Signature
tau2_med = exp(m_LN); tau2_low = exp(m_LN - 1.96 * s_LN); tau2_high = exp(m_LN + 1.96 * s_LN)
| Inputs | Definition | Unit |
|---|---|---|
m_LN | Mean of the logarithm of tau2 under the prior, the first parameter of LN(m_LN, s_LN^2) | log of squared effect units |
s_LN | Standard deviation of the logarithm of tau2 under the prior; its square is the second parameter of LN(m_LN, s_LN^2) | log of squared effect units |
tau2_med | Median of the log-normal prior for tau2 | squared units of the effect scale |
|---|---|---|
tau2_low | Value of tau2 below which the prior places 2.5% of its probability | squared units of the effect scale |
tau2_high | Value of tau2 above which the prior places 2.5% of its probability | squared units of the effect scale |
Function
Bayesian random-effects posterior for a pairwise meta-analysis
Maps each trial's effect estimate y_i and standard error s_i, together with priors for the mean effect mu and the between-study standard deviation tau, to a joint posterior distribution for mu and tau. The trial-level true effects are integrated out, so each estimate contributes a normal likelihood with variance s_i^2 + tau^2. The posterior of the mean effect, shrunken trial estimates and the predictive distribution of the effect in a new setting all follow from this joint posterior, and posterior draws can pass directly into a probabilistic cost-effectiveness model. Frequentist pooling of the same data, including the DerSimonian and Laird estimate, Cochran's Q, I-squared and the prediction interval, is covered under aggregate data meta-analysis (HE-FN-ADMA-001).
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Implementations
Excel
Log-normal prior median and 95% limits in three cells
With the log-scale mean in a cell named PriorLogMean and the log-scale standard deviation in PriorLogSD, the three formulas return the median and the 2.5% and 97.5% points of tau2. LOGNORM.INV(0.975,PriorLogMean,PriorLogSD) gives the upper limit with the exact normal quantile in place of 1.96.
=EXP(PriorLogMean); =EXP(PriorLogMean-1.96*PriorLogSD); =EXP(PriorLogMean+1.96*PriorLogSD)
Assumptions
Log-normal heterogeneity prior stated for tau2 on a named effect scale
Each published prior applies to a stated effect scale and setting. The Turner priors are for the between-study variance of log odds ratios for binary outcomes, in settings defined by outcome type and type of comparison. Rhodes and colleagues give log-t priors for standardised mean differences, to which this formula does not apply.
Prior parameters on the log scale of the between-study variance
m_LN and s_LN are the mean and standard deviation of the logarithm of tau2, not of tau2 itself, and the factor 1.96 gives limits that leave 2.5% of the prior probability in each tail.
Worked examples
Turner prior for all-cause mortality in pharmacological against placebo comparisons
The 2015 prior for all-cause mortality in pharmacological against placebo or control comparisons, LN(-3.95, 1.34^2), has a median tau2 of about 0.0193 and a central 95% range of about 0.0014 to 0.266, as in the article. On the tau scale this is a median of about 0.139 and a range of about 0.037 to 0.516.
m_LN = -3.95; s_LN = 1.34; tau2_med = 0.0193; tau2_low = 0.0014; tau2_high = 0.266
Turner prior for a general healthcare setting
The 2015 prior for a general healthcare setting, LN(-2.56, 1.74^2), has a median tau2 of about 0.0773 and a central 95% range of about 0.0026 to 2.34, much wider than the prior tailored to all-cause mortality.
m_LN = -2.56; s_LN = 1.74; tau2_med = 0.0773; tau2_low = 0.0026; tau2_high = 2.34
Common errors
Placing a published tau2 prior on tau
The Turner priors are for tau2. Placing LN(-3.95, 1.34^2) on tau instead gives a prior median for tau of about 0.0193 rather than about 0.139, and with the article's four trials it pulls the posterior median of tau down to about 0.020 instead of 0.133.
Reading the squared log-scale parameter as the standard deviation
In LN(-3.95, 1.34^2) the standard deviation of the logarithm of tau2 is 1.34. Reading 1.34^2, about 1.80, as the standard deviation raises the upper 95% limit of tau2 from about 0.266 to about 0.650 and lowers the lower limit from about 0.0014 to about 0.0006.
Borrowing a heterogeneity prior from another effect scale or setting
A prior derived for log odds ratios is not automatically suitable for another measure, and the general healthcare prior is far wider than the one tailored to all-cause mortality. The NICE manual (PMG36, 2022, updated 2026) recommends distributions tailored to particular outcomes and disease areas, with the source noted and justified and a sensitivity analysis of candidate priors.
Sources
Log-normal predictive distributions for heterogeneity in Turner and colleagues 2015
Turner RM, Jackson D, Wei Y, Thompson SG, Higgins JPT. Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis. Statistics in Medicine. 2015;34(6):984-998. Section 2, which places a log-normal prior on tau2 with parameters assumed known; section 3, which gives the general healthcare prior LN(-2.56, 1.74^2); and Table IV, which gives LN(-3.95, 1.34^2) for all-cause mortality in pharmacological against placebo or control comparisons and states that the parameters are the mean and standard deviation on the log scale.
Log-normal heterogeneity distributions in Turner and colleagues 2012
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. Abstract, which presents log-normal predictive distributions for tau2 on the log odds ratio scale for nine settings and reports that LN(-2.13, 1.58^2) has a median of 0.12.
Canonical Identity
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