Integrating multiplatform genomic data using hierarchical Bayesian relevance vector machines
 Sanvesh Srivastava^{1},
 Wenyi Wang^{2},
 Ganiraju Manyam^{2},
 Carlos Ordonez^{3} and
 Veerabhadran Baladandayuthapani^{4}Email author
https://doi.org/10.1186/1687415320139
© Srivastava et al.; licensee Springer. 2013
Received: 7 March 2013
Accepted: 10 June 2013
Published: 28 June 2013
Abstract
Background
Recent advances in genome technologies and the subsequent collection of genomic information at various molecular resolutions hold promise to accelerate the discovery of new therapeutic targets. A critical step in achieving these goals is to develop efficient clinical prediction models that integrate these diverse sources of highthroughput data. This step is challenging due to the presence of highdimensionality and complex interactions in the data. For predicting relevant clinical outcomes, we propose a flexible statistical machine learning approach that acknowledges and models the interaction between platformspecific measurements through nonlinear kernel machines and borrows information within and between platforms through a hierarchical Bayesian framework. Our model has parameters with direct interpretations in terms of the effects of platforms and data interactions within and across platforms. The parameter estimation algorithm in our model uses a computationally efficient variational Bayes approach that scales well to large highthroughput datasets.
Results
We apply our methods of integrating gene/mRNA expression and microRNA profiles for predicting patient survival times to The Cancer Genome Atlas (TCGA) based glioblastoma multiforme (GBM) dataset. In terms of prediction accuracy, we show that our nonlinear and interactionbased integrative methods perform better than linear alternatives and nonintegrative methods that do not account for interactions between the platforms. We also find several prognostic mRNAs and microRNAs that are related to tumor invasion and are known to drive tumor metastasis and severe inflammatory response in GBM. In addition, our analysis reveals several interesting mRNA and microRNA interactions that have known implications in the etiology of GBM.
Conclusions
Our approach gains its flexibility and power by modeling the nonlinear interaction structures between and within the platforms. Our framework is a useful tool for biomedical researchers, since clinical prediction using multiplatform genomic information is an important step towards personalized treatment of many cancers. We have a freely available software at: http://odin.mdacc.tmc.edu/~vbaladan.
Keywords
Bayesian modeling; Multiple kernel learning; Genomics; Highdimensional data analysis; Prediction; Variational inference1 Introduction
Recent advances in genome technologies such as microarrays and nextgeneration sequencing have enabled the measurement of genomic activity at a very detailed resolution (e.g., base pair, singlenucleotide polymorphisms) as well as across multiple molecular levels: the epigenome, transcriptome and proteome. The collection of genomic information at various resolutions holds promise to accelerate the amalgamation of discovery science and clinical medicine [1]. One of the overarching goals of such studies is to relate these genomic data to relevant (patientspecific) clinical outcomes, not only to find significant biomarkers of disease progression/evolution but also to use the biomarkers to develop prediction models for deployment in future therapeutic studies. Furthermore, genomic data are now available from multiple platforms and resolutions for the same individual, thus allowing a researcher to simultaneously query these multiple sources of data to achieve these goals. Such motivating data have been collected under the aegis of The Cancer Genome Atlas (TCGA) project, wherein data from multiple genomic platforms such as gene/mRNA expression, DNA copy number, methylation and microRNA expression profiles are available for multiple tumor types (see http://cancergenome.nih.gov for more details). In addition, the available clinical information, such as stage of disease and survival times, motivates the analytic frameworks that integrate patientspecific data.
One of the main challenges in modeling the statistical dependence between such highthroughput studies is that a large number of measurements (usually in thousands) is available for a relatively small number (usually in tens or hundreds) of patient samples; therefore, classical statistical approaches based on linear models and hierarchical clustering are prone to overfitting [2, 3]. In these situations, [3] recommends accounting for highdimensionality by using approaches that borrow information across covariates to compensate for the limited information available across samples, which leads to better and more reliable inference. Several approaches have been developed to address these challenges in various contexts. Some examples include linear parametric models and hierarchical clustering for inferring the relation between phenotypes and genomic features [4], hierarchical Bayesian modeling approaches based on linear shrinkage estimators [5], linear canonical correlation analysis [6], intensitybased approaches for merging datasets [7], and regularized linear regression approaches [8].
Although these approaches are computationally efficient, interpretable, and simple, they make two unrealistic assumptions for practical data analysis. First, due to the parametric and linear assumptions, they might miss the underlying nonlinear patterns in the data. Second, and more importantly, these nonlinear patterns are further amplified in the presence of complex interactions within and between the different platforms that must be modeled while integrating data from these platforms. In this paper, we present a statistical machine learning approach called the hierarchical relevance vector machines (HRVM) to address these modeling and inferential challenges. Briefly, the framework presented here: (a) models the relation between a relevant clinical outcome (scalar) and highdimensional covariates/features through a dataadaptive and flexible nonparametric approach,(b) borrows information within and between platforms through a hierarchical Bayesian framework, (c) acknowledges and models the interaction between platforms through nonlinear kernelbased functionals, (d) has parameters that have explicit interpretation as the effects of the platforms and their interactions on the outcome, and (e) uses a computationally efficient variational Bayes approach that can be readily scaled to large datasets.
Our methods are motivated by and applied to a TCGA based glioblastoma multiforme (GBM) dataset, for which we integrate gene (mRNA) and microRNA (miRNA) expression profiles to predict patient survival times^{a}. There is an increasing interest in identifying subtypes of GBM based on its gene expression data. The ultimate goal of subtyping GBM is to identify gene expression profiles that are prognostic or predictive of treatment outcomes. The known subtypes of GBM samples in TCGA include proneural, neural, classical, and mesenchymal; with the first two classes of which are suspected to differ from the other two in the cell of origin, which is a critical determinant of effective treatment regimens [9]. Differential expressions of miRNAs were recently found to be associated with many diseases, including cancers [10, 11]. Previous studies have shown that combining multiple types of data, such as mRNA and miRNA expressions, could significantly improve the accuracy of detecting GBM subtypes, and thereby potentially predict the clinical outcomes [12]. However, methods are lacking to accurately model the effect of interactions between these data types directly on clinical outcomes. Here we show that our nonlinear and interactionbased integrative methods have better prediction accuracy than linear alternatives and nonintegrative methods that do not account for the interactions between the platforms. We also find several prognostic mRNAs and microRNAs that are related to tumor invasion and that are known to drive tumor metastasis and severe inflammatory response in GBM. In addition, our analysis reveals several interesting mRNAmiRNA interactions that have known implications in the etiology of GBM. The paper is structured as follows. The basic construction of HRVM is detailed in Section 2. The analysis of GBM data is presented in Section 3, and concluding remarks about the HRVM framework are presented in Section 4.
2 Hierarchical Relevance Vector Machine model
For ease of exposition, we illustrate the model building process of HRVM using data from two sources: gene/mRNA and miRNA expression measurements. The framework is easily extended to multiple platforms as discussed in Section 4. Suppose, we have data for N patients, and X and Y represent meancentered and standardized gene and miRNA expression matrices, with rows corresponding to patients and columns representing the G genes and M miRNAs, respectively ^{b}. Centering and standardizing the gene and miRNA expression matrices remove any systematic mean or scaling effects caused by the use of different data sources, and make them compatible for model fitting. We denote the gene and miRNA expression for the ith patient as row vectors ${\mathbf{x}}_{i}^{T}=({x}_{i1},\dots ,{x}_{\mathit{\text{iG}}})$ and ${\mathbf{y}}_{i}^{T}=({y}_{i1},\dots ,{y}_{\mathit{\text{iM}}})$. These covariates are highdimensional, that is, both G and M are much larger than N; for example, in the GBM data G≈12000,M≈540,N≈250. Based on these measurements, our aim is to predict a relevant clinical outcome, which in our case is the (logtransformed) survival time measured from time of diagnosis to death, denoted by the column vector t=(t_{1},…,t_{ N }) for the N patients.
2.1 Basic construction
where (x_{ i }⊗y_{ i })=(x_{i 1}y_{i 1},…,x_{i 1}y_{ i M },…,x_{ i G }y_{i 1},…,x_{ i G }y_{ i M }) models the first order interactions between genes and miRNAs and α_{0}=0 because of the centered covariates. Further, due to the highdimensional covariates x_{ i }’s and y_{ i }’s, a penalty is imposed on the regression coefficients α=(α_{1},α_{2},α_{3}) to avoid overfitting. The most popular of such penalties is the Lasso because it has many desirable properties for highdimensional linear regression and variable selection [13, 14]. Although (2) with a Lasso penalty is a popular choice for highdimensional regression, the linearity of the basis functions imposes serious restrictions on the flexibility of the model. For example, (2) does not model nonlinear covariate effects as well as second or higher order interactions between genes and miRNAs.
where (x⊗y)_{ i }=(x_{i 1},…,x_{ i G },y_{i 1},…,y_{ i M }) is a vector of length G+M and β=(β_{1},β_{2},β_{3}) is such that its components lie on a probability simplex i.e. β_{ m }>0 for m=1,2,3 and $\sum _{m=1}^{3}{\beta}_{m}=1$. HRVM posits different kernels for the data sources and combines them through weights β. The model parameters have the following interpretation:

The kernel functions f_{ x }() and f_{ y }() model all possible interactions among genes and among miRNAs, respectively, and f_{(x⊗y)}() models all possible interactions between genes and miRNAs. The three kernels together account for the highdimensionality and nonlinearity of the covariate effects of X and Y by embedding them in the space of kernels.

The mth component of β, β_{ m }, denotes the influence of the mth source on predicting the log survival time and has the following interpretation: if β={1,0,0}, then (3) corresponds to a functional regression model that predicts t (log survival time) with only X (gene expressions) as covariates. Conversely, if β={1/3,1/3,1/3}, then (3) corresponds to a regression model, with the platforms and their interactions contributing equally to the prediction of the survival time. In reality, we expect (and show) different contributions from each platform and estimate these weights from the data.
The task now remains to explicitly characterize the functions f_{ x }(•), f_{ y }(•) and f_{(x⊗y)}(•) using multiple kernels, as detailed below.
2.2 Multiple kernel learning
MKL improves the flexibility of KL by introducing L bandwidth parameters ${\left\{{\sigma}_{l}^{2}\right\}}_{l=1}^{L}$ and L weights for feature matrices β=(β_{1},…,β_{ L })^{ T }. A variety of approaches exist to learn ${\left\{{\sigma}_{l}^{2}\right\}}_{l=1}^{L}$, β, and α for MKL (for details see [14, 16, 17]). Note that in all these works the data source (i.e., X) remains the same for both KL and MKL. The HRVM framework developed in this article extends KL to include multiple data sources and their interactions, and uses a learning algorithm similar to the MKL framework.
We further constrain β such that its components lie on a probability simplex, i.e., $\sum _{m=1}^{3}{\beta}_{m}=1$. This constraint ensures that the joint (convolved) kernel, K_{ β }, is positive definite and that β_{ m } denotes the influence of the mth source in predicting the log survival time. Note that HRVM is a special case of (3) with ${f}_{\mathbf{x}}({\mathbf{x}}_{i},\mathit{\alpha})\equiv {\mathbf{k}}_{1,i}^{T}\mathit{\alpha}$, ${f}_{\mathbf{y}}({\mathbf{y}}_{i},\mathit{\alpha})\equiv {\mathbf{k}}_{2,i}^{T}\mathit{\alpha}$, and ${f}_{(\mathbf{x}\otimes \mathbf{y})}({(\mathbf{x}\otimes \mathbf{y})}_{i},\mathit{\alpha})\equiv {\mathbf{k}}_{3,i}^{T}\mathit{\alpha}$, where k_{m,i} is the ith column of K_{ m }. Given ${\left\{{\mathbf{K}}_{i}\right\}}_{i=1}^{3}$, MKL can be used to learn α and β.
Although similar to (5), (6) differs in two important ways. First, (6) obtains kernels using (4) for different data sources, namely gene expression, miRNA expression, and their interaction. Second, we allow for dependence between data sources via the interaction kernel (K_{3}), but MKL does not; instead MKL uses a convex combination of the different kernels from the same data source to aid prediction.
where ${\sigma}_{m}^{2}$ is the “bandwidth” parameter of the mth kernel matrix and is chosen a priori through crossvalidation (see [14] for details). The other choices of kernels include polynomial kernels and matern kernels [18]. To account for the overall mean (or intercept) in (1), an extra row of 1’s is appended to the feature matrices in (7); therefore, ${\left\{{\mathbf{K}}_{i}\right\}}_{i=1}^{3}$ hereafter have dimensions (N+1)×N.
2.3 Generative Bayesian model for HRVM
HRVM reformulates (6) as a hierarchical Bayesian model for greater flexibility and better interpretation of its parameters. This reformulation serves two important purposes. First, HRVM is interpreted as a hierarchical Bayesian extension of RVM [15], which is a special case of Bayesian KL. Second, instead of using MKL methods, HRVM learns parameters α and β from t,X, and Y using the variational learning algorithm of hierarchical kernel learning (HKL) [14, 16].
where $\mathcal{N}\left(.\right\mathit{\mu},\mathit{\Sigma})$ represents a multivariate Gaussian distribution with mean μ and covariance matrix Σ and Gamma(.c_{•},d_{•}) represents a Gamma distribution with respective shape and rate parameters c_{•} and d_{•}.
where $\mathit{\varphi}=({\varphi}_{0},{\varphi}_{1},\dots ,{\varphi}_{n})$. This setting forces many α_{ j }’s a posteriori to be near 0 with high precision. Most of the variance in t is explained by a small number of feature vectors that correspond to nonzero α_{ j }’s. These feature vectors are the “relevance vectors” of HRVM that have the following three characteristics: they prevent overfitting, represent a parsimonious description of the data, and correspond to feature vectors that are most predictive of the log survival time. An equivalent prior setting is found by marginalizing the ϕ_{ j }’s from the joint distribution of α and ϕ above, which imposes a multivariate Student’s t prior on α with mean 0.
where the mth component of β, β_{ m }, denotes the influence of mth source in predicting the log survival time.
The hierarchical Bayesian model (8) – (12) specifies a complete sampling model for the HRVM framework. It can also be interpreted as a probabilistic approach for combining the predictions of log survival times from the three RVMs respectively corresponding to gene expressions, miRNA expressions, and their interactions. HRVM introduces an additional hierarchy and combines the predictions of these three RVMs as a weighted average, with the weights generated from a Dirichlet distribution (12). The increased flexibility of HRVM over RVM comes at the cost of analytic intractability of the posterior distributions of the HRVM parameters. Estimation of the posterior distributions of the HRVM’s parameters can proceed via either simulationbased Markov chain Monte Carlo (MCMC) approaches or deterministic variational Bayes approaches. Given the complexity and size of highthroughput data in general and GBM data in particular, MCMC approaches tend to be computationally expensive and slow. We employ variational Bayes methods from HKL [16] and obtain the analytically tractable variational posterior distribution, q(α,β,ϕ,γt,X,Y,c_{ ϕ },d_{ ϕ },c_{ γ },d_{ γ },a_{1},a_{2},a_{3}), that approximates analytically intractable true posterior distribution, p(α,β,ϕ,γt,X,Y,c_{ ϕ },d_{ ϕ },c_{ γ },d_{ γ },a_{1},a_{2},a_{3}). This approximation achieves analytic tractability by assuming that α,β,ϕ, and γ are independent under the variational posterior distribution. The analytic tractability leads to improved computational efficiency of the variational Bayes approach over samplingbased MCMC approaches. The derivations for variational posterior distributions are provided in Appendix A Appendix: Variational inference for HRVM.
3 Data analysis
We apply the HRVM approach to the GBM data as introduced in Section 1. GBM was one of the first cancers evaluated by the TCGA. GBM data have multiple molecular measurements on over 500 samples that include gene expression, copy number, methylation and microRNA expression. TCGA datasets are available at http://tcgadata.nci.nih.gov/tcga/. The dataset we analyze here includes information about the gene expressions (11972 probes), miRNA expressions (534 probes), and (uncensored) survival times for matched patient samples (248).
To remove the irrelevant noise variables before model fitting, we prescreened the gene and miRNA probes as follows. We performed univariate regression of the log survival times on the gene expression values, obtained pvalues, and retained gene and miRNA probes that cross a liberal pvalue threshold (≤ 0.05 here) – to balance the practical and statistical significance. This preselection identifies 1747 and 43 gene expression and miRNA probes, respectively, for downstream modeling. All our analyses and comparisons were based on these selected gene and miRNA probes.
We compare the predictions of HRVM and three linear methods: penalized regressions (2) with the Lasso penalty [13] and with the following covariates: i. gene expressions (GeneLasso), ii. miRNA expressions (MiRNALasso), and iii. both gene and miRNA expressions, and their first order interactions (InteractionLasso). We randomly split the GBM survival data into a training data and a test data with 223 (90%) and 25 (10%) patients, respectively. HRVM, GeneLasso, MiRNALasso, and Interaction Lasso are fit using the gene and miRNA expressions and log survival times in the training data. The variational inference algorithm is used for fitting HRVM (see Appendix A Appendix: Variational inference for HRVM). The R package glmnet is used for the three penalized linear regressions [19, 20]. The log survival times of the test data are predicted for the four methods using the model fits on the training data. The mean squared prediction errors (MSPEs) are respectively calculated for the four models as the average of the squared difference between the observed and predicted values for the test data. This process of randomly splitting the GBM survival data into training+test data and fitting the four models is repeated 50 times. The results are summarized below.
To gain biological insights into our results, we performed a functional analysis of our model fitting results. We used Ingenuity Pathway Analysis software to perform functional analysis on selected significant genes used in fitting HRVM. We used targetHub [22] to find the known and predicted interactions between significant genes and miRNAs. mirTarBase, a curated database of experimentally validated miRNA targets, was our choice as a source of known gene and miRNA interactions [23]. To identify the predicted gene and miRNA interactions, we used targetScan data [24] to filter out miRNAgene interactions in which the miRNA and gene effects on survival were concordant, since discordant behavior is expected in biological systems for a direct interaction between miRNA and its targets.
List of known genemicroRNA interactions identified as significant in the HRVM model using target analysis
Gene symbols  microRNA 

FOXP3, YY1, KLF13, ETS1  hsamir31 
FOXO3, DDIT4  hsamir221 
ATAT1  hsamir23a 
FOXO3  hsamir222 
FGG, CPEB3, FGB, PIK3R1  hsamir29a 
PDGFB  hsamir146b 
PDCD4, TOPORS, BASP1, MARCKS, TP53BP2  hsamir21 
SIRT1, YY1, E2F3, CDC25C  hsamir34a 
4 Conclusions and future work
We have presented an integrative framework, HRVM, that generalizes the multiple kernel learning framework for integrating highdimensional data from multiple sources, incorporating within and between platform interactions to develop a prediction model for clinical outcomes. We applied HRVM to a highdimensional TCGA GBM data to predict patient survival times using two data sources: gene and miRNA expressions, and found that the predictive performance of HRVM is better than those of linear methods that do not model nonlinear effects and interactions. We hypothesize that HRVM gains flexibility and power by modeling the nonlinear interaction structures between gene and miRNA expressions. HRVM will be a useful tool for biomedical researchers, as clinical prediction using multiplatform genomic information is an important step towards identifying personalized treatments for many cancers. We have code for fitting HRVM that is freely available at the corresponding author’s website (see Additional file 2).
Although we have presented the application of HRVM in the context of two platforms, the framework is general and can be extended and adapted to data from multiple platforms with different distributional assumptions. This will essentially entail a generalization of the HRVM model by assuming additional terms for the different platforms. One key issue that warrants further investigation is an increase in the number of (multiplicative) betweenplatform interaction terms. We used the computationally efficient variational Bayes methods, which are extremely useful for handling large datasets from projects such as TCGA. In addition, [17] presents more scalable versions of HKL and MKL that can be adapted to our framework. Our future work will concentrate on extending the HRVM framework using Bayesian spike and slab priors to select variables from the interacting covariates before embedding the data in the space of kernels, as well as incorporating uncertainty estimations of the scale parameters – thus aiding the joint model building process.
Endnotes
^{a} We use gene and mRNA interchangeably to mean transcriptlevel expression.
^{b} We use bold lowercase and uppercase alphabets to denote column vectors and matrices, respectively.
A Appendix: Variational inference for HRVM
where all expectations are with respect to the variational posterior distributions. Hereafter, we will denote ${\mathbb{E}}_{\u2022}\left[f\right]$ as 〈f〉_{•} for notational simplicity.
which is the type II maximum likelihood procedure as recommended in [16]. The kernel parameters ${\left\{{\sigma}_{i}^{2}\right\}}_{i=1}^{3}$ are learned respectively from three RVMs for each of the three sources using crossvalidation as recommended by [15].
Declarations
Acknowledgements
VB’s research is partially supported by NIH grant R01 CA160736; NSF grant IIS915196 and the Cancer Center Support Grant (CCSG) (P30 CA016672). WW’s work is in part funded by 5U24 CA14388304 and P30 CA016672. CO is supported by NSF grant IIS914861. The content is solely the responsibility of the authors and does not necessarily represent the official views of the U.S. National Cancer Institute, the National Institutes of Health, or the National Science Foundation. We also thank LeeAnn Chastain for editorial assistance with the manuscript.
Authors’ Affiliations
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