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Many studies in biomedical and health sciences involve small sample sizes dueto logistic or financial constraints. Often, identifying weak (butscientifically interesting) associations between a set of predictors and aresponse necessitates pooling datasets from multiple diverse labs or groups.While there is a rich literature in statistical machine learning to addressdistributional shifts and inference in multi-site datasets, it is less clear${\it when}$ such pooling is guaranteed to help (and when it does not) --independent of the inference algorithms we use. In this paper, we present ahypothesis test to answer this question, both for classical and highdimensional linear regression. We precisely identify regimes where poolingdatasets across multiple sites is sensible, and how such policy decisions canbe made via simple checks executable on each site before any data transfer everhappens. With a focus on Alzheimer's disease studies, we present empiricalresults showing that in regimes suggested by our analysis, pooling a localdataset with data from an international study improves power.

When can Multi-Site Datasets be Pooled for Regression? Hypothesis Tests, $\ell_2$-consistency and Neuroscience Applications