Selecting dissimilar genes for multi-class classification, an application in cancer subtyping
© Cai et al; licensee BioMed Central Ltd. 2007
Received: 28 January 2007
Accepted: 16 June 2007
Published: 16 June 2007
Gene expression microarray is a powerful technology for genetic profiling diseases and their associated treatments. Such a process involves a key step of biomarker identification, which are expected to be closely related to the disease. A most important task of these identified genes is that they can be used to construct a classifier which can effectively diagnose disease and even recognize the disease subtypes. Binary classification, for example, diseased or healthy, in microarray data analysis has been successful, while multi-class classification, such as cancer subtyping, remains challenging.
We target on the challenging multi-class classification in microarray data analysis, especially on the cancer subtyping using gene expression microarray. We present a novel class discrimination strength vector to represent individual genes and introduce a new measurement to quantify the class discrimination strength difference between two genes. Such a new distance measure is employed in gene clustering, and subsequently the gene cluster information is exploited to select a set of genes which can be used to construct a sample classifier.
We tested our method on four real cancer microarray datasets each contains multiple subtypes of cancer patients. The experimental results show that the constructed classifiers all achieved a higher classification accuracy than the previously best classification results obtained on these four datasets. Additional tests show that the selected genes by our method are less correlated and they all contribute statistically significantly to the more accurate cancer subtyping.
The proposed novel class discrimination strength vector is a better representation than the gene expression vector, in the sense that it can be used to effectively eliminate highly correlated but redundant genes for classifier construction. Such a method can build a classifier to achieve a higher classification accuracy, which is demonstrated via cancer subtyping.
DNA microarray technology enables the measurement of expression levels of thousands of genes simultaneously. This unique feature has a fundamental role in a wide range of current biological and medical research. One of the most common applications is to compare the gene expression levels in tissues extracted from different conditions, such as healthy versus diseased tissues, and thus to characterize the genetic profiles of these conditions. Two main genetic profiling tasks have been investigated extensively in the past two decades: identification of biomarker genes, which are the genes differentially expressed under various experimental conditions; and the construction of classifiers based on these identified genes, to effectively recognize the experimental conditions. For example, recognizing the primary anatomical site of tumor origin is a fundamental requirement for optimal treatment for cancer patients.
Early research on genetic profiling employed gene and sample class simultaneous clustering to recognize expression patterns associated with samples inside a class, as well as those genes whose expression levels characterize the class [1, 2]. Nevertheless, in general, among the thousands of genes examined, only a small number of them are significantly associated with the experimental conditions. These genes, or biomarkers, would have their expression levels increase or decrease under certain conditions, compared to normal expression levels. Such discriminatory genes are very important in genetic disease studies as they often are set as targets for drug design [3–5]. However, identification of discriminatory genes is not an easy task, because there are usually only a small number of experiments (chips, arrays, or samples) available for a typical application, largely due to the high experimental cost. The huge number of genes versus a tiny number of experiments is a familiar machine learning challenge, often labeled as "the curse of dimensionality".
There are many algorithms proposed in the past two decades for gene selection, most of which directly extend ideas in the more general problem of feature selection/extraction, which is a long-standing research topic in statistics and machine learning. Nevertheless, one should keep in mind that gene selection in gene expression microarray data analysis differs from the general feature selection in many aspects, such as the dimensionality issue and the subsequent data overfitting; consequently, many general feature selection algorithms will not work satisfactorily on microarray data. On the other hand, gene selection is closely related to the other main task in genetic profiling: to build a good sample classifier. Besides validating selected genes via other biological experiments, for example in which they are set as drug targets, another way to confirm that the selected genes are biologically correlated to various experimental conditions is to use them to build a sample classifier and to test its classification accuracy. Clearly, high classification accuracy suggests the good quality of the selected genes. In fact, much research has followed this rule to compare their gene selection methods, in addition to considering the detailed biological annotations for the selected genes.
To name a few gene selection methods, Golub et al.  developed a measure of correlation that emphasizes the "signal-to-noise" (S2N) ratio in using the gene as a class membership (acute myeloid leukemia or acute lymphoblastic leukemia) predictor, and selected a number of top ranked genes as discriminatory genes. The S2N ratio captures the basic rule of gene selection: a discriminatory gene must have close expression levels in samples within a class, but significantly different expression levels in samples across different classes. Other approaches that follow the same rule, with certain modification and refinement, include t-test , regularized t-test , supervised gene shaving , a method focusing on expression homogeneity in a certain class , a method which uses both small within-class expression variance and large between-class expression variance , as well as many other so-called single gene scoring or univariate methods.
To further exploit the correlations amongst genes, Xiong et al.  first ranked genes using their individual classification accuracy and then selected a subset of genes with an overall near maximal classification accuracy through sequential (floating) forward selections (SFS, SFFS). Guyon et al.  did not rank genes but suggested a gene selection method to use support vector machines (SVMs) combined with recursive feature elimination (RFE) to select a subset of genes with an overall near maximal classification accuracy. Li et al.  combined a genetic algorithm (GA) and a k-nearest neighbor (KNN) method to identify a fixed number of genes that can correctly classify binary samples, based on the occurrence frequency of the gene in many gene subsets. Lee et al.  and Zhou et al.  proposed to first constrain the number of genes to be selected and then use a Markov Chain Monte Carlo (MCMC) based stochastic search algorithm to discover important genes. Shevade and Keerthi  proposed a logistic regression for class membership estimation based on a linear combination of all the genes, then further reduced the density by setting up a sensitivity threshold to prune genes. The resulting regression model is thus sparse and the remaining genes are considered informative. Díaz-Uriarte and Alvarez de Andres  defined a measure of gene importance with the random forest and iteratively removed the gene with the smallest importance until the smallest out-of-bag error rate is yielded.
Jaeger et al.  proposed a pre-filtering approach where fuzzy clustering was applied, using the gene expression levels across all samples, followed by selecting a varying number of representatives per cluster.
These representative genes were then ranked by the aforementioned t-test to select the top-ranked as feature genes. A similar approach was proposed by Hanczar et al.  to cluster genes using their expression levels across all samples and then to represent the clusters using their mean expression levels (called prototype genes); this was followed with single gene scoring methods on the prototype genes to select a pre-specified number of top-ranked genes. Noticing that in general the gene clustering algorithms need the number of clusters as an input parameter (which is often difficult to tune), the HykGene proposed by Wang et al.  first applied single gene scoring methods to select a set of top-ranked genes, then performed a hierarchical clustering on them, and lastly selected one representative per cluster to form the final marker genes. The number of clusters in HykGene was determined using the leave-one-out cross validation classification accuracy.
It is worth pointing out that most of the proposed gene selection methods are classifier independent, or they can be combined with any classification methods, such as linear discrimination analysis [4, 7, 8, 14], nearest neighbor models [4, 5, 10, 14, 17], support vector machines [5, 8, 9, 14–17], and logistic regression models . These gene selection methods typically produce a small set of biomarker genes which can be used in classifier construction. However, there are some other gene selection methods which are bound with specific classifiers [11–14].
It is also equally important to point out that most of these gene selection methods, and the associated classifiers, work only on two-class datasets, though through one-versus-all (OVA) they can be theoretically extended to multi-class datasets. The methods that have been explicitly tested on multi-class dataset(s) include [4, 5, 7, 14, 16, 17]. In addition, Ramaswamy et al. , Su et al. , and Pomeroy et al.  specifically dealt with classification of multiple classes of human tumors through identification of a set of tumor genes (see Results for more details).
In the context of gene selection and the subsequent classification, the performance of one method is normally validated through a training stage that tunes the parameters in the method, followed by a testing stage which estimates the quality of the selected genes and the performance of the resultant classifier. For some of the above mentioned methods, such as the ones in  and , a given gene expression dataset is partitioned into two parts, one called training dataset and the other testing dataset. The class memberships of training samples are used in the training process while the class memberships of testing samples are blinded to the classifier for estimating its classification accuracy. Other methods adopt cross validation schemes for performance evaluation. There are two popular cross validation schemes, one is ℓ-fold and the other is leave-one-out (LOOCV). In ℓ-fold cross validation [4, 5, 7], the whole given dataset is (randomly) partitioned into ℓ equal parts, and (ℓ - 1) parts of them are used to form the training dataset while the other one to form the testing dataset; the process is done when every part has been used as a testing dataset. The classification accuracy is defined as the ratio between the number of correctly predicted samples and the total number of testing samples. Most of the methods that adopt this cross validation scheme repeat the (random) partition several times and the average classification accuracy is reported. However, most of the methods, for example, Ramaswamy et al. , Su et al. , and Wang et al. , adopt the LOOCV scheme, in which only one sample is used as the testing sample while all the others are used to form the training dataset; the scheme goes over every sample in the given dataset. Again, the ratio between the number of correctly predicted samples and the total number of samples is defined as the classification accuracy. In this work, on each real cancer microarray dataset, we adopt the same cross validation scheme to the methods we are comparing with. We also report the LOOCV results on all the datasets and some 5-fold cross validation results.
While binary classification has been extensively explored, multi-class classification remains challenging in microarray data analysis. In this work, we focus on gene selection for the multi-class classification and we demonstrate the strength of the proposed method by applying it on cancer subtype identification. Our main goal is to select genes that, as a whole, have superior class discrimination strength. To this purpose, for each gene we define its class discrimination strength vector, and based on these vectors we measure the similarity between two genes. Such a similarity measurement is adopted in the k-means algorithm to cluster genes. Subsequently, a single gene scoring method is used to rank all the genes. Our method then walks through this ordered gene list and picks up one gene per cluster until a pre-specified number is reached. These selected genes are then used to construct a classifier whose performance is evaluated and measured by the LOOCV (and 5-fold cross validation) classification accuracy. Through experiments on four real multi-class human cancer microarray datasets, we demonstrate that our method can achieve significantly higher classification accuracies than the best previously published. In this sense, our method can serve as a useful addition to existing methods for highly accurate human tumor classification.
Assume in the given multi-class microarray dataset there are n samples on p genes, and these n samples belong to m classes. In the following, we define a novel vector representation for genes, which can differentiate their class recognition strength.
Let g ij denote the expression level of the i-th gene in the j-th sample. That is, in the gene expression matrix, each row represents a gene, G i = ⟨gi 1, gi 2,...,g in ⟩, and each column represents a sample. For the i-th gene, its mean expression level in the k-th class is denoted as h ik , for k = 1, 2,...,m. The value |h ik - h il | captures the difference between the mean expression levels of the i-th gene in the k-th class and in the l-th class. Obviously, if this value is small, then the i-th gene would not be effective in discriminating samples from these two classes, but it could be effective otherwise. Therefore, we define the class discrimination strength vector for the i-th gene as
H i = ⟨|hi 1- hi 2|, |hi 1- hi 3|,...,|hi 1- h im |, |hi 2- hi 3|, |hi 2- hi 4|,...,|hi 2- h im |, |hi 3- hi 4|,...,|hi,m-1- h im |.
to be the distance between the i1-th and the i2-th genes.
We can calculate the Euclidean distance by Eq. (1) between every pair of genes, and then call the k-means algorithm to cluster genes, where the default value for the number of gene clusters k is 160 (such a value is set based on extensive empirical study, see Discussion). Essentially, k-means is a centroid-based clustering algorithm that partitions the genes into k clusters based on their pairwise distances, to ensure that intra-cluster similarity is high and inter-cluster similarity is low.
At the same time, we run a gene ranking method to sort all the genes into a decreasing order of their class discrimination strength. Such a method essentially assigns a score for each gene, which approximates its class discrimination strength. The gene scoring functions can be the classification accuracy of individual genes , or can be designed to capture the basic rule that discriminatory genes are those having close expression levels in samples in a common class but significantly different expression levels in samples from different classes [4–7]. The latter category of gene ranking methods are also called single gene scoring methods, which do not consider the correlation between genes when assigning the scores. In our study, we adopt the F-test [4, 6] and the GS method  as our base gene ranking methods.
Walking through the gene order and using the gene cluster information obtained by the k-means algorithm, our method picks up a pre-specified number of genes under the constraint that at most T genes per cluster are included. For ease of presentation, we call our gene selection method, which targets at selecting genes having dissimilar class discrimination strength, the Disc-based method, and use Disc-F-test and Disc-GS to denote the facts that the base gene ranking method is the F-test and the GS method, respectively. We remark that the Disc-based method is generic, in that it can use any other single gene scoring method, such as the Cho's gene ranking method, to create the Disc-Cho's gene selection method.
To computationally evaluate the quality of the selected genes as a whole, we use the classification accuracy of the classifier built on the selected genes, under the LOOCV scheme. In other words, for each gene selection method (in this paper, they are F-test, GS, Disc-F-test, and Disc-GS), the selected genes are used in the k nearest neighbor model or the support vector machine model to construct a classifier (the KNN-classifier and the SVM-classifier, respectively), and then use the classifier to predict the class memberships of the testing samples. The percentage of correctly predicted memberships is defined as the classification accuracy of the classifier. Essentially, the KNN-classifier  predicts the class membership of a testing sample by a majority vote using its k nearest neighbors in the training dataset; the linear kernel SVM-classifier [21, 22] finds decision planes to best separate the set of training samples having different class memberships, then uses these planes to predict the class memberships of the testing samples. For each class membership prediction, we have also calculated the associated confidence score as follows. Su et al.  adopt the Dixon metric to assign confidence scores, where the distances between the testing sample and all the training samples are calculated, then the "class distance" between the testing sample and a class is derived to be the average distance over all the distances between the testing sample and all the training samples in that particular class. These class distances are then sorted in increasing order, say for example, D1 ≤ D2 ≤...≤ D m where m is the number of classes, to compute the value c = (D2 - D1)/(D m - D1) which is the confidence value for assigning the closest class to the testing sample. A Dixon threshold of 0.1 is generally accepted as conservative boundary for high confidence prediction. Associated with our KNN-classifier for k = 5, assuming the closest 5 neighbors for the testing sample are at increasing distances d1, d2,...,d5 and the k1-, k2-, k3-th neighbors form a majority vote, then is assigned as the confidence value. The prediction is considered as highly confident if the value is greater than 0.75. The SVM-classifier  we adopt implements a decision directed acyclic graph to combine several binary classifiers into a multiclass classifier and does not produce all the distances for confidence evaluation. We also calculated the covariance of the selected genes, and conducted permutation tests, to measure the extent of dissimilar class discrimination strength of the selected genes as a whole.
3 Computational Results
3.1 The Carcinomas Dataset
The Carcinomas dataset contains in total 174 samples (U95a GeneChip) in 11 classes: prostate, bladder/ureter, breast, colorectal, gastroesophagus, kidney, liver, ovary, pancreas, lung adenocarcinomas, and lung squamous cell carcinoma, which have 26, 8, 26, 23, 12, 11, 7, 27, 6, 14, 14 samples, respectively . Each sample contains 9,183 genes, whose maximum hybridization intensity is ≥ 200 in at least one sample. All hybridization intensity values < 20 were raised to 20. The data were subsequently log transformed. We obtained this dataset through the website provided by Su et al. [19, 23].
On this dataset, Su et al.  applied a Wilcoxon score to rank genes and selected the top ranked 100 genes for each of the 11 classes. Using a subset of 100 samples, under the OVA/LOOCV scheme, an SVM-classifier was used to further select 110 best genes (10 genes per class), among the 1100 genes, which achieved the highest LOOCV classification accuracy (97 out of the 100 predictions were correct; 94 out of the 100 predictions were confident, among which 92 were correct). The classifier trained on these 100 samples was then tested independently on the other 74 samples (no Bladder/Ureter or Pancreas samples), and achieved 70 correct predictions, amongst which 64 were confident.
The detailed predictions on the Carcinomas dataset. Carcinomas dataset: testing result of the Disc-F-test-KNN classifier trained on the 100 samples by setting 160 gene clusters to select 80 genes. 72 out of the 74 predictions are correct (in bold).
Lung Adeno. (LA)
Lung Squamous (LS)
3.2 The Embryonal Dataset
The Embryonal dataset  contains a total of 42 patient samples analyzed with oligonucleotide microarrays containing probes for 7,129 genes (HuGeneFL arrays; dataset A in ). These 42 samples include 10 medulloblastomas, 5 CNS AT/RTs, 5 renal and extrarenal rhabdoid tumours, and 8 supratentorial PNETs, as well as 10 non-embryonal brain tumours (malignant glioma) and 4 normal human cerebella. Note that there is a slight difference between this dataset and the one mentioned in the original paper , which contains 6,817 genes. Note also that CNS AT/RTs and rhabdoid tumours are together considered as a class, so there are 42 samples in 5 separate classes.
On this dataset, Pomeroy et al.  applied OVA S2N statistics (and the standard t-statistics) to select a number of genes, and then built a weighted KNN (k = 5) to predict class memberships. This method obtained 35 correct predictions of the 42 under the LOOCV scheme.
The detailed predictions on the Embryonal dataset. Embryonal dataset: LOOCV result of the Disc-GS-KNN classifier by setting 100 gene clusters to select 45 genes. 36 out of the 42 predictions were correct (in bold).
CNS, renal and extrarenal rhabdoid (CRE)
Supratentorial PNETs (S)
Malignant glioma (MG)
3.3 The Lung Carcinomas Dataset
The detailed predictions on the Lung Carcinomas dataset. Lung Carcinomas dataset: LOOCV result of the Disc-GS-KNN classifier by setting 160 gene clusters to select 80 genes. 197 out of the 203 predictions were correct (in bold).
Squamous cell lung carcinomas (SQ)
Pulmonary carcinoids (P)
Small-cell lung carcinomas (SM)
Hanczar et al.  proposed a method that first uses the k-means algorithm to cluster genes and then defines a progene as the mean expression vector of the genes in a cluster. Subsequently, a single gene scoring method is run on the progenes to select a subset of them to build an SVM-classifier. Under the 3-fold cross validation (averaged over several times of random partition), a highest classification accuracy of 95.2% was obtained in , where 100 gene clusters were formed and 100 progenes were selected.
Similarly, under the 3-fold cross validation in which 20 random partitions have been performed, our Disc-GS-KNN and Disc-F-test-KNN classifiers were able to achieve 96.5% classification accuracy, when no more than 80 genes were selected.
3.4 The All Tumor Dataset
The detailed predictions on the All Tumor dataset. All Tumor dataset: testing result of the Disc-GS-KNN classifier by setting 160 gene clusters to select 80 genes. 36 out of the 46 predictions were correct (in bold).
Adopting OVA scheme, the authors proposed to use a linear SVM algorithm (a KNN algorithm was also tested) to recursively eliminate the bottom 10% genes that show low importance. Under the LOOCV scheme, on the training dataset, their method was able to make 115 (or 79.9%) confident predictions of which 103 (or 89.6%) were correct. Only 8 (or 27.6%) of the other 29 (or 20.1%) predictions of low confidence were correct, which gave an overall training classification accuracy of 78.5%. The classifier thus trained on the 144 samples was tested on the independent 46 samples, and achieved a classification accuracy of 78.3% (among which, 30 of the 36 confident predictions were correct, 6 of the 10 low-confidence predictions were correct). The LOOCV classification accuracy of our Disc-GS-KNN classifier on the training dataset reached 78.5% too (the detailed predictions in Table 4), the same as that in . Nonetheless, with respect to confident prediction, our Disc-GS-KNN classifier performed much better on the testing dataset, where it made 41 (or 89.1%) confident predictions, of which 34 (or 82.9%) were correct.
4.1 The Number of Gene Clusters and the Maximum Number of Genes per Cluster
4.2 The Impact of Class Discrimination Strength Vector
4.3 Covariance of Selected Genes and Their Permutation Tests
Statistical significance analysis of the selected genes on the Lung Carcinomas dataset. Covariance and standard deviation of the genes selected by the Disc-F-test, F-test, Disc-GS, and GS methods, on the Lung Carcinomas dataset.
Permutation test result on the Embryonal dataset. The p-values of the classification accuracies, under 10,000 permutation tests, of the KNN- and SVM-classifiers built using the 80 genes selected by the Disc-F-test, F-test, Disc-GS, and GS methods, on the Embryonal dataset.
6.1533 × 10-10
1.9375 × 10-11
5.1585 × 10-11
2.1727 × 10-13
1.2128 × 10-9
9.0832 × 10-7
1.3323 × 10-15
1.0000 × 10-15
In this paper, we aimed at solving the much more challenging multi-class classification problem in microarray data analysis. We have examined a novel method to incorporate gene cluster information to identify many biologically relevant discriminatory genes, and to use that information to construct classifiers for such a purpose. We define a distance measurement between two genes to approximate their difference in the class discrimination strength, based on a novel class discrimination strength vector representation. This Disc-based gene selection method is generic, in that it can be combined with any gene ranking method to identify genes that have dissimilar strength in class discrimination, but together they would have superior class discrimination strength. Our experiments on four real human cancer microarray gene expression datasets showed that the Disc-based gene selection methods achieved significantly higher classification accuracies, compared to the corresponding non-Disc-based gene selection methods.
The authors are grateful to the research support from AICML, CFI, iCORE, NSERC, and the University of Alberta.
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