Consider samples from two different data sources {x_s~i} ~ P_(source) and {x_t~i} ~ P_(target). We only observe their transformed versions h(x_s~i) and g(x_t~i), for some known function class h(·) and g(·). Our goal is to perform a statistical test checking if P_(source) = P_(target) while removing the distortions induced by the transformations. This problem is closely related to domain adaptation, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches - a fairly common impediment in conducting analyses with much larger sample sizes. We address this problem using ideas from hypothesis testing on the transformed measurements, wherein the distortions need to be estimated in tandem with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for Alzheimer's disease, our framework is competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.
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