Summary. We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together.The elastic net is particularly useful when the number of predictors (p) is much bigger than the number of observations (n). By contrast, the lasso is not a very satisfactory variable selection method in the

/pdf/regularization-and-variable-selection-via-the-elastic-net-4z8q6j5o0i.pdf

Regularization and variable selection via the elastic net

Variable and feature selection have become the focus of much research in areas of application for which datasets with tens or hundreds of thousands of variables are available. These areas include text processing of internet documents, gene expression array analysis, and combinatorial chemistry. The objective of variable selection is three-fold: improving the prediction performance of the predictors, providing faster and more cost-effective predictors, and providing a better understanding of the underlying process that generated the data. The contributions of this special issue cover a wide range of aspects of such problems: providing a better definition of the objective function, feature construction, feature ranking, multivariate feature selection, efficient search methods, and feature validity assessment methods.

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An introduction to variable and feature selection

The problem of identifying differentially expressed genes in designed microarray experiments is considered. Lonnstedt and Speed (2002) derived an expression for the posterior odds of differential expression in a replicated two-color experiment using a simple hierarchical parametric model. The purpose of this paper is to develop the hierarchical model of Lonnstedt and Speed (2002) into a practical approach for general microarray experiments with arbitrary numbers of treatments and RNA samples. The model is reset in the context of general linear models with arbitrary coefficients and contrasts of interest. The approach applies equally well to both single channel and two color microarray experiments. Consistent, closed form estimators are derived for the hyperparameters in the model. The estimators proposed have robust behavior even for small numbers of arrays and allow for incomplete data arising from spot filtering or spot quality weights. The posterior odds statistic is reformulated in terms of a moderated t-statistic in which posterior residual standard deviations are used in place of ordinary standard deviations. The empirical Bayes approach is equivalent to shrinkage of the estimated sample variances towards a pooled estimate, resulting in far more stable inference when the number of arrays is small. The use of moderated t-statistics has the advantage over the posterior odds that the number of hyperparameters which need to estimated is reduced; in particular, knowledge of the non-null prior for the fold changes are not required. The moderated t-statistic is shown to follow a t-distribution with augmented degrees of freedom. The moderated t inferential approach extends to accommodate tests of composite null hypotheses through the use of moderated F-statistics. The performance of the methods is demonstrated in a simulation study. Results are presented for two publicly available data sets.

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Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments

Statistical method for testing the neutral mutation hypothesis by DNA polymorphism.

The purpose of this study was to classify breast carcinomas based on variations in gene expression patterns derived from cDNA microarrays and to correlate tumor characteristics to clinical outcome. A total of 85 cDNA microarray experiments representing 78 cancers, three fibroadenomas, and four normal breast tissues were analyzed by hierarchical clustering. As reported previously, the cancers could be classified into a basal epithelial-like group, an ERBB2-overexpressing group and a normal breast-like group based on variations in gene expression. A novel finding was that the previously characterized luminal epithelial/estrogen receptor-positive group could be divided into at least two subgroups, each with a distinctive expression profile. These subtypes proved to be reasonably robust by clustering using two different gene sets: first, a set of 456 cDNA clones previously selected to reflect intrinsic properties of the tumors and, second, a gene set that highly correlated with patient outcome. Survival analyses on a subcohort of patients with locally advanced breast cancer uniformly treated in a prospective study showed significantly different outcomes for the patients belonging to the various groups, including a poor prognosis for the basal-like subtype and a significant difference in outcome for the two estrogen receptor-positive groups.

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Gene expression patterns of breast carcinomas distinguish tumor subclasses with clinical implications

Microarrays can measure the expression of thousands of genes to identify changes in expression between different biological states. Methods are needed to determine the significance of these changes while accounting for the enormous number of genes. We describe a method, Significance Analysis of Microarrays (SAM), that assigns a score to each gene on the basis of change in gene expression relative to the standard deviation of repeated measurements. For genes with scores greater than an adjustable threshold, SAM uses permutations of the repeated measurements to estimate the percentage of genes identified by chance, the false discovery rate (FDR). When the transcriptional response of human cells to ionizing radiation was measured by microarrays, SAM identified 34 genes that changed at least 1.5-fold with an estimated FDR of 12%, compared with FDRs of 60 and 84% by using conventional methods of analysis. Of the 34 genes, 19 were involved in cell cycle regulation and 3 in apoptosis. Surprisingly, four nucleotide excision repair genes were induced, suggesting that this repair pathway for UV-damaged DNA might play a previously unrecognized role in repairing DNA damaged by ionizing radiation.

/pdf/significance-analysis-of-microarrays-applied-to-the-ionizing-10j4beasgw.pdf

Significance analysis of microarrays applied to the ionizing radiation response

Microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels. A typical microarray experiment can produce millions of data points, raising serious problems of data reduction, and simultaneous inference. We consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes. A simple nonparametric empirical Bayes model is introduced, which is used to guide the efficient reduction of the data to a single summary statistic per gene, and also to make simultaneous inferences concerning which genes were affected by the radiation. Although our focus is on one specific experiment, the proposed methods can be applied quite generally. The empirical Bayes inferences are closely related to the frequentist false discovery rate (FDR) criterion.

/pdf/empirical-bayes-analysis-of-a-microarray-experiment-2gyv398qm5.pdf

Empirical Bayes analysis of a microarray experiment

Microarrays can measure the expression of thousands of genes and thus identify changes in expression between different biological states. Methods are needed to determine the significance of these changes, while accounting for the enormous number of genes. We describe a new method, Significance Analysis of Microarrays (SAM), that assigns a score to each gene based on the change in gene expression relative to the standard deviation of repeated measurements. For genes with scores greater than an adjustable threshold, SAM uses permutations of the repeated measurements to estimate the percentage of such genes identified by chance, the false discovery rate (FDR). When the transcriptional response of human cells to ionizing radiation was measured by microarrays, SAM identified 34 genes that changed at least 1.5-fold with an estimated FDR of 12%, compared to FDRs of 60% and 84% using conventional methods of analysis. Of the 34 genes, 19 were involved in cell cycle regulation, and 3 in apoptosis. Surprisingly, 4 nucleotide excision repair genes were induced, suggesting that this repair pathway for UV-damaged DNA might play a heretofore unrecognized role in repairing DNA damaged by ionizing radiation.

Significance analysis of microarrays

Toxicity from radiation therapy is a grave problem for cancer patients. We hypothesized that some cases of toxicity are associated with abnormal transcriptional responses to radiation. We used microarrays to measure responses to ionizing and UV radiation in lymphoblastoid cells derived from 14 patients with acute radiation toxicity. The analysis used heterogeneity-associated transformation of the data to account for a clinical outcome arising from more than one underlying cause. To compute the risk of toxicity for each patient, we applied nearest shrunken centroids, a method that identifies and cross-validates predictive genes. Transcriptional responses in 24 genes predicted radiation toxicity in 9 of 14 patients with no false positives among 43 controls (P = 2.2 × 10-7). The responses of these nine patients displayed significant heterogeneity. Of the five patients with toxicity and normal responses, two were treated with protocols that proved to be highly toxic. These results may enable physicians to predict toxicity and tailor treatment for individual patients.

https://www.pnas.org/content/pnas/101/17/6635.full.pdf

Toxicity from radiation therapy associated with abnormal transcriptional responses to DNA damage

ATM, the gene that is mutated in ataxia-telangiectasia, is associated with cerebellar degeneration, abnormal proliferation of small blood vessels, and cancer. These clinically important manifestations have stimulated interest in defining the sequence variation in the ATM gene. Therefore, we undertook a comprehensive survey of sequence variation in ATM in diverse human populations. The protein-encoding exons of the gene (9,168 bp) and the adjacent intron and untranslated sequences (14,661 bp) were analyzed in 93 individuals from seven major human populations. In addition, the coding sequence was analyzed in one chimpanzee, one gorilla, one orangutan, and one Old World monkey. In human ATM, 88 variant sites were discovered by denaturing high-performance liquid chromatography, which is 96%–100% sensitive for detection of DNA sequence variation. ATM was compared to 14 other autosomal genes for nucleotide diversity. The noncoding regions of ATM had diversity values comparable to other genes, but the coding regions had very low diversity, especially in the last 29% of the protein sequence. A test of the neutral evolution hypothesis, through use of the Hudson/Kreitman/Aguade statistic, revealed that this region of the human ATM gene was significantly constrained relative to that of the orangutan, the Old World monkey, and the mouse, but not relative to that of the chimpanzee or the gorilla. ATM displayed extensive linkage disequilibrium, consistent with suppression of meiotic recombination at this locus. Seven haplotypes were defined. Two haplotypes accounted for 82% of all chromosomes analyzed in all major populations; two others carrying the same D126E missense polymorphism accounted for 33% of chromosomes in Africa but were never observed outside of Africa. The high frequency of this polymorphism may be due either to a population expansion within Africa or to selective pressure.

Virginia Goss Tusher

Papers

Significance analysis of microarrays applied to the ionizing radiation response

Empirical Bayes analysis of a microarray experiment

Significance analysis of microarrays

Toxicity from radiation therapy associated with abnormal transcriptional responses to DNA damage

Global analysis of ATM polymorphism reveals significant functional constraint