Signature
w_i = 1 / (v_i + tau2)
| Inputs | Definition | Unit |
|---|---|---|
v_i | Sampling variance of the estimate from study i, the square of its standard error | the squared effect scale |
tau2 | Estimated residual between-study variance from the meta-regression with its covariates, 0 for a fixed-effect meta-regression | the squared effect scale |
w_i | Weight of study i in the weighted least squares fit of the meta-regression | the inverse of the squared effect scale |
|---|
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.
Try this function
Implementations
Excel
Meta-regression weights from a variance range and the residual tau2
With the within-study variances in a range named StudyVar and the residual between-study variance in a cell named ResidTau2, the formula returns the weights as an array, or one weight per row when written against a single variance cell.
=1/(StudyVar+ResidTau2)
Assumptions
Residual variance taken from the fitted meta-regression
The tau2 in the weight is the residual estimate from the model that contains the covariates, estimated by a method such as restricted maximum likelihood or DerSimonian and Laird, and then treated as known while the coefficients are estimated.
Worked examples
Meta-regression weight of a smaller illustrative trial
A trial with a within-study variance of 0.04 and a residual tau2 of 0.03, as in the article's illustrative fit, has a weight of about 14.29.
v_i = 0.04; tau2 = 0.03; w_i = 14.2857
Meta-regression weight of a larger illustrative trial
A trial with a within-study variance of 0.01 and the same residual tau2 has a weight of 25. It carries 1.75 times the weight of the smaller trial, against four times under fixed-effect weights of 100 and 25.
v_i = 0.01; tau2 = 0.03; w_i = 25
Fixed-effect meta-regression weight with zero residual variance
With tau2 set to zero the weight reduces to the inverse of the within-study variance, 100 for a variance of 0.01.
v_i = 0.01; tau2 = 0; w_i = 100
Common errors
Weighting a meta-regression with the tau2 from the meta-analysis without covariates
Using the between-study variance before the covariate is added, 0.09 in the article's example, instead of the residual 0.03 gives the larger trial a weight of 10 instead of 25 and the smaller one about 7.69 instead of 14.29. The weights become too equal and the analysis ignores the heterogeneity the covariate has explained.
Adding tau instead of tau2 to a meta-regression weight
Adding the between-study standard deviation to a variance mixes scales. With a variance of 0.01 and a residual tau2 of 0.03, tau is about 0.173 and the weight falls to about 5.46 instead of 25.
Sources
Weights for fixed-effect and mixed-effects models in metafor
Viechtbauer W. Conducting meta-analyses in R with the metafor package. Journal of Statistical Software. 2010;36(3). Section 2.3 (fixed-effects weights typically 1/v_i) and section 2.4 (coefficients estimated by weighted least squares with weights 1/(v_i + tau2) once tau2 has been estimated).
Precision weighting in meta-regression 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 larger studies have more influence because studies are weighted by the precision of their effect estimates, and that residual heterogeneity should be allowed for.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0