Volume 12 Supplement 12
A new method for identifying bivariate differential expression in high dimensional microarray data using quadratic discriminant analysis
© Arevalillo and Navarro; licensee BioMed Central Ltd. 2011
Published: 24 November 2011
One of the drawbacks we face up when analyzing gene to phenotype associations in genomic data is the ugly performance of the designed classifier due to the small sample-high dimensional data structures (n ≪ p) at hand. This is known as the peaking phenomenon, a common situation in the analysis of gene expression data. Highly predictive bivariate gene interactions whose marginals are useless for discrimination are also affected by such phenomenon, so they are commonly discarded by state of the art sequential search algorithms. Such patterns are known as weak/marginal strong bivariate interactions. This paper addresses the problem of uncovering them in high dimensional settings.
We propose a new approach which uses the quadratic discriminant analysis (QDA) as a search engine in order to detect such signals. The choice of QDA is justified by a simulation study for a benchmark of classifiers which reveals its appealing properties. The procedure rests on an exhaustive search which explores the feature space in a blockwise manner by dividing it in blocks and by assessing the accuracy of the QDA for the predictors within each pair of blocks; the block size is determined by the resistance of the QDA to peaking. This search highlights chunks of features which are expected to contain the type of subtle interactions we are concerned with; a closer look at this smaller subset of features by means of an exhaustive search guided by the QDA error rate for all the pairwise input combinations within this subset will enable their final detection. The proposed method is applied both to synthetic data and to a public domain microarray data. When applied to gene expression data, it leads to pairs of genes which are not univariate differentially expressed but exhibit subtle patterns of bivariate differential expression.
We have proposed a novel approach for identifying weak marginal/strong bivariate interactions. Unlike standard approaches as the top scoring pair (TSP) and the CorScor, our procedure does not assume a specified shape of phenotype separation and may enrich the type of bivariate differential expression patterns that can be uncovered in high dimensional data.
The development of high-throughput technologies, such as gene or proteinn microarrays, has provided the scenario of the state of cells by monitoring the expression levels of hundreds or thousands of biological inputs (p) for a few number (n) of experimental units measured under different clinical conditions. A challenging problem within this domain is the identification of inputs or interactions of them highly correlated to the outcome. The low sample-high dimensional (n ≪ p) structure of the data we handle makes this challenge a difficult task, in particular when we are concerned with the detection of bivariate interactions. Some papers that tackle this problem using indexes of pairwise feature association are [1, 2], which introduced the TSP score, and  which defined the CorScor index by the changes in the intra-class correlation coefficient and explores the feature space looking for gap/substitution and on/off association patterns. These approaches assume a specified shape for the interaction.
In this paper we address the problem by evaluating the performance of a classification rule trained on the data at hand; hopefully, this will enrich the typology of interactions that might be hidden in the data. One of the main drawbacks for facing up this problem is the well known peaking phenomenon; it consists of the deterioration of the performance of the designed classifier when the number of inputs increases and many noisy variables are involved in fitting the classifier, so the signal gets masked and the classifier confuses it with the noise. There is a great deal of literature discussing this phenomenon; some recent papers [4, 5] study the problem within a general framework and  tackles it in the context of feature selection.
The peaking phenomenon is more acute for weak marginal/strong bivariate signals as pointed out in , that is, for highly predictive interactions having useless marginal distributions for classifying the outcome. This paper studies the peaking phenomenon for this type of bivariate interaction patterns. We propose a search procedure which utilizes the error rate of the quadratic discriminant analysis (QDA) classifier to carry out an exploration of the feature space in order to find such signals. The use of the QDA classifier will hopefully enlarge the type of patterns of bivariate differential expression uncovered by the aforementioned methods TSP and CorScor.
This section gives a detailed description of how weak marginal/strong bivariate interactions are lost by different classification rules as the dimension of the feature space increases and many noise predictors are involved in training the classifier. For the sake of simplicity we confine ourselves to the binary classification problem, where n0 and n1 observations are drawn from each one of the categories of the outcome variable.
Weak marginal/strong bivariate interactions
Four examples of weak marginal/strong bivariate interaction patterns are given by the following synthetic scenarios.
Scenario 2 (XOR)
The observations for the XOR pair are drawn from a bivariate random variable X = (X1, X2) in accordance with the following scheme: the conditional distribution of the random variable X|Y = 0 —black labels— is uniform over the quadrants . On the other hand, X|Y = 1 —red labels— has a uniform distribution over .
Scenario 3 (circular pattern)
Cases are simultated from a bivariate normal distribution R = (R1, R2) with vector means µ = (0, 0) and covariance matrix the indentity I. The labels are assigned in accordance to the following rules: if then Y = 0 —black labels— and if then Y = 1 —red labels.
Scenario 4 (V-shaped pattern)
Observations in this situation are drawn from uniform distributions confined to the domain . The interaction between the pair V = (V1, V2) and the outcome variable Y is given by the following rules: V|Y = 0 —black labels— has a uniform distribution over , with . On the other hand, V|Y = 1 —red labels— has a uniform distribution over , with .
The peaking phenomenon
Note that the error rate of the polynomial kernel in scenario 1 is high, around 0.5, when only the variables of the pair (Z1, Z2) are used as predictors. This ugly performance is related to the type of polynomial kernel we used, a second order polynomial kernel, which is well suited for tracing nonlinear quadratic patterns but poor for identifying linear decision boundaries as in scenario 1; see the improvement of SVM classifier in scenarios 2, 3 and 4 where it makes a better job with the non linear interaction patterns (X1, X2), (R1, R2) and (V1, V2) tracing the separation between the classes.
In addition, we can observe that the error rate deteriorates as the number of features increases; this shows the peaking phenomenon for weak marginal/strong bivariate interactions. The QDA resistance to peaking compares to RF and Adaboost, with the exception of scenario 3 when p < 20; see that the error rates are nearly similar, specially for a number of inputs under 10 or 15. In addition, the QDA has the appealing property of requiring a low computational cost for training the classifier; this fact is crucial in the design of the search strategy since our procedure will explore the feature space in an almost exhaustive way by fitting thousands of times the QDA classifier.
Comparative study for a benchmark of classification rules
The CMA package  from Bioconductor project repositories in  provides an interface for the analysis of genomic data. One of the utilities of CMA is the possibility to carry out a comparative study of the performance of classifiers for a benchmark of classification rules.
In this section we revisit the effect of the peaking phenomenon for a selection of classifiers from the CMA package: k-nearest neighbors (knn) and neural networks (nnet) , diagonal (DLDA), linear (LDA) and quadratic (QDA) discriminant analysis as in , partial least squares with lda, logistic regresion and RF variants (pls_lda, pls_lr, pls_rf) as in , PAM classifier (scDA) as introduced in , random forests , the componentwise boosting (compBoost) introduced in , the ElasticNet  and two versions of the SVM (svm, svm2) with second order polynomial and radial kernels respectively. The error rate was estimated by 10-fold cross validation.
The function compare gives a picture of how these classifiers compare one with each other. For each one, it displays the boxplot of the error rate over the 10 validation sets.
A closer look at QDA and its resistance to peaking
Now we explore with more detail the resistance of QDA classification rule to peaking by means of a simulation study.
with and π1, π2 the a priori class probabilities. When we are dealt with balanced classes π1 = π2 = 0.5.
When the sample estimates , , , of the covariances and the means are plugged in the expression above, we obtain the QDA designed classifier.
The previous decision boundary defines an hyperquadric whose shape depends on the elements involved in the difference of inverses , more specifically on the product of its eigenvalues. This yields to elliptical, hyperbolic, parabolic or linear boundaries; see  for details. Thus, the variety of patterns recognized by QDA is rich enough to consider it a good classification rule for pattern discovery. We now carry out a simulation experiment in order to study its resistance to peaking for weak marginal/strong bivariate interactions.
We have drawn 80 observations (n0 = n1 = 40) according to patterns in scenarios 1, 2, 3 and 4, along with 80 cases from p – 2 independent standard normal variables, which are uninformative features for class prediction. On the order hand, we have generated n0 + n1 = 80 samples from p independent standard normal variables and obtain a data set with only noisy features. The error rate of the QDA classifier was estimated by 10-fold cross validation for each data set. We repeated the experiment B = 100 times in order to get both populations of error rates: with the signal and with only noisy features. We have considered feature spaces with p = 2, 5, 10, 15, 20, 30 predictors.
The QDA interaction detector procedure
The results of the simulations have shown that QDA resistance threshold to peaking can be set at p = 10 (or at most p = 15), when we are concerned with the detection of weak marginal/strong bivariate interactions in high dimensional data sets. This finding is crucial and puts the basis of the search strategy we will design to uncover this type of interaction patterns. The rationale behind this strategy is as follows. The naive solution would explore the feature space in an exhaustive way by fitting a QDA classifier to each pair of variables; a high accurate classification would be highlighting the presence of a signal. Obviously, this alternative is time consuming prohibitive since it would require a total of p(p – 1)/2 QDA fits; for example, if p = 2000 then 1999000 fits are needed.
Our search strategy proceeds in a nearly exhaustive way by dividing the feature space in small blocks of features of a specified size bsize and by fitting the QDA for all pairs of blocks. As we know that QDA is resistant to peaking while the number of features ranges between p* = 10 and p* = 15, we propose to take bsize such that 2 ∗ bsize ≤ p*; in this way we protect ourselves against the danger of peaking when the QDA classifier is fit with all the features belonging to the union of both blocks. Once the QDA classifiers are obtained for all the possible matchings of blocks, we know that for a matching containing a bivariate interaction pattern, the classifier will give a very low error rate; meanwhile, for a block matching with only noisy features we will obtain a high error rate. Thus, we can construct a ranking of block matchings, where the top ranked matchings will contain the informative bivariate interaction patterns, and the matchings at the bottom of the ranking carrying on only noisy features. Now, at a second stage we can restrict the search to the subset of features belonging to the top ranked matchings of blocks. For example if we confine the search to the 2 ∗ bsize features of the first block matching, we would need to explore bsize × bsize interactions in order to find out which one of them is responsible for the observation of such a low error rate in the QDA; usually this search is very low time consuming since bsize is smaller than 7.
Searching in the feature space in a blockwise manner has an enormous advantage with respect to the exhaustive search; for example, if p = 2000 and we take bsize = 5, we would obtain 400 blocks; so the search would need only 79800 QDA fits, much less than the 1999000 fits of the naive solution.
It is recommended to carry out a first screening step in order to filter the strong marginal features highly correlated with the outcome before applying the search procedure. Recall that our search strategy was designed to uncover the weak marginal/strong bivariate interactions which are usually rejected by traditional sequential search procedures or pre-screening filtering tools (see  for a detailed explanation of this fact).
Results and discussion
A simulation example for synthetic data
Let n0 = n1 = 40 be the class sizes. The cases were drawn from p-dimensional random vectors, (Z1, Z2, E), (X1, X2, E), (R1, R2, E) and (V1, V2, E) corresponding to scenarios 1, 2, 3 and 4, with E = (E1, …, E p –2) a vector of independent noisy standard normal variables added to the signal. For p = 200 the signal represents 1% of a 200-dimensional feature space.
Note that the procedure has found interaction patterns (Z1, Z2), (X1, X2) and (R1, R2). The red hot squares of scenarios 1, 2, 3 and 4 in Figure 8 highlight the weak marginal/strong bivariate interaction hidden in the matching. These matrices may also provide a useful tool for identifying different types of weak marginal/strong bivariate signals: for example the first row of red squares in scenario 4 is highlighting the not so weak behavior of component V2 in the V-shaped pattern. On the other hand, the isolated red hot spot in scenario 2, surrounded by light yellow and orange spots, might be explaining the weak predictive power of the components in the XOR interaction, where both variables behave as noisy features.
An application to real data: the colon cancer dataset
The colon cancer data set is a publicly available experiment which can be obtained from the package colonCA in . Gene expression levels for 2000 genes across 40 tumor and 22 normal tissue samples were collected with Affymetrix oligonucleotide arrays . The data were preprocessed by a log transformation and standardization across genes.
Random Forests (RF) outlier detector utility identified cases 18, 20, 52, 55 and 58 as outliers. These were previously identified in  as aberrant observations and will be removed from the analysis.
Data analysis and findings
Subset of genes retained from a previous Random Forests screening step
Application of the QDA interaction detector procedure
After putting aside the biomarkers identified in the screening step and eliminating a few duplicated columns, we end up with a data set containing 1891 features along with the binary outcome.
Note that none of the genes in the scatterplots exhibit univariate differential expression since both classes of the outcome do overlap when projected on the axes. However, if both genes are considered together, they discriminate the binary outcome; such discrimination stems from the bivariate association between them. They are four cases of weak marginal/strong bivariate gene interaction patterns uncovered by our procedure.
Comparative analysis with TSP and CorScor methods
Selection of genes given by the methods: QDA interaction detector procedure, TSP and CorScor
QDA interaction detector procedure
TSP = 0.11 CorScor = 0.73
TSP = 0.92 CorScor = 0.77
TSP = 0.92 CorScor = 1.18
TSP = 0.84 CorScor = 1.05
TSP = 0.92 CorScor = 1.18
TSP = 0.81 CorScor = 1.14
TSP = 0.89 CorScor = 0.81
TSP = 0.81 CorScor = 1.10
TSP = 0.76 CorScor = 0.80
TSP = 0.84 CorScor = 1.09
DNA-(APURINIC OR APYRIMIDINIC SITE) LYASE (HUMAN)
Human mRNA (KIAA0052) for ORF, partial cds
ATP SYNTHASE COUPLING FACTOR 6, MITOCHONDRIAL PRECURSOR (HUMAN)
HETEROGENEOUS NUCLEAR RIBONUCLEOPROTEINS C1/C2 (HUMAN)
PEPTIDYL-PROLYL CIS-TRANS ISOMERASE A (HUMAN)
CYTOCHROME C OXIDASE POLYPEPTIDE VIA-LIVER (HUMAN)
Human mRNA for calmodulin, complete cds
VASCULAR ENDOTHELIAL GROWTH FACTOR (Cavia porcellus)
ATP SYNTHASE B CHAIN, MITOCHONDRIAL PRECURSOR (HUMAN)
GTP-BINDING NUCLEAR PROTEIN RAN (Homo sapiens)
Human mRNA (KIAA0089) for ORF (mouse glycerophosphate dehydrogenase-related), partial cds
H.sapiens CL 100 mRNA for protein tyrosine phosphatase
This paper has explored the peaking phenomenon in the context of detecting marginal/strong bivariate interactions in high dimensional settings. The appealing properties of the QDA classifier and its resistance to peaking has justified its use as a search engine of a procedure that explores the feature space in order to look for this type of signals in high dimensional data.
The method was applied both to artificial data and to a real microarray gene expression experiment, the colon cancer data set. The application to real data has led to promising results providing gene interactions that exhibit bivariate differential expression but are not differentially expressed when considered marginally. The results show the usefulness of QDA interaction detector procedure, which is expected to become an efficient tool for biologists and bioinformaticians for the discovery of new gene to gene interactions.
The proposed method has been developed for binary classification; the analysis for multi-class problems is a natural extension for conducting future research efforts. Some research regarding the computational cost involved in the QDA interaction detector search strategy is also an issue for further improvements.
This work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion grant MTM 2008-03282.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 12, 2011: Selected articles from the 9th International Workshop on Data Mining in Bioinformatics (BIOKDD). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/12?issue=S12.
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