Every random-effects meta-analysis rests on an estimate of tau-squared, the between-study variance that quantifies how much true effect sizes differ across your included studies. The estimator you choose to compute tau-squared is not a technical footnote. It determines the weights assigned to each study, the width of your pooled confidence interval, and ultimately the precision of your summary effect.

Two estimators dominate the field: DerSimonian-Laird (DL), introduced in 1986 and still the most commonly used, and REML (Restricted Maximum Likelihood), which simulation studies consistently show outperforms DL in most realistic meta-analysis scenarios.

Try our free Forest Plot Generator to visualize pooled effects under both estimators and compare how your forest plot changes.

How Tau-Squared Estimation Shapes Your Entire Random-Effects Analysis

In a random-effects synthesis, each study receives a weight proportional to the inverse of its total variance, which is the sum of within-study sampling variance and the between-study variance tau-squared.

When tau-squared is small, large studies dominate. When tau-squared is large, weights become more equal across studies. An overestimated tau-squared makes your weights artificially uniform and widens confidence intervals unnecessarily. An underestimated tau-squared produces overconfident results and artificially narrow intervals.

The DerSimonian-Laird Method: Strengths and Known Limitations

DerSimonian and Laird (1986) proposed a method-of-moments estimator that became the default in almost every meta-analysis software package. Its appeal is computational simplicity with a closed-form solution.

However, simulation studies identified consistent problems. The estimator is biased downward, tending to underestimate the true tau-squared, particularly when the number of studies is small (k less than 30) or when heterogeneity is moderate to high. Studies by Viechtbauer (2005) and Langan and colleagues (2019) showed that DL produces 95% confidence intervals with actual coverage closer to 92-93%.

REML: Fisher Scoring, Iterations, and Why It Performs Better

Restricted Maximum Likelihood estimation treats tau-squared as a variance component estimated via maximum likelihood applied to a restricted log-likelihood function. Unlike DL, REML uses iterative optimization with Fisher scoring iterations, updating the estimate at each step based on the gradient and curvature of the likelihood until convergence.

Simulation results consistently favor REML. Viechtbauer's comprehensive comparison showed REML has lower mean squared error than DL across most scenarios, particularly when k is small (5-20 studies). The metafor package in R uses REML as its default estimator, reflecting the methodological consensus.

When DerSimonian-Laird Remains Acceptable

When k is large (30 or more studies), DL's bias diminishes substantially. When tau-squared is very small or zero, both estimators agree by definition. For exploratory preliminary analyses where computational simplicity aids rapid iteration, DL remains useful.

However, if you are writing for a high-quality clinical journal, if your synthesis has fewer than 20 studies, or if reviewers are likely to scrutinize your heterogeneity estimate, REML is the defensible choice.

See our Sensitivity Analysis Tool to examine whether your pooled estimate is stable across individual study exclusions.

Other Estimators Worth Knowing

Paule-Mandel (PM) avoids DL's downward bias by iterating to find tau-squared such that the expected Q equals its observed value. Simulation studies suggest PM performs comparably to REML.

Hedges' estimator applies a bias correction to the DL estimate and outperforms DL when k is small.

ML (maximum likelihood) without the restriction component underestimates tau-squared more than REML and is generally not recommended.

Our Funnel Plot Generator complements this analysis by letting you visualize how publication bias might interact with your heterogeneity estimate.

Key Takeaways

FAQ

Why does REML produce wider confidence intervals than DerSimonian-Laird?

REML typically estimates a larger tau-squared than DL because it corrects for the downward bias in the method-of-moments approach. A larger tau-squared means more between-study variance is attributed to true heterogeneity, which widens the pooled confidence interval. These wider intervals are more accurate reflections of genuine uncertainty.

Does the choice of tau-squared estimator affect the I-squared statistic?

Yes. When tau-squared enters the calculation of study weights in iterative estimators, the Q statistic itself can differ from its DL-based counterpart. In practice, the effect on I-squared is usually modest.

How many Fisher scoring iterations does REML typically need to converge?

For most meta-analysis datasets, REML converges in 10 to 30 iterations. The metafor package uses a default maximum of 100 iterations. Non-convergence is rare but can occur with very few studies or extreme outliers.

Should I use REML for subgroup analyses as well?

Yes. If you are running within-subgroup random-effects models, REML is appropriate for each subgroup, provided each subgroup has at least 3 to 4 studies.

Is there a consensus recommendation on tau-squared estimators from Cochrane?

The Cochrane Handbook (version 6) recommends REML as the preferred estimator for random-effects meta-analyses in most circumstances, while acknowledging DL as a widely used alternative.

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