Trevor Hastie

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

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

Least angle regression

DISCUSSION OF “LEAST ANGLE REGRESSION” BY EFRONET AL.By Jean-Michel Loubes and Pascal MassartUniversit´e Paris-SudThe issue of model selection has drawn the attention of both applied andtheoretical statisticians for a long time. Indeed, there has been an enor-mous range of contribution in model selection proposals, including work byAkaike (1973), Mallows (1973), Foster and George (1994), Birg´e and Mas-sart (2001a) and Abramovich, Benjamini, Donoho and Johnstone (2000).Over the last decade, modern computer-driven methods have been devel-oped such as All Subsets, Forward Selection, Forward Stagewise or Lasso.Such methods are useful in the setting of the standard linear model, wherewe observe noisy data and wish to predict the response variable using onlya few covariates, since they provide automatically linear models that fit thedata. The procedure described in this paper is, on the one hand, numeri-cally very efficient and, on the other hand, very general, since, with slightmodifications, it enables us to recover the estimates given by the Lasso andStagewise.1. Estimation procedure. The “LARS” method is based on a recursiveprocedure selecting, at each step, the covariates having largest absolute cor-relation with the response y. In the case of an orthogonal design, the esti-mates can then be viewed as an lDISCUSSION OF “LEAST ANGLE REGRESSION” BY EFRONET AL.By Berwin A. TurlachUniversity of Western AustraliaI would like to begin by congratulating the authors (referred to belowas EHJT) for their interesting paper in which they propose a new variableselection method (LARS) for building linear models and show how their newmethod relates to other methods that have been proposed recently. I foundthe paper to be very stimulating and found the additional insight that itprovides about the Lasso technique to be of particular interest.My comments center around the question of how we can select linearmodels that conform with the marginality principle [Nelder (1977, 1994)and McCullagh and Nelder (1989)]; that is, the response surface is invariantunder scaling and translation of the explanatory variables in the model.Recently one of my interests was to explore whether the Lasso techniqueor the nonnegative garrote [Breiman (1995)] could be modified such that itincorporates the marginality principle. However, it does not seem to be atrivial matter to change the criteria that these techniques minimize in such away that the marginality principle is incorporated in a satisfactory manner.On the other hand, it seems to be straightforward to modify the LARStechnique to incorporate this principle. In their paper, EHJT address thisissue somewhat in passing when they suggest toward the end of Section 3that one first fit main effects only and interactions in a second step to controlthe order in which variables are allowed to enter the model. However, sucha two-step procedure may have a somewhat less than optimal behavior asthe following, admittedly artificial, example shows.Assume we have a vector of explanatory variables X =(XThe purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principled choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.

Repeated observation of breast tumor subtypes in independent gene expression data sets

Characteristic patterns of gene expression measured by DNA microarrays have been used to classify tumors into clinically relevant subgroups. In this study, we have refined the previously defined subtypes of breast tumors that could be distinguished by their distinct patterns of gene expression. A total of 115 malignant breast tumors were analyzed by hierarchical clustering based on patterns of expression of 534 “intrinsic” genes and shown to subdivide into one basal-like, one ERBB2-overexpressing, two luminal-like, and one normal breast tissue-like subgroup. The genes used for classification were selected based on their similar expression levels between pairs of consecutive samples taken from the same tumor separated by 15 weeks of neoadjuvant treatment. Similar cluster analyses of two published, independent data sets representing different patient cohorts from different laboratories, uncovered some of the same breast cancer subtypes. In the one data set that included information on time to development of distant metastasis, subtypes were associated with significant differences in this clinical feature. By including a group of tumors from BRCA1 carriers in the analysis, we found that this genotype predisposes to the basal tumor subtype. Our results strongly support the idea that many of these breast tumor subtypes represent biologically distinct disease entities.

Diagnosis of multiple cancer types by shrunken centroids of gene expression

We have devised an approach to cancer class prediction from gene expression profiling, based on an enhancement of the simple nearest prototype (centroid) classifier. We shrink the prototypes and hence obtain a classifier that is often more accurate than competing methods. Our method of “nearest shrunken centroids” identifies subsets of genes that best characterize each class. The technique is general and can be used in many other classification problems. To demonstrate its effectiveness, we show that the method was highly efficient in finding genes for classifying small round blue cell tumors and leukemias.

Missing value estimation methods for DNA microarrays

MOTIVATION Gene expression microarray experiments can generate data sets with multiple missing expression values. Unfortunately, many algorithms for gene expression analysis require a complete matrix of gene array values as input. For example, methods such as hierarchical clustering and K-means clustering are not robust to missing data, and may lose effectiveness even with a few missing values. Methods for imputing missing data are needed, therefore, to minimize the effect of incomplete data sets on analyses, and to increase the range of data sets to which these algorithms can be applied. In this report, we investigate automated methods for estimating missing data. RESULTS We present a comparative study of several methods for the estimation of missing values in gene microarray data. We implemented and evaluated three methods: a Singular Value Decomposition (SVD) based method (SVDimpute), weighted K-nearest neighbors (KNNimpute), and row average. We evaluated the methods using a variety of parameter settings and over different real data sets, and assessed the robustness of the imputation methods to the amount of missing data over the range of 1--20% missing values. We show that KNNimpute appears to provide a more robust and sensitive method for missing value estimation than SVDimpute, and both SVDimpute and KNNimpute surpass the commonly used row average method (as well as filling missing values with zeros). We report results of the comparative experiments and provide recommendations and tools for accurate estimation of missing microarray data under a variety of conditions.

Estimating the number of clusters in a data set via the gap statistic

We propose a method (the ‘gap statistic’) for estimating the number of clusters (groups) in a set of data. The technique uses the output of any clustering algorithm (e.g. K-means or hierarchical), comparing the change in within-cluster dispersion with that expected under an appropriate reference null distribution. Some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature.

A working guide to boosted regression trees

1. Ecologists use statistical models for both explanation and prediction, and need techniques that are flexible enough to express typical features of their data, such as nonlinearities and interactions. 2. This study provides a working guide to boosted regression trees (BRT), an ensemble method for fitting statistical models that differs fundamentally from conventional techniques that aim to fit a single parsimonious model. Boosted regression trees combine the strengths of two algorithms: regression trees (models that relate a response to their predictors by recursive binary splits) and boosting (an adaptive method for combining many simple models to give improved predictive performance). The final BRT model can be understood as an additive regression model in which individual terms are simple trees, fitted in a forward, stagewise fashion. 3. Boosted regression trees incorporate important advantages of tree-based methods, handling different types of predictor variables and accommodating missing data. They have no need for prior data transformation or elimination of outliers, can fit complex nonlinear relationships, and automatically handle interaction effects between predictors. Fitting multiple trees in BRT overcomes the biggest drawback of single tree models: their relatively poor predictive performance. Although BRT models are complex, they can be summarized in ways that give powerful ecological insight, and their predictive performance is superior to most traditional modelling methods. 4. The unique features of BRT raise a number of practical issues in model fitting. We demonstrate the practicalities and advantages of using BRT through a distributional analysis of the short-finned eel (Anguilla australis Richardson), a native freshwater fish of New Zealand. We use a data set of over 13 000 sites to illustrate effects of several settings, and then fit and interpret a model using a subset of the data. We provide code and a tutorial to enable the wider use of BRT by ecologists.

Statistical Models in S

The interactive data analysis and graphics language S (Becker, Chambers and Wilks, 1988) has become a popular environment for both data analysts and research statisticians. A common complaint, however, has concerned the lack of statistical modeling tools, such as those provided by GLIM© or GENSTAT©.

Sparse Principal Component Analysis

Principal component analysis (PCA) is widely used in data processing and dimensionality reduction. However, PCA suffers from the fact that each principal component is a linear combination of all the original variables, thus it is often difficult to interpret the results. We introduce a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings. We first show that PCA can be formulated as a regression-type optimization problem; sparse loadings are then obtained by imposing the lasso (elastic net) constraint on the regression coefficients. Efficient algorithms are proposed to fit our SPCA models for both regular multivariate data and gene expression arrays. We also give a new formula to compute the total variance of modified principal components. As illustrations, SPCA is applied to real and simulated data with encouraging results.

An Introduction to Statistical Learning

Statistics An Intduction to Stistical Lerning with Applications in R An Introduction to Statistical Learning provides an accessible overview of the fi eld of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fi elds ranging from biology to fi nance to marketing to astrophysics in the past twenty years. Th is book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classifi cation, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fi elds, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical soft ware platform. Two of the authors co-wrote Th e Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. Th is book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. Th e text assumes only a previous course in linear regression and no knowledge of matrix algebra.