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Analysing complex interventions using component network meta-analysis. Systematic reviews with network meta-analysis frequently evaluate complex interventions combining multiple healthcare interventions (known as components). Components may act separately of each other or in conjunction with other components, synergistically or antagonistically. Component effect estimation is crucial to produce relevant and clinically meaningful evidence. However, standard network meta-analysis cannot quantify individual component effects of complex interventions. This article presents methods for modelling complex interventions and highlights the advantages and limitations of component network meta-analysis (CNMA). CNMA enables the estimation of individual component effects, whether additive or interactive. Interaction CNMA can be considered an extension of the additive CNMA model that allows component effects to vary in the presence of each other. This article presents practical guidance on how to carry out these analyses via empirical examples, which showcase both the strengths and limitations of CNMA. Implementing CNMA models is complex and requires the skills of a multidisciplinary team including clinicians, methodologists, and statisticians.

Our illustrative example focuses on prehabilitation, a unimodal or multimodal intervention (typically including one or more of exercise (EXE), nutrition (NUT), cognitive (COG) or psychosocial (PSY) components) that aims to help patients build reserve before surgery to improve postoperative recovery. Key questions in the field of prehabilitation relate to what components provide greatest efficacy in improving outcomes, which could inform design of clinical programmes and future study interventions. Our example analysis uses data from 106 randomised controlled trials of 8816 adults undergoing surgery that assessed prehabilitation interventions for the prevention of postoperative complications.4 In these trials, individuals were allocated to receive a prehabilitation intervention comprised of one or more of the following components: EXE, NUT, COG, or PSY (an active comparator intervention); or usual care (UC; an inactive comparator; reference group). Each intervention is different and has different implementation requirements, making the decision for the patient and clinician challenging. The outcome of interest was binary, defined as the occurrence of any postoperative medical or surgical complication during the initial hospital stay or within 30 days after surgery. This outcome was negative, so odds ratios (OR) lower than 1 indicated a beneficial effect of the intervention. Seven active interventions were compared to UC and formed a network of trials with one loop owing to a single, three arm study (see section below—What makes component network meta-analysis useful). In addition to the four single component interventions (EXE, NUT, COG, PSY), three complex interventions were assessed in the eligible randomised controlled trials: EXE+NUT, EXE+PSY, and EXE+NUT+PSY (appendix 1). The evidence graph and selection of components were informed by data available in the literature and input from the research team. Team-wide questionnaires were used to identify key prehabilitation components and prioritise critical outcomes for analysis, while a broader taxonomy was prespecified at the protocol stage.4 5

The complexity of an intervention lies in its included components, which may act independently of or in conjunction with each other. Components reflecting the nature of the intervention may interact with each other or with other intervention characteristics, such as behaviours of those delivering (eg, clinician, nurse, lay person) or receiving the intervention, groups targeted by or delivering the intervention, and the intensity (eg, daily, weekly), setting (eg, hospital, home, community), and mode of delivery (eg, virtually, in-person) of the intervention.1 In our prehabilitation example, consider a randomised controlled trial assessing the efficacy of a complex intervention consisting of two components, EXE and NUT, compared with UC in preventing postoperative complications. Although we can estimate the effect size of this complex intervention by considering it as a single treatment, a question of perhaps greater clinical interest may pertain to the relative merits of the individual components (EXE and NUT). A potential design to answer this question would be a factorial randomised controlled trial, wherein participants are randomised to receive one of four potential options: EXE, NUT, both EXE and NUT (ie, EXE+NUT), or neither EXE nor NUT (ie, UC). However, this design is not commonly encountered because of the increased resources required, reducing feasibility.6

would be a factorial randomised controlled trial, wherein participants are randomised to receive one of four potential options: EXE, NUT, both EXE and NUT (ie, EXE+NUT), or neither EXE nor NUT (ie, UC). However, this design is not commonly encountered because of the increased resources required, reducing feasibility.6 If multiple randomised controlled trials exist, we typically conduct a standard NMA to evaluate the relative efficacy of the interventions of interest.7 8 However, standard NMA cannot quantify the effects of individual components from complex interventions. Consider a scenario in which an NMA of interventions including EXE, NUT, and COG for the prevention of postoperative complications suggests that EXE has a large effect, EXE+NUT has a small effect, and EXE+NUT+COG has a large effect, where the same components (EXE and NUT) are included in the most and least efficacious interventions. This increases the difficulty of relating components to efficacy. To develop tailored recommendations to optimise patient care and better allocate finite healthcare resources, it is useful to disentangle the effects of each component of the prehabilitation interventions using CNMA.2 CNMA can estimate a separate effect for each individual component of complex interventions.9 For example, the efficacy of the EXE and NUT components can be estimated separately and in combination, without requiring direct evidence from previous randomised controlled trials by borrowing information from trials sharing each of the components.

parate effect for each individual component of complex interventions.9 For example, the efficacy of the EXE and NUT components can be estimated separately and in combination, without requiring direct evidence from previous randomised controlled trials by borrowing information from trials sharing each of the components. CNMA allows us to derive estimates of the effects of individual components, even when delivered in combination. Exploring intervention components in a network of trials can help with exploring clinical heterogeneity, gaining insights into how an intervention works and the drivers of its effect, identifying effective components, and informing the design of future studies, particularly regarding what components and their combinations may be most promising to assess. A key question asks how we should analyse complex interventions; this depends on the research question. Box 1 and figure 1 exemplify this process, using the example of prehabilitation interventions. Box 2 presents assumptions and key concepts in CNMA. Question: “Do prehabilitation interventions overall work compared with usual care?” Use a single effect (pairwise) meta-analysis.1 10 11 Appropriate when estimating an overall effect for a class of interventions. All interventions are combined into one group (eg, prehabilitation v usual care; fig 1A). Limitations: Provides limited information for clinical practice or implementation decisions. Assumes intervention (and component) effects are equivalent.

Use a single effect (pairwise) meta-analysis.1 10 11 Appropriate when estimating an overall effect for a class of interventions. All interventions are combined into one group (eg, prehabilitation v usual care; fig 1A). Limitations: Provides limited information for clinical practice or implementation decisions. Assumes intervention (and component) effects are equivalent. Increases or masks heterogeneity (eg, combining strong and weak intervention effects can increase heterogeneity and hide valuable information). Question: “Which prehabilitation interventions work?” Use a standard network meta-analysis (NMA).11 12 13 14 Appropriate when assessing which types of interventions are most effective and how they compare to each other.15 16 Each unique combination of components is treated as a separate intervention (node) (fig 1B). Reduces heterogeneity compared with a single effect model.17 18 Limitations: Treats combinations as unrelated, assumed to act independently, and ignores shared components (eg, exercise and nutrition together (EXE+NUT), EXE only, NUT only). Cannot estimate individual component effects and their contribution to the network effects. May result in sparse networks with limited data points per intervention comparison, an increased number of parameters in the model, and decreased statistical power. Question: “Which components of prehabilitation interventions work?” Use a component network meta-analysis (CNMA; fig 1C).10 15 19 CNMA extends NMA by disaggregating interventions into components.

May result in sparse networks with limited data points per intervention comparison, an increased number of parameters in the model, and decreased statistical power. Question: “Which components of prehabilitation interventions work?” Use a component network meta-analysis (CNMA; fig 1C).10 15 19 CNMA extends NMA by disaggregating interventions into components. Additive effects: The effect of a complex intervention is the sum of the effects of its components: for example, EXE+NUT has an intervention effect equal to the sum of the individual component effects of EXE and NUT (box 2). Interaction effects: positive (synergistic), EXE+NUT + EXE*NUT (where the asterisk corresponds to the interaction between EXE and NUT); negative (antagonistic), EXE+NUT – EXE*NUT. Including interactions relaxes the additivity assumption. CNMA with all possible interactions (ie, all unique combinations of components with direct evidence), called the full interaction model, is equivalent to the standard NMA.15 Borrows strength across studies with shared components; reduces the number of model parameters and increases precision; explains heterogeneity via component level characteristics (eg, nature, setting, provider, intensity, mode of delivery of intervention); creates model based links between interventions sharing components, enhancing connectivity; estimates component specific effects as well as intervention level effects, increasing interpretability.15

geneity via component level characteristics (eg, nature, setting, provider, intensity, mode of delivery of intervention); creates model based links between interventions sharing components, enhancing connectivity; estimates component specific effects as well as intervention level effects, increasing interpretability.15 CNMA can reconstruct disconnected networks when subnetworks share common components and clinical evidence for additivity is strong (fig 1D); a network is disconnected when it consists of two or more independent subnetworks of interventions that do not share a common comparator. Network diagrams depicting different research questions on the prevention of postoperative complications. Nodes represent different combinations of components evaluated within each model and edges depicting studies comparing the different combinations of components they connect. Solid line thickness is proportional to the number of studies included in the group comparison, and node size is proportional to the number of patients included in the underlying group. Dashed lines represent links added by component network meta-analysis (CNMA) model between different combinations of components sharing common components. COG=cognitive component; EXE=exercise component; NUT=nutrition component; PSY=psychosocial component; UC=usual care

ber of patients included in the underlying group. Dashed lines represent links added by component network meta-analysis (CNMA) model between different combinations of components sharing common components. COG=cognitive component; EXE=exercise component; NUT=nutrition component; PSY=psychosocial component; UC=usual care Transitivity: Assumes that the distribution of any effect modifiers (ie, any variables that may change the true effect of an intervention) is similar on average across intervention comparisons. Under this assumption, participants included in different trials could, in principle, have been randomised to any of the interventions being compared. Consistency: The statistical equivalent of transitivity. It assumes that the different sources of evidence (ie, direct and indirect evidence) agree. Homogeneity: Assumes that studies contributing to each pairwise comparison are sufficiently similar in their clinical and methodological characteristics to be meaningfully combined. Additivity: The effect of a complex intervention equals the sum of its individual component effects (eg, log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC; where OR=odds ratio; EXE=exercise component; NUT=nutrition component; UC=usual care). In additive CNMA, components act independently. Connectivity: Standard network meta-analysis (NMA) requires a connected network of interventions to allow indirect comparisons. CNMA can reconnect disconnected subnetworks that share common components, provided there is robust clinical evidence supporting the additivity assumption.20

UC; where OR=odds ratio; EXE=exercise component; NUT=nutrition component; UC=usual care). In additive CNMA, components act independently. Connectivity: Standard network meta-analysis (NMA) requires a connected network of interventions to allow indirect comparisons. CNMA can reconnect disconnected subnetworks that share common components, provided there is robust clinical evidence supporting the additivity assumption.20 Component: A distinct element of an intervention (eg, exercise, nutrition, psychological support). Complex intervention: An intervention arm comprising two or more components delivered together (eg, exercise and nutrition). Additive CNMA model: A model that disentangles complex interventions into their individual component effects, assuming additivity. It estimates the effect of each component assuming that components act without interaction. Interaction CNMA model: A model that includes one or more interaction terms, allowing for synergistic (positive) or antagonistic (negative) deviations from additivity. Allows assessment of whether certain component combinations modify each other’s effects. Full interaction model: A CNMA model including all possible component combinations; mathematically equivalent to the standard NMA. It is the most complex model (with the greatest number of parameters), whereas the additive CNMA is the most parsimonious model (fewest parameters). Standard NMA: Considers each unique combination of components as a distinct intervention node without disentangling the interventions into components.

Full interaction model: A CNMA model including all possible component combinations; mathematically equivalent to the standard NMA. It is the most complex model (with the greatest number of parameters), whereas the additive CNMA is the most parsimonious model (fewest parameters). Standard NMA: Considers each unique combination of components as a distinct intervention node without disentangling the interventions into components. Q statistic: A heterogeneity measure used to test the assumption of additivity by comparing model fit between additive and interaction CNMA models.15 Comparing the Q statistics for the NMA model and the additive CNMA model (QAdd v QNMA) helps evaluate whether the additivity assumption holds; a statistically significant difference in Q indicates evidence of interaction.

he assumption of additivity by comparing model fit between additive and interaction CNMA models.15 Comparing the Q statistics for the NMA model and the additive CNMA model (QAdd v QNMA) helps evaluate whether the additivity assumption holds; a statistically significant difference in Q indicates evidence of interaction. In standard NMA, the effect of one intervention compared with another can be estimated indirectly through a common comparator. For example, if trials compare EXE+PSY with UC and compare PSY with UC, we can indirectly estimate the effect of EXE+PSY versus PSY through UC. However, standard NMA ignores any potential associations between combinations of components (eg, EXE+PSY) and the individual components (eg, EXE, PSY) that are part of the combination. Component effects may be additive (ie, combined effect equals the sum of the component effects; box 2) or interactive, where different sets of components may interact with each other to decrease or increase the effect (ie, the combined effect is greater or smaller than the sum of their component effects). Also, certain components may be effective in some combinations but not in others, which increases the difficulty in making inferences about component effects.

mponents may interact with each other to decrease or increase the effect (ie, the combined effect is greater or smaller than the sum of their component effects). Also, certain components may be effective in some combinations but not in others, which increases the difficulty in making inferences about component effects. In CNMA, because each intervention is broken into its individual components, and EXE+PSY and PSY share the PSY component, the model creates a direct connection between the two interventions, allowing the estimation of the PSY effect rather than only relying on the indirect comparison through UC. This modelling approach effectively creates an additional link between those interventions (see dashed lines in fig 1C and fig 1D), beyond the usual indirect path through UC; the additional link is not an observed direct comparison from data, but a model based connection that uses shared components to strengthen estimation. CNMA models can also be performed to reconstruct (or fuse) disconnected networks by modelling component effects, as long as complex interventions in each network share common individual components and there is strong clinical evidence of additivity.20

sed connection that uses shared components to strengthen estimation. CNMA models can also be performed to reconstruct (or fuse) disconnected networks by modelling component effects, as long as complex interventions in each network share common individual components and there is strong clinical evidence of additivity.20 As an example, consider a network of 10 unique combinations of components (including UC), presented in figure 1D . This network includes all randomised controlled trials from the illustrative example of prehabilitation interventions, along with one additional trial comparing two unique combinations of components (COG+PSY v NUT+PSY). The two arm study comparing COG+PSY with NUT+PSY is not connected to the original network of trials, and hence two disconnected networks are formed. CNMA models can reconnect the disconnected networks, as long as there are common components across the two networks. Because the components COG, PSY, and NUT are common within the two networks, their efficacy can be estimated using the CNMA model. However, strong clinical evidence for additivity must exist because the additivity assumption cannot be assessed statistically in disconnected networks.15 For example, previous studies or expert consensus may suggest that the effects of the individual components in the two subnetworks (eg, PSY and NUT in prehabilitation) are independent and expected to be combined without interaction. Additivity can be assessed by comparison of the CNMA with the standard NMA, if the network is connected. However, standard NMA cannot be performed on a disconnected network.20

CNMA allows us to derive estimates of the effects of individual components, even when delivered in combination. Exploring intervention components in a network of trials can help with exploring clinical heterogeneity, gaining insights into how an intervention works and the drivers of its effect, identifying effective components, and informing the design of future studies, particularly regarding what components and their combinations may be most promising to assess. A key question asks how we should analyse complex interventions; this depends on the research question. Box 1 and figure 1 exemplify this process, using the example of prehabilitation interventions. Box 2 presents assumptions and key concepts in CNMA. Question: “Do prehabilitation interventions overall work compared with usual care?” Use a single effect (pairwise) meta-analysis.1 10 11 Appropriate when estimating an overall effect for a class of interventions. All interventions are combined into one group (eg, prehabilitation v usual care; fig 1A). Limitations: Provides limited information for clinical practice or implementation decisions. Assumes intervention (and component) effects are equivalent. Increases or masks heterogeneity (eg, combining strong and weak intervention effects can increase heterogeneity and hide valuable information). Question: “Which prehabilitation interventions work?” Use a standard network meta-analysis (NMA).11 12 13 14 Appropriate when assessing which types of interventions are most effective and how they compare to each other.15 16

Increases or masks heterogeneity (eg, combining strong and weak intervention effects can increase heterogeneity and hide valuable information). Question: “Which prehabilitation interventions work?” Use a standard network meta-analysis (NMA).11 12 13 14 Appropriate when assessing which types of interventions are most effective and how they compare to each other.15 16 Each unique combination of components is treated as a separate intervention (node) (fig 1B). Reduces heterogeneity compared with a single effect model.17 18 Limitations: Treats combinations as unrelated, assumed to act independently, and ignores shared components (eg, exercise and nutrition together (EXE+NUT), EXE only, NUT only). Cannot estimate individual component effects and their contribution to the network effects. May result in sparse networks with limited data points per intervention comparison, an increased number of parameters in the model, and decreased statistical power. Question: “Which components of prehabilitation interventions work?” Use a component network meta-analysis (CNMA; fig 1C).10 15 19 CNMA extends NMA by disaggregating interventions into components. Additive effects: The effect of a complex intervention is the sum of the effects of its components: for example, EXE+NUT has an intervention effect equal to the sum of the individual component effects of EXE and NUT (box 2). Interaction effects: positive (synergistic), EXE+NUT + EXE*NUT (where the asterisk corresponds to the interaction between EXE and NUT); negative (antagonistic), EXE+NUT – EXE*NUT. Including interactions relaxes the additivity assumption.

Additive effects: The effect of a complex intervention is the sum of the effects of its components: for example, EXE+NUT has an intervention effect equal to the sum of the individual component effects of EXE and NUT (box 2). Interaction effects: positive (synergistic), EXE+NUT + EXE*NUT (where the asterisk corresponds to the interaction between EXE and NUT); negative (antagonistic), EXE+NUT – EXE*NUT. Including interactions relaxes the additivity assumption. CNMA with all possible interactions (ie, all unique combinations of components with direct evidence), called the full interaction model, is equivalent to the standard NMA.15 Borrows strength across studies with shared components; reduces the number of model parameters and increases precision; explains heterogeneity via component level characteristics (eg, nature, setting, provider, intensity, mode of delivery of intervention); creates model based links between interventions sharing components, enhancing connectivity; estimates component specific effects as well as intervention level effects, increasing interpretability.15 CNMA can reconstruct disconnected networks when subnetworks share common components and clinical evidence for additivity is strong (fig 1D); a network is disconnected when it consists of two or more independent subnetworks of interventions that do not share a common comparator.

Borrows strength across studies with shared components; reduces the number of model parameters and increases precision; explains heterogeneity via component level characteristics (eg, nature, setting, provider, intensity, mode of delivery of intervention); creates model based links between interventions sharing components, enhancing connectivity; estimates component specific effects as well as intervention level effects, increasing interpretability.15 CNMA can reconstruct disconnected networks when subnetworks share common components and clinical evidence for additivity is strong (fig 1D); a network is disconnected when it consists of two or more independent subnetworks of interventions that do not share a common comparator. Network diagrams depicting different research questions on the prevention of postoperative complications. Nodes represent different combinations of components evaluated within each model and edges depicting studies comparing the different combinations of components they connect. Solid line thickness is proportional to the number of studies included in the group comparison, and node size is proportional to the number of patients included in the underlying group. Dashed lines represent links added by component network meta-analysis (CNMA) model between different combinations of components sharing common components. COG=cognitive component; EXE=exercise component; NUT=nutrition component; PSY=psychosocial component; UC=usual care

Transitivity: Assumes that the distribution of any effect modifiers (ie, any variables that may change the true effect of an intervention) is similar on average across intervention comparisons. Under this assumption, participants included in different trials could, in principle, have been randomised to any of the interventions being compared. Consistency: The statistical equivalent of transitivity. It assumes that the different sources of evidence (ie, direct and indirect evidence) agree. Homogeneity: Assumes that studies contributing to each pairwise comparison are sufficiently similar in their clinical and methodological characteristics to be meaningfully combined. Additivity: The effect of a complex intervention equals the sum of its individual component effects (eg, log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC; where OR=odds ratio; EXE=exercise component; NUT=nutrition component; UC=usual care). In additive CNMA, components act independently. Connectivity: Standard network meta-analysis (NMA) requires a connected network of interventions to allow indirect comparisons. CNMA can reconnect disconnected subnetworks that share common components, provided there is robust clinical evidence supporting the additivity assumption.20

Component: A distinct element of an intervention (eg, exercise, nutrition, psychological support). Complex intervention: An intervention arm comprising two or more components delivered together (eg, exercise and nutrition). Additive CNMA model: A model that disentangles complex interventions into their individual component effects, assuming additivity. It estimates the effect of each component assuming that components act without interaction. Interaction CNMA model: A model that includes one or more interaction terms, allowing for synergistic (positive) or antagonistic (negative) deviations from additivity. Allows assessment of whether certain component combinations modify each other’s effects. Full interaction model: A CNMA model including all possible component combinations; mathematically equivalent to the standard NMA. It is the most complex model (with the greatest number of parameters), whereas the additive CNMA is the most parsimonious model (fewest parameters). Standard NMA: Considers each unique combination of components as a distinct intervention node without disentangling the interventions into components. Q statistic: A heterogeneity measure used to test the assumption of additivity by comparing model fit between additive and interaction CNMA models.15 Comparing the Q statistics for the NMA model and the additive CNMA model (QAdd v QNMA) helps evaluate whether the additivity assumption holds; a statistically significant difference in Q indicates evidence of interaction.

Depending on how the effects of individual components are related to each other (eg, additive or interactive), CNMA may be categorised as an additive CNMA or an interaction CNMA. Interaction CNMA can be considered as an extension of the additive CNMA model that includes interaction terms. These models have been described both in bayesian10 and frequentist settings.15 The additive CNMA model assumes that a separate and independent effect for each of the components of a complex intervention. Under the additivity assumption, the total intervention effect is the sum of the relevant component effects. That is, the effect of the combination of components EXE+NUT (including two components EXE and NUT) against UC is equal to the sum of the individual component effect of EXE versus UC plus the individual component effect of NUT versus UC.10 12 14 19 21 If the effect estimate is an OR, the additivity assumption applies on the log scale: log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC (fig 2). Equally, this assumption implies a product on the original ratio scale, and not a sum: OREXE+NUT v UC = OREXE v UC × ORNUT v

The additive CNMA model assumes that a separate and independent effect for each of the components of a complex intervention. Under the additivity assumption, the total intervention effect is the sum of the relevant component effects. That is, the effect of the combination of components EXE+NUT (including two components EXE and NUT) against UC is equal to the sum of the individual component effect of EXE versus UC plus the individual component effect of NUT versus UC.10 12 14 19 21 If the effect estimate is an OR, the additivity assumption applies on the log scale: log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC (fig 2). Equally, this assumption implies a product on the original ratio scale, and not a sum: OREXE+NUT v UC = OREXE v UC × ORNUT v UC. The log additive principle applies to any ratio measure (eg, risk ratio, hazard ratio). For absolute measures (eg, mean difference, risk difference), additivity applies on the original scale, where the combined effect equals the sum of the component effects. This approach can readily be extended to combinations of components consisting of more than two components. While this model can theoretically predict the effects of component combinations not evaluated in the included trials, such predictions should be viewed as exploratory and interpreted with caution.15

ent effects. This approach can readily be extended to combinations of components consisting of more than two components. While this model can theoretically predict the effects of component combinations not evaluated in the included trials, such predictions should be viewed as exploratory and interpreted with caution.15 Illustration of the additive and interaction effects in the illustrative example of the prevention of postoperative complications. The stacked bars show the log odds ratios (logOR) for exercise (EXE), nutrition (NUT), their additive combination (EXE+NUT = log(0.53) + log(0.66) = −0.635 + −0.416 = −1.050 = log(0.35)), and the model accounting for their interaction (EXE+NUT + EXE*NUT = log(0.53) + log(0.66) + 0.32 = log(0.48)). The light blue region within the bar on the further right (EXE+NUT + EXE*NUT) represents the interaction component (EXE*NUT), which was found to be antagonistic

g(0.66) = −0.635 + −0.416 = −1.050 = log(0.35)), and the model accounting for their interaction (EXE+NUT + EXE*NUT = log(0.53) + log(0.66) + 0.32 = log(0.48)). The light blue region within the bar on the further right (EXE+NUT + EXE*NUT) represents the interaction component (EXE*NUT), which was found to be antagonistic Under the additivity assumption, the comparative effects of common components in pairwise comparisons against the same component (eg, NUT) cancel out.11 15 Therefore, the effect of EXE+NUT versus NUT is identical to the incremental effect of EXE alone. If desirable, it is possible to define a single component as inactive, which means that adding it to any combination of active components does not affect the underlying intervention effect. For example, assuming that UC is an inactive component, the effect of EXE+NUT versus UC is identical to the incremental effect of EXE+NUT because the effect of UC is assumed to be zero. If UC is considered an inactive component, intervention effects will be relative to UC, which is assumed to have no impact when combined with other interventions. However, if no inactive component (eg, UC, control, placebo) is chosen or available in the randomised controlled trials, incremental intervention effects will be relative to no intervention. The choice of an inactive component is important because it affects the parameterisation of component effects, and it is assumed that the comparison of an inactive component with an active component leads to the active component’s net effect.15 Whether an inactive component is considered in a CNMA is at the discretion of the review authors and should be guided by clinical expertise.

ts the parameterisation of component effects, and it is assumed that the comparison of an inactive component with an active component leads to the active component’s net effect.15 Whether an inactive component is considered in a CNMA is at the discretion of the review authors and should be guided by clinical expertise. Additive CNMA assumes no interaction between components. However, clinically, components may interact with each other (ie, the effect of a component may differ in the presence of another component), and additive CNMA may lead to biased estimates of component effects. If the additive CNMA does not fit the data well (ie, the additivity assumption is not met), then relaxing the additivity assumption and assuming interactions between components in an interaction CNMA model may be appropriate to explore which sets of component effects are synergistic or antagonistic.

component effects. If the additive CNMA does not fit the data well (ie, the additivity assumption is not met), then relaxing the additivity assumption and assuming interactions between components in an interaction CNMA model may be appropriate to explore which sets of component effects are synergistic or antagonistic. The interaction CNMA model is an extension of the additive CNMA model that includes additional terms for component interactions that may account for higher (synergistic) or lower (antagonistic) intervention effects than the sum of the component effects alone.10 11 12 21 A two way interaction CNMA model allows pairs of components within interventions to interact, resulting in potentially lower or higher component effects than the additive effects of the two components. In our illustrative example, a two way interaction model answers the question “Are prehabilitation interventions containing specific pairs of components violating the additivity assumption?”. For example, the effect of the combination of components EXE+NUT, where components EXE and NUT have an interaction effect, is: log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC + log(OR)EXE*NUT v UC (fig 2). In other words, when exercise is implemented along with nutrition to prevent postoperative complications, their combined effect is amplified (or suppressed) by a specified amount (ie, the effect of the interaction term). Overall, there are three possible situations regarding the association between EXE and NUT components for prevention of postoperative complications:

g with nutrition to prevent postoperative complications, their combined effect is amplified (or suppressed) by a specified amount (ie, the effect of the interaction term). Overall, there are three possible situations regarding the association between EXE and NUT components for prevention of postoperative complications: there is minimal or no interaction between EXE and NUT, leading to log(OR)EXE*NUT being approximately zero (or OREXE*NUT=1); the interaction is positive (synergistic) between EXE and NUT, where log(OR)EXE*NUT is negative (or OREXE*NUT<1), suggesting that the log odds of postoperative complications are reduced, and hence EXE and NUT together are more protective than expected from their individual effects; or the interaction is negative (antagonistic) between EXE and NUT, where log(OR)EXE*NUT is positive (or OREXE*NUT>1, as also shown in fig 2), suggesting that the log odds of postoperative complications are increased, and hence the combined effect is less protective than expected. The interpretation of the direction of effect of the interaction would be the opposite for beneficial outcomes (eg, treatment response). Specifically, log(OR)EXE*NUT <0 indicates a negative interaction, meaning that EXE and NUT together are less beneficial than expected (eg, their combined use reduces the log odds of response to treatment). However, log(OR)EXE*NUT>0 indicates a positive interaction, with the EXE and NUT combination enhancing benefits more than either component alone.

indicates a negative interaction, meaning that EXE and NUT together are less beneficial than expected (eg, their combined use reduces the log odds of response to treatment). However, log(OR)EXE*NUT>0 indicates a positive interaction, with the EXE and NUT combination enhancing benefits more than either component alone. In principle, the interaction CNMA model can incorporate interaction terms for each unique combination of components found in the dataset and, hence, can include interactions between two or more components. As explained above, including all estimable interactions of all orders found in the dataset would lead to the standard NMA model (ie, the full interaction model).15 However, there is a trade-off between model fit and estimation precision; as more interaction terms are added to the model, the number of parameters to be estimated increases, decreasing precision and eventually statistical power. Ideally, interaction terms should be defined a priori on the basis of their clinical plausibility as determined by clinical expertise and the literature; an entirely data driven approach is discouraged (ie, testing all possible interaction models).

mated increases, decreasing precision and eventually statistical power. Ideally, interaction terms should be defined a priori on the basis of their clinical plausibility as determined by clinical expertise and the literature; an entirely data driven approach is discouraged (ie, testing all possible interaction models). The ability to estimate and the power to detect statistically significant interactions will be limited by the availability of data on the unique combinations of components. In a frequentist setting, adding interaction terms for component combinations that are missing from the dataset is not possible, because these are not estimable (ie, the design matrix on the active components cannot be defined for a non-existent combination of components).9 These interactions cannot be validated because there is no direct evidence for examination.15 In a bayesian setting, the estimation of interaction effects for component combinations not observed in the dataset could, in principle, be achieved by incorporating informative priors. However, careful consideration would be needed when selecting these priors, because they can be expected to strongly influence the posterior distribution, provided that there is no direct evidence from the data. In this article, we focus on the inclusion of identifiable interaction terms, applicable to both frequentist and bayesian settings. Overall, interaction CNMA is an extension of additive CNMA and a compromise between additive CNMA and standard NMA. Figure 3 presents the model options for the postoperative complications example.

article, we focus on the inclusion of identifiable interaction terms, applicable to both frequentist and bayesian settings. Overall, interaction CNMA is an extension of additive CNMA and a compromise between additive CNMA and standard NMA. Figure 3 presents the model options for the postoperative complications example. Hierarchy of component network meta-analysis (CNMA) models in illustrative example of postoperative complications. Diagram illustrates the nested relationship between the additive, interaction, and full interaction (standard network meta-analysis (NMA)) models. The top-down structure shows how the additive CNMA model (no interactions) can be extended by adding one or more interaction terms (EXE*NUT, EXE*PSY). The two interaction model (EXE*NUT + EXE*PSY) and the three interaction model (EXE*NUT + EXE*PSY + EXE*NUT*PSY) increase model complexity, with the three interaction model being equivalent to standard NMA for this example. Red box indicates models that are not identifiable owing to an absence of supporting data (eg, no NUT+PSY studies). EXE=exercise; NUT=nutrition; PSY=psychosocial

Additive CNMA assumes no interaction between components. However, clinically, components may interact with each other (ie, the effect of a component may differ in the presence of another component), and additive CNMA may lead to biased estimates of component effects. If the additive CNMA does not fit the data well (ie, the additivity assumption is not met), then relaxing the additivity assumption and assuming interactions between components in an interaction CNMA model may be appropriate to explore which sets of component effects are synergistic or antagonistic. The interaction CNMA model is an extension of the additive CNMA model that includes additional terms for component interactions that may account for higher (synergistic) or lower (antagonistic) intervention effects than the sum of the component effects alone.10 11 12 21 A two way interaction CNMA model allows pairs of components within interventions to interact, resulting in potentially lower or higher component effects than the additive effects of the two components. In our illustrative example, a two way interaction model answers the question “Are prehabilitation interventions containing specific pairs of components violating the additivity assumption?”. For example, the effect of the combination of components EXE+NUT, where components EXE and NUT have an interaction effect, is: log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC + log(OR)EXE*NUT v

The interaction CNMA model is an extension of the additive CNMA model that includes additional terms for component interactions that may account for higher (synergistic) or lower (antagonistic) intervention effects than the sum of the component effects alone.10 11 12 21 A two way interaction CNMA model allows pairs of components within interventions to interact, resulting in potentially lower or higher component effects than the additive effects of the two components. In our illustrative example, a two way interaction model answers the question “Are prehabilitation interventions containing specific pairs of components violating the additivity assumption?”. For example, the effect of the combination of components EXE+NUT, where components EXE and NUT have an interaction effect, is: log(OR)EXE+NUT v UC = log(OR)EXE v UC + log(OR)NUT v UC + log(OR)EXE*NUT v UC (fig 2). In other words, when exercise is implemented along with nutrition to prevent postoperative complications, their combined effect is amplified (or suppressed) by a specified amount (ie, the effect of the interaction term). Overall, there are three possible situations regarding the association between EXE and NUT components for prevention of postoperative complications: there is minimal or no interaction between EXE and NUT, leading to log(OR)EXE*NUT being approximately zero (or OREXE*NUT=1);

UC (fig 2). In other words, when exercise is implemented along with nutrition to prevent postoperative complications, their combined effect is amplified (or suppressed) by a specified amount (ie, the effect of the interaction term). Overall, there are three possible situations regarding the association between EXE and NUT components for prevention of postoperative complications: there is minimal or no interaction between EXE and NUT, leading to log(OR)EXE*NUT being approximately zero (or OREXE*NUT=1); the interaction is positive (synergistic) between EXE and NUT, where log(OR)EXE*NUT is negative (or OREXE*NUT<1), suggesting that the log odds of postoperative complications are reduced, and hence EXE and NUT together are more protective than expected from their individual effects; or the interaction is negative (antagonistic) between EXE and NUT, where log(OR)EXE*NUT is positive (or OREXE*NUT>1, as also shown in fig 2), suggesting that the log odds of postoperative complications are increased, and hence the combined effect is less protective than expected. The interpretation of the direction of effect of the interaction would be the opposite for beneficial outcomes (eg, treatment response). Specifically, log(OR)EXE*NUT <0 indicates a negative interaction, meaning that EXE and NUT together are less beneficial than expected (eg, their combined use reduces the log odds of response to treatment). However, log(OR)EXE*NUT>0 indicates a positive interaction, with the EXE and NUT combination enhancing benefits more than either component alone.

A variety of approaches for model selection are possible, and we describe one such approach in detail here, and other possible approaches below. Initially, an additive CNMA model can be fitted and assessed to determine whether the additivity assumption is met. The additivity assumption can then be assessed by comparing the effects of different combinations of component pairs: for example, does it appear that the effect of EXE versus NUT equals the effect of EXE+COG versus NUT+COG and the effect of EXE+COG+PSY versus NUT+COG+PSY? A statistical test to assess the additivity assumption, proposed by Rücker et al,15 compares the Q heterogeneity statistics from the standard NMA model with those from the additive CNMA model. The Q statistic15 is computed as the weighted sum of squared differences between the observed and model fitted treatment effects across all comparisons. Under the assumption of homogeneity, the Q statistic for the NMA model (QNMA) follows a χ2 distribution with degrees of freedom determined by the total number of treatment arms across all studies, the number of studies, and the number of treatments in the network. For the additive CNMA model’s Q statistic (QAdd), the degrees of freedom depend on the rank of the design matrix, which represents the number of independent treatment contrasts that can be formed given the network structure and component combinations.

the number of studies, and the number of treatments in the network. For the additive CNMA model’s Q statistic (QAdd), the degrees of freedom depend on the rank of the design matrix, which represents the number of independent treatment contrasts that can be formed given the network structure and component combinations. The difference between the two statistics (Qdiff = QAdd – QNMA) follows a χ2 distribution, with degrees of freedom equal to the difference in the number of parameters between the models.22 Under the null hypothesis that the additive CNMA adequately explains the data and that component effects combine additively without interactions, Qdiff is expected to be small (ie, QAdd and QNMA are numerically similar). A statistically significant difference in Q statistics (ie, Qdiff) suggests that the additivity assumption is not met, and further model exploration with interaction terms should be considered. This test can only be applied to connected networks (eg, fig 1C).15 20 This approach is analogous to a test of subgroup differences in pairwise meta-analysis; for instance, comparing the effect of EXE versus NUT across three subgroups: no additional therapy, COG added, and COG+PSY added. When the additivity assumption is met, compared to standard NMA, the additive CNMA model can increase precision of estimated component and intervention effects, while also yielding results that are expected to be unbiased.12

of EXE versus NUT across three subgroups: no additional therapy, COG added, and COG+PSY added. When the additivity assumption is met, compared to standard NMA, the additive CNMA model can increase precision of estimated component and intervention effects, while also yielding results that are expected to be unbiased.12 When the additivity assumption is not met, suggesting the need for an interaction CNMA model, model selection can be guided by changes in the Q statistic in a frequentist setting or deviance information criterion in a bayesian setting. In a frequentist setting, similar to the test of the additivity assumption, the difference between the Q statistics (ie, Q for additive CNMA model (QAdd) and Q for the interaction CNMA model (QInt)) can be used to explore if adding single interaction terms improves the fit over the additive model. This corresponds to a likelihood ratio test and is related to the Akaike information criterion,20 which can be used for model selection. An improvement in model fit over the additive CNMA occurs when the Q statistic of the interaction CNMA is statistically significantly lower than that of the additive CNMA, using an adjusted P value according to the difference in degrees of freedom of the two models (eg, P<0.16 for one degree of freedom)22 23 to account for the lower power of the test to detect violations of additivity. However, this test is not well powered to identify minor violations of the additivity assumption.20 Similar to the Q test for heterogeneity, its statistical power is expected to be low in sparse networks with few studies across intervention comparisons.24 Heterogeneity should also be evaluated across models, a reduction of which may suggest that the interaction model explains some of the variability observed across studies.

to the Q test for heterogeneity, its statistical power is expected to be low in sparse networks with few studies across intervention comparisons.24 Heterogeneity should also be evaluated across models, a reduction of which may suggest that the interaction model explains some of the variability observed across studies. The Q statistics of CNMA models with two or more interaction terms (called multi-interaction models) can also be compared to those of nested interaction CNMA models with fewer interaction terms (ie, lower order). However, the process of determining the best fitting model should be prespecified and clearly explained and justified. For example, one approach could be to apply groups in the following order: models with all a priori determined and clinically relevant two way interaction terms; models with all a priori determined and clinically relevant three way interaction terms; and subsequently combine the best fitting models to create more complex ones.

The Q statistics of CNMA models with two or more interaction terms (called multi-interaction models) can also be compared to those of nested interaction CNMA models with fewer interaction terms (ie, lower order). However, the process of determining the best fitting model should be prespecified and clearly explained and justified. For example, one approach could be to apply groups in the following order: models with all a priori determined and clinically relevant two way interaction terms; models with all a priori determined and clinically relevant three way interaction terms; and subsequently combine the best fitting models to create more complex ones. Overall, various strategies exist, in terms of both the models considered and the statistical metrics used to assess model fit (eg, Q statistics, P values, heterogeneity, deviance information criteria25), which can lead to different results. In our illustrative and empirical examples, we focused on the Q statistics and corresponding P values. However, this is just one approach and others are possible. For example, in cases with notable differences in heterogeneity observed across models, differences in heterogeneity estimates (eg, τ2) across models might instead be used as a model selection criterion. These statistical approaches should be used in combination with clinical expertise regarding how the components are likely to work together. Different approaches and component taxonomies may yield different results, because they answer distinct questions. As such, findings may not be generalisable across all populations or settings.

cal approaches should be used in combination with clinical expertise regarding how the components are likely to work together. Different approaches and component taxonomies may yield different results, because they answer distinct questions. As such, findings may not be generalisable across all populations or settings. Returning to our illustrative example, we demonstrate a model selection approach using the network of trials on prevention of postoperative complications (fig 1C). As shown in table 1, standard NMA model yielded QNMA=144.20 (degrees of freedom (df)=100; P=0.003), while the additive CNMA model yielded QAdd=147.72 (df=103; P=0.003). The difference between the two models (Qdiff=3.52, df=3; P=0.32) was not statistically significant, suggesting no strong evidence that the additivity assumption was violated. However, for demonstration purposes, we will explore all interaction models. Interactions to be explored in statistical models should be selected using clinical judgment from the possible interaction terms identifiable in the available data. In particular, the final selection of interactions should be guided by clinical judgment, where clinicians assess their clinical plausibility a priori for inclusion in the models. Model selection for prehabilitation interventions example Model selection uses a dataset with combinations of intervention components, including: exercise (EXE), nutrition (NUT), psychosocial (PSY), cognitive (COG), EXE+NUT, EXE+PSY, EXE+NUT+PSY, and usual care.

Returning to our illustrative example, we demonstrate a model selection approach using the network of trials on prevention of postoperative complications (fig 1C). As shown in table 1, standard NMA model yielded QNMA=144.20 (degrees of freedom (df)=100; P=0.003), while the additive CNMA model yielded QAdd=147.72 (df=103; P=0.003). The difference between the two models (Qdiff=3.52, df=3; P=0.32) was not statistically significant, suggesting no strong evidence that the additivity assumption was violated. However, for demonstration purposes, we will explore all interaction models. Interactions to be explored in statistical models should be selected using clinical judgment from the possible interaction terms identifiable in the available data. In particular, the final selection of interactions should be guided by clinical judgment, where clinicians assess their clinical plausibility a priori for inclusion in the models. Model selection for prehabilitation interventions example Model selection uses a dataset with combinations of intervention components, including: exercise (EXE), nutrition (NUT), psychosocial (PSY), cognitive (COG), EXE+NUT, EXE+PSY, EXE+NUT+PSY, and usual care. CNMA=component network meta-analysis; df=difference in numbers of parameters between models; NMA=network meta-analysis; Qdiff=difference between Q statistics of NMA and CNMA models. Compared to a P value of 0.11 (obtained in R as P=1−Pr(χdf 2 ≤2×df), with df=difference in the numbers of parameters between models (df=3)). Compared to a P value of 0.14 (obtained in R as P=1−Pr(χ2 2 ≤2×2)).

CNMA=component network meta-analysis; df=difference in numbers of parameters between models; NMA=network meta-analysis; Qdiff=difference between Q statistics of NMA and CNMA models. Compared to a P value of 0.11 (obtained in R as P=1−Pr(χdf 2 ≤2×df), with df=difference in the numbers of parameters between models (df=3)). Compared to a P value of 0.14 (obtained in R as P=1−Pr(χ2 2 ≤2×2)). Compared to a P value of 0.16 (obtained in R as P=1−Pr(χ1 2 ≤2×1)). Compared to model 1. Compared to model 3. Statistically significant difference. Model 4 can be seen to be equivalent to the standard (ie, full interaction) NMA. This small network has two estimable, two way interactions (EXE*NUT and EXE*PSY) and one three way interaction (EXE*NUT*PSY), which all were considered clinically meaningful. In more complex networks, developing interaction models for all identifiable interactions is not advisable. With entirely data driven approaches that make no use of clinical judgment, multiple testing increases the likelihood of finding a model that appears to perform well by chance alone, given the large number of comparisons with the additive CNMA model, the standard NMA model, and other nested interaction models. To determine whether an interaction CNMA model improves on all previously fitted models, its Q statistic can be compared to that of the standard NMA; the additive CNMA; and a previous nested, best fitting, interaction CNMA model, for multi-interaction models.

the standard NMA model, and other nested interaction models. To determine whether an interaction CNMA model improves on all previously fitted models, its Q statistic can be compared to that of the standard NMA; the additive CNMA; and a previous nested, best fitting, interaction CNMA model, for multi-interaction models. In our illustrative example, the most parsimonious model with a Q statistic not statistically significant from that of the standard NMA was preferred. A forward selection, model building approach was used, with all clinically relevant interactions individually added to the additive model, creating multiple separate models with one interaction term each (table 1, models 1-2). Note that a single interaction NUT*PSY model could not be fit because no studies evaluated the NUT+PSY combination, rendering the underlying interaction term unidentifiable. Hence, no Q statistic could be computed for a NUT*PSY interaction model. Additionally, a model with the three way interaction term EXE*NUT*PSY alone was not generated because two lower order interaction terms (EXE*NUT and EXE*PSY) were identified, and had to be added to the model. In this small network, this model was equivalent to the full interaction (standard) NMA (model 4). Without including interactions of lower order, if the EXE*NUT*PSY interaction was found to have a beneficial or adverse effect, we would not know if the effect was driven by just two of the three components. Identifiable lower order interactions allow us to disentangle the effects and understand the contributions of each component. Also, the financial cost of performing all three components, rather than just two, is an important factor to consider. Therefore, when interactions of higher order are included in the model, our position is that it is good practice to also incorporate their corresponding lower order interactions, whenever possible.

nt. Also, the financial cost of performing all three components, rather than just two, is an important factor to consider. Therefore, when interactions of higher order are included in the model, our position is that it is good practice to also incorporate their corresponding lower order interactions, whenever possible. Although the additivity assumption held, for demonstration purposes here, interaction models were fit. Among single interaction CNMA models, the EXE*NUT interaction produced the largest reduction in Q (QInt=144.69, df=102, P=0.004), and provided a better fit than the additive CNMA model (table 1, model 1: Qdiff=3.03, df=1, P=0.08 <0.16 representing adjusted P value for one degree of freedom). In light of the additivity assumption being met, this finding should be interpreted with caution, because multiple testing may spuriously identify statistically better fit models. In the context of this example, the effect estimate of the interaction term was found to be opposite to currently held clinical expertise (ie, the EXE and NUT components were found in modelling to be antagonistic, while clinically there is strong belief that the effects of exercise and nutrition are either additive or synergistic).4 Thus, given that the additivity assumption held, selection of model 1 was deemed to be spurious.

ly held clinical expertise (ie, the EXE and NUT components were found in modelling to be antagonistic, while clinically there is strong belief that the effects of exercise and nutrition are either additive or synergistic).4 Thus, given that the additivity assumption held, selection of model 1 was deemed to be spurious. Model fit testing using the Q statistic demonstrated that the single interaction model also did not fit better than the standard NMA model (table 1, model 1: Qdiff=0.49, df=2, P=0.78 >0.14; model 2: Qdiff=3.44, df=2, P=0.18 >0.14, representing adjusted P value for two degrees of freedom). For demonstration purposes, we fit all possible models with two identifiable interaction terms, of which there was one model (model 3), and we compared its model fit against the standard NMA, the additive CNMA, and model 1 (nested single interaction CNMA model with the lowest Q statistic). No gains in model fit were observed. For completeness, the full interaction model has been reported (model 4), demonstrating its equivalence to the standard NMA. Overall, the additive CNMA explained the data well, suggesting that treatment components combine additively.

The fundamental NMA assumptions of transitivity and consistency (both corresponding to the assumption of exchangeability16) are also required assumptions for CNMA models (box 2).8 Numerous approaches have been developed for evaluating transitivity and consistency in NMA,8 26 27 28 29 30 but these approaches have not been extended to CNMA. Heterogeneity also remains a critical concern in CNMA, as in standard NMA, yet current methods to estimate and interpret heterogeneity in component effects remain limited. The use of prediction intervals around component estimates (eg, assuming a normal distribution with means equal to the component effects and variance equal to the heterogeneity between studies)31 32 may provide insight on the extent of heterogeneity. Overall, moving from a single effect model (ie, pairwise meta-analysis) to an additive CNMA, then an interaction CNMA, and finally a standard NMA (full interaction model) increases model complexity and reduces assumptions, but does not negate important considerations, such as inconsistency and heterogeneity in the network. While this progression may reduce precision of intervention effects, it offers a more detailed understanding of how individual components or combinations contribute to overall effects.

mplexity and reduces assumptions, but does not negate important considerations, such as inconsistency and heterogeneity in the network. While this progression may reduce precision of intervention effects, it offers a more detailed understanding of how individual components or combinations contribute to overall effects. Studies contributing to each pairwise comparison were clinically and methodologically sufficiently similar to allow meaningful synthesis in NMA and CNMA. Both models rely on the key assumptions of transitivity and consistency, that require clinical considerations to inform assessment of these assumptions. Transitivity was evaluated using a multifaceted approach, including review of study and patient characteristics based on evidence tables along with graphical approaches, such as inspection of box plots and bar plots, to examine characteristics of treatment comparisons within the evidence network. As reported in the original publication, no evidence of intransitivity was identified when mean age, female proportion, control group risk, year of publication, and surgery type were assessed as potential effect modifiers (see also supplementary file, appendix 14 of original publication).4

within the evidence network. As reported in the original publication, no evidence of intransitivity was identified when mean age, female proportion, control group risk, year of publication, and surgery type were assessed as potential effect modifiers (see also supplementary file, appendix 14 of original publication).4 The design-by-treatment interaction model26 was used as an extension of the standard NMA model and assuming unique combinations of components as different nodes, and suggested no statistical evidence for inconsistency (Qbetween designs=3.41, df=2, P=0.18, τ2=0.14). However, local assessment of inconsistency using the back-calculation method27 indicated that the direct evidence in EXE versus UC was borderline inconsistent with the remaining network at the 5% significance level (P=0.07). Exploration of evidence of inconsistency was conducted through network meta-regression for control group risk and surgery type (a summary of these explorations is included in supplementary file, appendix 15 of original publication). The certainty rating for that comparison was downgraded, owing to inconsistency. Assessments of both transitivity and consistency were informed by team discussions, incorporating clinical considerations relevant to prehabilitation. These discussions also guided the interpretation of results once the analyses were complete.

Studies contributing to each pairwise comparison were clinically and methodologically sufficiently similar to allow meaningful synthesis in NMA and CNMA. Both models rely on the key assumptions of transitivity and consistency, that require clinical considerations to inform assessment of these assumptions. Transitivity was evaluated using a multifaceted approach, including review of study and patient characteristics based on evidence tables along with graphical approaches, such as inspection of box plots and bar plots, to examine characteristics of treatment comparisons within the evidence network. As reported in the original publication, no evidence of intransitivity was identified when mean age, female proportion, control group risk, year of publication, and surgery type were assessed as potential effect modifiers (see also supplementary file, appendix 14 of original publication).4

When reporting the findings from a CNMA, the results must be aligned with the review question and relevant to clinicians, policy makers, and patient partners. Authors of CNMA research publications are encouraged to adhere the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) extension for NMA reporting guideline to ensure transparent and comprehensive reporting.33 The selection of the primary analysis model (eg, additive CNMA, interaction CNMA, or standard NMA) should be guided by both methodological considerations (eg, model fit, transitivity, heterogeneity) and knowledge user priorities, including the clinical plausibility of additivity and component interactions. For the presentation of CNMA results, a reference intervention is typically defined as the comparator. The reference intervention could be the current, standard intervention or inactive placebo/UC in settings where no standard intervention has been established. Graphical approaches can be used to present and compare the findings of intervention and component effects across NMA and CNMA models.34 35

ed as the comparator. The reference intervention could be the current, standard intervention or inactive placebo/UC in settings where no standard intervention has been established. Graphical approaches can be used to present and compare the findings of intervention and component effects across NMA and CNMA models.34 35 In this example, we applied random effects, additive and interaction CNMA models. We also fitted standard NMA models for completeness. The heterogeneity common within networks was estimated using the DerSimonian and Laird method.36 Results were expressed as ORs with corresponding 95% confidence intervals (CIs) for each model. We calculated P scores37 to rank interventions and used a rank heat plot38 for their presentation across the different models. Additional information on the individual components as obtained from the NMA model are presented in appendix 7. All analyses, and across all empirical examples, were conducted in a frequentist setting using R packages netmeta39 and viscomp.40 The publication of the prehabilitation interventions example4 reported both standard NMA and CNMA findings, but prioritised NMA because patient partners and clinical collaborators felt strongly that clinically meaningful interactions between components made standard NMA more appropriate for capturing the distinct effects of each intervention combination.

on interventions example4 reported both standard NMA and CNMA findings, but prioritised NMA because patient partners and clinical collaborators felt strongly that clinically meaningful interactions between components made standard NMA more appropriate for capturing the distinct effects of each intervention combination. Figure 4 presents a forest plot comparing findings from standard NMA, additive CNMA, and interaction CNMA, using UC as the reference group. The interaction CNMA shown corresponds to model 1 from table 1, which includes a single interaction term, EXE*NUT, and yielded a likely spuriously statistically significant reduction in the Q statistic. Based on these results, had this interaction model been selected over the additive model, the EXE*NUT interaction term (OREXE*NUT=1.52, 95% CI 0.86 to 2.70) would have acted to decrease the combined impact of EXE and NUT in the EXE+NUT intervention (ie, EXE and NUT acted antagonistically, so that the EXE+NUT combination then appears less effective than in the additive model; CNMA, OREXE+NUT=0.35, 0.26 to 0.46; interaction CNMA, OREXE+NUT=0.48, 95% CI 0.29 to 0.80; fig 2). This finding would have been counter to current clinical thinking, potentially reinforcing the preference for the additive model. However, given that the assumption of additivity held, the estimated effects of combinations of components could be calculated from the incremental effects (OREXE=0.53, 0.42 to 0.66; ORNUT=0.66, 0.54 to 0.81), after converting ORs to log ORs. After consulting with the team and incorporating clinical and patient partner input, parsimony was prioritised over marginal decreases in the Q statistic. This decision was guided by the understanding that component effects estimated from an interaction model cannot be generalised beyond the specific combinations observed in the dataset.

th the team and incorporating clinical and patient partner input, parsimony was prioritised over marginal decreases in the Q statistic. This decision was guided by the understanding that component effects estimated from an interaction model cannot be generalised beyond the specific combinations observed in the dataset. Interval plot of intervention effects (odds ratios) with corresponding 95% confidence intervals for the prevention of postoperative complications. Results are shown for three models: standard network meta-analysis (NMA), additive component network meta-analysis (CNMA), and interaction CNMA with EXE*NUT interaction term (model 1 in table 1). COG=cognitive; EXE=exercise; NUT=nutrition; PSY=psychosocial; CI=confidence interval

In this example, we applied random effects, additive and interaction CNMA models. We also fitted standard NMA models for completeness. The heterogeneity common within networks was estimated using the DerSimonian and Laird method.36 Results were expressed as ORs with corresponding 95% confidence intervals (CIs) for each model. We calculated P scores37 to rank interventions and used a rank heat plot38 for their presentation across the different models. Additional information on the individual components as obtained from the NMA model are presented in appendix 7. All analyses, and across all empirical examples, were conducted in a frequentist setting using R packages netmeta39 and viscomp.40 The publication of the prehabilitation interventions example4 reported both standard NMA and CNMA findings, but prioritised NMA because patient partners and clinical collaborators felt strongly that clinically meaningful interactions between components made standard NMA more appropriate for capturing the distinct effects of each intervention combination.

Complex interventions are increasingly evaluated in healthcare research with growing methodological interest. The comparative efficacy of intervention components should be determined in consideration of their potential interactions and contextual factors. Component effect estimation produces relevant and clinically meaningful evidence synthesis results for knowledge users. Provided that there is an increased clinical interest in the assessment of the efficacy of individual components in complex interventions,19 41 and given the freely available R package netmeta,39 we expect a rise in the use of CNMA models. CNMA should be considered when interventions share clearly defined components and there is clinical plausibility for assuming additive or interactive effects among components. CNMA allows for the assessment and disentanglement of individual component effects: the additive CNMA model assumes full additivity of component effects. When additivity is violated, an interaction CNMA model is suggested, including terms that account for interactions between components, to generate less biased component effects. Interaction CNMA provides a compromise between additive CNMA and standard NMA, especially when component effects may be synergistic or antagonistic. The full interaction model, considering all possible interactions between components, is the standard NMA model.16

n components, to generate less biased component effects. Interaction CNMA provides a compromise between additive CNMA and standard NMA, especially when component effects may be synergistic or antagonistic. The full interaction model, considering all possible interactions between components, is the standard NMA model.16 CNMA has several benefits over standard NMA: it estimates fewer parameters, borrows strength from studies sharing components, can provide more powerful and precise results,15 and can connect disconnected networks with common components, when there is strong clinical evidence that additivity can be assumed.20 Ultimately, the choice between the models depends on the clinical relevance, structure of the evidence base, model complexity, and data sufficiency. Implementing CNMA requires multidisciplinary expertise and careful clinical, statistical, and methodological planning, including protocol development,42 model selection,20 and presentation of results.34 35

Although the use of NMA has increased in the past two decades,16 43 CNMA has not been widely used. Challenges in CNMA remain that may affect the validity, interpretability, and certainty of CNMA results. These challenges include decisions on component definition and selection, coding strategies to build a network graph, data availability (eg, interaction terms cannot be coded when certain component combinations are not observed), data scarcity (ie, few studies v many components or component combinations), differences in usual care (or control) across studies, heterogeneity, and inconsistency in the network. Component selection in CNMA is analogous to node making in standard NMA and represents a critical methodological decision that requires clear clinical justification and transparency.44 45 46 47 Key considerations include the process for identifying components, consistency of component reporting in the primary literature, and the number of components selected (eg, too many components may result in sparse data). Available structured frameworks can help guide consistent and replicable component classification.48 49 50

nsiderations include the process for identifying components, consistency of component reporting in the primary literature, and the number of components selected (eg, too many components may result in sparse data). Available structured frameworks can help guide consistent and replicable component classification.48 49 50 In this article, we have illustrated key CNMA concepts using the prehabilitation intervention example.4 Two additional examples, presented in appendices 1-9, demonstrate real world complexities of applying CNMA to complex interventions: one focusing on the prevention of fractures related to falls3 and the other on improving quality of life through knowledge translation strategies.51 All data and R code are available in web appendices 2-5. The nature of these two additional examples is quite different, highlighting both the strengths and limitations of CNMA. The example of fall prevention interventions3 shows where CNMA is particularly helpful in a simplified analysis that performs well. In contrast, the example of knowledge translation interventions51 suggests that CNMA is less likely to offer improvements over NMA, given the complex interactions between components. In this case, higher order interactions seem necessary, bringing the CNMA model closer to the standard NMA.

analysis that performs well. In contrast, the example of knowledge translation interventions51 suggests that CNMA is less likely to offer improvements over NMA, given the complex interactions between components. In this case, higher order interactions seem necessary, bringing the CNMA model closer to the standard NMA. While CNMA can streamline some analyses, it may not be feasible in networks with sparse data (eg, few studies per component or combination), poorly reported components, or when strong interactions between components violate the additivity assumption without sufficient data to model interactions. Also, interactions can be hard to identify. As noted in our empirical examples of fall related fractures and quality of life through knowledge translation strategies, usually the amount of data in a network was too sparse to test for many interactions. However, different CNMA approaches and decisions (eg, about interactions) may generate important variations in results so, when possible, decisions in CNMA model development and selection of clinically relevant interactions should be made a priori at the protocol stage.

ork was too sparse to test for many interactions. However, different CNMA approaches and decisions (eg, about interactions) may generate important variations in results so, when possible, decisions in CNMA model development and selection of clinically relevant interactions should be made a priori at the protocol stage. Currently, frameworks that assess the certainty of evidence in standard NMAs, such as CINEMA52 (Confidence in Network Meta-Analysis) and GRADE53 (Grading of Recommendations, Assessment, Development, and Evaluations), have not yet been formally extended to CNMA, and can be applied only to overall intervention effects derived from standard NMA rather than individual component estimates. The certainty of evidence for component effects remains an open methodological question, as does the assessment of transitivity and consistency in the network.

et been formally extended to CNMA, and can be applied only to overall intervention effects derived from standard NMA rather than individual component estimates. The certainty of evidence for component effects remains an open methodological question, as does the assessment of transitivity and consistency in the network. Study level characteristics may influence which components are included in an intervention and may also modify the effects of those components. For instance, studies conducted in higher socioeconomic status populations may be more likely to implement intensive or resource-demanding components, which could also yield larger effects. Such factors may challenge the transitivity assumption in CNMA and complicate the interpretation of component specific estimates. Future advancements in CNMA may resolve current limitations, including the assessment of component consistency (eg, between additive effects and direct comparisons), ranking of components, component contribution to intervention effects, and component effect modelling in multivariate CNMA, along with inherent challenges these bring. For instance, intervention rankings in NMA can vary depending on the ranking metric used, because each metric answers a different type of ranking question.54

ng of components, component contribution to intervention effects, and component effect modelling in multivariate CNMA, along with inherent challenges these bring. For instance, intervention rankings in NMA can vary depending on the ranking metric used, because each metric answers a different type of ranking question.54 Furthermore, current ranking approaches do not incorporate assessments of the certainty of evidence, which is particularly important in cases where an intervention with a small sample size or high risk of bias ranks highest, potentially resulting in misleading or overly confident interpretations of treatment hierarchies.55 56 We anticipate that similar challenges apply to CNMA, and future developments should aim to integrate certainty assessments into both component and intervention rankings. Future simulation studies should provide guidance on the minimum number of studies required for reliable CNMA estimation across varying numbers of components and under different modelling assumptions (eg, additivity or interaction). Ultimately, future research to guide selection of CNMA models will be critical to develop a consensus based approach and to advance evidence synthesis methods incorporating CNMA.