This paper employs a progressive framework as shown in Fig. 2 for clustering the genes. Unlike the traditional two-way methods which cluster the genes into specified number of clusters, the progressive framework clusters the genes into all possible resolutions. A resolution is a measure of compactness of clusters. The Higher the resolution, the compact are the clusters and vice versa. If the discriminative ability of clusters is to be studied, it must be performed at various levels of granularities. A progressive clustering algorithm provides a flexible way to achieve this goal. For example, as shown in Fig. 2, resolution 2 has more granularity than resolution 1 and so on. The clustering of the data progresses into various levels of granularity ranging from macro to micro clusters. The resolution level at which the progressive framework terminates is determined using Davies-Bouldin indices 25. The two-way clustering framework employs the gene clusters at each resolution to cluster the samples as shown in the Fig. 3.

The sample cluster groups are compared with the sample class label information. The score for each gene cluster 'CSC_k' as shown in the Eq. 2 is determined by finding the number of correctly identified samples.

C S C k k = 1 M = ∑ i = 1 t ∑ j = 1 L ( C i ∩ G j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfWaqaaiabdoeadjabdofatjabdoeadnaaBaaaleaacqWGRbWAaeqaaaqaaiabdUgaRjabg2da9iabigdaXaqaaiabd2eanbaakiabg2da9maaqahabaWaaabCaeaadaqadaqaaiabdoeadnaaBaaaleaacqWGPbqAaeqaaOGaeyykICSaem4raC0aaSbaaSqaaiabdQgaQbqabaaakiaawIcacaGLPaaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdYeambqdcqGHris5aaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemiDaqhaniabggHiLdaaaa@4D84@

Here, 'M' is the total number of gene clusters, 'L' is the number of different labels according to the ground truth, C_i are the samples that are part of cluster 'i', G_j is the group of samples having label 'j' according to the ground truth and (C_i ∩ G_j) is the maximum consistency between any of the sample clusters C_i and the samples 'G' with labels 'j' according to the ground truth.

The adaptive subspace iteration (ASI) is a subspace based method to cluster the data. It involves an optimization process that iteratively identifies the subspace structure. The following notations are used in the algorithm:

• D_nxm is the data matrix that contains the microarray data with 'n' genes and 'm' samples. Also, assume that each macro cluster is divided into 'k' number of micro clusters at each resolution level (cf. Fig. 2).

• The matrix M_nxk is the membership matrix. Each gene has 'k' memberships corresponding to the 'k' clusters. The cluster to which the gene belongs has membership of 1 and rest of the memberships are 0. This enables hard clustering of the genes.

• Let S_mxk be the subspace structure associated with each gene cluster. This subspace has adequate information about the most informative genes in the cluster. The columns of S determine the relevance of each sample in the formation of a cluster. Hence, (DS)_nxk represents the projection of the data onto the subspaces.

• Let 'C' be the projection of centroid of each gene cluster onto the subspaces given by S_mxk. This enables calculating the distances between the genes in the subspace and to each of the centroids in the subspace to determine the relevance of each gene with each of the clusters. The relationship between the 'C', 'S' and 'M' is given by,

C = (M^T M)^-1 M^T DS.

Here, (M^T M) provides the size of the clusters (number of genes in a cluster). The diagonal elements provide the size of each cluster and off diagonal elements are zero. Hence 'C' matrix calculates the mean of each gene cluster to estimate the centroids. These centroids are projected to subspace as shown in Eq. 3.

The objective function 'O' is given by,

O = 1 2 ‖ D S − M C ‖ F . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGpbWtcqGH9aqpdaWcaaqaaiabigdaXaqaaiabikdaYaaadaqbdaqaaiabdseaejabdofatjabgkHiTiabd2eanjabdoeadbGaayzcSlaawQa7amaaBaaaleaacqWGgbGraeqaaOGaeiOla4caaa@3B80@

Here, ||.||_F is called the Frobenius norm defined as ||A||_F = tr(AA^T) where, 'tr' is the trace of a matrix. The objective function optimizes by rendering the distance between the gene cluster centroid and each of the genes in that cluster as small as possible thereby making the clusters compact. Therefore,

O = 1 2 t r [ ( D S − M C ) ( D S − M C ) T ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGpbWtcqGH9aqpdaWcaaqaaiabigdaXaqaaiabikdaYaaacqWG0baDcqWGYbGCdaWadaqaaiabcIcaOiabdseaejabdofatjabgkHiTiabd2eanjabdoeadjabcMcaPiabcIcaOiabdseaejabdofatjabgkHiTiabd2eanjabdoeadjabcMcaPmaaCaaaleqabaGaemivaqfaaaGccaGLBbGaayzxaaaaaa@4525@

= 1 2 [ t r ( D S S T D T ) − 2 t r ( D S C T M T ) + t r ( M C C T M T ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGH9aqpdaWcaaqaaiabigdaXaqaaiabikdaYaaadaWadaqaaiabdsha0jabdkhaYnaabmaabaGaemiraqKaem4uamLaem4uam1aaWbaaSqabeaacqWGubavaaGccqWGebardaahaaWcbeqaaiabdsfaubaaaOGaayjkaiaawMcaaiabgkHiTiabikdaYiabdsha0jabdkhaYnaabmaabaGaemiraqKaem4uamLaem4qam0aaWbaaSqabeaacqWGubavaaGccqWGnbqtdaahaaWcbeqaaiabdsfaubaaaOGaayjkaiaawMcaaiabgUcaRiabdsha0jabdkhaYnaabmaabaGaemyta0Kaem4qamKaem4qam0aaWbaaSqabeaacqWGubavaaGccqWGnbqtdaahaaWcbeqaaiabdsfaubaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@5752@

= tr(DSS^T D^T) - tr(D^T S^T MC)

Here, the first component (DSS^T D^T) = (DS)(DS)^T gives the total deviance of the data in the subspace. The second component (D^T S^T MC) = (((D^T S^T)M)C) first projects the data onto the subspace as given by (D^T S^T). Further, the sum of distance between the centroids is estimated using (((D^T S^T)M)C). The objective function shown in Eq. 7 is minimized by maximizing distances between the centroids of individual clusters.

The objective function in (7) is minimized by considering first 'k' Eigen vectors of (D^T(M(M^T M)^-1 M^T - I)D)_1:k 26. Therefore, 'S 'is updated using Eq. 8

S = (D^T(M(M^T M)^-1 M^T - I)D)_k.

Please note that this feature provides dimensionality reduction and further computations are all performed in the reduced sub space. The output of the algorithm is 'M' and 'S'. Here, 'M' offers the cluster memberships and 'S'offers the weights of the samples forming the clusters defined by the matrix 'M'. Based on the membership, the relevance of the gene with a cluster may be estimated. If the membership is 0, there is no relevance and membership 1 indicates the gene belongs to that cluster.

Begin clustering

Step 1: Begin Initialization

Initialize 'M' with zeros and randomly place 1 in each row.

Initialize 'S' with Random values such that each column adds up to 1;

End Initialization

Step 2: Project the centroids of each cluster onto the subspaces using Eq. (3);

Step 3: Compute the initial optimization value 'O₀' using the objective function of Eq. (7);

Step 4: Perform optimization by iteratively updating D, F, S;

Begin Optimization

While (O₁ <O₀)   //Continue as long as the cluster compactness increases

Step 4-1: Update 'M' given by the formula in equation (5)

Begin Loop   //Iterate over all the features

//update the memberships by finding the relevance of a gene with each of the updated cluster centroids as shown in Eq. 9

M(i, j) = ((DS)_i,j - C_:,j);   j = 1...k

Min(M(i, j)) = 1;   j = 1...k

End loop

Step 4-3: Update 'S 'given by the formula in equation (8);

Step 4-4: Compute Step 2;

Step 4-5: Compute 'O₁' using equation (7);

Step 4-6: If (O₁ <O₀);   //Check for the terminating condition//.

O₀ = O₁;

End optimization

End Clustering

Davies-Bouldin index is a measure of cluster validation metric25. It measures the homogeneity of the clusters by finding the ratio of the sum of intra-cluster scatter to inter-cluster scatter. The intra cluster scatter is a measure of spread of a cluster. The inter cluster scatter on the other hand is a measure of distinctiveness of the clusters. Therefore, lower the ratio of intra cluster scatter to inter cluster scatter, the better.

The intra-cluster scatter is given by

S i , q = ( 1 | A i | ∑ x ∈ A i ‖ x − v i ‖ 2 q ) 1 q , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaGaemyAaKMaeiilaWIaemyCaehabeaakiabg2da9maabmaabaWaaSaaaeaacqaIXaqmaeaadaabdaqaaiabdgeabnaaBaaaleaacqWGPbqAaeqaaaGccaGLhWUaayjcSdaaamaaqafabaWaauWaaeaacqWG4baEcqGHsislcqWG2bGDdaWgaaWcbaGaemyAaKgabeaaaOGaayzcSlaawQa7amaaDaaaleaacqaIYaGmaeaacqWGXbqCaaaabaGaemiEaGNaeyicI4Saemyqae0aaSbaaWqaaiabdMgaPbqabaaaleqaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeGymaedabaGaemyCaehaaaaakiabcYcaSaaa@5160@

and the inter-cluster scatter is given by,

d i j , t = { ∑ s = 1 p | v s i − v s j | t } 1 t = ‖ v i − v j ‖ t . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5C8C@

Where, v_i is the centroid of i^th cluster, q,t ≥ 1, q,t can be selected independently of each other. For example, when t = 2, d_ij,t is the Euclidean distance between 'v_si' and 'v_sj'. The |A_i| is the number of elements in the cluster A_j and 'x's are the elements of cluster A_j.

Define  R i , q t = max ⁡ j ∈ c , j ≠ i { S i , q + S j , q d i j , t } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGebarcqqGLbqzcqqGMbGzcqqGPbqAcqqGUbGBcqqGLbqzcqqGGaaicqWGsbGudaWgaaWcbaGaemyAaKMaeiilaWIaemyCaeNaemiDaqhabeaakiabg2da9maaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqWGQbGAcqGHiiIZcqWGJbWycqGGSaalcqWGQbGAcqGHGjsUcqWGPbqAaeqaaOWaaiWabeaadaWcaaqaaiabdofatnaaBaaaleaacqWGPbqAcqGGSaalcqWGXbqCaeqaaOGaey4kaSIaem4uam1aaSbaaSqaaiabdQgaQjabcYcaSiabdghaXbqabaaakeaacqWGKbazdaWgaaWcbaGaemyAaKMaemOAaOMaeiilaWIaemiDaqhabeaaaaaakiaawUhacaGL9baacqGGUaGlaaa@5F5B@

Here, 'R' is a measure of compactness and distinctiveness of the clusters formulated as the ratio of intra cluster scatter and inter cluster scatter.

Davies-Bouldin index is defined as  D B = 1 c ∑ i = 1 c R i , q t . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGebarcqqGHbqycqqG2bGDcqqGPbqAcqqGLbqzcqqGZbWCcqqGTaqlcqqGcbGqcqqGVbWBcqqG1bqDcqqGSbaBcqqGKbazcqqGPbqAcqqGUbGBcqqGGaaicqqGPbqAcqqGUbGBcqqGKbazcqqGLbqzcqqG4baEcqqGGaaicqqGPbqAcqqGZbWCcqqGGaaicqqGKbazcqqGLbqzcqqGMbGzcqqGPbqAcqqGUbGBcqqGLbqzcqqGKbazcqqGGaaicqqGHbqycqqGZbWCcqqGGaaicqWGebarcqWGcbGqcqGH9aqpdaWcaaqaaiabigdaXaqaaiabdogaJbaadaaeWbqaaiabdkfasnaaBaaaleaacqWGPbqAcqGGSaalcqWGXbqCcqWG0baDaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabdogaJbqdcqGHris5aOGaeiOla4caaa@6BE3@

Here, 'c' is the number of clusters and 'DB' is the measure of homogeneity of the clusters. Lower the 'DB', more homogenous are the clusters.

The adaptive ranking based method adopts a modification of the classical t-statistic based ranking function 24. Let 'D_n,(m+k)' be the data matrix with 'n' genes and 'm' samples under one condition (say tumor class) and 'k' samples under the other condition (say non-tumor class). The bootstrapping procedure is employed on the original dataset D_n,(m+k) and j <min(m,k) samples are randomly selected from both cases and pooled to form the data 'D1_n,2j' ('j' samples from each condition). The process is repeated to construct another dataset 'D2_n,2j'. The readers are encouraged to read 24 for further information on bootstrapping procedure. The ranking function of the Eq. 14 is applied independently on these two datasets to obtain the scores of the marker genes describing their differential expression. These scores are ranked and sorted from highest to lowest resulting in R₁ and R₂ in the Eq. 15. Since these two datasets are the subset of the original dataset 'D_n,(m+k)', they must produce similar ranking. The optimized set of parameters θs which provide high consistency (Eq. 16) between the rankings are obtained using Monte Carlo simulation 24. Since it is adequate to test the consistency using a few high ranked genes 'h', h = 100 is employed in this paper. The first 'h' high ranked genes are obtained from these two rankings resulting in two sets given by Eq. 15,

R ( θ 1 , θ 2 , θ 3 ) = d + θ 1 θ 2 σ ^ + θ 3 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGucqGGOaakiiGacqWF4oqCdaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=H7aXnaaBaaaleaacqaIYaGmaeqaaOGaeiilaWIae8hUde3aaSbaaSqaaiabiodaZaqabaGccqGGPaqkcqGH9aqpdaWcaaqaaiabdsgaKjabgUcaRiab=H7aXnaaBaaaleaacqaIXaqmaeqaaaGcbaGae8hUde3aaSbaaSqaaiabikdaYaqabaGccuWFdpWCgaqcaiabgUcaRiab=H7aXnaaBaaaleaacqaIZaWmaeqaaaaakiabc6caUaaa@494A@

Here,'d' is the difference of means for 'mean method' and Hausdorff distance between different samples for the 'Hausdorff distance method'. The σ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFdpWCgaqcaaaa@2E86@ is the square root of the sample variance.

S₁ = R₁(1: h) and S₂ = R₂(1: h).

A consistency measure is obtained by comparing these two sets

Co = S₁ ∩ S₂.

The ranking 'R' which produces highest consistency 'Co' is considered as the best ranking. The distance measure in this ranking function was initially based on absolute difference of means 3. Mean is not a good representative of the sample expressions and may be driven by outliers. A robust distance measure called K^th Hausdorff distance is supplemented with the mean method by Shaik et al. 4. The rankings R_M and R_H are obtained for mean method and Hausdorff distance method respectively using Eq. 14 and combined to develop a fused ranking method 4 as shown in Eq. 17,

S C 2 = W 1 R M + W 2 R H W 1 + W 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWucqWGdbWqdaWgaaWcbaGaeGOmaidabeaakiabg2da9maalaaabaGaem4vaC1aaSbaaSqaaiabigdaXaqabaGccqWGsbGudaWgaaWcbaGaemyta0eabeaakiabgUcaRiabdEfaxnaaBaaaleaacqaIYaGmaeqaaOGaemOuai1aaSbaaSqaaiabdIeaibqabaaakeaacqWGxbWvdaWgaaWcbaGaeGymaedabeaakiabgUcaRiabdEfaxnaaBaaaleaacqaIYaGmaeqaaaaakiabc6caUaaa@424A@

Here, W₁ and W₂ represent the confidence in the rankings R_M and R_H obtained using the consistency 'Co' obtained from the Eq. 16.

The R-test followed in 20 is followed in this paper to convert scores to p-values. This is formulated as a hypothesis testing problem. Let 'I' be the informative genes and 'UI' be the non-informative genes. The null hypothesis is that the gene is not informative and the alternate hypothesis is that the gene is informative. The distribution of statistics under null hypothesis is obtained as follows (Please see 20 for more details):

• Obtain the ranks of the genes using the scores for 'I' iterations using bootstrapping. The value I = 25 is often adequate as indicated in 20.

• Construct the distribution of statistics under null hypothesis from consistently high ranked (insignificant) genes.

• The median rank (r) of each gene is obtained (in 20 mean rank was computed).

• The p-value of each gene is obtained by finding p(r_i/g ∈ UI) i.e. the probability of the ranking of gene 'r_i' given the gene belongs to null-hypothesis. The null hypothesis is that gene is uninformative therefore lower the probability under null hypothesis, more significant is the gene.

The FDR analysis is the process of selecting the DEGs such that there are minimum possible expected false positives. Let 'S_g' be the number of selected DEGs at significance level α and let V be the number of false positives among the selected DEGs. The FDR as proposed by Storey and Tibshirani 17 is given by,

F D R = E ( V S g | S g > 0 ) p ( S g > 0 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrcqWGebarcqWGsbGucqGH9aqpcqWGfbqrdaqadaqaamaalaaabaGaemOvayfabaGaem4uam1aaSbaaSqaaiabdEgaNbqabaaaaOGaeiiFaWNaem4uam1aaSbaaSqaaiabdEgaNbqabaGccqGH+aGpcqaIWaamaiaawIcacaGLPaaacqWGWbaCcqGGOaakcqWGtbWudaWgaaWcbaGaem4zaCgabeaakiabg6da+iabicdaWiabcMcaPiabc6caUaaa@4685@

The FDR provides the expected proportion of false positives among the selected DEGs where the number of selected genes is greater than 0. In this paper α is selected such that FDR is minimized. If the ground truth information about the DEGs is available, the performance of ranking algorithms may be compared using the performance analysis curves.

A lognormal distribution-based model is used to generate artificial microarray datasets as proposed in 27. The artificial microarray datasets are generated based on a multivariate lognormal model. The means under both conditions for the DEGs, are set to a fixed value and for the non-DEGs, means under one condition are set to zero and for other condition are drawn from N (3, 1). Unequal variances following a Gamma distribution are used for DEGs as reported in the literature 327. The parameters used for generating the artificial microarray datasets are shown in table 1.

The artificial microarray datasets are created using the same procedure employed for lognormal distribution but by using an Asymmetric Laplace distribution as reported in 28. The mean and variance of the DEGs and Non-DEGs are approximated using the parameters in table 1. The sample size was set to 12.

The various well known gene selection methods are applied on the training set and the p-values of the genes are obtained. For the gene selection methods which offer only ranking, the R-test is employed to obtain the p-values. The Table 2 shows the FDR analysis for the leukemia training dataset. As shown in Table 2, the unified framework recorded less percentage of false positives at various levels of α (6.8%–16.92%). This indicates the improved performance of the unified framework over other gene selection methods. The top 300 low ranked genes obtained using the unified framework is provided as a supplementary document (See additional file 2). The first 52 genes are selected at significance level 0.001 as shown in the Table 2 because it offered minimum expected percentage of false positives (6.8%).

The 52 significantly expressed genes obtained using the unified framework are compared to the DEGs obtained by Golub et al. 30 (see additional file 3). There are 24 genes common to the genes found by Golub et al. This shows that the genes obtained by the unified framework are not significantly different from those obtained by Golub et al. It also shows that there are many genes selected by the unified framework that were not considered significant by Golub et al. It has already been statistically validated that unified framework offered less percentage of expected false positives and hence the genes selected using unified framework are considered to be relevant.

The ASI-based clustering algorithm 21 and the steps outlined in the background section are employed for validation using clustering. The training samples are clustered using the DEGs obtained for individual methods. The obtained sample clusters are compared with the class labels of the samples. The Table 3, row 2 shows the number of correctly identified samples. As shown in the Table 3, the DEGs obtained by the unified framework offered 100% accuracy in the identification of training sample classes. The two-way framework also offered 100% accuracy in identification of the labels of training samples. Additional validations are performed to assess the performance of the individual methods.

The ASI algorithm is further applied to cluster the test samples using the DEGs obtained through respective methods. It is evident from the row 3 of Table 3 that the DEGs obtained using the unified framework classified the AML and ALL samples better (97.06%) than the DEGs obtained using the other methods. This also shows the improved performance of unified framework over the other methods as shown in Table 3.

The idea behind visualization-based cross validation is that if the genes obtained using a gene selection method are differentially expressed, they should separate the sample cases of different classes in the projected space 24. The Fig. 5 shows the visualization of samples using DEGs as features using a 3D star coordinate projection algorithm (3DSCP). Comparing the Figs. 5(a) to 5(f) it may be seen that the unified framework offered clear differentiation between different sample cases. Although the two-way clustering and unified approach identified all the samples correctly as shown in the Table 3 for training samples, comparing the Figs. 5(e) and 5(f), it may be seen that the unified framework offered much clear separation between the samples of different cases. This evidence suggests that better gene selection is achieved using the unified framework.

The gene selection is performed such that there is minimum percentage of expected false positives. As shown in the Table 4, the unified framework recorded less percentage of false positives (2.48%-11.98%) than the other gene selection methods at various levels of α. The full list of better ranked genes can be accessed from additional file 4.

The DEGs found using the unified framework are compared against the DEGs found by Chen et al. 29. The 204 genes out of 210 genes found by unified algorithm are common to the DEGs found by Chen et al. 29. The list of common genes may be accessed through additional file 5. It may be seen that most of the genes found using the unified framework were present in the list of 3000 genes found significant by Chen et al. The improved performance of the unified framework may be attributed to the rejection of most of the genes deemed significant by Chen et al. This is one of the advantages of FDR analysis which focuses not only on the selection of DEGs but also on the rejection of the insignificant genes.

The method similar to clustering based validation for leukemia dataset is followed for gastric cancer dataset. As shown in Table 5, row 2, the DEGs obtained by the unified framework offered 100% accuracy in the identification of the training samples which was not achieved by DEGs obtained by other methods. It may also be seen from the row 3 of the Table 5, that DEGs obtained using the unified framework identified the primary and neoplastic samples from the test set better than the DEGs obtained using the other gene selection methods (97.5%).

The training samples from the Gastric cancer dataset are projected using DEGs as features for visual validation of the DEGs. The Fig. 6 shows that the unified framework offered much clear separation between the samples of different cases with no overlap between the elements of two classes. This evidence suggests the better gene selection obtained using the unified framework.

The Table 6 shows the percentage of expected false positives for various gene selection methods for different values of α. The unified framework as shown in the Table 6 offered improved performance in the selection of DEGs. The full list of better ranked genes can be accessed from additional file 6. These genes are obtained for α = 0.001 which offered minimum expected percentage of false positives (3.03%).

A list of significantly differentially expressed genes is not available from Alon et. al for comparison. However, the comprehensive analysis on this dataset is performed by Su et al. 33. The procedure involves ranking the genes using 8 different measures viz. t-test, information gain, sum of variances, twoing rule, gini index, sum minority, max minority and ID SVM. The rankings are then fused to obtain a list of 100 better ranked genes 33. This list of 100 ranked genes is compared with the list of 66 genes obtained using the unified framework. The 51 genes out of 66 genes were among the top 100 genes obtained using the 'rankgene' method. The rank gene method did not employ any FDR analysis for gene selection, it merely lists the top 100 genes.

The Table 7, row 2 shows the number of samples identified correctly by the gene selection methods. As shown in the Table 7, the unified framework performed relatively better in the identification of training samples. The performance of the gene selection methods is also compared by using the test set. The Table 7, row 3 shows that DEGs obtained using the Unified framework performed better than DEGs obtained using other gene selection methods.

The training samples from the colon cancer dataset are projected using DEGs as features for visual validation of the DEGs. The Fig. 7 shows that the unified framework performed better in the separation of colon cancer samples using the DEGs as features. The validation using 3DSCP shows the better performance of unified framework in the robust selection of DEGs.