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In classical CCA and PLS regression, all variables from both data sets are included in the fitted linear combinations or variates. However, in the context of high throughput biological data, the number of variables often exceeds tens of thousands. In this case, linear combinations of the entire set of features make biological interpretability difficult as they contain too many variables to perform further tests or to generate biological hypotheses. Most importantly, the high dimensionality and the insufficient sample size lead to computational problems as CCA requires the computation of the inverse of the covariance matrices of X and Y. To circumvent this problem, regularized CCA (rCCA) has been recently proposed by [1] when dealing with ill-conditioned covariance matrices by adding a regularization term on their diagonal. Sparse PLS (sPLS) has been recently proposed to perform simultaneous variable selection in the two data sets [4, 7]. sPLS includes Lasso penalization terms on the loading vectors (the vectors which weight are used in the determination of the PLS variates) to shrink some of the coefficients towards zero.
Correlation Circle plots for the simulation study. Correlation Circle plots for dimensions 1 and 2 (a), and 2 and 3 (b). The X and Y variables are represented by thick points and triangles respectively. The subsets of correlated variables are colored according to the legend. Expression profiles of some positively and negatively correlated variables across samples (c).
Correlation Circle plots were found to supplement pair wise correlation approaches [25]. In the high dimensional case, the interpretation of the correlation structure between variables from two data sets can be difficult, and a threshold can be chosen to remove some weaker associations.
A conceptually simple approach for modelling net-like correlation structures between two data sets is to use Relevance Networks. This concept was introduced by [10] as a tool to study associations between pair of variables coming from several types of genomic data. This method generates a graph where nodes represent variables, and edges represent variable associations. The Relevance Network is built in a simple manner. First, the correlation matrix is inferred from the data. Second, for every estimated correlation coefficients exceeding (in absolute value) a prespecified threshold between two variables (say 0.6), an edge is drawn between these two variables, otherwise, no edge is drawn and these two variables are considered not associated for this threshold, and the variables/nodes with no link are not represented in the graph.
Correlation Circles plots, Relevance Networks and Clustered Image Maps are implemented in the R package mixOmics[12] to be applied to a variety of integrative approaches implemented in the package, such as rCCA and sPLS methodologies. Full tutorials on how to analyse data sets with different methodologies and how to obtain specific graphical outputs with desired legends and colors are available on the website -toulouse.fr/biostat/mixOmics. For users not familiar with the R programming language, an associated web application is available at and provides a Cytoscape plugin to display the Relevance Networks in an attractive manner.
We investigate the relevance of Correlation Circle plot, Relevance Networks and CIM representations, firstly on a simulated data set to assess if the proposed graphical outputs are able to highlight pair-wise association structure between two data sets, and secondly on two biological data sets to assess the biological relevance of such graphical tools.
We generated two data sets X and Y with an equal number of 30 observations in each data set. A subset of relevant variables in X were associated with a subset of relevant variables in Y according to the model described below, and the remaining variables were simulated as noise. This simulation study enables to assess if the proposed graphical representations allow differentiate the associated groups of relevant variables from the noisy variables.
The rCCA approach was also applied to these data sets with regularization parameters λ1=0.889 and λ2=0.889 for the first three dimensions (canonical values obtained were of 0.959, 0.925, and 0.881 on each dimension respectively, followed by much lower values). As expected, the graphical outputs were very similar to those with PLS-can.
This simulation study shows that Correlation Circle plots, Relevance Networks and CIM are able to highlight the relevant variables amongst the noisy ones and pinpoint the pair-wise association structure between the two data sets. In the following, we illustrate the use of such graphical outputs on real data sets and discuss the biological relevancy of the obtained results.
These data sets are publicly available in the mixOmics package [12] and provide good illustrative examples for this Section. However, much larger biological data sets could be analysed through mixOmics as the integrative approaches rCCA and sPLS have been specifically developed to handle large data sets (several thousands of variables in both data sets).
In the Liver Toxicity data, we applied the methodology sPLS-reg as the aim is to highlight a subset of correlated genes which expression can predict the clinical chemistry measurements [33]. This analysis was performed in a previous paper to demonstrate the numerical good results of the sPLS-reg approach but no focus was made on the biological relevance of the results or on the use of variable graphical outputs. In this paper, we focus instead on the biological relevancy of the resulting Relevance Networks. In both studies, using these integrative methodologies and associated graphical outputs, the biological questions we ask are: which subsets of variables from both types are strongly positively or negatively correlated with each other? Do these selected features bring any relevant insight in relation to system under study?
Two parameters need to be tune in sPLS: the number of dimensions and the number of variables to select on each dimension. For both data sets, three dimensions were chosen (see numerical results presented in [1, 7]). To illustrate the use of the proposed graphical outputs, we arbitrarily chose to select 50 transcripts or genes on each dimension. This rather large selection size (150 transcript or genes) is justified by the Gene Ontology (GO) analysis which require a sufficient number of variables to assess their biological relevance. The similarity matrices were computed from the sPLS method on the basis of the selected variables.
Preliminary analysis comparing the different graphical outputs. In order to illustrate the usefulness of the variable graphical outputs in a real case study, we first discuss the outputs obtained on the first two components, where 50 genes were selected on each dimension. The Correlation Circle plot (Figure 5) displays all fatty acids and the genes selected on each component (a 100 in total in this plot). It highlights subsets of variables that are important to define each component. For example C18:2 ω 6, C20:2 ω 6 and C16:0 are the fatty acids which variation mainly participate to the definition of the sPLS component 2 (top and bottom of the y-axis). Similarly, genes such as CAR1, ACOTH, SIAT4C, SR.BI, Ntop are positively correlated to each other, and to the fatty acid C16:1 ω 9 and their variation participate to the definition of the sPLS component 1 (left-hand side of the x-axis).
While the CIM better highlights different clusters of variables and their degree of correlation (indicated by the colour code) than the Correlation Circle plots (Figure 6), the visualisation of the correlation within variables sets is more difficult to observe. For example, the Correlation Circle plot highlights a negative correlation between [C18:2 ω 6, C20:2 ω 6] and C16:0, which is less obvious in the CIM.
Finally, the relevance network representation (Figure 7) adds another layer of information as it allows the visualisation of variable groups in the network. In this case, the network highlights two main subsets of genes and fatty acids (top and bottom) which seem to contain very specific information in each of these groups. This information is slightly suggested on the CIM after a careful interpretation of the dendrograms, but is barely observable in the Correlation Circle plot. This comparison demonstrates the usefulness of such graphical outputs, as well as their complementarity to unravel the complex relationship structure between these different biological features. In the following, we discuss the biological relevance of the subsets of genes and fatty acids highlighted by the full sPLS-can analysis (with 3 sPLS components) using GeneGo and topGO. For an easier visualisation, the cytoscape software was used to represent the networks, but the similarity matrix was estimated with mixOmics. 2ff7e9595c
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