Signature
OR_BC = exp(d_BC); OR_BC_L = exp(d_BC - 1.96 * sqrt(V_BC)); OR_BC_U = exp(d_BC + 1.96 * sqrt(V_BC))
| Inputs | Definition | Unit |
|---|---|---|
d_BC | Indirect log odds ratio of C relative to B | log odds ratio |
V_BC | Variance of d_BC | squared log odds ratio |
OR_BC | Odds ratio for C compared with B from the indirect comparison | ratio, no unit |
|---|---|---|
OR_BC_L | Lower limit of the 95% confidence interval for OR_BC | ratio, no unit |
OR_BC_U | Upper limit of the 95% confidence interval for OR_BC | ratio, no unit |
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.
Implementations
Excel
Indirect odds ratio and limits in three cells
With the indirect log odds ratio in dBC and its variance in VBC, the three formulas return the odds ratio and its 95% limits.
=EXP(dBC); =EXP(dBC-1.96*SQRT(VBC)); =EXP(dBC+1.96*SQRT(VBC))
Assumptions
Approximate normality on the log scale
The indirect log odds ratio is approximately normally distributed, so a 95% interval is the estimate plus or minus 1.96 standard errors on the log scale.
Worked examples
Back-transforming the TSD 4 indirect estimate
An indirect log odds ratio of minus 1.37 with variance 0.4292 gives an odds ratio of about 0.2541 with a 95% confidence interval of about 0.0704 to 0.9177.
d_BC = -1.37; V_BC = 0.4292; OR_BC = 0.2541; OR_BC_L = 0.0704; OR_BC_U = 0.9177
Back-transforming the placebo-anchored example
An indirect log odds ratio of minus 0.3364 with variance 0.0625 gives an odds ratio of about 0.7143 with a 95% confidence interval of about 0.4376 to 1.1660, so the indirect evidence is compatible with no difference between B and C.
d_BC = -0.3364; V_BC = 0.0625; OR_BC = 0.7143; OR_BC_L = 0.4376; OR_BC_U = 1.1660
Common errors
Building the interval on the odds ratio scale
Adding and subtracting 1.96 standard errors to an odds ratio gives a symmetric interval that can include negative values. The interval is built on the log scale and exponentiated.
Sources
Indirect odds ratio with a confidence interval
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 indirect comparison is reported as an odds ratio with a 95% confidence interval (0.37, 0.21 to 0.65, in the Pneumocystis prophylaxis example).
Canonical Identity
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