Alexander Y. Gordon

Control of the mean number of false discoveries, Bonferroni and stability of multiple testing

The Bonferroni multiple testing procedure is commonly perceived as being overly conservative in large-scale simultaneous testing situations such as those that arise in microarray data analysis. The objective of the present study is to show that this popular belief is due to overly stringent requirements that are typically imposed on the procedure rather than to its conservative nature. To get over its notorious conservatism, we advocate using the Bonferroni selection rule as a procedure that controls the per family error rate (PFER). The present paper reports the first study of stability properties of the Bonferroni and Benjamini--Hochberg procedures. The Bonferroni procedure shows a superior stability in terms of the variance of both the number of true discoveries and the total number of discoveries, a property that is especially important in the presence of correlations between individual

Assessing stability of gene selection in microarray data analysis.

p

Multivariate search for differentially expressed gene combinations

-values. Its stability and the ability to provide strong control of the PFER make the Bonferroni procedure an attractive choice in microarray studies.

The choice of optimal distance measure in genome-wide datasets

BackgroundThe number of genes declared differentially expressed is a random variable and its variability can be assessed by resampling techniques. Another important stability indicator is the frequency with which a given gene is selected across subsamples. We have conducted studies to assess stability and some other properties of several gene selection procedures with biological and simulated data.ResultsUsing resampling techniques we have found that some genes are selected much less frequently (across sub-samples) than other genes with the same adjusted p-values. The extent to which this type of instability manifests itself can be assessed by a method introduced in this paper. The effect of correlation between gene expression levels on the performance of multiple testing procedures is studied by computer simulations.ConclusionResampling represents a tool for reducing the set of initially selected genes to those with a sufficiently high selection frequency. Using resampling techniques it is also possible to assess variability of different performance indicators. Stability properties of several multiple testing procedures are described at length in the present paper.

The L 1 -version of the Cramér-von mises test for two-sample comparisons in microarray data analysis

BackgroundTo identify differentially expressed genes, it is standard practice to test a two-sample hypothesis for each gene with a proper adjustment for multiple testing. Such tests are essentially univariate and disregard the multidimensional structure of microarray data. A more general two-sample hypothesis is formulated in terms of the joint distribution of any sub-vector of expression signals.ResultsBy building on an earlier proposed multivariate test statistic, we propose a new algorithm for identifying differentially expressed gene combinations. The algorithm includes an improved random search procedure designed to generate candidate gene combinations of a given size. Cross-validation is used to provide replication stability of the search procedure. A permutation two-sample test is used for significance testing. We design a multiple testing procedure to control the family-wise error rate (FWER) when selecting significant combinations of genes that result from a successive selection procedure. A target set of genes is composed of all significant combinations selected via random search.ConclusionsA new algorithm has been developed to identify differentially expressed gene combinations. The performance of the proposed search-and-testing procedure has been evaluated by computer simulations and analysis of replicated Affymetrix gene array data on age-related changes in gene expression in the inner ear of CBA mice.

MEASURABLE ENUMERATION OF EIGENELEMENTS

MOTIVATION Many types of genomic data are naturally represented as binary vectors. Numerous tasks in computational biology can be cast as analysis of relationships between these vectors, and the first step is, frequently, to compute their pairwise distance matrix. Many distance measures have been proposed in the literature, but there is no theory justifying the choice of distance measure. RESULTS We examine the approaches to measuring distances between binary vectors and study the characteristic properties of various distance measures and their performance in several tasks of genome analysis. Most distance measures between binary vectors turn out to belong to a single parametric family, namely generalized average-based distance with different exponents. We show that descriptive statistics of distance distribution, such as skewness and kurtosis, can guide the appropriate choice of the exponent. On the contrary, the more familiar distance properties, such as metric and additivity, appear to have much less effect on the performance of distances. AVAILABILITY R code GADIST and Supplementary material are available at http://research.stowers-institute.org/bioinfo/

Imperfectly grown periodic medium: absence of localized states

Distribution-free statistical tests offer clear advantages in situations where the exact unadjusted -values are required as input for multiple testing procedures. Such situations prevail when testing for differential expression of genes in microarray studies. The Cramér-von Mises two-sample test, based on a certain -distance between two empirical distribution functions, is a distribution-free test that has proven itself as a good choice. A numerical algorithm is available for computing quantiles of the sampling distribution of the Cramér-von Mises test statistic in finite samples. However, the computation is very time- and space-consuming. An counterpart of the Cramér-von Mises test represents an appealing alternative. In this work, we present an efficient algorithm for computing exact quantiles of the -distance test statistic. The performance and power of the -distance test are compared with those of the Cramér-von Mises and two other classical tests, using both simulated data and a large set of microarray data on childhood leukemia. The -distance test appears to be nearly as powerful as its counterpart. The lower computational intensity of the -distance test allows computation of exact quantiles of the null distribution for larger sample sizes than is possible for the Cramér-von Mises test.

A spectral alternative for continuous families of self-adjoint operators

We prove that for eigenelements of a measurable family of linear self-adjoint operators in a separable Hilbert space there exists a measurable enumeration. We also prove a similar result for measurable families of bounded linear operators having at most countably many eigenvalues (under certain restrictions on the parameter space). The proof of the latter result is based on descriptive set theory, while in the case of self-adjoint (and some more general) operators the proof is constructive.

On a paradoxical property of the Kolmogorov–Smirnov two-sample test

We consider a discrete model of the d-dimensional medium with Hamiltonian ∆ + v; the lattice potential v is constructed recursively on a nested sequence of cubes Qn obtained by successive inflations with integer coefficients. Initially, the potential is defined on the cube Q0. At the nth step the potential, which is already constructed on the cube Qn−1, gets extended Qn−1-periodically to the cube Qn; then its values at mn randomly chosen points of Qn are arbitrarily changed. This alternating process of periodic extension and introduction of impurities goes on, resulting in an (in general, unbounded) potential v. We show that if the size of the cube Qn grows fast enough with n while the sequence mn grows not too fast, then the Schrodinger operator ∆ + v almost surely does not have eigenvalues.

Occupancy Numbers in Testing Random Number Generators

We consider a continuous family of self-adjoint operators As in a separable Hilbert space, the parameter s being a point of a complete metric space S. It is well known that isolated simple eigenvalues (assuming that the operators are bounded and the mapping s 7→ As is continuous in the norm sense) behave “well”: under small changes of the parameter they do not disappear and change continuously. Unlike this, the eigenvalues embedded in the essential spectrum can display a “very bad” behavior. It turns out, nevertheless, that if the set of eigenvalues is non-empty for a topologically rich (e.g., open) set of values of the parameter, then the (multi-valued) eigenvalue function has continuous branches. One application is as follows. Suppose a one-dimensional quasiperiodic Schrodinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues. 2010 Mathematics Subject Classification: 47B25