Identification of differentially expressed/methylated genes using Statistical tests.

Our proposed method basically depends on statistical analysis. As we know that in case of big gene expression/methylation dataset, there may exist 10,000 or more genes. Among them, most of the genes are non-differentially expressed/methylated (i.e., nDE/nDM), and only some of them are differentially expressed/methylated (i.e., DE/DM). When a rule is generated, then these two types of genes may occur together in the rule. According to biological scenario, DE/DM genes can only make sense in a rule relating to specific disease, where the other type of genes is irrelevant to the disease. Therefore, we have initially used a novel statistical strategy on the dataset to identify the set of statistically significant non-redundant DE/DM genes in such way that significance level must rely on the data distribution property (viz., either normal distribution or non-normal distribution). For doing this, at first, the genes which have low variance are eliminated from the gene expression/methylation dataset. Thereafter, we have used zero-mean normalization on the data of these genes to adjust the values measured on different scales to a common scale. The zero-mean normalization can be stated as: (1) where μ and σ denote mean and standard deviation of the expression/methylation data of a gene i before normalization respectively; and v_ij and $v_{ij}^{\prime }$ refer to the value of i-th gene at j-th condition before and after normalization, respectively.

It is well known that the parametric statistical tests [25] are appropriate for normally distributed data, and non-parametric statistical tests [25] are appropriate for non-normally distributed data, respectively. Therefore, Jarque-Bera normality test [12], [26] is utilized on the normalized data to determine the pattern of distribution of the data whether it is normally distributed or non-normally distributed. The Jarque-Bera normality test is defined as follows: (2) where d denotes the degree of freedom, S is the skewness of the sample, and K refers to the kurtosis of the sample. Hence, depending on the resulting distribution patterns, the whole normalized dataset is partitioned into two sub-datasets, where one sub-dataset has all normally distributed data, and remaining one contains all non-normally distributed data.

Thereafter, we have applied four parametric statistical tests (viz., t-test [11], Welch’s t-test [11], modified Bayes’ t-test by Fox and Dimmic [13] in 2006 and Pearson’s correlation test (Corr) [11]) on the normally distributed data to obtain differentially expressed genes for the normally distributed sub-dataset. Similarly, four non-parametric tests (viz., Limma [11], Significant analysis of microarrays (SAM) [11], Wilcoxon ranksum test (Wcox) [11] and permute t-test (Perm) [11]) are applied on the non-normally distributed data to obtain differentially expressed genes for the non-normally distributed sub-dataset.

Before further proceeding, we have shortly discussed in the followings about some of the statistical tests mentioned above.

The “2-sample t-test” makes comparison between means of the two groups with the variation in the data. From the test statistic, we compute a measure (i.e., p-value). The p-value indicates the probability of observing a t-value as large or larger than the actually observed t-value where the null hypothesis is given true. By convention, if the p-value of a gene (item) is less than 5%, then the gene is statistically called as differentially expressed/methylated gene. Now, suppose, for each gene g, group 1: n₁ treated samples, with mean ${\bar{x}}_{1g}$ and standard deviation s_1g; and group 2: n₁ controlled samples, with mean ${\bar{x}}_{2g}$ and standard deviation s_2g. (3) Here, se_g denotes the standard error of the groups’ mean, thus, (4) where sPooled is the pooled estimate of the population standard deviation; i.e., (5) Here, df is degree of freedom of the test. It is stated as df = (n₁ + n₂ − 2). This strategy is used assuming that variance of two groups are equal.

For Welch’s t-test, the variance of two groups are checked whether they are equal to each other or not. If equal, then use earlier mentioned t-statistic in Equation 3, otherwise use the following t-statistic: (6) Here we use unpooled estimates of the population standard deviations.

Pearson’s correlation coefficient (commonly denoted as ρ) between two variables is described as the covariance of the two variables divided by the product of their standard deviations, i.e., (7) where (8) where samplesize for the two groups are n1 and n2, respectively (here, n1 = n2). This test can predict whether two variables are related or not.

The moderated t-statistic in Limma [27] can be demonstrated as: (9) where samplesize n = n₁ + n₂, ${\widehat{\beta }}_g$ and ${\tilde{s}}_g^2$ denote the contrast estimator and posterior sample variance for the gene g respectively. The statistic for calculating contrast estimator for gene g is: (10) where, N is normal distribution, and the statistic for estimating posterior sample variance for the gene g is: (11) Where, d₀ (< ∞) and $s_0^2$ refer to the prior degrees of freedom and variance respectively, and d_g (> 0) and $s_g^2$ denote the experimental degrees of freedom and the sample variance of a particular gene g, respectively.

SAM chooses to add a small positive constant s₀ (stated as “fudge factor”) to solve small variance problem. The SAM statistic by Tusher et al.(2001) is: (12) where se_g is the standard error of the groups’ mean (see Equation 4). Here, sPooled is the pooled estimate of the population standard deviation (see Equation 5). Here, df is degree of freedom of the test. It is stated as df = (n₁ + n₂ − 2).

In Wilcoxon ranksum test, a list of ranks of the gene expression values for each gene is prepared in ascending for each group, and then tests for equality of means of the two ranked samples. The z-statistic of the test is: (13) where (14) (15) and (16)

A permuted t-test is a kind of t-test in which an rearrangement is conducted in the labels on the observed data-points of each gene (item).

However, as stated earlier that the four parametric statistical tests are applied on the normally distributed dataset, thus different number of up-regulated and down-regulated genes are coming out from the different parametric tests. Thereafter, we have performed intersection of the up-regulated genes to identify set of common up-regulated genes (denoted by UPDESET_N) for the normally distributed sub-dataset. Similarly, we have got set of common down-regulated genes (denoted by DOWNDESET_N). We have then made a list (denoted by TOTALDESET_N) containing all the common up-regulated genes and all the common down-regulated genes; i.e., TOTALDESET_N = UPDESET_N + DOWNDESET_N. Similarly, as stated earlier that the four non-parametric statistical tests are applied on the non-normally distributed dataset, thus different number of up-regulated and down-regulated genes are coming out from the different non-parametric tests. Then, we have made intersection of the up-regulated genes to identify set of common up-regulated genes (denoted by UPDESET_NN) for the non-normally distributed sub-dataset. Similarly, we have got set of common down-regulated genes (denoted by DOWNDESET_NN). We have then made another list (denoted by TOTALDESET_NN) containing all the common up-regulated genes and all the common down-regulated genes; i.e., TOTALDESET_NN = UPDESET_NN + DOWNDESET_NN.

Finally, we have produced a final list (denoted by TOTALDESET_(N+NN)) containing all the common up-regulated and down-regulated genes from the normally distributed and non-normally distributed datasets; i.e., TOTALDESET_(N+NN) = TOTALDESET_N + TOTALDESET_NN. Hence, the final list of genes (i.e., TOTALDESET_(N+NN)) are utilized in the next step. Similar steps are performed to obtain HYPERDMSET_N, HYPODMSET_N, TOTALDMSET_N, HYPERDMSET_NN, HYPODMSET_NN, TOTALDMSET_NN and TOTALDMSET_(N+NN) instead of UPDESET_N, DOWNDESET_N, TOTALDESET_N, UPDESET_NN, DOWNDESET_NN, TOTALDESET_NN, TOTALDESET_(N+NN) respectively.

Discretization and Post-discretization.

Suppose, the data matrix of the resulting list of genes, TOTALDESET_(N+NN) is denoted by I. Now, first of all, I whose rows denote genes and columns denote samples, is transposed. Suppose, PIT is the transposed matrix. As the PIT matrix is already normalized by zero-mean normalization, therefore the following step is utilized for binary discretization of the matrix: (17) where PITij denotes the expression/methylation value of i-th row and j-th column (1 ≤ i ≤ m, 1 ≤ j ≤ n), m and n are number of rows (samples) and number of columns (genes) of PIT_ij matrix, respectively, and IT is the resulting discretized matrix. Now, let us assume that in the discretized boolean matrix, a up-regulated gene and a down-regulated gene are denoted by DE_up and DE_down, respectively. In the matrix IT, ‘1’ and ‘0’ refer to presence of up-regulated gene (DE_up), and presence of down-regulated gene (DE_down), respectively (see part (b) of Fig. 3). After discretization, we will apply Bimax biclustering for finding all-1 biclusters. As the Bimax biclustering rectifies only ‘1’, not ‘0’, thus we need to do post-discretization in such way where ‘1’ will represent both DE_up and DE_down properties. Therefore, after discretization, number of columns is doubled where the first half is a domain for DE_up property, and remaining half is another domain for DE_down property (see part (c) of Fig. 3). E.g., the column denoted by g1 in part (b) of Fig. 3 is divided into the two columns denoted by g1+ and g1− in part (c) of Fig. 3, where for the g1+ column, ‘1’ denotes presence of up-regulated gene (DE_up) and ‘0’ denotes absence of up-regulated gene (∼ DE_up), and for the g1− column, ‘1’ denotes presence of down-regulated gene (DE_down) and ‘0’ denotes absence of down-regulated gene (∼ DE_down). Hence, for methylation data, DM_hyper and DM_hypo are used instead of DE_up and DE_down, respectively. Note that in this paper, ‘+’ and ‘-’ denote up-regulation/hyper-methylation and down-regulation/hypo-methylation, respectively.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. An example of generating special rules from data matrix of the differentially expressed genes.

Here, up-regulation (i.e., ‘+’) and down-regulation (‘-’) are denoted by ‘1’ and ‘0’ in (b), and red and green colors in (c), respectively. Here, s_tr and s_nr denote experimental/diseased/treated and control/normal samples respectively.

https://doi.org/10.1371/journal.pone.0119448.g003

Dividing whole data into training and test sets.

Let us assume that the post-discretized matrix is denoted by ITb. For classification of the matrix ITb, we have applied 4-fold cross-validations (CVs) on the matrix to divide it into test and training data, where one-fold of ITb will be used as test set and remaining three fold will be considered as training set. This procedure will be repeated for four times as it is 4-fold CV.

Finding maximal biclusters and extracting special rules.

We have transposed the training boolean dataset, and applied Bimax biclustering to identify maximal frequent closed homogeneous itemsets (MFCHOIs). Before further proceeding, the fundamental method of BiMax biclustering is discussed in short.

Suppose, a boolean matrix e has size of n × m, where n is number of genes and m is the number of samples. A cell e_ij is 1 if gene i expresses differentially in the sample/condition j and otherwise, e_ij is 0. A bicluster (G, S) is a subset of genes G ⊆ {1, 2, …, n} which express differently together under a subset of samples S ⊆ {1, 2, …, m}; i.e., the pair (G, S) refers to a subset of the matrix e whose all elements have 1. The biclusters which are inclusion-maximal (i.e., the biclusters that are not entirely part of any other bicluster), are only interesting. The pair (G, S) ∈ 2^{1,2,…,n} × 2^{1,2,…,m} can be stated as a bicluster of the type inclusion-maximal [10] if and only if (i) e_ij = 1, ∀iεG, jεS, and (ii) ∄ (G′, S′) ∈ 2^{1,2,…,n} × 2^{1,2,…,m} with (a) e_ij = 1, ∀i′ ∈ G′, j′ ∈ S′ and (b) G ⊆ G′∧S ⊆ S′∧(G′, S′) ≠ (G, S). When there is no proper superset of an itemset have been found at the same support value, then the itemset is called closed itemset. Finding the set of frequent itemsets is totally equivalent to get a set of all-1 biclusters each having at least number of conditions/samples (i.e., satisfying minimum support).

In our experiment, for the biclustering, we have set a fixed minimum cutoff of items/genes (viz., 2), and different minimum cutoffs of sample/condition for determining itemsets at different minimum support of each rule. The BiMax biclustering can generate all maximal biclusters. The items/genes of maximal biclusters represent a (maximal) closed itemset. Thus, all extracted biclusters that are satisfying minimum support condition, produce the set of (maximal) frequent closed itemsets with their class-labels (i.e., conditions/types of samples). Thereby, we have to filter the biclusters depending on their conditions. We have selected such (maximal) biclusters which have the group of homogeneous (non-contradictory) conditions. In other words, we have to identify maximal frequent closed homogeneous itemsets (MFCHOIs). E.g., Bicluster 1 is a MFCHOI which has three genes g1+, g2− and g3+, and two homogeneous conditions/samples s_tr1 and s_tr3 (presented in part (d) of Fig. 3). Similarly, Bicluster 2 is another MFCHOI which has three genes g1+, g2+ and g3−, and two homogeneous conditions/samples s_nr1 and s_nr3 (presented in part (e) of Fig. 3). Hence, we have omitted the biclusters that have the group of heterogeneous (contradictory) conditions. E.g., Bicluster 3 is such type of heterogeneous (contradictory) bicluster which has three genes g1−, g2+ and g3+, and two heterogeneous conditions s_tr2 and s_nr2 (presented in part (f) of Fig. 3).

From each selected bicluster of genes, we can extract an association rule. Each resulting rule must be of special type, i.e., consequent of the rule consists of its class-label (i.e., either treated/diseased/experimental class-label or normal/control class-label) only. E.g., from the Bicluster 1 (depicted in part (d) of Fig. 3), rule id 1 (i.e., {g1+, g2−, g3+} ⇒ disease) is produced. It states that if both of gene1 and gene3 are up-regulated/hyper-methylated and gene2 is down-regulated/hypo-methylated simultaneously, then ‘disease’ occurs (see part (d) and part (g).(i) of Fig. 3). Similarly, rule id 2 (i.e., {g1+, g2+, g3−} ⇒ normal) is generated from the Bicluster 2 (see part (e) and part (g).(ii) of Fig. 3).

Ranking of rules.

We have evaluated each evolved rule based on 24 rule-interestingness measures. Support, confidence, coverage, prevalance, sensitivity (or, recall), specificity, accuracy, lift (or, interest), leverage, added value, relative risk, Jaccard, Yules’ Q, klosgen, Laplace correction, Gini index, two-way support, linear correlation coefficient (or, ϕ-coefficient), cosine, least contradiction, Zhang, liverage2 (or, Piatetsky-Shapiro) and kappa [14, 15] are already included among the 24 measures (see S1 Text). The last and novel measure is the number of satisfiable conditions/samples to each evolved rule. E.g., according to part (g).(i) of Fig. 3, the value of the measure of rule id 1 (i.e., {g1+, g2−, g3+} ⇒ disease) is 2 as its corresponding bicluster (in part (d) of Fig. 3) has two conditions (s_tr1 and s_tr3). Similarly, according to part (g).(ii) of Fig. 3, the value of the measure of rule id 2 (i.e., {g1+, g2+, g3−} ⇒ normal) is 2. The rank of the rule is proportional to the value of the measure of it (i.e., if a rule that has higher value of the measure than other rule, then the rank of the rule will be better than the second rule).

Thereafter, the evolved rules are ranked according to each of the 24 rule-interestingness measures individually using Fractional ranking [16–18]. In the fraction ranking, items which compare equal, hold the same rank. This rank is the mean of ranking numbers that are received in ordinal ranking. E.g., suppose, a data set is {1 2 2}. Here, only two different numbers are available, so there should be two different ranks. If 2 and 2 are actually different numbers, then they should hold ranks 2 and 3, respectively. As these two numbers are same, thus we should calculate their rank by making the average of their ranks as follows: (2+3)/2 = 2.5; therefore, the fractional ranks will be: 1 2.5 2.5.

Hence, the final ranking of each rule is determined by average ranking on the resulting fractional rankings of the rule. The rules are then rearranged in ascending order (i.e., from best to worst rank).

Assigning weights to the rules w.r.t. their final ranking.

For classification, we have to apply a majority voting technique on each test data point to identify its class-label through weighted-sum method. Thus, we firstly assign some weight on the final list of rules in such a way that the topmost rule gets the highest weight, 2nd topper gets 2nd highest weight and so on. The weight of the first ranked rule is always 1. The ranges of weight lie in between 0 and 1. The weight of each rule (denoted by w_j, 1 ≤ j ≤ p) is estimated from a function of its final rank of the rule (denoted by r_j) and the total number of rules (viz., p) as described below: (18) Here, the weight-interval between any two consecutive ranked rules is same. Thus, the calculated weights of the rules are normalized using zero-mean normalization (in Equation 1).

Majority voting and classification.

Consider one test data point (ts). For determining of the predicted class-label of it, we have applied ‘majority voting’ technique. At first, we have identified the rules whose all the items (genes) in their antecedent sides exist in ts. The weights of the rules (i.e., trR_ts number of rules) which have only the class-label ‘disease’ in their consequent sides are then summed up (viz., Ws_tr_ts). Similar summation (viz., Ws_nr_ts) is performed for the rules (i.e., nrR_ts number of rules) having only the class-label ‘normal’ in their consequent sides. The two weighted-sum are then compared, and the class-label with higher weighted-sum becomes the predicted class-label of ts (viz., PredCls_ts). But, if both the weighted-sum are equal to each other, then the class-label of the top rule which satisfies ts (i.e., ClsTopR_ts), becomes the predicted class-label of it. In case, if there is no such rule which satisfies that ts, then the class-label of the rule which satisfies maximum number of test points (i.e., Cls_R′) becomes the predicted class-label of it. An example of majority voting technique is presented in Fig. 4. Repeat this step for other test points. This process is then repeated 4 times for 4 sub-matrices of the test data as here 4-fold CV is used. Using this technique, we have calculated true positive (TP), true negative (TN), false positive (FP), false negative (FN), sensitivity [7], specificity [7], accuracy [7] and Mathews correlation coefficient (MCC) [7] for the proposed classification. Sensitivity, specificity, accuracy and MCC are defined in the followings, respectively: (19) (20) (21) (22) we have repeated the 4-fold CV for 10 times, and then their average sensitivity, average specificity, average accuracy, and average MCC are calculated with the standard deviations based on the results of cross-validations. Thereafter, a comparative performance analysis has been conducted between our proposed method (i.e., Prop) and other popular rule-based classifiers (i.e., ConjunctiveRule (CJR) [28], DecisionTable (DT) [28], JRip [28], OneR [28], PART [28] and Ridor [28] implemented in Weka 3.6 software) based on their average sensitivity, average specificity, average accuracy, and average MCC. Note that the other rule-based classifiers are also started with the same post-discretized matrix (i.e., ITb). We have also performed significance test (viz., One-way Anova) on the accuracies of the classifiers pairwise to know the level of significance (i.e., p-value) of the test for each (pairwise) comparison.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. An example of classification of evolved rules by the majority voting using weighted-sum.

Here, ‘r’ and ‘w’ denote rank and weight of the rule (computed by Equation 18), respectively. Tickmark/crossmark in ‘Q’ column states that test-point (ts) is satisfied/non-satisfied by the corresponding rule.

https://doi.org/10.1371/journal.pone.0119448.g004

The above steps have been described the proposed methodology for the classification.

Performance comparison with other rule mining algorithms.

For the purpose of rule mining only, the whole post-discretized data matrix (i.e., ITb) is used directly as a input of the Bimax biclustering. In this case, we have not performed any cross-validation since there is no need to use classification. Hence, we have compared our proposed rule mining algorithms with the other existing popular rule mining algorithms (i.e., AprioriTid [21], Eclat [22], Tao et al. [23] and H-mine [24]). It should be noted that same input binary matrix (i.e., ITb) is utilized for the other rule mining algorithms.

Biological significance of evolved rules, and Biomarker identification.

As all the real datasets are microarray/beadchip (biological) datasets, so the evolved top rules should have biological significance. The information about the relation between the genes and any disease can be determined from pathway and Gene Ontology (GO) analyses. If all the genes (except the class-label) of a rule occur together in any pathway/GO-term and if the occurrence is statistically significant (i.e., p-value is less than 0.05), then the rule becomes the (statistically) biologically significant rule. If the pathway/GO-term relates to the corresponding disease, then the rule becomes important for diagnosing the disease. Therefore, KEGG pathway and GO analyses have been performed on the genes of the evolved rules using David Database to identify top significant rules with their involved KEGG pathways or GO-terms. The top rules occupied in a significant number of pathways/GOs are obtained. Frequency of occurrence of the genes in the evolved rules for experimental/treated class is performed to identify potential biomarkers.