Juliane Schäfer

Inferring large-scale covariance matrices from sparse genomic data is an ubiquitous problem in bioinformatics. Clearly, the widely used standard covariance and correlation estimators are ill-suited for this purpose. As statistically efficient and computationally fast alternative we propose a novel shrinkage covariance estimator that exploits the Ledoit-Wolf (2003) lemma for analytic calculation of the optimal shrinkage intensity. Subsequently, we apply this improved covariance estimator (which has guaranteed minimum mean squared error, is well-conditioned, and is always positive definite even for small sample sizes) to the problem of inferring large-scale gene association networks. We show that it performs very favorably compared to competing approaches both in simulations as well as in application to real expression data.

MOTIVATION: Genetic networks are often described statistically using graphical models (e.g. Bayesian networks). However, inferring the network structure offers a serious challenge in microarray analysis where the sample size is small compared to the number of considered genes. This renders many standard algorithms for graphical models inapplicable, and inferring genetic networks an 'ill-posed' inverse problem. METHODS: We introduce a novel framework for small-sample inference of graphical models from gene expression data. Specifically, we focus on the so-called graphical Gaussian models (GGMs) that are now frequently used to describe gene association networks and to detect conditionally dependent genes. Our new approach is based on (1) improved (regularized) small-sample point estimates of partial correlation, (2) an exact test of edge inclusion with adaptive estimation of the degree of freedom and (3) a heuristic network search based on false discovery rate multiple testing. Steps (2) and (3) correspond to an empirical Bayes estimate of the network topology. RESULTS: Using computer simulations, we investigate the sensitivity (power) and specificity (true negative rate) of the proposed framework to estimate GGMs from microarray data. This shows that it is possible to recover the true network topology with high accuracy even for small-sample datasets. Subsequently, we analyze gene expression data from a breast cancer tumor study and illustrate our approach by inferring a corresponding large-scale gene association network for 3883 genes.

BACKGROUND: Graphical Gaussian models are popular tools for the estimation of (undirected) gene association networks from microarray data. A key issue when the number of variables greatly exceeds the number of samples is the estimation of the matrix of partial correlations. Since the (Moore-Penrose) inverse of the sample covariance matrix leads to poor estimates in this scenario, standard methods are inappropriate and adequate regularization techniques are needed. Popular approaches include biased estimates of the covariance matrix and high-dimensional regression schemes, such as the Lasso and Partial Least Squares. RESULTS: In this article, we investigate a general framework for combining regularized regression methods with the estimation of Graphical Gaussian models. This framework includes various existing methods as well as two new approaches based on ridge regression and adaptive lasso, respectively. These methods are extensively compared both qualitatively and quantitatively within a simulation study and through an application to six diverse real data sets. In addition, all proposed algorithms are implemented in the R package "parcor", available from the R repository CRAN. CONCLUSION: In our simulation studies, the investigated non-sparse regression methods, i.e. Ridge Regression and Partial Least Squares, exhibit rather conservative behavior when combined with (local) false discovery rate multiple testing in order to decide whether or not an edge is present in the network. For networks with higher densities, the difference in performance of the methods decreases. For sparse networks, we confirm the Lasso's well known tendency towards selecting too many edges, whereas the two-stage adaptive Lasso is an interesting alternative that provides sparser solutions. In our simulations, both sparse and non-sparse methods are able to reconstruct networks with cluster structures. On six real data sets, we also clearly distinguish the results obtained using the non-sparse methods and those obtained using the sparse methods where specification of the regularization parameter automatically means model selection. In five out of six data sets, Partial Least Squares selects very dense networks. Furthermore, for data that violate the assumption of uncorrelated observations (due to replications), the Lasso and the adaptive Lasso yield very complex structures, indicating that they might not be suited under these conditions. The shrinkage approach is more stable than the regression based approaches when using subsampling.

In many studies, particularly in the field of systems biology, it is essential that identical protein sets are precisely quantified in multiple samples such as those representing differentially perturbed cell states. The high degree of reproducibility required for such experiments has not been achieved by classical mass spectrometry-based proteomics methods. In this study we describe the implementation of a targeted quantitative approach by which predetermined protein sets are first identified and subsequently quantified at high sensitivity reliably in multiple samples. This approach consists of three steps. First, the proteome is extensively mapped out by multidimensional fractionation and tandem mass spectrometry, and the data generated are assembled in the PeptideAtlas database. Second, based on this proteome map, peptides uniquely identifying the proteins of interest, proteotypic peptides, are selected, and multiple reaction monitoring (MRM) transitions are established and validated by MS2 spectrum acquisition. This process of peptide selection, transition selection, and validation is supported by a suite of software tools, TIQAM (Targeted Identification for Quantitative Analysis by MRM), described in this study. Third, the selected target protein set is quantified in multiple samples by MRM. Applying this approach we were able to reliably quantify low abundance virulence factors from cultures of the human pathogen Streptococcus pyogenes exposed to increasing amounts of plasma. The resulting quantitative protein patterns enabled us to clearly define the subset of virulence proteins that is regulated upon plasma exposure. In many studies, particularly in the field of systems biology, it is essential that identical protein sets are precisely quantified in multiple samples such as those representing differentially perturbed cell states. The high degree of reproducibility required for such experiments has not been achieved by classical mass spectrometry-based proteomics methods. In this study we describe the implementation of a targeted quantitative approach by which predetermined protein sets are first identified and subsequently quantified at high sensitivity reliably in multiple samples. This approach consists of three steps. First, the proteome is extensively mapped out by multidimensional fractionation and tandem mass spectrometry, and the data generated are assembled in the PeptideAtlas database. Second, based on this proteome map, peptides uniquely identifying the proteins of interest, proteotypic peptides, are selected, and multiple reaction monitoring (MRM) transitions are established and validated by MS2 spectrum acquisition. This process of peptide selection, transition selection, and validation is supported by a suite of software tools, TIQAM (Targeted Identification for Quantitative Analysis by MRM), described in this study. Third, the selected target protein set is quantified in multiple samples by MRM. Applying this approach we were able to reliably quantify low abundance virulence factors from cultures of the human pathogen Streptococcus pyogenes exposed to increasing amounts of plasma. The resulting quantitative protein patterns enabled us to clearly define the subset of virulence proteins that is regulated upon plasma exposure. A key element of the experimental framework for systems biology is the comprehensive, quantitative measurement of whole biological systems in differentially perturbed states (1Ideker T. Thorsson V. Ranish J.A. Christmas R. Buhler J. Eng J.K. Bumgarner R. Goodlett D.R. Aebersold R. Hood L. Integrated genomic and proteomic analyses of a systematically perturbed metabolic network.Science. 2001; 292: 929-934Crossref PubMed Scopus (1646) Google Scholar). Among the different types of measurements possible, protein quantification is particularly informative because proteins catalyze or control the majority of cellular functions. Currently the most widely applied quantitative proteome analysis technologies consist of the labeling of the samples by stable isotopes, the reproducible separation of complex peptide mixtures, usually by capillary LC, and the identification and quantification of selected peptides by tandem mass spectrometry and sequence database searching (2Ong S.E. Mann M. Mass spectrometry-based proteomics turns quantitative.Nat. Chem. Biol. 2005; 1: 252-262Crossref PubMed Scopus (1309) Google Scholar, 3Aebersold R. Mann M. Mass spectrometry-based proteomics.Nature. 2003; 422: 198-207Crossref PubMed Scopus (5562) Google Scholar). Relative quantitative values are generated by these methods if two or more samples are being compared, and absolute quantification can be achieved if suitable, calibrated reference samples are available (4Gerber S.A. Rush J. Stemman O. Kirschner M.W. Gygi S.P. Absolute quantification of proteins and phosphoproteins from cell lysates by tandem MS.Proc. Natl. Acad. Sci. U. S. A. 2003; 100: 6940-6945Crossref PubMed Scopus (1537) Google Scholar). Using such shotgun methods, in each measurement only a fraction of the analytes present in a complex sample is identified and quantified. Peptide ions are selected by the mass spectrometer automatically based on precursor ion signal intensities. Due to a multitude of factors, including interference between analytes and variations in precursor ion spectra, the selection of peptides is not reproducible in consecutive runs in particular for peptides of lower signal intensities. As a critical consequence of this undersampling effect comprehensive analyses of whole systems are not supported by these technologies rendering them poorly suitable for systems biology and other experiments that depend on the comparison of complete or at least reproducible data sets. To overcome these fundamental technical limitations confronting proteomics we have suggested in the past a substantially different approach that emulates successful genomics strategies (5Aebersold R. Constellations in a cellular universe.Nature. 2003; 422: 115-116Crossref PubMed Scopus (78) Google Scholar, 6Domon B. Aebersold R. Mass spectrometry and protein analysis.Science. 2006; 312: 212-217Crossref PubMed Scopus (1593) Google Scholar, 7Kuster B. Schirle M. Mallick P. Aebersold R. Scoring proteomes with proteotypic peptide probes.Nat. Rev. Mol. Cell Biol. 2005; 6: 577-583Crossref PubMed Scopus (303) Google Scholar). It depends on the generation of deep, ideally complete proteome maps followed by the targeted analysis of peptides that collectively represent the proteins that constitute the system under investigation. We have termed peptides that are typically observed in a mass spectrometer and that uniquely a particular protein proteotypic protein multiple reaction Identification for Quantitative Analysis by MRM. proteotypic protein multiple reaction Identification for Quantitative Analysis by MRM. P. Schirle M. Ranish J. B. R. T. B. Aebersold R. of proteotypic peptides for quantitative PubMed Scopus Google Scholar). has been achieved with of this First, we have the PeptideAtlas Mallick P. Eng J. S. J. Aebersold R. The PeptideAtlas 2006; PubMed Scopus Google a of high peptide and software types of including the identification of present data sets for the Ranish J.A. Mallick P. Eng J. M. B. B. Aebersold R. Analysis of the proteome with Biol. 2006; PubMed Scopus Google Mallick P. Eng J.K. A. R. S. Hood L. B. Ranish J.A. B. L. Aebersold R. with the human of peptide by mass Biol. 6: PubMed Google human Mallick P. Eng J.K. A. R. S. Hood L. B. Ranish J.A. B. L. Aebersold R. with the human of peptide by mass Biol. 6: PubMed Google and human plasma Eng J.K. B. R. Aebersold R. 2005; PubMed Scopus Google have been in the Second, multiple reaction monitoring (MRM) has as the of for the targeted and quantification of peptides and in complex B. Aebersold R. and in proteomics data 2006; PubMed Scopus Google Scholar, A. S. reaction monitoring for quantitative proteomic analysis of cellular Natl. Acad. Sci. U. S. A. PubMed Scopus Google Scholar, J. V. R. Aebersold R. B. sensitivity of plasma proteins by multiple reaction monitoring of 6: PubMed Scopus Google Scholar, J. of for quantification of proteins by mass Chem. 2006; PubMed Scopus Google Scholar). sensitivity and a have been achieved mass in The of first at the of the precursor ions and at the of the the increasing the the of transitions and the of peptides in a has been by the of for the transitions J. V. R. Aebersold R. B. sensitivity of plasma proteins by multiple reaction monitoring of 6: PubMed Scopus Google Scholar). these ideally for the quantification of low abundance proteins in complex samples. these technical limitations at present the of the targeted proteomics described are the of target selection and the of the measurement it is critical that for each protein in the targeted protein set the peptides are and that for these peptides the transitions are identified and To this we a suite of software termed TIQAM (Targeted Identification for Quantitative Analysis by with a protein set in the system the PeptideAtlas and other of to the and a of are for experiments MS2 acquisition. 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In non-inferiority trials, where non-inferiority of a new experimental drug compared to an active control has to be shown, it may be advisable to use an additional placebo group for internal validation if ethically justifiable. The focus of this paper is on such designs. Assuming normality and homogeneity of variances we will derive a statistical test procedure which turns out to be equivalent to the assessment based on Fieller's confidence interval. Based on the power function of this test, sample size calculations are carried out to achieve a given power. Additionally, the optimal allocation of the total sample size is derived. As an alternative to this parametric procedure, the bootstrap percentile interval is discussed and finally compared with Fieller's confidence interval in a study on mildly asthmatic patients.