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

如今，现代生命科学技术产生了高维数据，其中数据的维数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。在这类应用程序中，隐藏效应（标记为信号）通常很少出现。这就是统计学中的稀疏性。在基因组研究中，通常只有少数基因组风险位点发生。为了找到这些类型的影响，统计学家使用多个测试。它们的质量是由错误发现率（FDR）来判断的。FDR是被拒绝的错误假设数的比率除以所有被拒绝的数的期望值。最著名的测试是1995年发表的Benjamini/Hochberg多重测试。在我们上一个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计划的主题是一个多重测试的质量研究。它基于未知有效FDR的估计和重采样过程。我们考虑到多重测试和风险估计的解通常有很大的方差。在实践中，稀疏效应隐藏在高维噪声中。我们首先研究了有效FDR的各种估计量，这些估计量是文献中给出的估计量的轻微修改。在稀疏性条件下，它们的可变性将由非参数置信区间控制。为此，我们将通过重抽样方法估计估计量的分布。为此，我们提出了一种改进的自举过程，它与所谓的低重采样自举有关。我们研究小组的早期出版物可以用来解决这个问题。然而，由于信号的稀疏性问题，由于普通的Efron自举可能失败，必须开发新的方法。我们喜欢研究我们的估计对于不同的多重测试的一致性。然后检查我们重采样程序的质量。该项目将伴随着大型蒙特卡罗模拟。此外，我们愿意继续与分子生物学和生物识别科学的同事合作，他们为应用提供了重要的提示。

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.