i.e., the conditional log odds ratio) is typically assumed to be independent of the baseline risk (e.g., the "fixed" intercept models 1 and 2 as in White et al., 2019), and is assumed to be "portable" across different populations.However, this assumption may not hold in most meta-analyses.
Recently, Xiao, Chen, et al. (2022) and Xiao, Chu, et al. (2022) assessed the correlation of the study-specific conditional odds ratios with the baseline risks using the Cochrane Database of Systematic Reviews updated to January 2020.In nearly half of the 40,243 meta-analyses containing at least three studies, the absolute values of Spearman's correlation were larger than 0.5, a threshold corresponding to moderate correlation, and the 75th percentile of Spearman's correlation for a given number of studies in a meta-analysis is mostly negative suggesting that as the baseline risk increases, the effect size in odds ratio scale is likely to decrease.Previously, Schmid et al. (1998Schmid et al. ( , 2004) also found a high correlation between baseline risk and odds ratios using the baseline risk model of McIntosh (1996).Control risk regression has been discussed explicitly in the Handbook of Meta-analysis (Schmid et al., 2020), but its extension to NMA awaits further development.
Furthermore, based on the CB-NMA, exp d 1t ð Þ represents the conditional median treatment effect on the odds ratio scale at a given baseline risk in the reference treatment 1 (denoted as π 10 ), not the conditional mean treatment effect.The conditional mean treatment effect (Zeger et al., 1988) is approximately equal to In addition, it is not the marginal treatment effect for all subjects included in the NMA.In addition, due to the "noncollapsibility" of OR, the interpretation is not straightforward, and it is hard to compare these conditional estimates across different meta-analyses, even using marginal standardization (Whitcomb & Naimi, 2021).Here, "noncollapsibility" refers to the circumstance where the measure of association conditioned on studies is unequal to the marginal measure collapsed over studies, in the absence of confounding.It is a property of the model rather than the estimation process.Described mathematically in terms of expectation, it means that E DjT, S ð Þ≠ E DjT ð Þ where D is the disease status, S is the study and T is the treatment (Whitcomb & Naimi, 2021).
Here we focused on the most commonly used CB-NMA with a logit link in Equation (2).In theory, one can also use other link functions such as log or identity if one can implement appropriate constraints such that the probabilities are within the range (Chu et al., 2011;Chu & Cole, 2010).It is worth mentioning that the correlations among effects might be different on different scales, and it might be challenging to implement appropriate constraints on the covariance matrix using other scales (Lu & Ades, 2009).

| The arm-based (AB) NMA approach
In AB-NMA, a multivariate normal distribution (MVN) is assumed for p it f g on a transformed scale (Hong et al., 2016;Lin, Chu, & Hodges, 2017;Lin, Zhang, et al., 2017;Zhang et al., 2014;Zhang et al., 2017).In the absence of any individual-level covariates, the model is specified as where g Á ð Þ is the link function such as the probit, logit or complementary log-log functions, μ 1 ,μ 2 ,…, μ T ð Þare treatment-specific fixed effects, R T is a positive definite correlation matrix, and σ t is the standard deviation for the random effects v it .If we use the logit link function, then μ j À μ k corresponds to the conditional log odds ratio comparing treatment j versus treatment k. Let Þbe a diagonal matrix with elements σ t , the covariance matrix is thus Σ ¼ Σ σ R T Σ σ , which is the same as model 4 in White et al. (2019).Here σ t captures trial-level heterogeneity in response to treatment t, and R T captures the within-study dependence among treatments.As demonstrated by Wang et al. (2020), specifying priors on the standard deviation Σ σ and correlation matrix R T based on the separation strategy proposed by Barnard et al. (2000) performs better compared to specifying the inverse-Wishart priors on the variance-covariance matrix Σ.
To reduce model complexity and strengthen model identifiability and estimation, an exchangeable structure for R T is recommended (Lin, Zhang, et al., 2017;Wang et al., 2020), where all off-diagonal elements ρ jk are assumed equal to a common value ρ.To keep R T positive definite, ρ must be larger than À 1 TÀ1 .Thus, we may use a vague uniform prior for ρ on À 1 TÀ1 ,1 À Á .Because the number of clinical trials involving each treatment is often small in an NMA, treatment-specific variance estimates are often unstable when non-or weakly informative priors are employed under unequal variance assumption.Wang, Lin, Hodges, et al. (2021) introduced a variance shrinkage method assuming different treatment-specific variances share a common prior with unknown hyper-parameters.This approach requires a weaker assumption compared to the homogeneous variance assumption and improves estimation by shrinking the variances in a data-dependent way.Rover et al. (2021) have provided some useful guidance on weakly informative prior specification for the heterogeneity parameter in Bayesian random effects meta-analysis, which can be extended to network meta-analysis.
Based on the model in Equation ( 4), the population-averaged (or marginal) treatment-specific event rate can be estimated as is the standard Gaussian density function.While the estimation of π t involves integration, one can use an explicit formula to compute π t in some cases, for example, π t ¼ for probit link function, and for logit link function.When there is no explicit formula, π t can be computed by numerical integration, for example, by the trapezoidal rule with 1000 equal space subintervals (Chu et al., 2012).
Using the marginal event rate π t , one can easily compute the marginal relative treatment effects of OR= π k 1Àπ k = π l 1Àπ l , RR=π k =π l and RD=π k À π l for a pairwise comparison between any two treatments k and l.Note that the marginal treatment effects are for all subjects included in the NMA.
Moreover, for any given baseline risk π 10 in a reference treatment 1, we can obtain the conditional median risk (Chu et al., 2012;Xiao, Chu, et al., 2022) Therefore, the conditional estimates of OR, RD, RR given a reference risk of π 10 are ÞÀp 0 , and M π t0 jπ 10 ð Þ=π 10 M p 1 jp 0 ð Þ=p 0 , respectively.Thus, the AB-NMA assumes that the (conditional) treatment effects depend on the underlying risks, and the magnitude of this dependence is impacted by the variances and the correlation parameter.
It is worth emphasizing that the CB-NMA discussed in the article assumes that the study-specific intercepts (which corresponds to study-specific baseline risk) are "fixed."In contrast, the AB-NMA assumes that the study-specific intercepts are "random."White et al. (2019) showed that if one assumes random intercepts in the CB-NMA, then the AB-NMA follows naturally from the CB model when the intercepts are allowed to be correlated with the relative contrasts.However, the parameterization of CB-NMA with random intercepts and correlated contrasts needs to carefully consider the constraints on the covariance matrix, which can be complex and cumbersome to implement (Lu & Ades, 2009).
Under some assumptions, one can even re-parameterize the CB-NMA to be equivalent to the AB-NMA model (i.e., model 4 in White et al., 2019).

| EVALUATION OF CB VERSUS AB NMA USING THREE HYPOTHETICAL DATA SETS
To compare the performance of the "arm-based" and "contrast-based" NMA, we create three realistic hypothetical NMA data sets under a homogenous relative risk (RR), a homogenous rate difference (RD), or a homogenous odds ratio (OR) assumption.Each NMA includes 15 trials and 3 treatment arms.Trials with arms A, B and C were simulated (i.e., the fully observed data) and then one of the arms was dropped to make the "partially" observed data.Specifically, because in a typical NMA, most trials only compare two treatments, we let 5 trials each compare A and B, B and C, A and C, respectively.
Next, the response rates in Treatment A are assigned from a uniform distribution as equally spaced points ranging from 0.06 to 0.20 in ascending order for the 15 trials.The corresponding response rates in Treatment B and C in each trial are assigned based on a fixed RR, a fixed RD or a fixed OR assumption.Specifically, the RR of B versus A is 2.0 and C versus A is 3.0 under the fixed RR assumption, the RD of B versus A is 5% and C versus A is 10% under the fixed RD assumption, and the OR of B versus A is 2.0 and C versus A is 3.0 under the fixed OR assumption.To simplify the illustration, we ignore the random sampling error and assume each arm in each trial has 1000 subjects, and the number of events is equal to the response rates multiplied by the number of subjects.This assumption allows us to calculate the true event rates for each treatment arm and the conditional effects given baseline risk from "fully observed" data (i.e., not from the underlying data generating mechanism) easily.
As "all models are wrong" (Box, 1976) and each approach has its own assumptions and focuses on different estimands, instead of comparing results from different models, we focus on comparing the results from the "partially observed" data versus the "fully observed" data from each model.Here the "full" data refers to the ideal scenario in which each trial would compare all three treatments, and it serves as the "gold standard" estimates that an NMA model can aim for with the "partially observed" data.The "partially observed" data refers to the realistic scenario in a typical NMA that each trial only compares a subset of treatments.To minimize the impact of missing data generating mechanisms, we assume that each trial will not include a treatment in rotating order.The hypothetical data are given in the web appendix wTable 1, wTable 2 and wTable 3, respectively.We analyzed the above three hypothetical data using three methods.The first is based on the contrast-based NMA with homogeneous variance.The second and third are the arm-based NMA methods under either homogeneous or heterogeneous variance assumption, both with equal correlation assumption.For simplicity, we used the logit link function for all models.Although Wang, Lin, Hodges, et al. (2021) had introduced a variance shrinkage method that can improve arm-based NMA, we consider both homogeneous and heterogeneous variance assumptions in the analyses as there are only three treatments in the hypothetical data.
Specifically, we conducted all analyses using JAGS through the R package "rjags" in R version 4.2.2.For our AB-NMA models, we adapted the source code from two R functions in the pcnetmeta package (Lin, Zhang, et al., 2017), and used vague priors N(0, 10 4 ) for the treatment-specific fixed effects, Unif(0.0001,10) for the standard deviation of the random effects (Gelman, 2006).For the CB-NMA model, we followed WinBUGS code provided in the appendix by Dias et al. (2013) with vague priors N(0, 10 4 ) for fixed intercepts and relative treatment effects, and Unif(0.0001,10) for the between-trial standard deviation.
The web appendix wTable 4 summarizes the Deviance Information Criterion (DIC) (Spiegelhalter et al., 2002) under constant RR, RD and OR assumptions comparing the three models for both observed and full data.It suggests that the arm-based NMA method with heterogeneous variance fits the observed and full data better under both fixed RR and RD, while the contrast-based NMA method fits the observed and full data better under fixed OR (because the logit link function is the proper one under the fixed OR).It suggests that the arm-based and contrast-based NMAs can complement each other in practice, or one might want to make inference based on the model that provides better goodness-of-fit.Table 1 presents the population-averaged treatment-specific event rate estimates from the observed data versus that from the full data based on the AB-NMA with homogenous or heterogeneous variance and equal correlation assumptions.This table indicates that the estimates of the population-averaged treatment-specific event rates are nearly unbiased under all three scenarios using both approaches.In addition, the information loss due to missing data is mostly recovered as evidenced by the similarity of the length of the posterior credible intervals comparing results from the observed data versus that from the full data.Table 2 presents the conditional odds ratio estimates from the observed data versus that from the full data based on the AB-NMA with homogenous or heterogeneous variance and equal correlation assumptions, and on CB-NMA with homogenous variance assumption.It indicates that the estimates of the conditional odds ratio are nearly unbiased under all three scenarios using all three approaches.In addition, it is worth mentioning that it seems that CB-NMA tends to give slightly narrower credible intervals under all scenarios, particularly under fixed OR assumption which favors the CB-NMA.
Table 3 presents the marginal (or population-averaged) treatment effects (measured by marginal RR, RD, 6) from the observed data versus that from the full data based on the AB-NMA with homogenous or heterogeneous variance and equal correlation assumptions.Consistent with Tables 1 and 2, Table 3 indicates that the estimates of the marginal treatment effects are nearly unbiased under all three scenarios using both approaches, and the information loss due to missing data is mostly recovered.
Figure 1 presents the conditional treatment effects (in terms of RR, RD, or OR) comparing treatment B versus the baseline treatment A given different baseline risk of treatment A for the observed and full data, based on the arm-based NMA with heterogeneous variance and equal correlation.The three rows correspond to the constant RR, RD and OR assumptions, respectively.It shows that within the range of the observed baseline risk, the arm-based NMA with heterogeneous variance seems to perform well, giving nearly unbiasedly conditional treatment effects.
The web appendix wFigures 1 and 2 present similar figures based on the arm-based NMA with homogenous variance and the contrast-based approaches.The wFigures 1 and 2 show that the magnitude of bias due to model misspecification can be substantial, particularly when the baseline risk is different from the average risk.In summary, when the transportability assumption of OR is not valid, the commonly used contrast-based approach can lead to substantial bias for estimating the conditional treatment effects given baseline risk, particularly if the baseline risk is beyond the range of our data.Thus, it can be dangerous to estimate the marginal effect of treatment using external information in a well-defined population assuming relative effect is transportable.

| DISCUSSION
Using three hypothetical NMA data sets under a homogenous relative risk (RR) assumption, a homogenous rate difference (RD) assumption, and a homogenous odds ratio (OR) assumption with well-balanced and connected networks, we demonstrated that both CB NMA and AB-NMA can provide nearly unbiased estimates by comparing results from the "observed" data versus that from the "full" data, and can recover most information loss due to missing data.We also demonstrated the benefit of AB-NMA with random intercepts can provide the patient-centered measures, including the treatment-specific event rates and risk differences (RDs), which in general cannot be provided by CB-NMA with fixed intercepts.Only reporting relative effects, such as odds ratio, may create some unnecessary obstacles for patients to comprehensively trade-off efficacy and safety outcomes (Zhang et al., 2014).
As reported by White et al. (2019), the most important difference between CB-NMA and AB-NMA models is whether they have fixed or random study intercepts.CB-NMA models which respect randomization with fixed study intercepts assumes that relative treatment effects, that is, OR when the outcome is binary, is transportable across population with different underlying risks.However, OR is a non-collapsible effect measure and its interpretation is not straightforward, particularly in network meta-regression (Phillippo et al., 2020;Shrier & Pang, 2015).It is hard to compare these conditional estimates across different meta-analyses or studies, even using marginal standardization (Whitcomb & Naimi, 2021).
In addition, in the presence of effect modification with underlying risks, models using fixed study intercepts can lead to inconsistent likelihood-based estimation and introduce important biases (van Houwelingen et al., 2002).When there are systematic differences between trials of different designs (e.g., missing not at random for treatment contrasts), the assumption that odds ratio is transportable across populations with different underlying risks itself is overoptimistic.As Xiao et al. recently demonstrated, the odds ratio is likely dependent on the underlying risks (Xiao, Chen, et al., 2022;Xiao, Chu, et al., 2022).At a minimum, we should report the range of baseline risks over which the relative effects are estimated because it is dangerous to extrapolate beyond the scope of any model.
On the other hand, the AB-NMA assumes random intercepts and as a consequence, the (conditional) treatment effects depend on the underlying risks.The main weakness of AB-NMA or models with random intercepts is that they do not entirely respect the principle of concurrent control.More importantly, the argument about a fixed versus random intercept may hold even in a meta-analysis of two treatments when many of the trials involve unequal randomization (Senn, 2010).Thus, AB-NMA may also introduce bias due to potentially breaking randomization when there are systematic differences between trials of different designs (e.g., missing not at random for treatment arms).More research is needed to identify any scenarios where this could be of practical importance (Senn, 2010).However, random intercepts are very attractive in NMA with disconnected networks or single-arm trials (Beliveau et al., 2017;Wang, Lin, Murray, et al., 2021;Zhang et al., 2019), in which fixed intercepts models are not useful.
In conclusion, when there are profound and systematic differences between trials of different designs (e.g., under the untestable missing not at random assumption either for treatment contrasts or treatment arms), it is important to consider both approaches in practice as complementary sensitivity analyses and to provide the totality of evidence, because each approach has its own assumptions and limitations and, more importantly, indirect treatment comparison is susceptible to risk of bias no matter which model is chosen in a given analysis.

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E Y W O R D S arm-based, Bayesian hierarchical model, binary outcome, contrast-based, network metaanalysis, patient-centered outcome research 1 | INTRODUCTION

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I G U R E 1 The conditional treatment effects comparing treatment B versus the baseline treatment A for the partially and fully observed data, based on arm-based NMA with heterogeneous variance and equal correlation assumption.The three rows correspond to the constant RR, RD, and OR assumptions, respectively.The red arrows indicate the magnitude of bias due to model misspecification.
Conditional odds ratio (cOR) with 95% credible intervals under constant RR, RD, and OR assumptions.Marginal (or population-averaged) treatment effect estimates with 95% credible intervals under constant RR, RD, and OR assumptions.The true marginal OR, RR and RD are computed using the marginal event rates reported in Table1.Abbreviations: OR, odds ratio; RD, risk difference; RR, risk ratio.
T A B L E 2Note: The true cOR is computed as the average of study-specific OR from the full data.Abbreviations: OR, odds ratio; RD, risk difference; RR, risk ratio.a The arm-based NMA with homogeneous variance and equal correlation assumption.b The arm-based NMA with heterogeneous variance and equal correlation assumption.c The contrast-based NMA with homogeneous variance.CHU ET AL.T A B L E 3 a The arm-based NMA with homogeneous variance and equal correlation assumption.b The arm-based NMA with heterogeneous variance and equal correlation assumption.