Predicting qualitative phenotypes from microarray data - the Eadgene pig data set.

BACKGROUND: The aim of this work was to study the performances of 2 predictive statistical tools on a data set that was given to all participants of the Eadgene-SABRE Post Analyses Working Group, namely the Pig data set of Hazard et al. (2008). The data consisted of 3686 gene expressions measured on 24 animals partitioned in 2 genotypes and 2 treatments. The objective was to find biomarkers that characterized the genotypes and the treatments in the whole set of genes.
METHODS: We first considered the Random Forest approach that enables the selection of predictive variables. We then compared the classical Partial Least Squares regression (PLS) with a novel approach called sparse PLS, a variant of PLS that adapts lasso penalization and allows for the selection of a subset of variables.
RESULTS: All methods performed well on this data set. The sparse PLS outperformed the PLS in terms of prediction performance and improved the interpretability of the results.
CONCLUSION: We recommend the use of machine learning methods such as Random Forest and multivariate methods such as sparse PLS for prediction purposes. Both approaches are well adapted to transcriptomic data where the number of features is much greater than the number of individuals.

Background

Often, an important goal of transcriptomic analyses is to identify differentially expressed genes; the expression level of each gene is explained by the phenotype in a linear model setting (either regression or ANOVA for a quantitative or a qualitative phenotype). Another important goal is to find biomarkers, i.e. genes that have a high predictive value for the phenotype. One statistical method that can be considered is discriminant analysis, where the phenotype is modelled as a linear combination of a subset of gene expressions. However, in the case of transcriptomic data, gene expressions are highly correlated, leading to multicollinearity problems in discriminant analysis. Furthermore, the high number of variables limits the use of this method. To circumvent this problem, one can pre-select the variables, or use approaches that can deal with a large number of variables.

In this study, the objective was to assess the behaviour of 2 prediction methods on one data set [1,2], where the phenotype is a set of factors, genotype or breed (Large White denoted LW, and Meishan denoted MS), and treatment (control and ACTH, coded c and a). Expression levels were available for 3686 genes on 24 animals. We first focus on the machine learning approach Random Forest (RF), and then on the modelling approach sparse Partial Least Squares (sPLS) to analyse this data set.

Methods

Random forests

Random forests [2] is a classification algorithm that takes advantage of the unstable property of Classification and Regression Trees (CART) classifiers [3] and their lack of accuracy by aggregating them. It combines two sources of randomness that improve the prediction accuracy: bagging (bootstrap aggregating) and random feature selection to construct each CART. This results in a low correlation of the individual trees as well as low bias and low variance. The individual trees Tk are constructed as follows:

- N bootstrap samples (B1,..., BN) are drawn from the original data.

- Each sample Bk (k = 1,..., N) is used as a training set to construct an unpruned tree Tk. Let p be the input variables of the tree. For each node of Tk, m variables are randomly selected (m<

The predictions of the N trees are then aggregated to predict new data by majority vote for classification or by average for regression. Random forest avoids the need to perform a separate cross validation test to estimate the prediction error of the forest when performing a classification or a regression. While the forest is constructed, it generates an internal estimation of the generalisation error as follows:

- While constructing each tree Tk, about one-third of the cases are left out of the bootstrap sample and are not used in its construction. These data are called "Out-of-bag" or OOB data and are used as an "internal" test set for each tree that is grown.

- The OOB predictions are then aggregated and the error rate, called "OOB error estimate" is computed for the whole forest and should lead to an accurate and unbiased generalisation error [2].

The Mean Decrease Accuracy measure was used in this study as a feature selection criterion, where the OOB data are used to obtain estimates of variable importance by evaluating their contribution to the prediction accuracy. The values of each variable in the OOB cases are randomly permuted and are run along the tree. The proportion of cases in the correct classes with permuted OOB data is then subtracted from the proportion of cases in the correct classes where OOB data have not been permuted. The Mean Decrease Accuracy averages the difference between these two accuracies over all trees in the forest and normalizes it by the standard error.

PLS and Sparse PLS

Let Y be the n × q matrix of phenotypes with the 4 columns indicating each combination (LWa, LWc, MSa, MSc) for the 24 animals, and let X be the n × p matrix of the 3686 gene expressions measured on the 24 animals. Partial Least Squares regression (PLS) was introduced by Wold [4], first with the NIPALS algorithm and then followed by numerous variants. The PLS is based on the simultaneous decomposition of the data sets X and Y into loading vectors and associated latent variables. The main idea is to perform successive regressions with projections onto latent structures to highlight hidden or latent underlying biological effects. As in Principal Component Analysis (PCA), the PLS components (latent variables) are linear combinations of the initial variables. However, the coefficients that define these components are not linear, as they are solved via successive local regressions on the latent variables. Furthermore, PLS goes beyond a simple regression problem, since X and Y are simultaneously modelled by successive decompositions. The objective function involves maximizing the covariance between each linear combination of the variables from both groups:

The loading vectors are the p- and q- dimensional vectors uh and vh for each PLS dimension h and are respectively associated to the X and Y data sets. The associated latent variables are defined as ξh = Xuh and ωh = Yvh. As in PCA, the loading vectors uh and vh are directly interpretable, as they indicate the importance of the variables from both data sets in relation with each other. The latent variables ξh and ω, that are n-dimensional vectors contain the information regarding the similarities or dissimilarities between the individuals or samples [5]. PLS is an iterative method that is suitable for high dimensional data sets and has a valuable stability property. However, this interesting approach does not allow feature selection, which renders the results difficult to interpret in the n<