Signature
PI_low = theta_R - t_crit * sqrt(tau2 + SE_R^2); PI_high = theta_R + t_crit * sqrt(tau2 + SE_R^2)
| Inputs | Definition | Unit |
|---|---|---|
theta_R | Random-effects pooled mean | the chosen effect scale |
t_crit | 97.5th percentile of a t distribution with k-1 degrees of freedom, for example 4.303 for three studies | none |
tau2 | Estimated between-study variance | squared units of the effect scale |
SE_R | Standard error of theta_R | the chosen effect scale |
PI_low | Lower limit of the approximate 95% prediction interval | the chosen effect scale |
|---|---|---|
PI_high | Upper limit of the approximate 95% prediction interval | the chosen effect scale |
Function
Inverse-variance pooling of aggregate study-level effect estimates
Maps the effect estimate and its variance from each of k studies to one pooled estimate, its standard error and measures of between-study heterogeneity. Each study is weighted by the inverse of the variance of its estimate, so more precise studies count for more. Under a common-effect model the weight uses the within-study variance alone; under a random-effects model it also includes the between-study variance tau2. In health technology assessment the pooled relative effect, usually a log hazard ratio or log odds ratio, is then applied to a baseline from another source in a decision model.
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Implementations
Excel
Prediction interval limits with the t quantile
With named cells PooledRE, SEPooledRE, BetweenVar and NStudies, T.INV(0.975, NStudies-1) returns the critical t value and the two formulas return the lower and upper limits on the pooling scale; EXP converts them to ratios.
=PooledRE-T.INV(0.975,NStudies-1)*SQRT(BetweenVar+SEPooledRE^2); =PooledRE+T.INV(0.975,NStudies-1)*SQRT(BetweenVar+SEPooledRE^2)
Assumptions
Normally distributed true effects for the prediction interval
The true effects are taken to be normally distributed around theta_R with variance tau2. The Cochrane Handbook warns that prediction intervals can be very problematic when studies are few, appearing spuriously wide or narrow.
Worked examples
Prediction interval for three illustrative trials
With a mean of -0.3147, a standard error of 0.1910 and a tau2 of 0.07625, the standard deviation is about 0.336. Multiplied by a t value of 4.303 on 2 degrees of freedom it gives limits of about -1.759 and 1.130, hazard ratios of about 0.17 to 3.10, as in the article. Three trials are too few for the range to be reliable.
theta_R = -0.3147; SE_R = 0.1910; tau2 = 0.07625; t_crit = 4.303; PI_low = -1.759; PI_high = 1.130
Common errors
Using the interval of the mean as the range of effects in a new setting
The confidence interval describes uncertainty in the mean only. In the three-trial example it runs from hazard ratios of 0.502 to 1.062, while the prediction interval runs from about 0.17 to 3.10. Sampling the pooled log hazard ratio with the standard error of the mean, 0.191, in a probabilistic analysis instead of the predictive standard deviation, 0.336, ignores the between-study variation.
Using 1.96 in place of the t quantile with few studies
With three studies the normal value of 1.96 gives limits of about -0.973 and 0.343, hazard ratios of about 0.378 to 1.410, much narrower than the 0.17 to 3.10 obtained with the t value of 4.303.
Sources
Prediction interval formula in the Cochrane Handbook
Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.10.4.3, which gives the prediction interval from the summary mean, a t distribution with k-1 degrees of freedom, Tau2 and the standard error of the mean, and warns that it can be very problematic when studies are few.
Predictive distribution of a treatment effect for decision making
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. Executive summary and section 2, which argue that the predictive distribution of the effect in a new trial may often be more relevant to decision making and approximate its variance as the between-trials variance plus the variance of the mean.
Canonical Identity
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