Signature
theta_new = mu + tau * z; o_0 = p_0 / (1 - p_0); p_1 = o_0 * exp(theta_new) / (1 + o_0 * exp(theta_new))
| Inputs | Definition | Unit |
|---|---|---|
mu | Mean effect sampled at the same iteration | log odds ratio |
tau | Between-study standard deviation sampled at the same iteration | log odds ratio |
z | Draw from a standard normal distribution, independent of mu and tau | none |
p_0 | Annual mortality risk on the comparator, taken from an external source | probability from 0 to 1 |
theta_new | True effect in a new trial or population at one iteration, on the log odds ratio scale | log odds ratio |
|---|---|---|
o_0 | Baseline odds of death, p_0 divided by one minus p_0 | odds |
p_1 | Annual mortality risk on treatment implied by the draw | probability from 0 to 1 |
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
Predictive draw and treated mortality risk in three cells
With one posterior iteration per row and its sampled values in cells named Mu and Tau, the first formula draws theta_new using NORM.S.INV(RAND()) as the standard normal draw. With the baseline risk in BaselineRisk, the second returns o_0, and the third returns p_1 from cells named ThetaNew and BaselineOdds. PERCENTILE.INC of the p_1 column at 0.025 and 0.975 gives the 95% predictive range.
=Mu+Tau*NORM.S.INV(RAND()); =BaselineRisk/(1-BaselineRisk); =BaselineOdds*EXP(ThetaNew)/(1+BaselineOdds*EXP(ThetaNew))
Assumptions
Mean effect and between-study standard deviation from the same iteration
mu and tau are taken from the same posterior iteration, so their dependence is kept. Drawing them separately, or holding them at point estimates, changes the predictive distribution.
Effect additive on the log odds scale with a separate baseline risk
The baseline risk comes from evidence relevant to the decision population, separate from the trials that inform the relative effect, and the odds ratio is applied on the odds scale rather than to the risk directly.
Predictive spread limited to true between-population variation
NICE DSU TSD 3 cautions that the predictive variance should contain only true variation between populations and that observed heterogeneity is likely to overestimate it. An implausibly large tau from a vague prior passes straight into the draws and widens the decision uncertainty.
Worked examples
Central predictive draw under the Turner heterogeneity prior
At z = 0 the draw equals mu. With mu = -0.238, the posterior median under the Turner prior (an odds ratio of 0.788), and a baseline risk of 0.10, the baseline odds are about 0.1111 and the treated risk about 0.0805, as in the article.
mu = -0.238; tau = 0.133; z = 0; p_0 = 0.10; theta_new = -0.238; o_0 = 0.1111; p_1 = 0.0805
Upper-tail predictive draw with posterior medians held fixed
Holding mu at -0.238 and tau at 0.133, the posterior medians under the Turner prior, a draw at z = 1.96 gives theta_new of about 0.0227 and a treated risk of about 0.1021. This single draw falls short of the 97.5th percentile of the full predictive sample, a risk of about 0.121 in the article, because the full sample also varies mu and tau across iterations.
mu = -0.238; tau = 0.133; z = 1.96; p_0 = 0.10; theta_new = 0.0227; o_0 = 0.1111; p_1 = 0.1021
Common errors
Building the predictive range from posterior medians alone
Holding mu and tau at their posterior medians and taking mu plus or minus 1.96 tau gives odds ratios of about 0.607 to 1.023 under the Turner prior and 0.452 to 1.355 under the vague prior, against the article's predictive intervals of 0.489 to 1.239 and 0.225 to 2.592. The shortcut omits the uncertainty in mu and tau, which with four trials is large.
Sampling the mean effect where the predictive effect is needed
Sampling mu alone describes the average effect across trial settings. Under the Turner prior the 95% credible interval for mu gives odds ratios of 0.593 to 1.032 and treated risks of about 0.062 to 0.103, while the predictive draws give 0.489 to 1.239 and risks of about 0.052 to 0.121. TSD 3 argues that the predictive distribution may in many cases be more relevant to decision making than the distribution of the mean.
Sources
Predictive draw applied to a baseline risk in NICE DSU TSD 3
Dias S, Sutton AJ, Welton NJ, Ades AE. NICE DSU Technical Support Document 3: Heterogeneity: subgroups, meta-regression, bias and bias-adjustment. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated April 2012. Section 2.1, which obtains the predictive distribution in MCMC by drawing delta_new from N(d, sigma^2) and cautions that its variance should contain only true variation between populations, and section 3.1, which draws d and sigma from their posteriors and adds the predictive effect to the logit of a baseline probability of mortality.
Bayesian predictive distribution for a new study in Higgins and colleagues
Higgins JPT, Thompson SG, Spiegelhalter DJ. A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A. 2009;172(1):137-159. Section 5.4, which obtains the predictive distribution for the effect in a new study by sampling theta_new from N(mu, tau^2) within an MCMC analysis and takes the 95% prediction interval from the 2.5% and 97.5% quantiles.
Canonical Identity
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