When can Multi-Site Datasets be Pooled for Regression? Hypothesis Tests, 2-consistency and Neuroscience Applications (Supplementary Material)

Remarks on transformations in pre-processing step: For all i ∈ {1, ..., k}, after applying the transformation (shift correction), we pool (Xi, yi) together to estimate β∗. Note that in general the transformation (shift correction) should not depend on the responses yi, otherwise we get a dependence on the noise. To see this, notice that yi = Xiβi + i where Xi is the transformed set of features. But when the transformation depends on yi, then Xi will also depend on i, which causes a poor estimation of β∗ (and βi). In situations where the transformations must involve yi, a sensible strategy is to separate each site’s dataset into two parts, where one part from each site is used to learn the transformation, and the other part (after applying the learned transformation) is used for pooling towards β∗ estimation and conducting our hypothesis test.