Tumor classification by partial least squares using microarray gene expression data

MOTIVATION: One important application of gene expression microarray data is classification of samples into categories, such as the type of tumor. The use of microarrays allows simultaneous monitoring of thousands of genes expressions per sample. This ability to measure gene expression en masse has resulted in data with the number of variables p(genes) far exceeding the number of samples N. Standard statistical methodologies in classification and prediction do not work well or even at all when N < p. Modification of existing statistical methodologies or development of new methodologies is needed for the analysis of microarray data. RESULTS: We propose a novel analysis procedure for classifying (predicting) human tumor samples based on microarray gene expressions. This procedure involves dimension reduction using Partial Least Squares (PLS) and classification using Logistic Discrimination (LD) and Quadratic Discriminant Analysis (QDA). We compare PLS to the well known dimension reduction method of Principal Components Analysis (PCA). Under many circumstances PLS proves superior; we illustrate a condition when PCA particularly fails to predict well relative to PLS. The proposed methods were applied to five different microarray data sets involving various human tumor samples: (1) normal versus ovarian tumor; (2) Acute Myeloid Leukemia (AML) versus Acute Lymphoblastic Leukemia (ALL); (3) Diffuse Large B-cell Lymphoma (DLBCLL) versus B-cell Chronic Lymphocytic Leukemia (BCLL); (4) normal versus colon tumor; and (5) Non-Small-Cell-Lung-Carcinoma (NSCLC) versus renal samples. Stability of classification results and methods were further assessed by re-randomization studies.     

Predicting patient survival from microarray data by accelerated failure time modeling using partial least squares and LASSO

We consider the problem of predicting survival times of cancer patients from the gene expression profiles of their tumor samples via linear regression modeling of log-transformed failure times. The partial least squares (PLS) and least absolute shrinkage and selection operator (LASSO) methodologies are used for this purpose where we first modify the data to account for censoring. Three approaches of handling right censored data-reweighting, mean imputation, and multiple imputation-are considered. Their performances are examined in a detailed simulation study and compared with that of full data PLS and LASSO had there been no censoring. A major objective of this article is to investigate the performances of PLS and LASSO in the context of microarray data where the number of covariates is very large and there are extremely few samples. We demonstrate that LASSO outperforms PLS in terms of prediction error when the list of covariates includes a moderate to large percentage of useless or noise variables; otherwise, PLS may outperform LASSO. For a moderate sample size (100 with 10,000 covariates), LASSO performed better than a no covariate model (or noise-based prediction). The mean imputation method appears to best track the performance of the full data PLS or LASSO. The mean imputation scheme is used on an existing data set on lung cancer. This reanalysis using the mean imputed PLS and LASSO identifies a number of genes that were known to be related to cancer or tumor activities from previous studies.     

Predicting survival from microarray data--a comparative study

MOTIVATION: Survival prediction from gene expression data and other high-dimensional genomic data has been subject to much research during the last years. These kinds of data are associated with the methodological problem of having many more gene expression values than individuals. In addition, the responses are censored survival times. Most of the proposed methods handle this by using Cox's proportional hazards model and obtain parameter estimates by some dimension reduction or parameter shrinkage estimation technique. Using three well-known microarray gene expression data sets, we compare the prediction performance of seven such methods: univariate selection, forward stepwise selection, principal components regression (PCR), supervised principal components regression, partial least squares regression (PLS), ridge regression and the lasso. RESULTS: Statistical learning from subsets should be repeated several times in order to get a fair comparison between methods. Methods using coefficient shrinkage or linear combinations of the gene expression values have much better performance than the simple variable selection methods. For our data sets, ridge regression has the overall best performance. AVAILABILITY: Matlab and R code for the prediction methods are available at http://www.med.uio.no/imb/stat/bmms/software/microsurv/.     

A quantile‐slicing approach for sufficient dimension reduction with censored responses

Sufficient dimension reduction (SDR) that effectively reduces the predictor dimension in regression has been popular in high‐dimensional data analysis. Under the presence of censoring, however, most existing SDR methods suffer. In this article, we propose a new algorithm to perform SDR with censored responses based on the quantile‐slicing scheme recently proposed by Kim et al. First, we estimate the conditional quantile function of the true survival time via the censored kernel quantile regression (Shin et al.) and then slice the data based on the estimated censored regression quantiles instead of the responses. Both simulated and real data analysis demonstrate promising performance of the proposed method.     

Sufficient dimension reduction for censored regressions

Methodology of sufficient dimension reduction (SDR) has offered an effective means to facilitate regression analysis of high-dimensional data. When the response is censored, however, most existing SDR estimators cannot be applied, or require some restrictive conditions. In this article, we propose a new class of inverse censoring probability weighted SDR estimators for censored regressions. Moreover, regularization is introduced to achieve simultaneous variable selection and dimension reduction. Asymptotic properties and empirical performance of the proposed methods are examined.     

Dimension reduction methods for microarrays with application to censored survival data

MOTIVATION: Recent research has shown that gene expression profiles can potentially be used for predicting various clinical phenotypes, such as tumor class, drug response and survival time. While there has been extensive studies on tumor classification, there has been less emphasis on other phenotypic features, in particular, patient survival time or time to cancer recurrence, which are subject to right censoring. We consider in this paper an analysis of censored survival time based on microarray gene expression profiles. RESULTS: We propose a dimension reduction strategy, which combines principal components analysis and sliced inverse regression, to identify linear combinations of genes, that both account for the variability in the gene expression levels and preserve the phenotypic information. The extracted gene combinations are then employed as covariates in a predictive survival model formulation. We apply the proposed method to a large diffuse large-B-cell lymphoma dataset, which consists of 240 patients and 7399 genes, and build a Cox proportional hazards model based on the derived gene expression components. The proposed method is shown to provide a good predictive performance for patient survival, as demonstrated by both the significant survival difference between the predicted risk groups and the receiver operator characteristics analysis. AVAILABILITY: R programs are available upon request from the authors. SUPPLEMENTARY INFORMATION: http://dna.ucdavis.edu/~hli/bioinfo-surv-supp.pdf.     

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.     

Using sufficient direction factor model to analyze latent activities associated with breast cancer survival

High-dimensional gene expression data often exhibit intricate correlation patterns as the result of coordinated genetic regulation. In practice, however, it is difficult to directly measure these coordinated underlying activities. Analysis of breast cancer survival data with gene expressions motivates us to use a two-stage latent factor approach to estimate these unobserved coordinated biological processes. Compared to existing approaches, our proposed procedure has several unique characteristics. In the first stage, an important distinction is that our procedure incorporates prior biological knowledge about gene-pathway membership into the analysis and explicitly model the effects of genetic pathways on the latent factors. Second, to characterize the molecular heterogeneity of breast cancer, our approach provides estimates specific to each cancer subtype. Finally, our proposed framework incorporates sparsity condition due to the fact that genetic networks are often sparse. In the second stage, we investigate the relationship between latent factor activity levels and survival time with censoring using a general dimension reduction model in the survival analysis context. Combining the factor model and sufficient direction model provides an efficient way of analyzing high-dimensional data and reveals some interesting relations in the breast cancer gene expression data.     

Principal component analysis based methods in bioinformatics studies

In analysis of bioinformatics data, a unique challenge arises from the high dimensionality of measurements. Without loss of generality, we use genomic study with gene expression measurements as a representative example but note that analysis techniques discussed in this article are also applicable to other types of bioinformatics studies. Principal component analysis (PCA) is a classic dimension reduction approach. It constructs linear combinations of gene expressions, called principal components (PCs). The PCs are orthogonal to each other, can effectively explain variation of gene expressions, and may have a much lower dimensionality. PCA is computationally simple and can be realized using many existing software packages. This article consists of the following parts. First, we review the standard PCA technique and their applications in bioinformatics data analysis. Second, we describe recent 'non-standard' applications of PCA, including accommodating interactions among genes, pathways and network modules and conducting PCA with estimating equations as opposed to gene expressions. Third, we introduce several recently proposed PCA-based techniques, including the supervised PCA, sparse PCA and functional PCA. The supervised PCA and sparse PCA have been shown to have better empirical performance than the standard PCA. The functional PCA can analyze time-course gene expression data. Last, we raise the awareness of several critical but unsolved problems related to PCA. The goal of this article is to make bioinformatics researchers aware of the PCA technique and more importantly its most recent development, so that this simple yet effective dimension reduction technique can be better employed in bioinformatics data analysis.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.     

Gene expression-based survival prediction in lung adenocarcinoma: a multi-site, blinded validation study

Although prognostic gene expression signatures for survival in early-stage lung cancer have been proposed, for clinical application, it is critical to establish their performance across different subject populations and in different laboratories. Here we report a large, training-testing, multi-site, blinded validation study to characterize the performance of several prognostic models based on gene expression for 442 lung adenocarcinomas. The hypotheses proposed examined whether microarray measurements of gene expression either alone or combined with basic clinical covariates (stage, age, sex) could be used to predict overall survival in lung cancer subjects. Several models examined produced risk scores that substantially correlated with actual subject outcome. Most methods performed better with clinical data, supporting the combined use of clinical and molecular information when building prognostic models for early-stage lung cancer. This study also provides the largest available set of microarray data with extensive pathological and clinical annotation for lung adenocarcinomas.     

Identifying subset of genes that have influential impacts on cancer progression: a new approach to analyze cancer microarray data

Cancer is a complex genetic disease, resulting from defects of multiple genes. Development of microarray techniques makes it possible to survey the whole genome and detect genes that have influential impacts on the progression of cancer. Statistical analysis of cancer microarray data is challenging because of the high dimensionality and cluster nature of gene expressions. Here, clusters are composed of genes with coordinated pathological functions and/or correlated expressions. In this article, we consider cancer studies where censored survival endpoint is measured along with microarray gene expressions. We propose a hybrid clustering approach, which uses both pathological pathway information retrieved from KEGG and statistical correlations of gene expressions, to construct gene clusters. Cancer survival time is modeled as a linear function of gene expressions. We adopt the clustering threshold gradient directed regularization (CTGDR) method for simultaneous gene cluster selection, within-cluster gene selection, and predictive model building. Analysis of two lymphoma studies shows that the proposed approach - which is composed of the hybrid gene clustering, linear regression model for survival, and clustering regularized estimation with CTGDR - can effectively identify gene clusters and genes within selected clusters that have satisfactory predictive power for censored cancer survival outcomes.     

A targeted maximum likelihood estimator for two-stage designs

We consider two-stage sampling designs, including so-called nested case control studies, where one takes a random sample from a target population and completes measurements on each subject in the first stage. The second stage involves drawing a subsample from the original sample, collecting additional data on the subsample. This data structure can be viewed as a missing data structure on the full-data structure collected in the second-stage of the study. Methods for analyzing two-stage designs include parametric maximum likelihood estimation and estimating equation methodology. We propose an inverse probability of censoring weighted targeted maximum likelihood estimator (IPCW-TMLE) in two-stage sampling designs and present simulation studies featuring this estimator.     

PCA 和PLS 应用于胃癌亚型分类研究

文章研究了基于微阵列基因表达数据的胃癌亚型分类。微阵列基因表达数据样本少、纬度高、噪声大的特点,使得数据降维成为分类成功的关键。作者将主成分分析(PCA) 和偏最小二乘（PLS）两种降维方法应用于胃癌亚型分类研究,以支持向量机（SVM）、K- 近邻法（KNN）为分类器对两套胃癌数据进行亚型分类。分类效果相比传统的医理诊断略高,最高准确率可达100%。研究结果表明,主成分分析和偏最小二乘方法能够有效地提取分类特征信息,并能在保持较高的分类准确率的前提下大幅度地降低基因表达数据的维数。     

Pathway analysis of microarray data via regression.

Pathway analysis of microarray data evaluates gene expression profiles of a priori defined biological pathways in association with a phenotype of interest. We propose a unified pathway-analysis method that can be used for diverse phenotypes including binary, multiclass, continuous, count, rate, and censored survival phenotypes. The proposed method also allows covariate adjustments and correlation in the phenotype variable that is encountered in longitudinal, cluster-sampled, and paired designs. These are accomplished by combining the regression-based test statistic for each individual gene in a pathway of interest into a pathway-level test statistic. Applications of the proposed method are illustrated with two real pathway-analysis examples: one evaluating relapse-associated gene expression involving a matched-pair binary phenotype in children with acute lymphoblastic leukemia; and the other investigating gene expression in breast cancer tissues in relation to patients' survival (a censored survival phenotype). Implementations for various phenotypes are available in R. Additionally, an Excel Add-in for a user-friendly interface is currently being developed.     

Interval mapping methods for detecting QTL affecting survival and time-to-event phenotypes

Quantitative trait loci (QTL) are usually searched for using classical interval mapping methods which assume that the trait of interest follows a normal distribution. However, these methods cannot take into account features of most survival data such as a non-normal distribution and the presence of censored data. We propose two new QTL detection approaches which allow the consideration of censored data. One interval mapping method uses a Weibull model (W), which is popular in parametrical modelling of survival traits, and the other uses a Cox model (C), which avoids making any assumption on the trait distribution. Data were simulated following the structure of a published experiment. Using simulated data, we compare W, C and a classical interval mapping method using a Gaussian model on uncensored data (G) or on all data (G'=censored data analysed as though records were uncensored). An adequate mathematical transformation was used for all parametric methods (G, G' and W). When data were not censored, the four methods gave similar results. However, when some data were censored, the power of QTL detection and accuracy of QTL location and of estimation of QTL effects for G decreased considerably with censoring, particularly when censoring was at a fixed date. This decrease with censoring was observed also with G', but it was less severe. Censoring had a negligible effect on results obtained with the W and C methods.     

On the performance of adaptive preprocessing technique in analyzing high‐dimensional censored data

Preprocessing for high‐dimensional censored datasets, such as the microarray data, is generally considered as an important technique to gain further stability by reducing potential noise from the data. When variable selection including inference is carried out with high‐dimensional censored data the objective is to obtain a smaller subset of variables and then perform the inferential analysis using model estimates based on the selected subset of variables. This two stage inferential analysis is prone to circularity bias because of the noise that might still remain in the dataset. In this work, I propose an adaptive preprocessing technique that uses sure independence screening (SIS) idea to accomplish variable selection and reduces the circularity bias by some popularly known refined high‐dimensional methods such as the elastic net, adaptive elastic net, weighted elastic net, elastic net‐AFT, and two greedy variable selection methods known as TCS, PC‐simple all implemented with the accelerated lifetime models. The proposed technique addresses several features including the issue of collinearity between important and some unimportant covariates, which is often the case in high‐dimensional setting under variable selection framework, and different level of censoring. Simulation studies along with an empirical analysis with a real microarray data, mantle cell lymphoma, is carried out to demonstrate the performance of the adaptive pre‐processing technique.     

A modification of the redistribution to the right algorithm using disease markers

Disease markers are time-dependent covariates which describeprogression towards development of disease. Traditional methodsin survival analysis do not make use of available data on thesemarkers to recover additional information from censored individuals.Using a heuristic modification of the redistribution to theright algorithm (Efron, 1967), a new approach for recoveringinformation for censored individuals using disease markers isproposed. Additionally, the statistical properties of the proposedmethod are examined. There are two possible advantages to thismodification: (i) bias reduction when censoring is informative,and (ii) an increase in efficiency in the case of truly noninformativecensoring.     

Consistent estimation of the expected Brier score in general survival models with right-censored event times

In survival analysis with censored data the mean squared error of prediction can be estimated by weighted averages of time-dependent residuals. Graf et al. (1999) suggested a robust weighting scheme based on the assumption that the censoring mechanism is independent of the covariates. We show consistency of the estimator. Furthermore, we show that a modified version of this estimator is consistent even when censoring and event times are only conditionally independent given the covariates. The modified estimators are derived on the basis of regression models for the censoring distribution. A simulation study and a real data example illustrate the results.