Testing against a high dimensional alternative

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2006-06, Vol.68 (3), p.477-493
• Main Authors: Goeman, Jelle J., Van De Geer, Sara A., Van Houwelingen, Hans C.
• Format: Article
• Language: English
• Subjects: Bayesian analysis; Bayesian networks; Decision theory; Empirical Bayes modelling; Empirical tests; Exact sciences and technology; F-test; Hierarchies; High dimensional data; High dimensional spaces; Hypotheses; Hypothesis; Hypothesis testing; Linear models; Locally most powerful test; Mathematics; Maximum likelihood estimators; Modelling; Multivariate analysis; Nonparametric inference; Null hypothesis; Parametric inference; Power; Power functions; Probability and statistics; Ratio test; Regression coefficients; Sciences and techniques of general use; Score test; Statistical methods; Statistical variance; Statistics; Studies
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2006.00551.x

As the dimensionality of the alternative hypothesis increases, the power of classical tests tends to diminish quite rapidly. This is especially true for high dimensional data in which there are more parameters than observations. We discuss a score test on a hyperparameter in an empirical Bayesian model as an alternative to classical tests. It gives a general test statistic which can be used to test a point null hypothesis against a high dimensional alternative, even when the number of parameters exceeds the number of samples. This test will be shown to have optimal power on average in a neighbourhood of the null hypothesis, which makes it a proper generalization of the locally most powerful test to multiple dimensions. To illustrate this new locally most powerful test we investigate the case of testing the global null hypothesis in a linear regression model in more detail. The score test is shown to have significantly more power than the F-test whenever under the alternative the large variance principal components of the design matrix explain substantially more of the variance of the outcome than do the small variance principal components. The score test is also useful for detecting sparse alternatives in truly high dimensional data, where its power is comparable with the test based on the maximum absolute t-statistic.