Network meta‐analysis including treatment by covariate interactions: Consistency can vary across covariate values

Background Many reviews aim to compare numerous treatments and report results stratified by subgroups (eg, by disease severity). In such cases, a network meta‐analysis model including treatment by covariate interactions can estimate the relative effects of all treatment pairings for each subgroup of patients. Two key assumptions underlie such models: consistency of treatment effects and consistency of the regression coefficients for the interactions. Consistency may differ depending on the covariate value at which consistency is assessed. For valid inference, we need to be confident of consistency for the relevant range of covariate values. In this paper, we demonstrate how to assess consistency of treatment effects from direct and indirect evidence at various covariate values. Methods Consistency is assessed using visual inspection, inconsistency estimates, and probabilities. The method is applied to an individual patient dataset comparing artemisinin combination therapies for treating uncomplicated malaria in children using the covariate age. Results The magnitude of the inconsistency appears to be decreasing with increasing age for each comparison. For one comparison, direct and indirect evidence differ for age 1 (P = .05), and this brings results for age 1 for all comparisons into question. Conclusion When fitting models including interactions, the consistency of direct and indirect evidence must be assessed across the range of covariates included in the trials. Clinical inferences are only valid for covariate values for which results are consistent.


Supplementary models
Details of the individual patient data network meta-analysis models including treatment by covariate interactions that were applied are given below. Let denote the treatment given in trial i in arm k where and NT is the number of treatments in the network. Also specify that the node being split is ( , ) where and For example, if one wants to split the node (3, 4) then and .

Model S1. NMA model including treatment by covariate interaction
Assuming no multi-arm trials exist, the random-effects model is given as follows: where is the log odds of an event in arm 1 of trial i; is a study-specific regression parameter that represents the difference in the log odds of an event in arm 1 of trial i per unit increase in the covariate ; represents the difference in the log odds ratio of vs.
per unit increase in the covariate and = -; and represents the trialspecific log odds ratio of vs. . The trial-specific log odds ratios, are assumed to be realisations from a normal distribution where and In this model, represents the log odds ratio of vs. . The fixed-effect model is given by setting Under a Bayesian framework, prior distributions are specified for , , , and The model can also be applied to datasets with multi-arm trials but the correlation between trial-specific treatment effects must be taken into account. For each multi-arm trial i with m arms, the trial-specific treatment effects are taken to be a realisation from a multivariate normal distribution that can be decomposed into a series of conditional univariate normal distributions.

Model S2. NMA node-splitting model including treatment by covariate interaction
When there are no multi-arm trials, the random-effects model is specified as follows: and where represents the difference in the log odds ratio of vs. per unit increase in the covariate estimated using direct evidence; represents the difference in the log odds Whereas, for the same trial, if one wants to split node (3, 4) instead, then we fix and the model is for treatment 3, and for treatment 4 where .

Site
Artemisinin-based combination therapies (number of patients that achieved treatment success/number of patients)  Table S1. Summary of the individual patient data (i.e. event rate of each treatment group of each site for treatment success at day 28) and covariate information. AQ+AS: amodiaquine-artesunate; AL: artemether-lumefantrine; CD+A: chlorproguanil-dapsone plus artesunate; DHAPQ: dihydroartemisininpiperaquine. Figure S2. Posterior distributions of log odds ratios at various ages for treatment success for CD+A versus AL.
The mean age was 2.5 years.