This paper is motivated by the comparison of genetic networks based on microarray samples. The aim is to test whether the differences observed between two inferred Gaussian graphical models come from real differences or arise from estimation uncertainties. Adopting a neighborhood approach, we consider a two-sample linear regression model with random design and propose a procedure to test whether these two regressions are the same. Relying on multiple testing and variable selection strategies, we develop a testing procedure that applies to high-dimensional settings where the number of covariates @math is larger than the number of observations @math and @math of the two samples. Both type I and type II errors are explicitely controlled from a non-asymptotic perspective and the test is proved to be minimax adaptive to the sparsity. The performances of the test are evaluated on simulated data. Moreover, we illustrate how this procedure can be used to compare genetic networks on Hess breast cancer microarray dataset.
We propose a new test, based on model selection methods, for testing that the expectation of a Gaussian vector with n independent components belongs to a linear subspace of R n against a nonparametric alternative. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are nonasymptotic and we prove that the test is rate optimal [up to a possible log(n) factor] over various classes of alternatives simultaneously. We also provide a simulation study in order to evaluate the procedure when the purpose is to test goodness-of-fit in a regression model. 1. Introduction. We consider the regression model Let @math be a zero mean Gaussian vector and @math be a subset of @math . Suppose we are given @math i.i.d. replications of the vector @math . We propose a new test for testing that @math is independent of @math conditionally to @math against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of @math or the variance of @math and applies in a high-dimensional setting. It straightforwardly extends to test the neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic properties of the test and we prove that it is rate optimal (up to a possible @math factor) over various classes of alternatives under some additional assumptions. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure. We propose novel methodology for testing equality of model parameters between two high-dimensional populations. The technique is very general and applicable to a wide range of models. The method is based on sample splitting: the data is split into two parts; on the first part we reduce the dimensionality of the model to a manageable size; on the second part we perform significance testing (p-value calculation) based on a restricted likelihood ratio statistic. Assuming that both populations arise from the same distribution, we show that the restricted likelihood ratio statistic is asymptotically distributed as a weighted sum of chi-squares with weights which can be efficiently estimated from the data. In high-dimensional problems, a single data split can result in a "p-value lottery". To ameliorate this effect, we iterate the splitting process and aggregate the resulting p-values. This multi-split approach provides improved p-values. We illustrate the use of our general approach in two-sample comparisons of high-dimensional regression models ("differential regression") and graphical models ("differential network"). In both cases we show results on simulated data as well as real data from recent, high-throughput cancer studies. 