Signature
R2_het = (tau2_RE - tau2_MR) / tau2_RE
| Inputs | Definition | Unit |
|---|---|---|
tau2_RE | Between-study variance from the random-effects meta-analysis of the same studies without covariates, above zero | the squared effect scale |
tau2_MR | Residual between-study variance from the meta-regression with the covariates | the squared effect scale |
R2_het | Proportion of the between-study variance accounted for by the covariates, multiplied by 100 for a percentage | a proportion, at most 1 |
|---|
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
Share of heterogeneity explained from two tau2 cells
With the between-study variance without covariates in a cell named Tau2NoCov and the residual variance in a cell named Tau2Resid, the formula returns the proportion explained.
=(Tau2NoCov-Tau2Resid)/Tau2NoCov
Assumptions
Same studies and same estimator for both between-study variances
Both variances come from the same set of studies on the same effect scale and from the same estimator of tau2, so that the fall reflects the covariates and not a change of method or of sample.
Worked examples
Illustrative baseline-risk meta-regression explaining two thirds of heterogeneity
In the article's illustrative example the between-study variance falls from 0.09 to a residual 0.03 when the baseline log-odds is added, so the covariate accounts for about 67% of the heterogeneity.
tau2_RE = 0.09; tau2_MR = 0.03; R2_het = 0.6667
BCG vaccine meta-regression with two moderators
In Viechtbauer's BCG vaccine example the restricted maximum likelihood estimate of tau2 falls from 0.3132 to 0.1108 when absolute latitude and year are added, so about 65% of the heterogeneity is accounted for.
tau2_RE = 0.3132; tau2_MR = 0.1108; R2_het = 0.6462
Common errors
Reading the explained share as a share of all variation in the estimates
The ratio describes the between-study variance only. A covariate that accounts for 67% of tau2 has not explained 67% of the spread in the observed estimates, part of which is sampling error. Both tau2 values are also uncertain with few studies, as the standard error of 0.0845 for the residual estimate of 0.1108 in the BCG example shows.
Comparing tau2 from different sets of studies
When studies without a value for the covariate drop out of the meta-regression, comparing its residual tau2 with the tau2 from the full meta-analysis mixes the effect of the covariate with the effect of removing studies. The reference model is refitted to the studies used in the meta-regression.
Sources
Heterogeneity accounted for by moderators in metafor
Viechtbauer W. Conducting meta-analyses in R with the metafor package. Journal of Statistical Software. 2010;36(3). Section 3.4 example, in which the residual tau2 of 0.1108 against 0.3132 means 65% of the heterogeneity is accounted for by the two moderators, and the likelihood ratio test example giving the share as 100% times the difference between the reduced-model and full-model tau2, divided by the reduced-model tau2.
Canonical Identity
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