Signature
d_BC = d_AC - d_AB; V_BC = V_AC + V_AB
| Inputs | Definition | Unit |
|---|---|---|
d_AC | Pooled or single-trial estimate of the effect of C relative to the common comparator A | effect scale, for example log odds ratio |
d_AB | Pooled or single-trial estimate of the effect of B relative to the common comparator A | effect scale, for example log odds ratio |
V_AC | Variance of d_AC, the square of its standard error | squared units of the effect scale |
V_AB | Variance of d_AB, the square of its standard error | squared units of the effect scale |
d_BC | Indirect estimate of the relative effect of C compared with B on the additive scale | same scale as the inputs, for example log odds ratio |
|---|---|---|
V_BC | Variance of the indirect estimate d_BC | squared units of the effect scale |
Function
Network consistency function
Expresses every pairwise relative effect in a connected network through effects relative to a common reference treatment A, on a scale where effects add, such as the log odds ratio. Here d_XY is the effect of Y relative to X, the NICE Decision Support Unit convention. Consistency is the assumption that direct and indirect evidence estimate the same d_XY.
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Implementations
Excel
Indirect estimate and variance in two cells
With the direct log odds ratios in dAC and dAB and their standard errors in seAC and seAB, the two formulas return the indirect estimate and its variance.
=dAC-dAB; =seAC^2+seAB^2
Assumptions
Transitivity across the two sets of trials
The A versus B and A versus C trials are similar in the distribution of effect modifiers such as severity, line of treatment and outcome timing, so that participants could in principle have been randomised to any of the three treatments.
Independent sources of direct evidence
d_AB and d_AC come from different trials. When a three-arm trial contributes to both, the two estimates are correlated and the variance sum understates uncertainty; the covariance is then modelled, for example in a full network meta-analysis.
Additive scale and one direction convention
Effects are combined on a scale where they add, such as the mean difference, log odds ratio, log risk ratio or log hazard ratio, and every contrast uses the same direction. The article's indirect-comparison section writes d_XY as X relative to Y; the relation d_AC = d_AB + d_BC holds under either convention if it is applied throughout.
Worked examples
HIV example in NICE DSU TSD 4
Direct log odds ratios of 2.79 (standard error 0.56) for B against A and 1.42 (standard error 0.34) for C against A give an indirect log odds ratio for C against B of minus 1.37 with variance 0.4292, as in Box 1 of TSD 4.
d_AC = 1.42; d_AB = 2.79; V_AC = 0.1156; V_AB = 0.3136; d_BC = -1.37; V_BC = 0.4292
Two odds ratios against placebo
Odds ratios of 0.70 for B and 0.50 for C against placebo A, with standard errors on the log scale of 0.15 and 0.20, give an indirect log odds ratio of about minus 0.3364 for C against B with variance 0.0625, an odds ratio of about 0.714, the same as 0.50 divided by 0.70. The figures are illustrative.
d_AC = -0.6931; d_AB = -0.3567; V_AC = 0.04; V_AB = 0.0225; d_BC = -0.3364; V_BC = 0.0625
Common errors
Comparing active arms across trials
Pooling the active-treatment arms of different trials and comparing them directly discards randomisation and is prone to bias. Bucher and colleagues describe this naive approach as offering no advantage over observational data.
Sources
Adjusted indirect comparison
Bucher HC, Guyatt GH, Griffith LE, Walter SD. The results of direct and indirect treatment comparisons in meta-analysis of randomized controlled trials. Journal of Clinical Epidemiology. 1997;50(6):683-691. The method evaluates the differences between treatment and placebo in two sets of trials and preserves the randomisation of the originally assigned groups, in contrast to pooling active arms.
Bucher indirect estimate and its variance in TSD 4
Dias S, Welton NJ, Sutton AJ, Caldwell DM, Lu G, Ades AE. NICE DSU Technical Support Document 4: Inconsistency in networks of evidence based on randomised controlled trials. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; May 2011, last updated April 2014. Section 3.1 Bucher method for single loops of evidence (equation 1, the indirect estimate of C versus B as d_AC minus d_AB, and its variance as the sum of the variances of the independent direct estimates) and Box 1 worked example.
Consistency equations in the NICE DSU framework
Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; August 2011, last updated September 2016. Section 5 Extension to indirect comparisons and network meta-analysis (equation 12, d_23 = d_13 minus d_12, and the consistency equations for basic and functional parameters) and section 7.3 (two pairwise meta-analyses as an implementation of the Bucher method).
Canonical Identity
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