More information, less precision: Meta-analysis through random effects

Given several studies (inputs) of some phenomenon of interest, each input presents an estimate of a key parameter with an associated estimated precision. The random-effects model used in meta-analysis estimates this parameter based on a decomposition of the error term into within-input noise and across-input noise. Our interest is in the precision of this estimator, which leads to a confidence interval of the parameter. But we shall also be interested in the precision when we transform the inputs into one input, which leads to a (much wider) prediction interval. We review and extend the meta-analysis framework in a maximum-likelihood context, paying special attention to conflict between the inputs, correlation between the inputs, and the difference between confidence and prediction intervals and the corresponding notions of precision. We illustrate our approach with two meta-analyses from the world of clinical trials and finance.