Signature
p_k = p_0 * OR_k / (1 - p_0 + p_0 * OR_k)
| Inputs | Definition | Unit |
|---|---|---|
p_0 | Probability of the event with the network reference treatment in the target population | probability from 0 to 1 |
OR_k | Odds ratio of treatment k relative to the reference, the exponent of the network log odds ratio d_k | ratio, no unit |
p_k | Absolute probability of the event with treatment k over the period of the baseline probability | probability from 0 to 1 |
|---|
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
Treatment probability from a baseline and a log odds ratio
With the baseline probability in BaselineProb and the network log odds ratio in LogOR, Excel adds the effect on the logit scale and converts back.
=1/(1+EXP(-(LN(BaselineProb/(1-BaselineProb))+LogOR)))
Assumptions
Separate baseline model and additivity on the logit scale
The baseline probability comes from evidence relevant to the target population and is not counted again as relative-effect evidence. The treatment effect is constant on the log odds scale across baseline risks; models linear in log risk ratios or log hazard rates use the corresponding transformation instead.
Worked examples
Baseline probability of 0.25 and an odds ratio of 0.8
A probability of 0.25 on standard care and an odds ratio of 0.8 give a treatment probability of about 0.2105, reported as 0.21 in the introduction of TSD 5.
p_0 = 0.25; OR_k = 0.8; p_k = 0.2105
Common errors
Applying the network odds ratio as a risk ratio
Multiplying 0.25 by 0.8 gives 0.20 rather than about 0.2105. The error grows as the baseline probability rises and can give probabilities above 1 for odds ratios above 1.
Sources
Baseline natural history combined with relative effects
Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 5: Evidence synthesis in the baseline natural history model. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; August 2011, last updated April 2012. Section 1 Introduction (the absolute probability on treatment from logit(p) = logit(0.25) + ln(0.8), giving 0.21, and separate baseline and relative effect models).
Absolute effects from a logit-link network model
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 2 (logit link for binomial data) and the appendix code for absolute effects (logit(T[k]) equals the baseline log odds A plus d[k]).
Canonical Identity
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