Optimal testing using combined test statistics across independent studies

Lasse Vuursteen · Botond Szabo · Aad van der Vaart · Harry van Zanten

Great Hall & Hall B1+B2 (level 1) #1007
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[ Paper [ Poster [ OpenReview
Tue 12 Dec 8:45 a.m. PST — 10:45 a.m. PST

Abstract: 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.

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