We develop efficient and powerful statistical methods for high-dimensional data, where the sample size is much smaller than the number of features (the so-called 'large p, small n' problem). We deal with three important problems. First, we develop a mixture-model approach for parallel testing for unequal variances in two-sample experiments. The treatment effect on the variance has received little attention in the statistical literature, which so far focused mostly on the effect on the mean. The effect on the variance is increasingly recognized in recent biological literature, and we develop an empirical Bayes approach for testing differences in variance when the number of tests is large. We show that the model is useful in a wide range of applications, that our method is much more powerful than traditional tests for unequal variances, and that it is robust to the normality assumption. Second, we extend these ideas and develop a novel bivariate normal model that tests for both differential expression and differential variation between the two groups. We show in simulations that this new method yields a substantial gain in power when differential variation is present. Through a three-step estimation approach, in which we apply the Laplace approximation and the EM algorithm, we get a computationally efficient method, which is particularly well-suited for 'large p, small n' situations. Third, we deal with the problem of variable selection where the number of putative variables is large, possibly much larger than the sample size. We develop a model-based, empirical Bayes approach. By treating the putative variables as random effects, we get shrinkage estimation, which results in increased power and significantly faster convergence, compared with simulation-based methods. Furthermore, we employ computational tricks which allow us to increase the speed of our algorithm, to handle a very large number of putative variables, and to control the multicollinearity in the model. The motivation for developing this approach is QTL analysis, but our method is applicable to a broad range of applications. We use two widely-studied data sets, and show that our model selection algorithm yields excellent results.
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