Signature
B_i = s_i^2 / (s_i^2 + tau2); theta_post_i = B_i * mu + (1 - B_i) * y_i
| Inputs | Definition | Unit |
|---|---|---|
s_i | Standard error of the estimate from trial i | the effect scale |
tau2 | Between-study variance, the square of tau | squared units of the effect scale |
mu | Mean of the distribution of true trial effects, a posterior mean or a value sampled at one iteration | the effect scale |
y_i | Effect estimate reported by trial i, for example a log odds ratio | the effect scale |
B_i | Weight given to the mean effect mu in the shrunken estimate of trial i; 1 when tau2 is zero and close to zero when tau2 is large relative to s_i^2 | none |
|---|---|---|
theta_post_i | Posterior mean of the true effect in trial i given mu and tau2 | 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
Shrinkage factor and shrunken trial estimate in two cells
With the trial's standard error in TrialSE, its estimate in TrialEst, the mean effect in PooledMean and the between-study variance in BetweenVar, the first formula returns B_i in a cell named ShrinkFactor and the second the shrunken estimate.
=TrialSE^2/(TrialSE^2+BetweenVar); =ShrinkFactor*PooledMean+(1-ShrinkFactor)*TrialEst
Assumptions
Mean effect and between-study variance held fixed for trial shrinkage
The formula conditions on mu and tau2. Plugging in point estimates gives an empirical Bayes estimate; a full Bayesian analysis averages the shrunken estimate over the joint posterior of mu and tau, so that each trial borrows strength from the others with the uncertainty in mu and tau carried through.
Normal random effects and known within-trial variance for trial shrinkage
The true effects follow N(mu, tau2) and each estimate y_i is normal around its true effect with known variance s_i^2. Under these conditions the posterior of theta_i is normal with the stated mean and a variance equal to one minus B_i, multiplied by s_i^2.
Worked examples
Shrinkage of the smallest mortality trial towards the pooled mean
For trial 3, the smallest trial with the largest effect, s_i^2 is 0.09 and the shrinkage factor at tau2 = 0.0193 is about 0.823. With mu = -0.239 its estimate moves from -0.60 to about -0.303, as in the article.
s_i = 0.30; y_i = -0.60; mu = -0.239; tau2 = 0.0193; B_i = 0.823; theta_post_i = -0.303
Shrinkage of the most precise mortality trial towards the pooled mean
Trial 2, the most precise trial, has s_i^2 of 0.0225 and a shrinkage factor of about 0.538 at the same tau2 and mu, so its estimate of -0.10 moves only to about -0.175.
s_i = 0.15; y_i = -0.10; mu = -0.239; tau2 = 0.0193; B_i = 0.538; theta_post_i = -0.175
Common errors
Using tau in place of tau2 in the shrinkage factor
Using tau, about 0.139, in place of tau2 = 0.0193 in the denominator gives a shrinkage factor of about 0.393 for trial 3 and a shrunken estimate of about -0.458 instead of -0.303, so the small trial keeps most of its extreme effect.
Treating plug-in shrunken trial estimates as fully Bayesian
Plugging point estimates of mu and tau2 into the formula treats them as known. Higgins and colleagues note that these empirical Bayes expressions can be improved to account for uncertainty in the estimates, while fully Bayesian shrunken estimates come from the posterior distributions of the trial effects. With four trials a single plug-in tau2 can understate or overstate the shrinkage.
Sources
Conditional distribution of a study effect 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 5.3, which gives the conditional distribution of the effect in study i as normal with mean lambda_i mu plus one minus lambda_i times the study estimate, with lambda_i equal to sigma_i^2 divided by sigma_i^2 plus tau^2, and describes fully Bayesian shrunken estimates as borrowing strength from the other studies.
Canonical Identity
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