Probabilistic multi-resolution scanning for two-sample differences

We propose a multi-resolution scanning approach to identifying two-sample differences. Windows of multiple scales are constructed through nested dyadic partitioning on the sample space and a hypothesis regarding the two-sample difference is defined on each window. Instead of testing the hypotheses on different windows independently, we adopt a joint graphical model, namely a Markov tree, on the null or alternative states of these hypotheses to incorporate spatial correlation across windows. The induced dependence allows borrowing strength across nearby and nested windows, which we show is critical for detecting high resolution local differences. We evaluate the performance of the method through simulation and show that it substantially outperforms other state of the art two-sample tests when the two-sample difference is local, involving only a small subset of the data. We then apply it to a flow cytometry data set from immunology, in which it successfully identifies highly local differences. In addition, we show how to control properly for multiple testing in a decision theoretic approach as well as how to summarize and report the inferred two-sample difference. We also construct hierarchical extensions of the framework to incorporate adaptivity into the construction of the scanning windows to improve inference further.