We thank Krahn et al for their letter¹ and welcome the opportunity to discuss this further.

When assessing inconsistency, interest lies in the difference between the direct evidence and the network evidence. We believe the difference between our paper and the correction that Krahn et al¹ have suggested arises due to the possibility of interpreting “detaching a design” in one of two ways.

In their original paper, when describing the approach for detaching a single design, Krahn et al state “This procedure is equivalent to a ‘leave one out’ approach: Once per fit, studies with one design are left out of the network estimate to obtain an independent estimate of the treatment effect in design d and to obtain a network model fit independent of studies with design d”.² Our interpretation of this statement³ is that, when assessing the impact of detaching design d on the network estimate for design d, the network estimate is calculated from the evidence that remains when trials of design d are left out. If design d is left out then the network evidence can only be informed by what remains that is, the indirect evidence. This leads to our original result.

However, Krahn et al² also describe their approach in a subtly different way: as a design-by-treatment interaction. Following the design-by-treatment interaction approach, instead of leaving out design d, a new parameter is assigned to design d so that the model now includes two parameters for design d representing the direct and the indirect evidence. Krahn et al have now clarified (in their second paragraph) that they consider the former (ie, the direct evidence) to be the network evidence. Following this approach, we agree with the correction proposed by Krahn et al. However, we also incline to the view that the interpretation of the original paper² outlined in the preceding paragraph is more natural.

Nevertheless, accepting the correction proposed by Krahn et al¹ simplifies the argument in our paper³ because with $Q_{\mathrm{AC}(c)}^{\mathrm{inc}}=0$, our equation (7) is equal to our equation (6). Further, because Q^diff is still a squared and scaled version of the inconsistency parameter, our observation that this is correlated with the formal inconsistency test statistic, while awkward to interpret, remains valid. The question remains, why use a scaled version when the unscaled version has a known distribution?

While acknowledging the authors' important contributions in this area, we note that they have not disputed our key finding which is that the net heat plot displays a somewhat arbitrary weighting of the loop inconsistency statistics, which does not lend itself to statistical testing, and which should therefore be interpreted cautiously.

我们感谢 Krahn 等人的来信¹ ，并欢迎有机会进一步讨论这一问题。

在评估不一致时，兴趣在于直接证据和网络证据之间的差异。我们相信我们的论文与 Krahn 等人¹所建议的修正之间的差异是由于以两种方式之一解释“分离设计”的可能性而产生的。

Krahn 等人在其原始论文中描述分离单个设计的方法时指出，“此过程相当于‘留一法’方法：每次拟合后，使用一种设计的研究将被排除在网络估计之外，以获得对设计d中的治疗效果进行独立估计，并获得独立于设计d的研究的网络模型拟合”。 ²我们对此陈述³的解释是，在评估分离设计d对设计d的网络估计的影响时，网络估计是根据省略设计d的试验时剩余的证据计算的。如果设计d被遗漏，那么网络证据只能由剩下的，即间接证据来告知。这导致了我们最初的结果。

然而，Krahn 等人²也以一种略有不同的方式描述了他们的方法：作为治疗交互设计。遵循按治疗设计的交互方法，不是忽略设计d ，而是为设计d分配一个新参数，以便模型现在包括代表直接和间接证据的设计d的两个参数。 Krahn 等人现在（在第二段中）澄清，他们认为前者（即直接证据）是网络证据。按照这种方法，我们同意 Krahn 等人提出的修正。然而，我们也倾向于认为，对前一段概述的原始论文²的解释更为自然。

然而，接受 Krahn 等人提出的修正¹简化了我们论文³中的论点，因为 $Q_{\mathrm{AC}(c)}^{\mathrm{inc}}=0$ ，我们的方程（7）等于我们的方程（6）。此外，由于Q ^diff仍然是不一致性参数的平方和缩放版本，因此我们观察到这与形式的不一致性测试统计量相关，虽然难以解释，但仍然有效。问题仍然存在，当未缩放版本具有已知分布时，为什么要使用缩放版本？

在承认作者在这一领域的重要贡献的同时，我们注意到他们并没有对我们的主要发现提出异议，即净热图显示了循环不一致性统计数据的某种任意权重，这不适合统计测试，并且因此应谨慎解读。