Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks

We present a procedure for effective estimation of entropy and mutual information from small-sample data, and apply it to the problem of inferring high-dimensional gene association networks. Specifically, we develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically as well as computationally. Despite its simplicity, we show that it outperforms eight other entropy estimation procedures across a diverse range of sampling scenarios and data-generating models, even in cases of severe undersampling. We illustrate the approach by analyzing E. coli gene expression data and computing an entropy-based gene-association network from gene expression data. A computer program is available that implements the proposed shrinkage estimator.Comment: 18 pages, 3 figures, 1 tabl

Gene ranking and biomarker discovery under correlation

Biomarker discovery and gene ranking is a standard task in genomic high throughput analysis. Typically, the ordering of markers is based on a stabilized variant of the t-score, such as the moderated t or the SAM statistic. However, these procedures ignore gene-gene correlations, which may have a profound impact on the gene orderings and on the power of the subsequent tests. We propose a simple procedure that adjusts gene-wise t-statistics to take account of correlations among genes. The resulting correlation-adjusted t-scores ("cat" scores) are derived from a predictive perspective, i.e. as a score for variable selection to discriminate group membership in two-class linear discriminant analysis. In the absence of correlation the cat score reduces to the standard t-score. Moreover, using the cat score it is straightforward to evaluate groups of features (i.e. gene sets). For computation of the cat score from small sample data we propose a shrinkage procedure. In a comparative study comprising six different synthetic and empirical correlation structures we show that the cat score improves estimation of gene orderings and leads to higher power for fixed true discovery rate, and vice versa. Finally, we also illustrate the cat score by analyzing metabolomic data. The shrinkage cat score is implemented in the R package "st" available from URL http://cran.r-project.org/web/packages/st/Comment: 18 pages, 5 figures, 1 tabl

Partial Least Squares: A Versatile Tool for the Analysis of High-Dimensional Genomic Data

Partial Least Squares (PLS) is a highly efficient statistical regression technique that is well suited for the analysis of high-dimensional genomic data. In this paper we review the theory and applications of PLS both under methodological and biological points of view. Focusing on microarray expression data we provide a systematic comparison of the PLS approaches currently employed, and discuss problems as different as tumor classification, identification of relevant genes, survival analysis and modeling of gene networks

A Simple Data-Adaptive Probabilistic Variant Calling Model

Background: Several sources of noise obfuscate the identification of single nucleotide variation (SNV) in next generation sequencing data. For instance, errors may be introduced during library construction and sequencing steps. In addition, the reference genome and the algorithms used for the alignment of the reads are further critical factors determining the efficacy of variant calling methods. It is crucial to account for these factors in individual sequencing experiments. Results: We introduce a simple data-adaptive model for variant calling. This model automatically adjusts to specific factors such as alignment errors. To achieve this, several characteristics are sampled from sites with low mismatch rates, and these are used to estimate empirical log-likelihoods. These likelihoods are then combined to a score that typically gives rise to a mixture distribution. From these we determine a decision threshold to separate potentially variant sites from the noisy background. Conclusions: In simulations we show that our simple proposed model is competitive with frequently used much more complex SNV calling algorithms in terms of sensitivity and specificity. It performs specifically well in cases with low allele frequencies. The application to next-generation sequencing data reveals stark differences of the score distributions indicating a strong influence of data specific sources of noise. The proposed model is specifically designed to adjust to these differences.Comment: 19 pages, 6 figure

Modeling gene expression measurement error: a quasi-likelihood approach

BACKGROUND: Using suitable error models for gene expression measurements is essential in the statistical analysis of microarray data. However, the true probabilistic model underlying gene expression intensity readings is generally not known. Instead, in currently used approaches some simple parametric model is assumed (usually a transformed normal distribution) or the empirical distribution is estimated. However, both these strategies may not be optimal for gene expression data, as the non-parametric approach ignores known structural information whereas the fully parametric models run the risk of misspecification. A further related problem is the choice of a suitable scale for the model (e.g. observed vs. log-scale). RESULTS: Here a simple semi-parametric model for gene expression measurement error is presented. In this approach inference is based an approximate likelihood function (the extended quasi-likelihood). Only partial knowledge about the unknown true distribution is required to construct this function. In case of gene expression this information is available in the form of the postulated (e.g. quadratic) variance structure of the data. As the quasi-likelihood behaves (almost) like a proper likelihood, it allows for the estimation of calibration and variance parameters, and it is also straightforward to obtain corresponding approximate confidence intervals. Unlike most other frameworks, it also allows analysis on any preferred scale, i.e. both on the original linear scale as well as on a transformed scale. It can also be employed in regression approaches to model systematic (e.g. array or dye) effects. CONCLUSIONS: The quasi-likelihood framework provides a simple and versatile approach to analyze gene expression data that does not make any strong distributional assumptions about the underlying error model. For several simulated as well as real data sets it provides a better fit to the data than competing models. In an example it also improved the power of tests to identify differential expression

A unified approach to false discovery rate estimation

<p>Abstract</p> <p>Background</p> <p>False discovery rate (FDR) methods play an important role in analyzing high-dimensional data. There are two types of FDR, tail area-based FDR and local FDR, as well as numerous statistical algorithms for estimating or controlling FDR. These differ in terms of underlying test statistics and procedures employed for statistical learning.</p> <p>Results</p> <p>A unifying algorithm for simultaneous estimation of both local FDR and tail area-based FDR is presented that can be applied to a diverse range of test statistics, including <it>p</it>-values, correlations, <it>z</it>- and <it>t</it>-scores. This approach is semipararametric and is based on a modified Grenander density estimator. For test statistics other than <it>p</it>-values it allows for empirical null modeling, so that dependencies among tests can be taken into account. The inference of the underlying model employs truncated maximum-likelihood estimation, with the cut-off point chosen according to the false non-discovery rate.</p> <p>Conclusion</p> <p>The proposed procedure generalizes a number of more specialized algorithms and thus offers a common framework for FDR estimation consistent across test statistics and types of FDR. In comparative study the unified approach performs on par with the best competing yet more specialized alternatives. The algorithm is implemented in R in the "fdrtool" package, available under the GNU GPL from <url>http://strimmerlab.org/software/fdrtool/</url> and from the R package archive CRAN.</p

Over-optimism in bioinformatics: an illustration

In statistical bioinformatics research, different optimization mechanisms potentially lead to "over-optimism" in published papers. The present empirical study illustrates these mechanisms through a concrete example from an active research field. The investigated sources of over-optimism include the optimization of the data sets, of the settings, of the competing methods and, most importantly, of the method’s characteristics. We consider a "promising" new classification algorithm that turns out to yield disappointing results in terms of error rate, namely linear discriminant analysis incorporating prior knowledge on gene functional groups through an appropriate shrinkage of the within-group covariance matrix. We quantitatively demonstrate that this disappointing method can artificially seem superior to existing approaches if we "fish for significance”. We conclude that, if the improvement of a quantitative criterion such as the error rate is the main contribution of a paper, the superiority of new algorithms should be validated using "fresh" validation data sets

A nonparametric empirical Bayes framework for large-scale multiple testing

We propose a flexible and identifiable version of the two-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the non-null cases. We use a computationally efficient predictive recursion marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real-data examples demonstrate that, compared to existing approaches, PRtest's careful handling of the non-null density can give a much better fit in the tails of the mixture distribution which, in turn, can lead to more realistic conclusions.Comment: 18 pages, 4 figures, 3 table

A general modular framework for gene set enrichment analysis

<p>Abstract</p> <p>Background</p> <p>Analysis of microarray and other high-throughput data on the basis of gene sets, rather than individual genes, is becoming more important in genomic studies. Correspondingly, a large number of statistical approaches for detecting gene set enrichment have been proposed, but both the interrelations and the relative performance of the various methods are still very much unclear.</p> <p>Results</p> <p>We conduct an extensive survey of statistical approaches for gene set analysis and identify a common modular structure underlying most published methods. Based on this finding we propose a general framework for detecting gene set enrichment. This framework provides a meta-theory of gene set analysis that not only helps to gain a better understanding of the relative merits of each embedded approach but also facilitates a principled comparison and offers insights into the relative interplay of the methods.</p> <p>Conclusion</p> <p>We use this framework to conduct a computer simulation comparing 261 different variants of gene set enrichment procedures and to analyze two experimental data sets. Based on the results we offer recommendations for best practices regarding the choice of effective procedures for gene set enrichment analysis.</p