We describe the E-RFE method for gene ranking, which is useful for the identification of markers in the predictive classification of array data. The method supports a practical modeling scheme designed to avoid the construction of classification rules based on the selection of too small gene subsets (an effect known as the selection bias, in which the estimated predictive errors are too optimistic due to testing on samples already considered in the feature selection process).
With E-RFE, we speed up the recursive feature elimination (RFE) with SVM classifiers by eliminating chunks of uninteresting genes using an entropy measure of the SVM weights distribution. An optimal subset of genes is selected according to a two-strata model evaluation procedure: modeling is replicated by an external stratified-partition resampling scheme, and, within each run, an internal K-fold cross-validation is used for E-RFE ranking. Also, the optimal number of genes can be estimated according to the saturation of Zipf's law profiles.
Without a decrease of classification accuracy, E-RFE allows a speed-up factor of 100 with respect to standard RFE, while improving on alternative parametric RFE reduction strategies. Thus, a process for gene selection and error estimation is made practical, ensuring control of the selection bias, and providing additional diagnostic indicators of gene importance.
When confronted with a small sample, feature-selection algorithms often fail to find good feature sets, a problem exacerbated for high-dimensional data and large feature sets. The problem is compounded by the fact that, if one obtains a feature set with a low error estimate, the estimate is unreliable because training-data-based error estimators typically perform poorly on small samples, exhibiting optimistic bias or high variance. One way around the problem is limit the number of features being considered, restrict features sets to sizes such that all feature sets can be examined by exhaustive search, and report a list of the best performing feature sets. If the list is short, then it greatly restricts the possible feature sets to be considered as candidates; however, one can expect the lowest error estimates obtained to be optimistically biased so that there may not be a close-to-optimal feature set on the list. This paper provides a power analysis of this methodology; in particular, it examines the kind of results one should expect to obtain relative to the length of the list and the number of discriminating features among those considered. Two measures are employed. The first is the probability that there is at least one feature set on the list whose true classification error is within some given tolerance of the best feature set and the second is the expected number of feature sets on the list whose true errors are within the given tolerance of the best feature set. These values are plotted as functions of the list length to generate power curves. The results show that, if the number of discriminating features is not too small—that is, the prior biological knowledge is not too poor—then one should expect, with high probability, to find good feature sets.
Availability: companion website at http://gsp.tamu.edu/Publications/supplementary/zhao09a/
classification; feature ranking; ranking power
Cross-validation (CV) is an effective method for estimating the prediction error of a classifier. Some recent articles have proposed methods for optimizing classifiers by choosing classifier parameter values that minimize the CV error estimate. We have evaluated the validity of using the CV error estimate of the optimized classifier as an estimate of the true error expected on independent data.
We used CV to optimize the classification parameters for two kinds of classifiers; Shrunken Centroids and Support Vector Machines (SVM). Random training datasets were created, with no difference in the distribution of the features between the two classes. Using these "null" datasets, we selected classifier parameter values that minimized the CV error estimate. 10-fold CV was used for Shrunken Centroids while Leave-One-Out-CV (LOOCV) was used for the SVM. Independent test data was created to estimate the true error. With "null" and "non null" (with differential expression between the classes) data, we also tested a nested CV procedure, where an inner CV loop is used to perform the tuning of the parameters while an outer CV is used to compute an estimate of the error.
The CV error estimate for the classifier with the optimal parameters was found to be a substantially biased estimate of the true error that the classifier would incur on independent data. Even though there is no real difference between the two classes for the "null" datasets, the CV error estimate for the Shrunken Centroid with the optimal parameters was less than 30% on 18.5% of simulated training data-sets. For SVM with optimal parameters the estimated error rate was less than 30% on 38% of "null" data-sets. Performance of the optimized classifiers on the independent test set was no better than chance.
The nested CV procedure reduces the bias considerably and gives an estimate of the error that is very close to that obtained on the independent testing set for both Shrunken Centroids and SVM classifiers for "null" and "non-null" data distributions.
We show that using CV to compute an error estimate for a classifier that has itself been tuned using CV gives a significantly biased estimate of the true error. Proper use of CV for estimating true error of a classifier developed using a well defined algorithm requires that all steps of the algorithm, including classifier parameter tuning, be repeated in each CV loop. A nested CV procedure provides an almost unbiased estimate of the true error.
PAM, a nearest shrunken centroid method (NSC), is a popular classification method for high-dimensional data. ALP and AHP are NSC algorithms that were proposed to improve upon PAM. The NSC methods base their classification rules on shrunken centroids; in practice the amount of shrinkage is estimated minimizing the overall cross-validated (CV) error rate.
We show that when data are class-imbalanced the three NSC classifiers are biased towards the majority class. The bias is larger when the number of variables or class-imbalance is larger and/or the differences between classes are smaller. To diminish the class-imbalance problem of the NSC classifiers we propose to estimate the amount of shrinkage by maximizing the CV geometric mean of the class-specific predictive accuracies (g-means).
The results obtained on simulated and real high-dimensional class-imbalanced data show that our approach outperforms the currently used strategy based on the minimization of the overall error rate when NSC classifiers are biased towards the majority class. The number of variables included in the NSC classifiers when using our approach is much smaller than with the original approach. This result is supported by experiments on simulated and real high-dimensional class-imbalanced data.
Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples, and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.
bootstrap aggregation; data-adaptive regression; resistant HIV; Deletion/Substitution/Addition algorithm
When analysing microarray and other small sample size biological datasets, care is needed to avoid various biases. We analyse a form of bias, stratification bias, that can substantially affect analyses using sample-reuse validation techniques and lead to inaccurate results. This bias is due to imperfect stratification of samples in the training and test sets and the dependency between these stratification errors, i.e. the variations in class proportions in the training and test sets are negatively correlated.
We show that when estimating the performance of classifiers on low signal datasets (i.e. those which are difficult to classify), which are typical of many prognostic microarray studies, commonly used performance measures can suffer from a substantial negative bias. For error rate this bias is only severe in quite restricted situations, but can be much larger and more frequent when using ranking measures such as the receiver operating characteristic (ROC) curve and area under the ROC (AUC). Substantial biases are shown in simulations and on the van 't Veer breast cancer dataset. The classification error rate can have large negative biases for balanced datasets, whereas the AUC shows substantial pessimistic biases even for imbalanced datasets. In simulation studies using 10-fold cross-validation, AUC values of less than 0.3 can be observed on random datasets rather than the expected 0.5. Further experiments on the van 't Veer breast cancer dataset show these biases exist in practice.
Stratification bias can substantially affect several performance measures. In computing the AUC, the strategy of pooling the test samples from the various folds of cross-validation can lead to large biases; computing it as the average of per-fold estimates avoids this bias and is thus the recommended approach. As a more general solution applicable to other performance measures, we show that stratified repeated holdout and a modified version of k-fold cross-validation, balanced, stratified cross-validation and balanced leave-one-out cross-validation, avoids the bias. Therefore for model selection and evaluation of microarray and other small biological datasets, these methods should be used and unstratified versions avoided. In particular, the commonly used (unbalanced) leave-one-out cross-validation should not be used to estimate AUC for small datasets.
We consider the problem of designing a study to develop a predictive classifier from high dimensional data. A common study design is to split the sample into a training set and an independent test set, where the former is used to develop the classifier and the latter to evaluate its performance. In this paper we address the question of what proportion of the samples should be devoted to the training set. How does this proportion impact the mean squared error (MSE) of the prediction accuracy estimate?
We develop a non-parametric algorithm for determining an optimal splitting proportion that can be applied with a specific dataset and classifier algorithm. We also perform a broad simulation study for the purpose of better understanding the factors that determine the best split proportions and to evaluate commonly used splitting strategies (1/2 training or 2/3 training) under a wide variety of conditions. These methods are based on a decomposition of the MSE into three intuitive component parts.
By applying these approaches to a number of synthetic and real microarray datasets we show that for linear classifiers the optimal proportion depends on the overall number of samples available and the degree of differential expression between the classes. The optimal proportion was found to depend on the full dataset size (n) and classification accuracy - with higher accuracy and smaller n resulting in more assigned to the training set. The commonly used strategy of allocating 2/3rd of cases for training was close to optimal for reasonable sized datasets (n ≥ 100) with strong signals (i.e. 85% or greater full dataset accuracy). In general, we recommend use of our nonparametric resampling approach for determing the optimal split. This approach can be applied to any dataset, using any predictor development method, to determine the best split.
To estimate a classifier’s error in predicting future observations, bootstrap methods have been proposed as reduced-variation alternatives to traditional cross-validation (CV) methods based on sampling without replacement. Monte Carlo (MC) simulation studies aimed at estimating the true misclassification error conditional on the training set are commonly used to compare CV methods. We conducted an MC simulation study to compare a new method of bootstrap CV (BCV) to k-fold CV for estimating clasification error.
For the low-dimensional conditions simulated, the modest positive bias of k-fold CV contrasted sharply with the substantial negative bias of the new BCV method. This behavior was corroborated using a real-world dataset of prognostic gene-expression profiles in breast cancer patients. Our simulation results demonstrate some extreme characteristics of variance and bias that can occur due to a fault in the design of CV exercises aimed at estimating the true conditional error of a classifier, and that appear not to have been fully appreciated in previous studies. Although CV is a sound practice for estimating a classifier’s generalization error, using CV to estimate the fixed misclassification error of a trained classifier conditional on the training set is problematic. While MC simulation of this estimation exercise can correctly represent the average bias of a classifier, it will overstate the between-run variance of the bias.
We recommend k-fold CV over the new BCV method for estimating a classifier’s generalization error. The extreme negative bias of BCV is too high a price to pay for its reduced variance.
Cross-validation; Bootstrap Cross-validation; Classification Error Estimation; Mean Squared Error
The prediction of gains from selection allows the comparison of breeding methods and selection strategies, although these estimates may be biased. The objective of this study was to investigate the extent of such bias in predicting genetic gain. For this, we simulated 10 cycles of a hypothetical breeding program that involved seven traits, three population classes, three experimental conditions and two breeding methods (mass and half-sib selection). Each combination of trait, population, heritability, method and cycle was repeated 10 times. The predicted gains were biased, even when the genetic parameters were estimated without error. Gain from selection in both genders is twice the gain from selection in a single gender only in the absence of dominance. The use of genotypic variance or broad sense heritability in the predictions represented an additional source of bias. Predictions based on additive variance and narrow sense heritability were equivalent, as were predictions based on genotypic variance and broad sense heritability. The predictions based on mass and family selection were suitable for comparing selection strategies, whereas those based on selection within progenies showed the largest bias and lower association with the realized gain.
predicted genetic gain; realized genetic gain; recurrent selection
To assess the extent of measurement error bias due to methods used to allocate nursing staff to the acute care inpatient setting and to recommend estimation methods designed to overcome this bias.
Data Sources/Study Setting
Secondary data obtained from the California Office of Statewide Health Planning and Development (OSHPD) and the Centers for Medicare and Medicaid Services' Healthcare Cost Report Information System for 279 general acute care hospitals from 1996 to 2001.
California OSHPD provides detailed nurse staffing data for acute care inpatients. We estimate the measurement error and the resulting bias from applying different staffing allocation methods. Estimates of the measurement errors also allow insights into the best choices for alternate estimation strategies.
The bias induced by the adjusted patient days method (and its modification) is smaller than for other methods, but the bias is still substantial: in the benchmark simple regression model, the estimated coefficient for staffing level on quality of care is expected to be one-third smaller than its true value (and the bias is larger in a multiple regression model). Instrumental variable estimation, using one staffing allocation measure as an instrument for another, addresses this bias, but only particular choices of staffing allocation measures and instruments are suitable.
Staffing allocation methods induce substantial attenuation bias, but there are easily implemented estimation methods that overcome this bias.
Nurse staffing; research methodologies; measurement error
In order to study the molecular biological differences between normal and diseased tissues,
it is desirable to perform classification among diseases and stages of disease using
microarray-based gene-expression values. Owing to the limited number of microarrays
typically used in these studies, serious issues arise with respect to the design, performance
and analysis of classifiers based on microarray data. This paper reviews some fundamental
issues facing small-sample classification: classification rules, constrained classifiers, error
estimation and feature selection. It discusses both unconstrained and constrained classifier
design from sample data, and the contributions to classifier error from constrained
optimization and lack of optimality owing to design from sample data. The difficulty with
estimating classifier error when confined to small samples is addressed, particularly
estimating the error from training data. The impact of small samples on the ability to
include more than a few variables as classifier features is explained.
Four applications of permutation tests to the single-mediator model are described and evaluated in this study. Permutation tests work by rearranging data in many possible ways in order to estimate the sampling distribution for the test statistic. The four applications to mediation evaluated here are the permutation test of ab, the permutation joint significance test, and the noniterative and iterative permutation confidence intervals for ab. A Monte Carlo simulation study was used to compare these four tests with the four best available tests for mediation found in previous research: the joint significance test, the distribution of the product test, and the percentile and bias-corrected bootstrap tests. We compared the different methods on Type I error, power, and confidence interval coverage. The noniterative permutation confidence interval for ab was the best performer among the new methods. It successfully controlled Type I error, had power nearly as good as the most powerful existing methods, and had better coverage than any existing method. The iterative permutation confidence interval for ab had lower power than do some existing methods, but it performed better than any other method in terms of coverage. The permutation confidence interval methods are recommended when estimating a confidence interval is a primary concern. SPSS and SAS macros that estimate these confidence intervals are provided.
Mediation; Permutation test
For the last eight years, microarray-based class prediction has been the subject of numerous publications in medicine, bioinformatics and statistics journals. However, in many articles, the assessment of classification accuracy is carried out using suboptimal procedures and is not paid much attention. In this paper, we carefully review various statistical aspects of classifier evaluation and validation from a practical point of view. The main topics addressed are accuracy measures, error rate estimation procedures, variable selection, choice of classifiers and validation strategy.
accuracy measures; classification; conditional and unconditional error rate; error rate estimation; validation data; variable selection; gene expression; high-dimensional data
DNA microarray technology provides a promising approach to the diagnosis and prognosis of tumors on a genome-wide scale by monitoring the expression levels of thousands of genes simultaneously. One problem arising from the use of microarray data is the difficulty to analyze the high-dimensional gene expression data, typically with thousands of variables (genes) and much fewer observations (samples), in which severe collinearity is often observed. This makes it difficult to apply directly the classical statistical methods to investigate microarray data. In this paper, total principal component regression (TPCR) was proposed to classify human tumors by extracting the latent variable structure underlying microarray data from the augmented subspace of both independent variables and dependent variables. One of the salient features of our method is that it takes into account not only the latent variable structure but also the errors in the microarray gene expression profiles (independent variables). The prediction performance of TPCR was evaluated by both leave-one-out and leave-half-out cross-validation using four well-known microarray datasets. The stabilities and reliabilities of the classification models were further assessed by re-randomization and permutation studies. A fast kernel algorithm was applied to decrease the computation time dramatically. (MATLAB source code is available upon request.)
Penalized regression incorporating prior dependency structure of predictors can be effective in high-dimensional data analysis (Li and Li 2008). Pan, Xie and Shen (2010) proposed a penalized regression method for better outcome prediction and variable selection by smoothing parameters over a given predictor network, which can be applied to analysis of microarray data with a given gene network. In this paper, we develop two modifications to their method for further performance enhancement. First, we employ convex programming and show its improved performance over an approximate optimization algorithm implemented in their original proposal. Second, we perform bias reduction after initial variable selection through a new penalty, leading to better parameter estimates and outcome prediction. Simulations have demonstrated substantial performance improvement of the proposed modifications over the original method.
Fused Lasso; Gene networks; Group variable selection; Lasso; Lγ-norm; L∞-norm; Microarray gene expression
For the last eight years, microarray-based classification has been a major topic in statistics, bioinformatics and biomedicine research. Traditional methods often yield unsatisfactory results or may even be inapplicable in the so-called "p ≫ n" setting where the number of predictors p by far exceeds the number of observations n, hence the term "ill-posed-problem". Careful model selection and evaluation satisfying accepted good-practice standards is a very complex task for statisticians without experience in this area or for scientists with limited statistical background. The multiplicity of available methods for class prediction based on high-dimensional data is an additional practical challenge for inexperienced researchers.
In this article, we introduce a new Bioconductor package called CMA (standing for "Classification for MicroArrays") for automatically performing variable selection, parameter tuning, classifier construction, and unbiased evaluation of the constructed classifiers using a large number of usual methods. Without much time and effort, users are provided with an overview of the unbiased accuracy of most top-performing classifiers. Furthermore, the standardized evaluation framework underlying CMA can also be beneficial in statistical research for comparison purposes, for instance if a new classifier has to be compared to existing approaches.
CMA is a user-friendly comprehensive package for classifier construction and evaluation implementing most usual approaches. It is freely available from the Bioconductor website at .
This article investigates the effects of measurement error on the estimation of nonparametric variance functions. We show that either ignoring measurement error or direct application of the simulation extrapolation, SIMEX, method leads to inconsistent estimators. Nevertheless, the direct SIMEX method can reduce bias relative to a naive estimator. We further propose a permutation SIMEX method which leads to consistent estimators in theory. The performance of both SIMEX methods depends on approximations to the exact extrapolants. Simulations show that both SIMEX methods perform better than ignoring measurement error. The methodology is illustrated using microarray data from colon cancer patients.
Heteroscedasticity; Local polynomial regression; Measurement error; Microarray; Nonparametric regression; Permutation; Simulation-extrapolation; Variance function estimation
The aim of many microarray experiments is to build discriminatory diagnosis and prognosis models. Given the huge number of features and the small number of examples, model validity which refers to the precision of error estimation is a critical issue. Previous studies have addressed this issue via the deviation distribution (estimated error minus true error), in particular, the deterioration of cross-validation precision in high-dimensional settings where feature selection is used to mitigate the peaking phenomenon (overfitting). Because classifier design is based upon random samples, both the true and estimated errors are sample-dependent random variables, and one would expect a loss of precision if the estimated and true errors are not well correlated, so that natural questions arise as to the degree of correlation and the manner in which lack of correlation impacts error estimation. We demonstrate the effect of correlation on error precision via a decomposition of the variance of the deviation distribution, observe that the correlation is often severely decreased in high-dimensional settings, and show that the effect of high dimensionality on error estimation tends to result more from its decorrelating effects than from its impact on the variance of the estimated error. We consider the correlation between the true and estimated errors under different experimental conditions using both synthetic and real data, several feature-selection methods, different classification rules, and three error estimators commonly used (leave-one-out cross-validation, -fold cross-validation, and .632 bootstrap). Moreover, three scenarios are considered: (1) feature selection, (2) known-feature set, and (3) all features. Only the first is of practical interest; however, the other two are needed for comparison purposes. We will observe that the true and estimated errors tend to be much more correlated in the case of a known feature set than with either feature selection or using all features, with the better correlation between the latter two showing no general trend, but differing for different models.
Critical to the development of molecular signatures from microarray and other high-throughput data is testing the statistical significance of the produced signature in order to ensure its statistical reproducibility. While current best practices emphasize sufficiently powered univariate tests of differential expression, little is known about the factors that affect the statistical power of complex multivariate analysis protocols for high-dimensional molecular signature development.
We show that choices of specific components of the analysis (i.e., error metric, classifier, error estimator and event balancing) have large and compounding effects on statistical power. The effects are demonstrated empirically by an analysis of 7 of the largest microarray cancer outcome prediction datasets and supplementary simulations, and by contrasting them to prior analyses of the same data.
The findings of the present study have two important practical implications: First, high-throughput studies by avoiding under-powered data analysis protocols, can achieve substantial economies in sample required to demonstrate statistical significance of predictive signal. Factors that affect power are identified and studied. Much less sample than previously thought may be sufficient for exploratory studies as long as these factors are taken into consideration when designing and executing the analysis. Second, previous highly-cited claims that microarray assays may not be able to predict disease outcomes better than chance are shown by our experiments to be due to under-powered data analysis combined with inappropriate statistical tests.
Gene set enrichment analysis (GSEA) is an analytic approach which simultaneously reduces the dimensionality of microarray data and enables ready inference of the biological meaning of observed gene expression patterns. Here we invert the GSEA process to identify class-specific gene signatures. Because our approach uses the Kolmogorov-Smirnov approach both to define class specific signatures and to classify samples using those signatures, we have termed this methodology “Dual-KS” (DKS).
The optimum gene signature identified by the DKS algorithm was smaller than other methods to which it was compared in 5 out of 10 datasets. The estimated error rate of DKS using the optimum gene signature was smaller than the estimated error rate of the random forest method in 4 out of the 10 datasets, and was equivalent in two additional datasets. DKS performance relative to other benchmarked algorithms was similar to its performance relative to random forests.
DKS is an efficient analytic methodology that can identify highly parsimonious gene signatures useful for classification in the context of microarray studies. The algorithm is available as the dualKS package for R as part of the bioconductor project.
gene set enrichment analysis (GSEA); gene expression; DKS algorithm; gene signatures
There has been increasing interest in predicting patients’ survival after therapy by investigating gene expression microarray data. In the regression and classification models with high-dimensional genomic data, boosting has been successfully applied to build accurate predictive models and conduct variable selection simultaneously. We propose the Buckley-James boosting for the semiparametric accelerated failure time models with right censored survival data, which can be used to predict survival of future patients using the high-dimensional genomic data. In the spirit of adaptive LASSO, twin boosting is also incorporated to fit more sparse models. The proposed methods have a unified approach to fit linear models, non-linear effects models with possible interactions. The methods can perform variable selection and parameter estimation simultaneously. The proposed methods are evaluated by simulations and applied to a recent microarray gene expression data set for patients with diffuse large B-cell lymphoma under the current gold standard therapy.
boosting; accelerated failure time model; Buckley-James estimator; censored survival data; LASSO; variable selection
Multiclass classification of microarray data samples with a reduced number of genes is a rich and challenging problem in Bioinformatics research. The problem gets harder as the number of classes is increased. In addition, the performance of most classifiers is tightly linked to the effectiveness of mandatory gene selection methods. Critical to gene selection is the availability of estimates about the maximum number of genes that can be handled by any classification algorithm. Lack of such estimates may lead to either computationally demanding explorations of a search space with thousands of dimensions or classification models based on gene sets of unrestricted size. In the former case, unbiased but possibly overfitted classification models may arise. In the latter case, biased classification models unable to support statistically significant findings may be obtained.
A novel bound on the maximum number of genes that can be handled by binary classifiers in binary mediated multiclass classification algorithms of microarray data samples is presented. The bound suggests that high-dimensional binary output domains might favor the existence of accurate and sparse binary mediated multiclass classifiers for microarray data samples.
A comprehensive experimental work shows that the bound is indeed useful to induce accurate and sparse multiclass classifiers for microarray data samples.
The selection of the reference to scale the data in a copy number analysis has paramount importance to achieve accurate estimates. Usually this reference is generated using control samples included in the study. However, these control samples are not always available and in these cases, an artificial reference must be created. A proper generation of this signal is crucial in terms of both noise and bias.
We propose NSA (Normality Search Algorithm), a scaling method that works with and without control samples. It is based on the assumption that genomic regions enriched in SNPs with identical copy numbers in both alleles are likely to be normal. These normal regions are predicted for each sample individually and used to calculate the final reference signal. NSA can be applied to any CN data regardless the microarray technology and preprocessing method. It also finds an optimal weighting of the samples minimizing possible batch effects.
Five human datasets (a subset of HapMap samples, Glioblastoma Multiforme (GBM), Ovarian, Prostate and Lung Cancer experiments) have been analyzed. It is shown that using only tumoral samples, NSA is able to remove the bias in the copy number estimation, to reduce the noise and therefore, to increase the ability to detect copy number aberrations (CNAs). These improvements allow NSA to also detect recurrent aberrations more accurately than other state of the art methods.
NSA provides a robust and accurate reference for scaling probe signals data to CN values without the need of control samples. It minimizes the problems of bias, noise and batch effects in the estimation of CNs. Therefore, NSA scaling approach helps to better detect recurrent CNAs than current methods. The automatic selection of references makes it useful to perform bulk analysis of many GEO or ArrayExpress experiments without the need of developing a parser to find the normal samples or possible batches within the data. The method is available in the open-source R package NSA, which is an add-on to the aroma.cn framework.
SNPs; Normalization; Scaling; High-throughput data; Batch effects
The antibody microarray is a powerful chip-based technology for profiling hundreds of proteins simultaneously and is used increasingly nowadays. To study humoral response in pancreatic cancers, Patwa et al. (2007) developed a two-dimensional liquid separation technique and built a two-dimensional antibody microarray. However, identifying differential expression regions on the antibody microarray requires the use of appropriate statistical methods to fairly assess the large amounts of data generated. In this paper, we propose a permutation-based test using spatial information of the two-dimensional antibody microarray. By borrowing strength from the neighboring differentially expressed spots, we are able to detect the differential expression region with very high power controlling type I error at 0.05 in our simulation studies. We also apply the proposed methodology to a real microarray dataset.
Antibody Microarray; Permutation; Spatial information
This article focuses on variable selection for partially linear models when the covariates are measured with additive errors. We propose two classes of variable selection procedures, penalized least squares and penalized quantile regression, using the nonconvex penalized principle. The first procedure corrects the bias in the loss function caused by the measurement error by applying the so-called correction-for-attenuation approach, whereas the second procedure corrects the bias by using orthogonal regression. The sampling properties for the two procedures are investigated. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle procedure (Fan and Li 2001). Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application.
Errors-in-variable; Error-free; Error-prone; Local linear regression; Quantile regression; SCAD