### Before we begin: systematic review and meta-analysis

### Introduction to NMA

_{A}, T

_{B}, T

_{C}and T

_{D}are used as abbreviated version of treatment A, B, C and D in the following manuscript and figures. In this figure, T

_{A}serves as an anchor that indirectly compares T

_{B}and T

_{C}or T

_{B}, T

_{C}, and T

_{D}. T

_{A}(anchor) is also called a common comparator. T

_{B}and T

_{C}or T

_{C}and T

_{D}are indirectly compared using anchor A. This type of comparison is called an ‘anchored indirect treatment comparison.' When the shape formed by direct comparisons is incomplete, it is also called an ‘open triangle.’

_{A}and T

_{B}; studies 5 and 6 directly compared T

_{A}and T

_{C}; and studies 3, 4, and 7 directly compared T

_{B}and T

_{C}. In addition to information from the direct comparison of T

_{A}and T

_{B}, information from the indirect comparison can also be used to compare the two treatments. T

_{A}and T

_{B}can be indirectly compared using T

_{C}as a common comparator in studies 3, 4, 5, and 6. As T

_{C}has also been investigated in study 7, T

_{A}and T

_{B}can also be indirectly compared via the common comparator T

_{C}in this study.

_{A}versus T

_{B}and T

_{A}versus T

_{C}(Fig. 2A). The red rectangle represents the difference between the treatment effects of T

_{A}and T

_{B}(T

_{AB}= T

_{B}− T

_{A}), and the sky blue rectangle represents the difference between the treatment effects of T

_{A}and T

_{c}(T

_{AC}= T

_{C}– T

_{A}). The difference between the treatment effects of T

_{B}and T

_{C}(T

_{BC}) may be obtained by subtracting the treatment effect of B from that of C (T

_{BC}= T

_{C}– T

_{B}). However, it may lead to bias when the treatment effect of T

_{A}in study AB (a study that compared T

_{A}and T

_{B}is named study AB) may be different from that in study AC. Therefore, the possibility of baseline differences between studies regarding the treatment effect of A as a common comparator should be considered (Fig. 2A).

### Steps of NMA

### Attempt to include all relevant RCTs

### Explore network geometry

_{A}and T

_{B}(study AB) and T

_{C}and T

_{D}(study CD) (Fig. 2B), when no connection between the treatments is noted, the relative treatment effect between T

_{AB}(difference in the treatment effect of T

_{A}and T

_{B}) and T

_{CD}(difference in the treatment effect of T

_{C}and T

_{D}) cannot be assumed. However, with a study comparing T

_{A}and T

_{C}(study AC), T

_{BC}(difference in the treatment effect of T

_{B}and T

_{C}) can be assumed through the common comparator T

_{A}(Fig. 2C). Moreover, T

_{AD}(difference in the treatment effect of T

_{A}and T

_{D}) can be assumed through the common comparator T

_{C}(Fig. 2C).

### Assess key assumptions

### Performance of analyses and NMA

### Key assumption in network meta-analysis

### Similarity and homogeneity for direct comparisons

### Transitivity for indirect comparisons

_{AB}and T

_{AC}, can be used to calculate the T

_{BC}, an indirect comparison. If the treatment effect of A is similar between the direct comparison of T

_{AB}and T

_{AC}, the common comparator A can be used from T

_{AB}and T

_{AC}, which is ‘transitive’ from treatments B through A to C.

_{AB}to T

_{AC}. If an imbalance in the distribution of effect modifiers exists between the studies, incorrect estimates may be obtained. Fig. 3 shows that the assumption of transitivity is violated when effect modifier D (between T

_{A}and T

_{B}) is not similar to effect modifier E (between T

_{A}and T

_{C}).

*jointly randomizable*’, which means that a trial including all treatments would be clinically reasonable. It is assumed that the investigators have included RCTs. Comparisons within an RCT are compared between the randomized groups, while those between RCTs are not randomized. However, comparisons between RCTs are not randomized. Therefore, the comparisons between RCTs should be assumed to be ‘jointly randomizable’ to perform an NMA. Thus, it is essential to consider this when conducting an evidence network. Transitivity may be violated if the intervention or treatment method has a different target patient group or indication between studies. For example, when T

_{A}is the primary treatment and T

_{B}and T

_{C}are both primary and secondary treatments, patients in study BC cannot be assumed to be randomly assigned to study AC.

### Consistency for mixed comparisons

#### Cochran’s Q statistics

*χ*

^{2}distribution [24].

^{2}, is provided to measure the degree of inconsistency. I

^{2}can be calculated as

*I*

^{2}= 100% × (

*Q*− df) /

*Q*, where

*Q*is Cochran’s heterogeneity statistic and df is the degree of freedom. A value of 0% indicates no heterogeneity, and larger values indicate increasing heterogeneity.

#### Loop inconsistency (Supplementary Fig. 2)

_{0}: IF = 0

_{1}: IF ≠ 0

#### Inconsistency parameter approach

_{ABC}) in each loop wherein inconsistency could occur.

_{BC}= μ

_{AC}- μ

_{AB}

_{BC}= μ

_{AC}- μ

_{AB}+ ω

_{ABC}

_{BC}is the treatment effect of BC,

_{AB}is the treatment effect of AB,

_{AC}is the treatment effect of AC, and

_{ABC}is the inconsistency parameter.

_{ABC}) is 0 in the inconsistency model, it can be classified as a consistency model. The distribution of the inconsistent variable is ω

_{j}~

*N*(0,

*σ*

^{2}).

#### Node-splitting (Supplementary Fig. 3)

#### Net heat plot (Supplementary Fig. 4)

#### Design-by-treatment interaction approach

### Statistical methods in NMA

### Result presentation

### Network plot

^{2}; Supplementary Figs. 1A and 1B).