Signature
tau2 = max(0, (Q - (k - 1)) / (sum_(i=1)^k [1 / v_i] - sum_(i=1)^k [1 / v_i^2] / sum_(i=1)^k [1 / v_i]))
| Inputs | Definition | Unit |
|---|---|---|
Q | Cochran's statistic computed with common-effect weights | none |
k | Number of studies, at least 2 | count |
v_i | Within-study variance of the estimate from study i; 1/v_i is its common-effect weight | squared units of the effect scale |
tau2 | Estimated variance of the true study effects around their mean | squared units of the effect scale |
|---|
Function
Inverse-variance pooling of aggregate study-level effect estimates
Maps the effect estimate and its variance from each of k studies to one pooled estimate, its standard error and measures of between-study heterogeneity. Each study is weighted by the inverse of the variance of its estimate, so more precise studies count for more. Under a common-effect model the weight uses the within-study variance alone; under a random-effects model it also includes the between-study variance tau2. In health technology assessment the pooled relative effect, usually a log hazard ratio or log odds ratio, is then applied to a baseline from another source in a decision model.
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Implementations
Excel
DerSimonian and Laird tau2 in one cell
With the variances in a range named StudyVar, Q in a cell named QStat and the number of studies in NStudies, the formula returns the truncated estimate.
=MAX(0,(QStat-(NStudies-1))/(SUMPRODUCT(1/StudyVar)-SUMPRODUCT(1/StudyVar^2)/SUMPRODUCT(1/StudyVar)))
Assumptions
Moment estimate truncated at zero
The estimate equates Q to its expected value under a random-effects model. When Q is at or below k-1 the estimate is set to 0 and random-effects pooling gives the common-effect result.
Point estimate of tau2 with few studies
With few studies the estimate of tau2 is very uncertain. The Cochrane Handbook records that other estimators, such as Paule and Mandel or restricted maximum likelihood, performed better in simulation studies, and the NICE manual (PMG36) notes that informative priors for the heterogeneity parameter may be preferable in networks with few studies.
Worked examples
DerSimonian and Laird tau2 for three illustrative trials
The weights 25, 100 and 16 sum to 141 and their squares to 10,881, so the scaling term is about 63.8298. The excess of Q over 2 is 4.867, giving a tau2 of about 0.07625 and a tau of about 0.2761, as in the article.
Q = 6.867; k = 3; v_i = [0.04,0.01,0.0625]; tau2 = 0.07625
Common errors
Omitting the truncation of a negative tau2
If Q were 1.5 with the article's three variances, the untruncated estimate would be about -0.00783. Adding it to the variances shrinks the variance of trial 2 to about 0.00217 and inflates its random-effects weight to about 462 instead of 100, and a larger negative value can make a variance negative.
Sources
Original moment estimator of between-study variance
DerSimonian R, Laird N. Meta-analysis in clinical trials. Controlled Clinical Trials. 1986;7(3):177-188. The random-effects model for combining trial results with a non-iterative, moment-based estimate of the between-study variance.
DerSimonian and Laird formula in a review of variance estimators
Veroniki AA, Jackson D, Viechtbauer W, Bender R, Bowden J, Knapp G, Kuss O, Higgins JPT, Langan D, Salanti G. Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research Synthesis Methods. 2016;7(1):55-79. Section on the DerSimonian and Laird estimator, which gives the truncated formula with common-effect weights, and the comparison with alternative estimators.
Choice of between-study variance estimator 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 10.10.4.4, which describes the DerSimonian and Laird method as moment-based and reports simulation studies favouring Paule and Mandel or restricted maximum likelihood.
Canonical Identity
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