A More Powerful Two-Sample Test in High Dimensions using Random Projection

By Miles E. Lopes, Laurent Jacob, Martin J. Wainwright. 2011

146 citations recorded by the source corpus.

We consider the hypothesis testing problem of detecting a shift between the means of two multivariate normal distributions in the high-dimensional setting, allowing for the data dimension p to exceed the sample size n. Our contribution is a new test statistic for the two-sample test of means that integrates a random projection with the classical Hotelling T2 statistic. Working within a high-dimensional framework that allows (p, n) → ∞, we first derive an asymptotic power function for our test, and then provide sufficient conditions for it to achieve greater power than other state-of-the-art tests. Using ROC curves generated from simulated data, we demonstrate superior performance against competing tests in the parameter regimes anticipated by our theoretical results. Lastly, we illustrate an advantage of our procedure with comparisons on a high-dimensional gene expression dataset involving the discrimination of different types of cancer.

Subfields: Artificial Intelligence, Molecular Biology, Statistics and Probability

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Citations in the corpus

Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing (2015, methodology): “This experimental setup is similar to one used in Lopes et al. [2011].”
Principal Differences Analysis: Interpretable Characterization of Differences between Distributions (2015, background): “The projections could also be chosen at random [1].”
Random matrix theory in statistics: A review (2014, methodology): “Under the same framework, Lopes et al. (2012) proposed a test constructed from averaged Hotelling's T2 statistics based on random projections of the data into lowerdimensional subspaces.”
Likelihood Ratio Tests for High‐Dimensional Normal Distributions (2013, methodology): “A projection method is used to investigate the two-sample test by Lopes et al. (2011).”
On the High-dimensional Power of Linear-time Kernel Two-Sample Testing under Mean-difference Alternatives (2014, unknown): “Parametric two-sample testing (for example, when P,Q are Gaussian) is also important, but will be out of the scope of our discussion; see Lopes et al. (2011) for a recent example. We assume equal number n of samples for simplicity; our results would also go through if n1/(n1 + n2)→ c ∈ (0, 1) as n1,”
On the High Dimensional Power of a Linear-Time Two Sample Test under Mean-shift Alternatives (2015, background): “This is the first explicit power derivation for a general nonparametric test in the high-dimensional setting, and the first analysis of how tests designed for general alternatives perform against easier ones.”
A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance (2017, methodology): “Random projections have been used in [28] to develop a powerful two-sample test in high dimensions.”