Tau-Squared

Tau-Squared and the Between-Study Variance in Random-Effects Meta-Analysis

Tau-squared, written $\tau^2$, is the variance of the true treatment effects that a random-effects meta-analysis assumes lie behind the trials it pools, in squared units of the analysis scale, such as the log odds ratio. Its square root $\tau$ is the standard deviation of those true effects, so it shows how far the effect in one trial setting is likely to sit from the average. In health technology assessment the estimate decides how much weight small trials receive and how widely the effect in a new population might range once a pooled relative effect enters a cost-effectiveness model. The estimate depends on the method used and is very imprecise when trials are few, so the NICE manual (PMG36) asks for external information on it. Tau-squared is unrelated to Kendall's tau, a rank correlation coefficient.

The variance of true effects behind a pooled estimate

A random-effects model treats each trial as estimating its own true effect, drawn from a distribution of effects across settings. The clinical and methodological sources of that spread are discussed under between-study heterogeneity, and the parameter appears in the model as follows.

$$y_i \sim N(\theta_i, v_i), \qquad \theta_i \sim N(\mu, \tau^2)$$

where $y_i$ is the observed effect in trial $i$, $\theta_i$ is its true effect, $v_i$ is its within-study sampling variance, $\mu$ is the mean of the true effects and $\tau^2$ is the between-study variance, zero when every trial estimates the same effect.

Its size is easier to judge through $\tau$. For a ratio measure pooled on the log scale, such as an odds ratio or a hazard ratio, about 95% of true ratios lie within a factor of $e^{1.96\tau}$ either side of the median ratio, as the table shows.

Higgins and colleagues noted in 2003 that its values depend on the treatment effect scale, and the Cochrane Handbook concedes that interpreting the estimate can be challenging, pointing to prediction intervals as a more intuitive presentation.

How the estimate changes the random-effects weights

Inverse-variance pooling in a random-effects model adds the estimated between-study variance to each trial's own sampling variance. That one term changes the weights and the precision of the pooled result.

$$w_i^{*} = \frac{1}{v_i + \hat{\tau}^2}$$

where $w_i^{*}$ is the random-effects weight of trial $i$ and $\hat{\tau}^2$ is the estimated between-study variance. When $\hat{\tau}^2$ is zero these are the weights of a fixed-effect model, and as it grows the weights move towards equality. The Cochrane Handbook explains that a random-effects analysis gives relatively more weight to smaller studies because they are more informative about the distribution of effects than about an assumed common effect. The standard error of the pooled mean also rises with $\hat{\tau}^2$, which carries through to any relative effect used in an economic model.

Estimators and why they give different values

Tau-squared is estimated from the scatter of trial results beyond what sampling variances explain. DerSimonian and Laird proposed in 1986 a simple non-iterative procedure, still the most familiar.

$$\hat{\tau}^2_{DL} = \max\left\{0, \frac{Q - (k - 1)}{\sum_{i} w_i - \sum_{i} w_i^2 / \sum_{i} w_i}\right\}, \qquad Q = \sum_{i=1}^{k} w_i \left(y_i - \hat{\mu}_{FE}\right)^2$$

where $w_i = 1/v_i$ is the fixed-effect weight, $k$ is the number of trials, $\hat{\mu}_{FE}$ is the fixed-effect pooled estimate and $Q$ is Cochran's heterogeneity statistic. Veroniki and colleagues, whose 2016 review identified 16 estimators, describe it as a special case of the method of moments. The Paule and Mandel estimator is iterative and chooses the value at which a generalised Q statistic, computed with random-effects weights, equals its expected value.

$$Q_{gen}(\tau^2) = \sum_{i=1}^{k} \frac{\left(y_i - \hat{\mu}_{RE}(\tau^2)\right)^2}{v_i + \tau^2} = k - 1$$

where $\hat{\mu}_{RE}(\tau^2)$ is the random-effects mean at that value of $\tau^2$. The estimate is zero when $Q_{gen}(0)$ is already below $k - 1$. Maximum likelihood estimation is asymptotically efficient but iterative, and restricted maximum likelihood (REML) corrects its negative bias.

Truncation at zero introduces positive bias into the DerSimonian and Laird estimator. Simulation studies find the estimator acceptable when the true variance is small and trials are many, but it can show significant negative bias when the true variance is large. Veroniki and colleagues judged Paule and Mandel a better alternative for dichotomous and continuous data and REML for continuous data, a recommendation based on a qualitative evaluation of the literature and expert consensus. The Cochrane Handbook reports that a large recent simulation study recommended REML while noting that no single approach is universally preferable, and REML is now the RevMan default.

The choice can matter. In a reanalysed meta-analysis of 11 trials of chemotherapy added to supportive care in advanced non-small cell lung cancer (log hazard ratios, I-squared 75%), Veroniki and colleagues obtained 0.132 with DerSimonian and Laird, 0.156 with REML and 0.190 with Paule and Mandel. The Handbook adds that estimators can differ substantially in heterogeneity but seldom have major implications for summary effects.

Uncertainty in the estimate and the frequent zero

An estimate of $\tau^2$ from a handful of trials says little without an interval. Veroniki and colleagues recommend the Q-profile method, which inverts the generalised Q statistic.

$$Q_{gen}(\tau^2_{L}) = \chi^2_{k-1,\,0.975}, \qquad Q_{gen}(\tau^2_{U}) = \chi^2_{k-1,\,0.025}$$

where $\tau^2_{L}$ and $\tau^2_{U}$ are the 95% limits and $\chi^2_{k-1,\,p}$ is the $p$ quantile of a chi-square distribution on $k - 1$ degrees of freedom. Negative values are not allowed, so the lower limit is zero whenever $Q_{gen}(0)$ falls below the upper quantile.

Zero estimates are common. Turner and colleagues analysed 14,886 meta-analyses of binary outcomes from the Cochrane Database of Systematic Reviews, with a median of three studies, and found the method of moments estimate set to zero in 57%. They noted that zero estimates often arise when true heterogeneity is small but positive, and that a conventional random-effects analysis does not acknowledge the often substantial uncertainty in the estimate.

Worked example with five trials and four estimators

All numbers in this example are illustrative. Five randomised trials compare a new drug with usual care for emergency admission within 12 months, each reporting a log odds ratio with its variance.

Step 1 (fixed effect and Q). The weights total 202.111 and the weighted effects -27.300, so the fixed-effect log odds ratio is -0.1351. The deviations of -0.4649, 0.0351, -0.3149, 0.2851 and -0.1649 give the weighted squared terms below.

$$Q = 5.4039 + 0.1230 + 1.1020 + 4.0634 + 0.4352 = 11.1275$$

where $Q$ has 4 degrees of freedom (P = 0.025), so I-squared is 7.1275/11.1275, or 64.1%.

Step 2 (DerSimonian and Laird). The squared weights total 13,504.457, and dividing by 202.111 gives 66.817, so the denominator is 202.111 minus 66.817, or 135.294.

$$\hat{\tau}^2_{DL} = \frac{11.1275 - 4}{135.294} = 0.0527$$

where 0.0527 is on the squared log odds ratio scale, so $\hat{\tau} = 0.2295$.

Step 3 (other estimators). Solving $Q_{gen}(\tau^2) = 4$ by iteration gives a Paule and Mandel estimate of 0.0546, REML gives 0.0614 and full maximum likelihood 0.0413.

Step 4 (pooled effect). With the DerSimonian and Laird value the weights become 10.790, 15.954, 7.009, 13.759 and 8.682, totalling 56.193, and the weighted effects total -11.764. The pooled log odds ratio is -11.764/56.193 = -0.2093, an odds ratio of 0.811, with standard error $\sqrt{1/56.193} = 0.1334$ and 95% confidence interval 0.624 to 1.054. Trial 2's share of the weight falls from 49.5% to 28.4%, and REML and Paule and Mandel give pooled odds ratios of 0.807 and 0.810.

Step 5 (reading $\tau$). 95% of true log odds ratios lie within $-0.2093 \pm 1.96 \times 0.2295$, or -0.659 to 0.241, odds ratios of 0.517 to 1.272. The true effect is harmful in settings more than 0.2093/0.2295 = 0.912 standard deviations above the mean, a share of 18.1%.

Step 6 (uncertainty in $\tau^2$). On 4 degrees of freedom the 97.5th chi-square percentile is 11.143, just above $Q_{gen}(0) = Q = 11.1275$, so the lower Q-profile limit is zero. Solving $Q_{gen}(\tau^2) = 0.484$ gives an upper limit of 0.683, so $\tau$ could lie anywhere between 0 and 0.83. At $\tau = 0.83$ the 97.5th percentile true odds ratio would be $e^{3.92 \times 0.8265} = 25.5$ times the 2.5th.

Step 7 (model input). At an illustrative baseline admission risk of 0.20 (odds 0.25), the pooled odds ratio gives odds of 0.2028 and a risk of 0.169, or 31 fewer admissions per 1,000. At the 2.5th and 97.5th percentiles of true effects the risks are 0.114 and 0.241, or 86 fewer and 41 more.

Interpretation. The estimates of $\tau^2$ range from 0.041 to 0.061, yet the pooled odds ratio barely moves between them, as the Handbook leads one to expect. The interval for $\tau^2$ is the real finding, since five trials cannot separate near agreement from a 25-fold spread of true effects. Step 5 also ignores uncertainty in the mean and in $\tau^2$, which the prediction interval adds.

External evidence and the decision model

The NICE technology appraisal and highly specialised technologies manual (PMG36, last updated 13 October 2026) asks in section 3.4.16 for external information to help estimate the between-study heterogeneity, and says that in networks with few studies informative prior distributions for the heterogeneity parameter may be preferable. Section 3.4.17 recommends distributions tailored to outcomes and disease areas, and section 3.4.18 asks for the source of the prior to be justified, with a sensitivity analysis using different candidate priors.

Turner and colleagues supplied such distributions on the log odds ratio scale, for use as prior distributions in a Bayesian meta-analysis. For a pharmacological intervention against placebo or control the predicted median was 0.017 (95% range 0.001 to 0.30) for all-cause mortality and 0.12 (0.005 to 2.63) for a subjectively measured outcome.

The NICE Decision Support Unit's technical support document 3 (TSD 3) argues that the predictive distribution of the effect in a new trial may often be more relevant to decision making than the distribution of the mean. In a probabilistic sensitivity analysis the estimate then enters twice, through the standard error of the mean and through the predictive variance, so a zero from three trials makes the draws as narrow as a fixed-effect analysis.

Tau-squared and I-squared

I-squared gives the share of observed variation attributed to heterogeneity and rises towards 100% as trials become more precise, while tau-squared gives the size of that variation on the effect scale, which Rücker and colleagues judged the appropriate measure of its clinical relevance. After covariates enter a meta-regression the remaining $\tau^2$ is residual heterogeneity. Variation between patients within a trial belongs to heterogeneous treatment effect.

Common misreadings

Most errors treat a noisy, scale-bound estimate as a fixed property of the evidence. Four are common.

1. Reading a zero as homogeneity. The Cochrane Handbook warns of much uncertainty in tau-squared when studies are few.
2. Comparing values across effect measures. Log odds ratios and mean differences are on different scales.
3. Reading the confidence interval of the mean as the spread of effects. The spread is described by $\tau$ and the prediction interval.
4. Treating the estimator as neutral. DerSimonian and Laird can understate large heterogeneity, so the method should be stated and checked.

Sources

• Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses (last updated November 2024). In: Higgins JPT, Thomas J, Chandler J, et al (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Sections 10.10 and 10.11.
• DerSimonian R, Laird N. Meta-analysis in clinical trials. Controlled Clinical Trials. 1986;7(3):177-188. doi:10.1016/0197-2456(86)90046-2
• Dias S, Sutton AJ, Welton NJ, Ades AE. NICE DSU Technical Support Document 3: Heterogeneity: subgroups, meta-regression, bias and bias-adjustment. Decision Support Unit; September 2011, last updated April 2012.
• Higgins JPT, Thompson SG, Deeks JJ, Altman DG. Measuring inconsistency in meta-analyses. BMJ. 2003;327(7414):557-560. doi:10.1136/bmj.327.7414.557
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 3.4.16 to 3.4.18.
• Rücker G, Schwarzer G, Carpenter JR, Schumacher M. Undue reliance on I2 in assessing heterogeneity may mislead. BMC Medical Research Methodology. 2008;8:79. doi:10.1186/1471-2288-8-79
• Turner RM, Davey J, Clarke MJ, Thompson SG, Higgins JPT. Predicting the extent of heterogeneity in meta-analysis, using empirical data from the Cochrane Database of Systematic Reviews. International Journal of Epidemiology. 2012;41(3):818-827. doi:10.1093/ije/dys041
• Veroniki AA, Jackson D, Viechtbauer W, Bender R, Bowden J, Knapp G, et al. Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research Synthesis Methods. 2016;7(1):55-79. doi:10.1002/jrsm.1164