  Combining test statistics from independent trials or experiments is a popular
method of meta-analysis. However, there is very limited theoretical
understanding of the power of the combined test, especially in high-dimensional
models considering composite hypotheses tests. We derive a mathematical
framework to study standard {meta-analysis} testing approaches in the context
of the many normal means model, which serves as the platform to investigate
more complex models.
  We introduce a natural and mild restriction on the meta-level combination
functions of the local trials. This allows us to mathematically quantify the
cost of compressing $m$ trials into real-valued test statistics and combining
these. We then derive minimax lower and matching upper bounds for the
separation rates of standard combination methods for e.g. p-values and
e-values, quantifying the loss relative to using the full, pooled data. We
observe an elbow effect, revealing that in certain cases combining the locally
optimal tests in each trial results in a sub-optimal {meta-analysis} method and
develop approaches to achieve the global optima. We also explore the possible
gains of allowing limited coordination between the trial designs. Our results
connect meta-analysis with bandwidth constraint distributed inference and build
on recent information theoretic developments in the latter field.
