Signature
RC = exp(beta_1 * Delta_x)
| Inputs | Definition | Unit |
|---|---|---|
beta_1 | Slope of the meta-regression of the log ratio measure on the covariate | log ratio per unit of the covariate |
Delta_x | Difference between two covariate values, 1 for the per-unit change | that of the covariate |
RC | Factor by which the odds ratio, risk ratio or hazard ratio is multiplied when the covariate changes by Delta_x | a ratio of ratio measures |
|---|
Function
Precision-weighted regression of study effect estimates on study characteristics
Maps the effect estimate y_i of each study in a meta-analysis, its within-study variance v_i and the study's value x_i of a characteristic such as dose, mean age or baseline risk to a regression line for the expected effect, together with the between-study variance tau2 that the characteristic leaves unexplained. Studies are weighted by the precision of their estimates, and the random-effects form adds the residual between-study variance to each study's own variance. Ratio measures enter as natural logarithms. In health technology assessment the fitted line gives the relative effect expected for the decision population instead of the average over the trials, and the slope is an association between studies, not a randomised comparison.
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Implementations
Excel
Relative change in a ratio measure from a meta-regression slope cell
With the log-scale slope in a cell named Slope and the change in the covariate in a cell named CovChange, the formula returns the multiplicative change in the ratio measure.
=EXP(Slope*CovChange)
Assumptions
Ratio measure regressed on the log scale
The relation is linear on the log scale, so on the ratio scale each unit of the covariate multiplies the ratio measure by the same factor. The Cochrane Handbook states that a ratio measure should always enter the regression as its logarithm.
Worked examples
Odds ratio change per unit of baseline log-odds
With the article's illustrative slope of 0.30, each unit increase in the baseline log-odds multiplies the odds ratio by about 1.35, so the protective effect weakens as baseline risk rises.
beta_1 = 0.30; Delta_x = 1; RC = 1.3499
Odds ratio change from the trials' mean risk to a 5% baseline risk
Moving from the trials' mean log-odds of -1.70 to the log-odds of a 5% risk, about -2.944, is a change of about -1.244. The factor is about 0.689, the ratio of the adjusted odds ratio of about 0.462 to the odds ratio of 0.670 at the mean.
beta_1 = 0.30; Delta_x = -1.244; RC = 0.6885
Common errors
Reading the exponentiated slope as the treatment odds ratio
A factor of 1.35 does not mean that treatment raises the odds of the event by 35%. It means the odds ratio is 1.35 times larger for each unit of the covariate, so an odds ratio of 0.670 at the mean would be about 0.905 one unit higher.
Regressing odds ratios or risk ratios on the ratio scale
A slope fitted to untransformed ratio measures cannot be exponentiated into a relative change and treats a halving and a doubling asymmetrically. The Cochrane Handbook advises the log-transformed value in every case.
Sources
Log scale and exponentiated meta-regression coefficients in the Cochrane Handbook
Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10, Analysing data and undertaking meta-analyses, in Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.11.4, which states that the log-transformed value of a ratio measure should always be used and that the exponential of the regression coefficient estimates the relative change in the effect per unit increase in the explanatory variable.
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