Estimation and resampling methods for judging the quality of multiple tests for high dimensional data

内容当今，生命科学中的现代技术产生了高维数据，其中数据的维度可以远远大于样本量。我们指的是生物学和医学中的基因组研究。在这类应用中，隐藏效果(标记为信号)通常很少出现。这就是统计中的稀疏性。在基因组研究方面，往往只出现少数几个基因组危险部位。为了找出这些类型的影响，统计学家使用了多种测试。它们的质量通过错误发现率(FDR)来判断。FDR是错误拒绝假设数除以所有拒绝数的期望值。最著名的测试是1995年发表的本贾米尼/霍奇伯格多重测试。在我们的上一个DFG项目中，我们研究了多个测试问题，其中提出了扩展的自适应解决方案。目前正在进行的研究方案的主题是多项测试的质量研究。它基于未知有效FDR的估计和重采样过程。我们考虑到多次测试和风险估计的解往往有很大的方差。实际上，稀疏效果隐藏在高维噪声中。我们首先研究了有效FDR的各种估计量，它们是对文献中给出的估计量的微小修改。在稀疏性条件下，它们的可变性也将由非参数置信度控制。为此，我们将通过重采样方法来估计估计量的分布。为此，我们提出了一种改进的Bootstrap过程，它与所谓的低重采样Bootstrap相关。我们研究小组的早期出版物可以用来回答这个问题。然而，由于信号的稀疏性问题，必须开发新的方法，因为普通的Efron自举可能会失败。我们喜欢研究我们的估计量对于不同的多重检验的一致性。然后检查我们重采样程序的质量。该项目将伴随着大规模的蒙特卡洛模拟。此外，我们还愿意继续与给出重要应用提示的分子生物学和生物识别科学的同行们合作。

Contents Nowadays, modern technologies in life sciences generate high dimensional data where the dimension of the data can be much larger than the sample size. We refer to genome studies in biology and medicine. Within these kind of applications hidden effects (labeled as signals) are typically rarely present. This is called the sparsity in statistics. In connection with genome studies often only a few genome risk positions occur. In order to find these type of effects the statisticians use multiple tests. Their quality is judged by the false discovery rate (FDR). The FDR is the expectation of the rate of the number of false rejected hypotheses divided by the number of all rejections. The most famous test is the Benjamini/Hochberg multiple test published in 1995. In our last DFG-project we studied multiple testing problems, where extended adaptive solutions were proposed. The topic of the present proceeding research proposal is a quality study of multiple tests. It is based on estimation and resampling procedures of the unknown effective FDR. We take into account that the solutions of multiple tests and risk estimation often have a large variance. In practice sparse effects are hidden within a high dimensional noise. We first study various estimators for the effective FDR which are slight modifications of estimators given in the literature. Their variability will be controlled by nonparametric confidence intervals also under sparsity. For this purpose, we will estimate the distribution of the estimators by resampling methods. To this end we propose a modified bootstrap procedure which is related to the so called low resampling bootstrap. Earlier publications of our research group can be used to attack this question. However, due to the sparsity problem given by the signals, new methods must be developed since the ordinary bootstrap of Efron may fail. We like to study the consistency of our estimators for different multiple tests. Then the quality of our resampling procedures is checked. The project will be accompanied by large Monte Carlo simulations. Moreover, we like to continue our cooperation with colleagues from molecular biology and biometrical sciences who gave important hints for applications.