We extended two adjustment methods for reporting bias from MAs to NMAs. The first method combined NMA and meta-regression models, with effect sizes regressed against their precision. The second one combined the NMA model with a logistic selection model estimating the probability that a trial was published or selected in the network. The former method basically adjusts for funnel plot asymmetry or small study effects, which may arise from causes other than publication bias. The latter adjusts for publication bias (ie, the suppression of an entire trial depending on results). The two models borrow strength from other trials in the network with the assumption that biases operate in a similar way in trials across the domain.

In a specific network of placebo-controlled trials of antidepressants, based on data already described and published previously by Turner et al., comparing the results of adjustment models applied to published data and those of the standard NMA model applied to published data allowed for assessing the robustness of efficacy estimates and ranking to publication bias or related small-study effects. Both models showed a decrease in all basic parameters (ie, the 12 effect sizes of drugs relative to placebo). The 66 contrasts for all possible pair-wise comparisons between drugs, the probabilities of being the best drug and the ranking were modified as well. The NMA of published data was not robust to publication bias and related small-study effects.

This specific dataset offered the opportunity to perform NMAs on both published and FDA data. The latter may be considered "an unbiased (but not the complete) body of evidence" for placebo-controlled trials of antidepressants [28]. The comparison of the results of the 2 models applied to published data and the standard NMA model applied to FDA data showed that the effect sizes of drugs relative to placebo were corrected for some but not all drugs. This observation led to differences in the 66 possible pair-wise comparisons between drugs, the probabilities of being the best drug and the ranking. It suggests that the 2 models should not be considered optimal; that is, the objective is not to produce definitive estimates adjusted for publication bias and related small-study effects but rather to assess the robustness of results to the assumption of bias.

Similar approaches have been used by other authors. Network meta-regression models fitted within a Bayesian framework were previously developed to assess the impact of novelty bias and risk of bias within trials [36, 37]. Network meta-regression to assess the impact of small-study effect was specifically used by Dias et al. in a re-analysis of a network of published head-to-head randomized trials of selective serotonin reuptake inhibitors [38]. Along the line of the regression-based approach of Moreno et al. in conventional MA, the authors introduced a measure of study size as a regression variable within the NMA model and identified a mean bias in pair-wise effect sizes. More recently, Moreno et al. used a similar approach to adjust for small-study effects in several conventional MAs of similar interventions and outcomes and illustrated their method using the dataset of Turner et al. [39]. Our approach differed in that we extended this meta-regression approach to NMAs. We used the standard error of treatment effect estimate as the regressor. As well, we specified an additive between-trial variance rather than a multiplicative overdispersion parameter. With the latter, the estimated multiplicative parameter may be < 1, which implies less heterogeneity than would be expected by chance alone. Selection model approaches have been considered recently. Chootrakool et al. introduced an approximated normal model based on empirical log-odds ratio for NMAs within a frequentist framework and applied Copas selection models for some groups of trials in the network selected according to funnel plot asymmetry [40]. Mavridis et al. presented a Bayesian implementation of the Copas selection model extended to NMA and applied their method on the network of Turner et al. [41]. In the Copas selection model, the selection probability depends on both the estimates of the treatment effects and their standard errors. In the extension to NMA, an extra correlation parameter ρ, assumed equal for all comparisons, needs to be estimated. When applied to published data of the network of Turner et al., the selection model we proposed and the treatment-specific selection model of Mavridis et al. yielded close results.

The 2 adjustment models rely on the assumption of exchangeability of selection processes across the network; that is, biases, if present, operate in a similar way in trials across the network. In this case study, all studies were, by construction, industry-sponsored, placebo-controlled trials registered with the FDA, and for all drugs, results of entire studies remained unreported depending on the results [23]. Thus, the assumption of exchangeability of selection processes is plausible. More generally, if we have no information to distinguish different reporting bias mechanisms across the network, an exchangeable prior distribution is plausible, "ignorance implies exchangeability" [42, 43]. However, the assumption may not be tenable in other contexts in which reporting biases may affect the network in an unbalanced way. It may operate differently in placebo-controlled and head-to-head trials [44], in older and more recent trials (because of trial registries), and for drug and non-drug interventions [7]. In more complex networks involving head-to-head trials, the 2 adjustment models could be generalized to allow the expected publication bias or small-study bias for active-active trials to differ from that of the expected bias in trials comparing active and inactive treatments [36]. In head-to-head trials, the direction of bias is uncertain but assumptions in defining
could be that the sponsored treatment is favored (sponsorship bias) [45, 46] or that the newest treatment is favored (optimism bias) [37, 47, 48]. If treatment *j* is the drug provided by the pharmaceutical that sponsored the trial and treatment *k* is not,
would be equal to 1. Or
would be equal to 1 if treatment *j* is newer than treatment *k*. However, disentangling the sources of bias operating on direct and indirect evidence would be difficult, especially if reporting bias and inconsistency are twisted together or if the assumed bias directions are in conflict on a loop.

The models we described have limitations. First, they would result in poor estimation of bias and effect sizes when the conventional MAs within the network include small numbers of trials [21]. Second, for the selection model, we specified the weight function. If the underlying assumptions (ie, a logistic link form and the chance of a trial being selected related to standard error) are wrong, the estimated selection model will be wrong. However, alternative weight functions (e.g., probit link) or conditioning (e.g., on the magnitude of effect size) could be considered. Finally, it was implemented with a weakly informative prior, which mainly suggested that the propensity for results to be published may decrease with increasing standard error. There is a risk that prior information overwhelms observed data, especially if the number of trials is low. Although they were somewhat arbitrarily set, our priors for the selection model parameters were in line with the values in previous studies using the Copas selection model [12, 49]. Different patterns of selection bias could be tested, for instance, by considering various prior modes for *p*
_{
min
} and *p*
_{
max
}, the probabilities of publication when the standard error takes its minimum and maximum values across the network [15].