Proteins are essential components of all living organisms and play a critical role in cellular survival. They have a broad range of applications, from clinical treatments to material engineering. This versatility has spurred the development of protein design, with amino acid sequence design being a crucial step in the process. Recent advancements in deep generative models have shown promise for protein sequence design. However, the scarcity of functional protein sequence data for certain types can hinder the training of these models, which often require large datasets. To address this challenge, we propose a hierarchical model named ProteinRG that can generate functional protein sequences using relatively small datasets. ProteinRG begins by generating a representation of a protein sequence, leveraging existing large protein sequence models, before producing a functional protein sequence. We have tested our model on various functional protein sequences and evaluated the results from three perspectives: multiple sequence alignment, t-SNE distribution analysis, and 3D structure prediction. The findings indicate that our generated protein sequences maintain both similarity to the original sequences and consistency with the desired functions. Moreover, our model demonstrates superior performance compared twith other generative models for protein sequence generation.

We consider the problem of density-ratio estimation using Bregman Divergence with Deep ReLU feedforward neural networks (BDD). We establish non-asymptotic error bounds for BDD density-ratio estimators, which are minimax optimal up to a logarithmic factor when the data distribution has finite support. As an application of our theoretical findings, we propose an estimator for the KL-divergence that is asymptotically normal, leveraging our convergence results for the deep density-ratio estimator and a data-splitting method. We also extend our results to cases with unbounded support and unbounded density ratios. Furthermore, we show that the BDD density-ratio estimator can mitigate the curse of dimensionality when data distributions are supported on an approximately low-dimensional manifold. Our results are applied to investigate the convergence properties of the telescoping density-ratio estimator proposed by Rhodes (2020). We provide sufficient conditions under which it achieves a lower error bound than a single-ratio estimator. Moreover, we conduct simulation studies to validate our main theoretical results and assess the performance of the BDD density-ratio estimator.

Sampling from probability distributions is a critical problem in statistics and machine learning, particularly in Bayesian inference, where direct integration over the posterior distribution is often infeasible, making sampling from the posterior essential for inference. This paper introduces the Schrödinger-Föllmer sampler (SFS), a novel approach for sampling from potentially unnormalized distributions. The SFS leverages the Schrödinger-Föllmer diffusion process on the unit interval, incorporating a time-dependent drift term that evolves the distribution from a degenerate form at time zero to the target distribution at time one. Unlike existing Markov chain Monte Carlo methods that rely on ergodicity, SFS operates independently of ergodicity. Computationally, SFS is straightforward to implement using the Euler-Maruyama discretization. In our theoretical analysis, we derive non-asymptotic error bounds for the SFS sampling distribution in the Wasserstein distance, subject to reasonable conditions. Numerical experiments demonstrate that SFS generates higher-quality samples than several established methods.

We consider parsimonious modeling of high-dimensional multivariate additive models using regression splines, with or without sparsity assumptions. The approach is based on treating the coefficients in the spline expansions as a third-order tensor. Note the data does not have tensor predictors or tensor responses, which distinguishes our study from the existing ones. A Tucker decomposition is used to reduce the number of parameters in the tensor. We also combined the Tucker decomposition with penalization to enable variable selection. The proposed method can avoid the statistical inefficiency caused by estimating a large number of nonparametric functions. We provide sufficient conditions under which the proposed tensor-based estimators achieve the optimal rate of convergence for the nonparametric regression components. We conduct simulation studies to demonstrate the effectiveness of the proposed novel approach in fitting high-dimensional multivariate additive models and illustrate its application on a breast cancer copy number variation and gene expression data set.

Gaussian denoising has emerged as a powerful method for constructing simulation-free continuous normalizing flows for generative modeling. Despite their empirical successes, theoretical properties of these flows and the regularizing effect of Gaussian denoising have remained largely unexplored. In this work, we aim to address this gap by investigating the well-posedness of simulation-free continuous normalizing flows built on Gaussian denoising. Through a unified framework termed Gaussian interpolation flow, we establish the Lipschitz regularity of the flow velocity field, the existence and uniqueness of the flow, and the Lipschitz continuity of the flow map and the time-reversed flow map for several rich classes of target distributions. This analysis also sheds light on the auto-encoding and cycle consistency properties of Gaussian interpolation flows. Additionally, we study the stability of these flows in source distributions and perturbations of the velocity field, using the quadratic Wasserstein distance as a metric. Our findings offer valuable insights into the learning techniques employed in Gaussian interpolation flows for generative modeling, providing a solid theoretical foundation for end-to-end error analyses of learning Gaussian interpolation flows with empirical observations.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >1$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests for nonparametric quantile regression.

The goal of supervised representation learning is to construct effective data representations for prediction. Among all the characteristics of an ideal nonparametric representation of high-dimensional complex data, sufficiency, low dimensionality and disentanglement are some of the most essential ones. We propose a deep dimension reduction approach to learning representations with these characteristics. The proposed approach is a nonparametric generalization of the sufficient dimension reduction method. We formulate the ideal representation learning task as that of finding a nonparametric representation that minimizes an objective function characterizing conditional independence and promoting disentanglement at the population level. We then estimate the target representation at the sample level nonparametrically using deep neural networks. We show that the estimated deep nonparametric representation is consistent in the sense that its excess risk converges to zero. Our extensive numerical experiments using simulated and real benchmark data demonstrate that the proposed methods have better performance than several existing dimension reduction methods and the standard deep learning models in the context of classification and regression.

We derive a novel approximation error bound with explicit prefactor for Sobolev-regular functions using deep convolutional neural networks (CNNs). The bound is non-asymptotic in terms of the network depth and filter lengths, in a rather flexible way. For Sobolev-regular functions which can be embedded into the H\"older space, the prefactor of our error bound depends on the ambient dimension polynomially instead of exponentially as in most existing results, which is of independent interest. We also establish a new approximation result when the target function is supported on an approximate lower-dimensional manifold. We apply our results to establish non-asymptotic excess risk bounds for classification using CNNs with convex surrogate losses, including the cross-entropy loss, the hinge loss (SVM), the logistic loss, the exponential loss and the least squares loss. We show that the classification methods with CNNs can circumvent the curse of dimensionality if input data is supported on a neighborhood of a low-dimensional manifold.

This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results establish the convergence rates of GANs under a collection of integral probability metrics defined through Hölder classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structures or have Hölder densities, when the network architectures are chosen properly. In particular, for distributions concentrated around a low-dimensional set, we show that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into the generator and discriminator approximation error and the statistical error, which may be of independent interest.

We derive nearly sharp bounds for the bidirectional GAN (BiGAN) estimation error under the Dudley distance between the latent joint distribution and the data joint distribution with appropriately specified architecture of the neural networks used in the model. To the best of our knowledge, this is the first theoretical guarantee for the bidirectional GAN learning approach. An appealing feature of our results is that they do not assume the reference and the data distributions to have the same dimensions or these distributions to have bounded support. These assumptions are commonly assumed in the existing convergence analysis of the unidirectional GANs but may not be satisfied in practice. Our results are also applicable to the Wasserstein bidirectional GAN if the target distribution is assumed to have a bounded support. To prove these results, we construct neural network functions that push forward an empirical distribution to another arbitrary empirical distribution on a possibly different-dimensional space. We also develop a novel decomposition of the integral probability metric for the error analysis of bidirectional GANs. These basic theoretical results are of independent interest and can be applied to other related learning problems.

MOTIVATION: Although genome-wide association studies (GWAS) have deepened our understanding of the genetic architecture of complex traits, the mechanistic links that underlie how genetic variants cause complex traits remains elusive. To advance our understanding of the underlying mechanistic links, various consortia have collected a vast volume of genomic data that enable us to investigate the role that genetic variants play in gene expression regulation. Recently, a collaborative mixed model (CoMM) was proposed to jointly interrogate genome on complex traits by integrating both the GWAS dataset and the expression quantitative trait loci (eQTL) dataset. Although CoMM is a powerful approach that leverages regulatory information while accounting for the uncertainty in using an eQTL dataset, it requires individual-level GWAS data and cannot fully make use of widely available GWAS summary statistics. Therefore, statistically efficient methods that leverages transcriptome information using only summary statistics information from GWAS data are required. RESULTS: In this study, we propose a novel probabilistic model, CoMM-S2, to examine the mechanistic role that genetic variants play, by using only GWAS summary statistics instead of individual-level GWAS data. Similar to CoMM which uses individual-level GWAS data, CoMM-S2 combines two models: the first model examines the relationship between gene expression and genotype, while the second model examines the relationship between the phenotype and the predicted gene expression from the first model. Distinct from CoMM, CoMM-S2 requires only GWAS summary statistics. Using both simulation studies and real data analysis, we demonstrate that even though CoMM-S2 utilizes GWAS summary statistics, it has comparable performance as CoMM, which uses individual-level GWAS data. AVAILABILITY AND IMPLEMENTATION: The implement of CoMM-S2 is included in the CoMM package that can be downloaded from https://github.com/gordonliu810822/CoMM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP)-penalized regression models are two important and widely used nonconvex sparse learning tools that can handle variable selection and parameter estimation simultaneously and thus have potential applications in various fields, such as mining biological data in high-throughput biomedical studies. Theoretically, these two models enjoy the oracle property even in the high-dimensional settings, where the number of predictors p may be much larger than the number of observations n . However, numerically, it is quite challenging to develop fast and stable algorithms due to their nonconvexity and nonsmoothness. In this article, we develop a fast algorithm for SCAD- and MCP-penalized learning problems. First, we show that the global minimizers of both models are roots of the nonsmooth equations. Then, a semismooth Newton (SSN) algorithm is employed to solve the equations. We prove that the SSN algorithm converges locally and superlinearly to the Karush-Kuhn-Tucker (KKT) points. The computational complexity analysis shows that the cost of the SSN algorithm per iteration is O(np) . Combined with the warm-start technique, the SSN algorithm can be very efficient and accurate. Simulation studies and a real data example suggest that our SSN algorithm, with comparable solution accuracy with the coordinate descent (CD) and the difference of convex (DC) proximal Newton algorithms, is more computationally efficient.

We propose a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the $\ell_0$-penalized least squares solutions. It generates a sequence of solutions iteratively, based on support detection using primal and dual information and root finding. We refer to the algorithm as SDAR for brevity. Under a sparse Riesz condition on the design matrix and certain other conditions, we show that with high probability, the $\ell_2$ estimation error of the solution sequence decays exponentially to the minimax error bound in $O(\log(R\sqrt{J}))$ iterations, where $J$ is the number of important predictors and $R$ is the relative magnitude of the nonzero target coefficients; and under a mutual coherence condition and certain other conditions, the $\ell_{\infty}$ estimation error decays to the optimal error bound in $O(\log(R))$ iterations. Moreover the SDAR solution recovers the oracle least squares estimator within a finite number of iterations with high probability if the sparsity level is known. Computational complexity analysis shows that the cost of SDAR is $O(np)$ per iteration. We also consider an adaptive version of SDAR for use in practical applications where the true sparsity level is unknown. Simulation studies demonstrate that SDAR outperforms Lasso, MCP and two greedy methods in accuracy and efficiency.

With accumulating research on the interconnections among different types of genomic regulations, researchers have found that multidimensional genomic studies outperform one-dimensional studies in multiple aspects. Among many sources of multidimensional genomic data, The Cancer Genome Atlas (TCGA) provides the public with comprehensive profiling data on >30 cancer types, making it an ideal test bed for conducting and comparing different analyses. In this article, the analysis goal is to apply several existing methods and associate multidimensional genomic measurements with cancer outcomes in particular prognosis, with special focus on the predictive power of genomic signatures. We exploit clinical data and four types of genomic measurement including mRNA gene expression, DNA methylation, microRNA and copy number alterations for breast invasive carcinoma, glioblastoma multiforme, acute myeloid leukemia and lung squamous cell carcinoma collected by TCGA. To accommodate the high dimensionality, we extract important features using Principal Component Analysis, Partial Least Squares and Least Absolute Shrinkage and Selection Operator (Lasso), which are representative of dimension reduction and variable selection techniques and have been extensively adopted, and fit Cox survival models with combined important features. We calibrate the predictive power of each type of genomic measurement for the prognosis of four cancer types and find that the results vary across cancers. Our analysis also suggests that for most of the cancers in our study and the adopted methods, there is no substantial improvement in prediction when adding other genomic measurement after gene expression and clinical covariates have been included in the model. This is consistent with the findings that molecular features measured at the transcription level affect clinical outcomes more directly than those measured at the DNA/epigenetic level.

MOTIVATION: Both gene expression levels (GEs) and copy number alterations (CNAs) have important biological implications. GEs are partly regulated by CNAs, and much effort has been devoted to understanding their relations. The regulation analysis is challenging with one gene expression possibly regulated by multiple CNAs and one CNA potentially regulating the expressions of multiple genes. The correlations among GEs and among CNAs make the analysis even more complicated. The existing methods have limitations and cannot comprehensively describe the regulation. RESULTS: A sparse double Laplacian shrinkage method is developed. It jointly models the effects of multiple CNAs on multiple GEs. Penalization is adopted to achieve sparsity and identify the regulation relationships. Network adjacency is computed to describe the interconnections among GEs and among CNAs. Two Laplacian shrinkage penalties are imposed to accommodate the network adjacency measures. Simulation shows that the proposed method outperforms the competing alternatives with more accurate marker identification. The Cancer Genome Atlas data are analysed to further demonstrate advantages of the proposed method. AVAILABILITY AND IMPLEMENTATION: R code is available at http://works.bepress.com/shuangge/49/.

Gene expression profiling has been extensively conducted in cancer research. The analysis of multiple independent cancer gene expression datasets may provide additional information and complement single-dataset analysis. In this study, we conduct multi-dataset analysis and are interested in evaluating the similarity of cancer-associated genes identified from different datasets. The first objective of this study is to briefly review some statistical methods that can be used for such evaluation. Both marginal analysis and joint analysis methods are reviewed. The second objective is to apply those methods to 26 Gene Expression Omnibus (GEO) datasets on five types of cancers. Our analysis suggests that for the same cancer, the marker identification results may vary significantly across datasets, and different datasets share few common genes. In addition, datasets on different cancers share few common genes. The shared genetic basis of datasets on the same or different cancers, which has been suggested in the literature, is not observed in the analysis of GEO data.

BACKGROUND: In high throughput cancer genomic studies, results from the analysis of single datasets often suffer from a lack of reproducibility because of small sample sizes. Integrative analysis can effectively pool and analyze multiple datasets and provides a cost effective way to improve reproducibility. In integrative analysis, simultaneously analyzing all genes profiled may incur high computational cost. A computationally affordable remedy is prescreening, which fits marginal models, can be conducted in a parallel manner, and has low computational cost. RESULTS: An integrative prescreening approach is developed for the analysis of multiple cancer genomic datasets. Simulation shows that the proposed integrative prescreening has better performance than alternatives, particularly including prescreening with individual datasets, an intensity approach and meta-analysis. We also analyze multiple microarray gene profiling studies on liver and pancreatic cancers using the proposed approach. CONCLUSIONS: The proposed integrative prescreening provides an effective way to reduce the dimensionality in cancer genomic studies. It can be coupled with existing analysis methods to identify cancer markers.

Development of high-throughput technologies makes it possible to survey the whole genome. Genomic studies have been extensively conducted, searching for markers with predictive power for prognosis of complex diseases such as cancer, diabetes and obesity. Most existing statistical analyses are focused on developing marker selection techniques, while little attention is paid to the underlying prognosis models. In this article, we review three commonly used prognosis models, namely the Cox, additive risk and accelerated failure time models. We conduct simulation and show that gene identification can be unsatisfactory under model misspecification. We analyze three cancer prognosis studies under the three models, and show that the gene identification results, prediction performance of all identified genes combined, and reproducibility of each identified gene are model-dependent. We suggest that in practical data analysis, more attention should be paid to the model assumption, and multiple models may need to be considered.

MOTIVATION: Although NHL (non-Hodgkin's lymphoma) is the fifth leading cause of cancer incidence and mortality in the USA, it remains poorly understood and is largely incurable. Biomedical studies have shown that genomic variations, measured with SNPs (single nucleotide polymorphisms) in genes, may have independent predictive power for disease-free survival in NHL patients beyond clinical measurements. RESULTS: We apply the CTGDR (clustering threshold gradient directed regularization) method to genetic association studies using SNPs, analyze data from an association study of NHL and identify prognosis signatures to diffuse large B cell lymphoma (DLBCL) and follicular lymphoma (FL), the two most common subtypes of NHL. With the CTGDR method, we are able to account for the joint effects of multiple genes/SNPs, whereas most existing studies are single-marker based. In addition, we are able to account for the 'gene and SNP-within-gene' hierarchical structure and identify not only predictive genes but also predictive SNPs within identified genes. In contrast, existing studies are limited to either gene or SNP identification, but not both. We propose using resampling methods to evaluate the predictive power and reproducibility of identified genes and SNPs. Simulation study and data analysis suggest satisfactory performance of the CTGDR method.

BACKGROUND: In cancer studies, it is common that multiple microarray experiments are conducted to measure the same clinical outcome and expressions of the same set of genes. An important goal of such experiments is to identify a subset of genes that can potentially serve as predictive markers for cancer development and progression. Analyses of individual experiments may lead to unreliable gene selection results because of the small sample sizes. Meta analysis can be used to pool multiple experiments, increase statistical power, and achieve more reliable gene selection. The meta analysis of cancer microarray data is challenging because of the high dimensionality of gene expressions and the differences in experimental settings amongst different experiments. RESULTS: We propose a Meta Threshold Gradient Descent Regularization (MTGDR) approach for gene selection in the meta analysis of cancer microarray data. The MTGDR has many advantages over existing approaches. It allows different experiments to have different experimental settings. It can account for the joint effects of multiple genes on cancer, and it can select the same set of cancer-associated genes across multiple experiments. Simulation studies and analyses of multiple pancreatic and liver cancer experiments demonstrate the superior performance of the MTGDR. CONCLUSION: The MTGDR provides an effective way of analyzing multiple cancer microarray studies and selecting reliable cancer-associated genes.

In bioinformatics studies, supervised classification with high-dimensional input variables is frequently encountered. Examples routinely arise in genomic, epigenetic and proteomic studies. Feature selection can be employed along with classifier construction to avoid over-fitting, to generate more reliable classifier and to provide more insights into the underlying causal relationships. In this article, we provide a review of several recently developed penalized feature selection and classification techniques--which belong to the family of embedded feature selection methods--for bioinformatics studies with high-dimensional input. Classification objective functions, penalty functions and computational algorithms are discussed. Our goal is to make interested researchers aware of these feature selection and classification methods that are applicable to high-dimensional bioinformatics data.

MOTIVATION: An important goal of microarray studies is to discover genes that are associated with clinical outcomes, such as disease status and patient survival. While a typical experiment surveys gene expressions on a global scale, there may be only a small number of genes that have significant influence on a clinical outcome. Moreover, expression data have cluster structures and the genes within a cluster have correlated expressions and coordinated functions, but the effects of individual genes in the same cluster may be different. Accordingly, we seek to build statistical models with the following properties. First, the model is sparse in the sense that only a subset of the parameter vector is non-zero. Second, the cluster structures of gene expressions are properly accounted for. RESULTS: For gene expression data without pathway information, we divide genes into clusters using commonly used methods, such as K-means or hierarchical approaches. The optimal number of clusters is determined using the Gap statistic. We propose a clustering threshold gradient descent regularization (CTGDR) method, for simultaneous cluster selection and within cluster gene selection. We apply this method to binary classification and censored survival analysis. Compared to the standard TGDR and other regularization methods, the CTGDR takes into account the cluster structure and carries out feature selection at both the cluster level and within-cluster gene level. We demonstrate the CTGDR on two studies of cancer classification and two studies correlating survival of lymphoma patients with microarray expressions. AVAILABILITY: R code is available upon request. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Microarray techniques survey gene expressions on a global scale. Extensive biomedical studies have been designed to discover subsets of genes that are associated with survival risks for diseases such as lymphoma and construct predictive models using those selected genes. In this article, we investigate simultaneous estimation and gene selection with right censored survival data and high dimensional gene expression measurements. RESULTS: We model the survival time using the additive risk model, which provides a useful alternative to the proportional hazards model and is adopted when the absolute effects, instead of the relative effects, of multiple predictors on the hazard function are of interest. A Lasso (least absolute shrinkage and selection operator) type estimate is proposed for simultaneous estimation and gene selection. Tuning parameter is selected using the V-fold cross validation. We propose Leave-One-Out cross validation based methods for evaluating the relative stability of individual genes and overall prediction significance. CONCLUSION: We analyze the MCL and DLBCL data using the proposed approach. A small number of probes represented on the microarrays are identified, most of which have sound biological implications in lymphoma development. The selected probes are relatively stable and the proposed approach has overall satisfactory prediction power.

BACKGROUND: A tremendous amount of efforts have been devoted to identifying genes for diagnosis and prognosis of diseases using microarray gene expression data. It has been demonstrated that gene expression data have cluster structure, where the clusters consist of co-regulated genes which tend to have coordinated functions. However, most available statistical methods for gene selection do not take into consideration the cluster structure. RESULTS: We propose a supervised group Lasso approach that takes into account the cluster structure in gene expression data for gene selection and predictive model building. For gene expression data without biological cluster information, we first divide genes into clusters using the K-means approach and determine the optimal number of clusters using the Gap method. The supervised group Lasso consists of two steps. In the first step, we identify important genes within each cluster using the Lasso method. In the second step, we select important clusters using the group Lasso. Tuning parameters are determined using V-fold cross validation at both steps to allow for further flexibility. Prediction performance is evaluated using leave-one-out cross validation. We apply the proposed method to disease classification and survival analysis with microarray data. CONCLUSION: We analyze four microarray data sets using the proposed approach: two cancer data sets with binary cancer occurrence as outcomes and two lymphoma data sets with survival outcomes. The results show that the proposed approach is capable of identifying a small number of influential gene clusters and important genes within those clusters, and has better prediction performance than existing methods.

BACKGROUND: An important application of microarrays is to discover genomic biomarkers, among tens of thousands of genes assayed, for disease diagnosis and prognosis. Thus it is of interest to develop efficient statistical methods that can simultaneously identify important biomarkers from such high-throughput genomic data and construct appropriate classification rules. It is also of interest to develop methods for evaluation of classification performance and ranking of identified biomarkers. RESULTS: The ROC (receiver operating characteristic) technique has been widely used in disease classification with low dimensional biomarkers. Compared with the empirical ROC approach, the binormal ROC is computationally more affordable and robust in small sample size cases. We propose using the binormal AUC (area under the ROC curve) as the objective function for two-sample classification, and the scaled threshold gradient directed regularization method for regularized estimation and biomarker selection. Tuning parameter selection is based on V-fold cross validation. We develop Monte Carlo based methods for evaluating the stability of individual biomarkers and overall prediction performance. Extensive simulation studies show that the proposed approach can generate parsimonious models with excellent classification and prediction performance, under most simulated scenarios including model mis-specification. Application of the method to two cancer studies shows that the identified genes are reasonably stable with satisfactory prediction performance and biologically sound implications. The overall classification performance is satisfactory, with small classification errors and large AUCs. CONCLUSION: In comparison to existing methods, the proposed approach is computationally more affordable without losing the optimality possessed by the standard ROC method.

MOTIVATION: An important application of microarrays is to discover genomic biomarkers, among tens of thousands of genes assayed, for disease classification. Thus there is a need for developing statistical methods that can efficiently use such high-throughput genomic data, select biomarkers with discriminant power and construct classification rules. The ROC (receiver operator characteristic) technique has been widely used in disease classification with low-dimensional biomarkers because (1) it does not assume a parametric form of the class probability as required for example in the logistic regression method; (2) it accommodates case-control designs and (3) it allows treating false positives and false negatives differently. However, due to computational difficulties, the ROC-based classification has not been used with microarray data. Moreover, the standard ROC technique does not incorporate built-in biomarker selection. RESULTS: We propose a novel method for biomarker selection and classification using the ROC technique for microarray data. The proposed method uses a sigmoid approximation to the area under the ROC curve as the objective function for classification and the threshold gradient descent regularization method for estimation and biomarker selection. Tuning parameter selection based on the V-fold cross validation and predictive performance evaluation are also investigated. The proposed approach is demonstrated with a simulation study, the Colon data and the Estrogen data. The proposed approach yields parsimonious models with excellent classification performance.

BACKGROUND: Normalization is a basic step in microarray data analysis. A proper normalization procedure ensures that the intensity ratios provide meaningful measures of relative expression values. METHODS: We propose a robust semiparametric method in a two-way semi-linear model (TW-SLM) for normalization of cDNA microarray data. This method does not make the usual assumptions underlying some of the existing methods. For example, it does not assume that: (i) the percentage of differentially expressed genes is small; or (ii) the numbers of up- and down-regulated genes are about the same, as required in the LOWESS normalization method. We conduct simulation studies to evaluate the proposed method and use a real data set from a specially designed microarray experiment to compare the performance of the proposed method with that of the LOWESS normalization approach. RESULTS: The simulation results show that the proposed method performs better than the LOWESS normalization method in terms of mean square errors for estimated gene effects. The results of analysis of the real data set also show that the proposed method yields more consistent results between the direct and the indirect comparisons and also can detect more differentially expressed genes than the LOWESS method. CONCLUSIONS: Our simulation studies and the real data example indicate that the proposed robust TW-SLM method works at least as well as the LOWESS method and works better when the underlying assumptions for the LOWESS method are not satisfied. Therefore, it is a powerful alternative to the existing normalization methods.