DATA MINING APPROACHES TO MULTIVARIATE BIOMARKER DISCOVERY

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1 DATA MINING APPROACHES TO MULTIVARIATE BIOMARKER DISCOVERY DARIUS M. DZIUDA Department of Mathematical Sciences Central Connecticut State University New Britain, USA Many biomarker discovery studies apply statistical and data mining approaches that are inappropriate for typical data sets generated by current high-throughput genomic and proteomic technologies. More sophisticated statistical methods should be used and combined with appropriate validation of their results as well as with methods allowing for linking biomarkers with existing or new biomedical knowledge. Although various supervised learning algorithms can be used to identify multivariate biomarkers, such biomarkers are often unstable and do not necessarily provide the insight into biological processes underlying phenotypic differences. In this paper, we will outline common misconceptions in biomarker discovery studies, and then focus on the methods and concepts leading to parsimonious multivariate biomarkers that are stable and biologically interpretable. First, we will describe the Informative Set of Genes defined as the set containing all of the information significant for class differentiation. Then, we will present the Modified Bagging Schema generating ensembles of classifiers that are used to vote for variables. Finally, we will show how to combine these methods to identify robust multivariate biomarkers with plausible biological interpretation. 1. Introduction Biomarker discovery studies based on current high-throughput genomic and proteomic technologies analyze data sets with thousands of variables and much fewer biological samples. To successfully deal with such data, bioinformaticians or biomedical researchers have to be familiar with intricacies of multivariate analysis, and particularly with approaches allowing to overcome the curse of dimensionality. One of the common misconceptions is to limit biomarker discovery studies to a univariate approach. Such an approach is a relic of the old one-gene-at-atime paradigm, and is based on the unrealistic assumption that gene expression (or protein expression) variables are independent of one another. Although it is possible that expression of a single gene may be all that is necessary to efficiently separate phenotypic classes, such situations are exceptions rather than a rule. In most situations, the assumptions of no correlations and no interactions among genes is unwarranted. Furthermore, limiting biomarker discovery investigations to some number of top univariately significant variables may remove from consideration the variables that are very important for class discrimination when their expression pattern is combined with that of some 1

2 other variables. 1,2 Such complementary combinations of variables can be identified only via multivariate approaches. Another quite common misconception is to preprocess training data using unsupervised methods, such as clustering or principal component analysis. The idea behind this approach is to reduce dimensionality of the problem by replacing the original variables by their subset or by their combinations. However, in an unsupervised environment there is no way to determine how much of the important discriminatory information is removed by such dimensionality reduction. For example, principal component analysis identifies directions of the most variation in the data. Such directions may be very different from the direction that best separates the differentiated classes. The latter can be identified only using supervised methods. 1,3 Nevertheless, it has to be stressed out that applying supervised learning algorithms indiscriminately even if they are powerful methods, such as support vector machines or discriminant analysis may lead to results that are due to random chance. In a multidimensional space of many variables, it is quite easy to perfectly or nearly perfectly separate sparse hyperareas representing different phenotypes. However, such results may just overfit the training data, and be useless as diagnostic, prognostic, or any other biomarkers. To avoid (or at least to minimize the danger of) overfitting the training data, we should look for the characteristic and repeatable expressions patterns, which when combined into diverse multivariate biomarkers can significantly separate the phenotypic classes. In the following sections, we will first describe how to use supervised feature selection to identify the Informative Set of Genes, which we define as the set containing all of the information significant for class differentiation. Then, we will present the Modified Bagging Schema, which utilizes the bootstrap and the ensemble approach paradigms to identify expression patterns that are most important for the class differentiation. Finally, we will show how to combine these methods to identify robust multivariate biomarkers with plausible biological interpretation. 2. The Informative Set of Genes By applying a heuristic approach to multivariate feature selection, we can identify a small subset of variables, which constitute a multivariate biomarker that can separate the differentiated phenotypic classes. However, such a single biomarker may be prone to overfitting, and it rarely provides sufficient insight into biological processes associated with class differences. To be able to find more robust biomarkers, and to have a better starting point for biological interpretation of the class differences, we identify the informative set of 2

3 variables. If we focus on gene expression data, we will call such a set the Informative Set of Genes. The Informative Set of Genes is defined as the set of genes whose expression data contain all of the information significant for the differentiation of the phenotypic classes represented in the training data. 1 To identify this set, we use heuristic feature selection to generate a sequence of multivariate biomarkers. After the first biomarker is found, its variables are removed from the training data, and the second alternative biomarker is identified. Then, its variables are also removed, and the next alternative marker is identified. This process continues until no alternative biomarkers with satisfactory discriminatory power can be identified (that is, when the remaining training data no longer includes any significant discriminatory information). The variables selected into the first biomarker and to all identified alternative biomarkers constitute the Informative Set of Genes. Since the Informative Set of Genes includes all significant discriminatory information, we assume that it also includes all gene expression patterns associated with biological processes underlying class differences. To identify these patterns (and thus facilitate biological interpretation), we may use clustering methods, such as hierarchical clustering or self-organizing maps. Please note that the unsupervised approach is applied here to the Informative Set of Genes, which includes only genes whose expressions have already been determined (by supervised methods) to be associated with the class differences. 3. The Modified Bagging Schema Although the Informative Set of Genes includes all of the significant discriminatory information, it is possible that some of the genes included in some of the alternative biomarkers were selected by random chance. To be able to find robust biomarkers, we need to identify the genes and the expression patterns that are most likely to be associated with the most important biological processes underlying class differences. The method we use is based on randomization of the training data and on the ensemble classifier approach. Bagging (bootstrap aggregating) is a popular method of combining classifiers based on randomized versions of the training data. 4 Typically, this method utilizes Efron s nonparametric bootstrap, 5,6 which makes no assumption about the underlying populations, selects with replacement, and generates randomized (bootstrap) training sets of the same size as the original training data set. However, sampling with replacement may cause problems when feature selection methods (applied to each of the bootstrap training sets) require the independence of training set observations. 3

4 Heuristic feature selection methods - such as stepwise forward selection, stepwise backward elimination, or stepwise hybrid methods that combine the two compare, at each step, discriminatory power of subsets that consist of the same number of variables. Thus, external cross-validation cannot be used for this purpose. The internal cross-validation, which uses the same set of variables and the same data for training and for validation, is unreliable, to say the least. Therefore, we use methods of subset evaluation that are based on a metric of class separation, specifically the Lawley-Hotelling T 2 criterion, which requires the independence of training set observations. To assure this independence, we modify bagging in a way that the randomized training sets are generated without replacement. The Modified Bagging Schema is defined as the algorithm that selects bootstrap training sets by utilizing stratified random sampling without replacement. 1 The parameter driving this selection is a proportion of out-of-bag samples γ OOB, that is, the proportion of the training observations (biological samples) that are not selected into a bootstrap training set. When using the Modified Bagging Schema, we will generate hundreds or thousands of bootstrap training sets. Each of them will include a specified proportion of observations randomly selected without replacement from the original training data. For each of the bootstrap training sets, we perform an independent feature selection, and then build a classifier. For example, if γ OOB = 0.8, then each classifier will be built on eighty percent of randomly selected training observations, with the remaining twenty percent of them constituting the out-of-bag samples that can be used to test the performance of a particular classifier. 4. Identification of Multivariate Biomarkers that are Robust and Biologically Interpretable By combining the randomization and ensemble approach (represented here by the Modified Bagging Schema) with the analysis of expression patterns within the Informative Set of Genes, we define a method that can be interpreted as regularization of feature selection leading to biomarkers that are more robust than a single biomarker selected from the entire training data. After the Informative Set of Genes is identified, we cluster its variables into groups of genes with similar shapes of their expression patterns across all biological samples. Then, we use the Informative Set of Genes as our base training data set, and utilize the Modified Bagging Schema to generate a large number of classifiers (an ensemble) based on bootstrap training sets selected from this base training set. By investigating the distribution of each cluster s genes among these classifiers, we identify the clusters whose genes are most frequently selected 4

5 into the ensemble classifiers. Alternatively, we may limit our investigation only to those of the ensemble classifiers that perfectly (or nearly perfectly) classify their out-of-bag samples. In any case, the clusters whose genes are most frequently used by the classifiers of the ensemble are deemed to be the primary clusters. We hypothesize that they represent the primary expression patterns, that is, the patterns associated with the most important biological processes associated with the differences among the investigated phenotypic classes. Furthermore, we assume that not all genes of the primary clusters are equally important for the class differentiation. Those genes of the primary clusters that are most frequently selected into these classifiers that perfectly or nearly perfectly classify their out-of-bag samples are called the frequent primary genes. They constitute the best starting point for elucidation of biological processes underlying the class differences. They are also the ones that are least likely to be selected by chance into the alternative markers constituting the Informative Set of Genes. Therefore, if we now perform heuristic feature selection based only on the frequent primary genes, the identified multivariate biomarker has the best chance to be robust as well as to have a plausible biological interpretation. References 1. D. M. Dziuda, Data Mining for Genomics and Proteomics: Analysis of Gene and Protein Expression Data, NJ: Wiley, Guyon and A. Elisseeff, "An introduction to variable and feature selection", Journal of Machine Learning Research, vol.3, March 2003, pp D. J. Hand, H. Mannila and P. Smyth, Principles of Data Mining. Cambridge, Mass.: MIT Press, L. Breiman, "Bagging predictors", Machine Learning, vol. 24, August 1996, pp B. Efron, "Bootstrap methods: another look at the jackknife", The Annals of Statistics, vol.7, January 1979, pp B. Efron and R. Tibshirani, An Introduction to the Bootstrap, New York: Chapman & Hall,

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