A Bayesian approach for inducing sparsity in generalized linear models with multi-category response
© Madahian et al. 2015
Published: 1 December 2015
The dimension and complexity of high-throughput gene expression data create many challenges for downstream analysis. Several approaches exist to reduce the number of variables with respect to small sample sizes. In this study, we utilized the Generalized Double Pareto (GDP) prior to induce sparsity in a Bayesian Generalized Linear Model (GLM) setting. The approach was evaluated using a publicly available microarray dataset containing 99 samples corresponding to four different prostate cancer subtypes.
A hierarchical Sparse Bayesian GLM using GDP prior (SBGG) was developed to take into account the progressive nature of the response variable. We obtained an average overall classification accuracy between 82.5% and 94%, which was higher than Support Vector Machine, Random Forest or a Sparse Bayesian GLM using double exponential priors. Additionally, SBGG outperforms the other 3 methods in correctly identifying pre-metastatic stages of cancer progression, which can prove extremely valuable for therapeutic and diagnostic purposes. Importantly, using Geneset Cohesion Analysis Tool, we found that the top 100 genes produced by SBGG had an average functional cohesion p-value of 2.0E-4 compared to 0.007 to 0.131 produced by the other methods.
Using GDP in a Bayesian GLM model applied to cancer progression data results in better subclass prediction. In particular, the method identifies pre-metastatic stages of prostate cancer with substantially better accuracy and produces more functionally relevant gene sets.
Using high-throughput microarray or massively parallel RNA sequencing technologies, the expression levels of several thousand genes can be measured across a number of samples simultaneously. Analysis of gene expression data obtained by these technologies is mathematically challenging because generally the number of samples are small (usually tens to hundreds) compared to thousands of variables . Several statistical methods in univariate analysis framework have been developed to address this problem [2–6]. However, single gene analysis is unable to identify weaker associations, especially for complex polygenic phenotypes for which the relevant variation is distributed across several genes . In order to address these limitations, several approaches for simultaneous analysis of multiple variables have been developed [7–9]. These approaches require an initial feature selection method to identify a smaller set of genes with the strongest effect and discriminating power. Some variable selection methods in a regression framework include backward elimination, forward selection, and stepwise selection. One of the shortcomings of these methods is that they are discrete processes which are very sensitive to the changes in the data. That is, a minor change in data can result in very different models [10–12]. Additionally, the computational complexity of these methods, when the number of variables is very large make them less attractive for gene expression analysis [10, 11]. Moreover in this setting, over-fitting is a major concern and may result in failure to identify important predictors. Thus, the data structure of typical gene expression experiments makes it difficult to use traditional multivariate regression analysis .
Several groups have developed methods to overcome drawbacks of multivariate regression analysis [7, 8, 10, 12, 13]. Various methods such as K-nearest neighbour classifiers , linear discriminant analysis , and classification trees  have been used for multi-class cancer classification and discovery [15–17]. However, gene selection and classification are treated as two separate steps which can limit their performance. One promising approach to analyse, predict, and classify binary or multi-category samples using gene expression data is Generalized Linear Models (GLM) [18–20]. However, due to the large number of variables, maximum likelihood estimates of parameters becomes computationally intensive and sometimes intractable. Additionally, since the sample size is much smaller than the number of variables, the maximum likelihood estimates may have large estimated variances and thus result in poor prediction accuracy. Finally, the maximization process may not converge to maximum likelihood estimates .
Previously, it was proposed that the prediction accuracy of GLMs can be improved by setting the parameters associated with unimportant variables to zero and thus obtaining more accurate prediction for the significant variables without over-fitting . Least Absolute Shrinkage and Selection Operator (LASSO) is a well-known method for inducing sparseness in the model while highlighting the relevant variables [11, 12, 21]. Later, a Bayesian LASSO method was proposed by [22, 23] in which double exponential prior is used on parameters in order to impose sparsity. However, these procedures may cause over-shrinkage of large coefficients due to the relatively light tails of the double exponential prior and introduce bias [24, 25]. A modification of this approach, which uses normal-Jeffreys prior with heavier tails than double exponential distribution, is able to shrink small coefficients to zero while minimally shrinking large coefficients reducing bias in the model. However it has no meaning from an inferential aspect as it leads to an improper posterior . An alternative class of hierarchical priors proposed in  uses Bayesian adaptive Lasso with non-convex penalization, but it lacks a simple analytic form. Others have proposed the Generalized Double Pareto (GDP) prior distribution, which has several advantages . The GDP distribution has a spike at zero alongside studentt like tails. While GDP resembles double exponential density in the neighbourhood of zero, it has heavier tails compared to the double exponential, which remedies unwanted bias resulting from over shrinkage of parameters toward zero . In addition, GDP has a simple analytic form and yields proper posteriors. In many of the approaches, the variables are assumed fixed, but in many cases where the predictor variables are random, such as gene expression data, assumptions can be made that result in the same formulation as in fixed case . One such assumptions is a joint multivariate normal distribution for response and predictors, other is an analysis of response conditioned upon the random predictors.
In our previous work, we implemented a sparse Bayesian generalized linear model with double exponential prior to classify different subtypes of prostate cancer using gene expression profiles . Given the limitations discussed above regarding this prior, in this study we aimed at using the GDP prior to overcome these issues. Here, we applied GDP for the first time into the Bayesian generalized linear model framework. The model was utilized to classify multi-category ordinal phenotypes based on gene expression data. We evaluated the model based on classification of progressive stages of prostate cancer using a publicly available microarray dataset . Our specific objectives were to test if the model can: 1) result in a smaller subset of genes with high discriminating power, 2) obtain high classification accuracy; 3) identify more biologically relevant genes compared to other classification methods.
In this way, the linear predictor is linked to the multi-category response variable y i . The function that links the linear predictor to the response variable is called a link function and in the multinomial Logistic model, this link function is cumulative distribution of a standard Logistic density as defined above [19, 20, 29]
Prior distributions and Baysian set up
As we can see, the fully conditional distributions of , u1, and u2 do not have closed form and thus we adopt the following embedded griddy gibbs sampling to sample from them [19, 24]. On a grid of k values (υ1, υ2, ..., υ k ) representing the degrees of freedom we consider, we perform the following procedure:
• Calculate the weights as r i = π(υ i |−) using fully conditional posterior obtained for υ.
• Normalize the weights
The conditional posterior distribution of γ s can be seen to be Uniform(δ1, δ2) in which δ1 = max[max i [l i |y i = s − 1], γs−1] and δ2 = min[min i [l i |y i = s], γ s ]. It should be noted that I() is the indicator function and its value is one if its argument is true and is zero otherwise .
Dataset and Feature Selection
The method was applied to a published dataset on prostate cancer progression downloaded from Gene Expression Omnibus at NCBI (GSE6099) . The dataset contains gene expression values for 20,000 probe sets and 101 samples corresponding to five prostate cancer progressive stages (subtypes): Benign, prostatic intraepithelial neoplasia (PIN), Proliferative inflammatory atrophy (PIA), localized prostate cancer (PCA), and metastatic prostate cancer (MET) . Since there were only two samples for PIA, we removed these samples from further analysis. Sample accession number and tumor types are listed in Additional file 1. Probes with null values in more than 10% of the samples were removed from the dataset. For the remaining probes, the null values were imputed by using the mean value of the probe across samples with non-null values. Before applying our model to this dataset, for each gene we performed logistic regression for ordinal response. This method enables us to take into account the ordinal nature of the response variable in the analysis and preparation of a gene list used as input to the model. Genes were ranked based on the p-value associated with the hypothesis H0 : θ i = 0 from the most significant to least significant. Here θ i is the parameter associated with gene i. We performed Benjamini and Hochberg FDR correction . An FDR cut-off value of 0.05 resulted in a list of 398 genes. Thus, the input to our model was 398 variables (genes) for 99 samples corresponding to four different prostate cancer subtypes (Additional files 1 and 2). The Gibbs sampling algorithm was implemented in R software and the program ran for 60k iterations and the first 20k was discarded as burn-in.
Simulation and Cross-validation Procedure
The dataset was randomly divided into training (N = 50) and test (N = 49) groups so that each group contained an equal number of prostate cancer subtypes Benign, PIN, PCA and MET. Genes were ranked based on the posterior mean of parameters and the top 10 or 50 genes obtained from the model were used for classification. In order to make the model more robust we performed 50 re-samplings on the selection of training and test groups and re-ran the model. Sample accession numbers for training and test sets for each of the 50 runs are listed in Additional file 3. The average performance of SBGG was compared to three well-known classification methods: Support Vector Machine (SVM), Random Forest, and the Sparse Bayesian Generalized Linear Model obtained by imposing double exponential prior (SBGDE) on parameters that we developed previously . SVM was implemented in R software using Kernlab library . Specifically, ksvm (y., data = dataset, kernel = "rbf dot", type =′ nu − svc′, prob.model = T RU E, kpar =′ automatic′) with automatic sigma estimation was used to fit SVM model. The Random Forest was implemented in R using default parameters in randomForest library , We implemented the SBGDE according to [25, 27] in R software.
Overall average accuracy and associated standard deviations (in parentheses) of SBGG, SBGDE, SVM and Random Forest models using 10 and 50 marker genes
Average classification accuracy and associated standard deviations (in parentheses) of prostate cancer subtypes in the test group using SBGG, BBGDE, SVM and Random Forest models for 50 marker genes
Average classification accuracy and associated standard deviations(in parentheses) of prostate cancer subtypes in the test group using SBGG, BBGDE, SVM and Random Forest models for using 10 marker genes
Literature based functional cohesion p-values (LPv) and associated standard deviations (in parentheses) of the top 100 genes obtained from SBGG, SBDE, logistic regression, and Random Forest models
Ordinal Logistic Regression
Microarray gene expression technology is commonly used to gain insights into the mechanisms of human disease and to develop classifiers for prediction of outcomes [40, 41]. Gene expression based classifiers can be used for diagnosis of disease as well as for specifically tailoring treatments for individuals [42, 43]. Developing robust classifiers is hampered because gene expression experiments measure thousands of genes across a few number of samples, known as the "large p, small n" situation in statistical modeling. Previous studies have shown that the correct selection of subsets of genes from microarray data is important for accurate classification of disease phenotypes, [44, 45]. However, statistical classifiers are prone to over-fitting to the specific cohort under investigation and may not be generalizable to other cohorts [46, 47]. From a biological perspective, classifiers are more generalizable if they focus on specific pathways that are mechanistically related to the disease phenotype. In this study, we have developed a sparse Bayesian generalized double pareto model which addresses the "large p, small n" problem and produces a more functionally cohesive set of genes.
The Generalized Double Pareto (GDP) prior distribution was proposed, in linear regression framework, as an alternative to induce sparseness in situations when we are faced with large number of variables compared to sample size . This prior has a simple analytic form, yields a proper posterior and possesses appealing properties, including a spike at zero, Student t-like tails, and a simple characterization as a scale mixture of normals leading to a straightforward Gibbs sampler for posterior inferences that makes Bayesian shrinkage estimation and regularization feasible . Utilizing this prior in a more general framework of generalized linear models, we presented a Bayesian hierarchical model to handle multi-category outcome situations when the number of variables is much larger that sample size. While shrinking small effects toward zero and producing sparse solutions, the over shrinkage problem caused by using light-tailed priors is remedied by the heavier tails obtained via mixing over the hyper parameters .
We used the Sparse Bayesian Generalized Linear Model (SBGG) model to do prediction of tumor type on the test dataset. We showed that the average classification accuracy of SBGG using 50 marker genes was substantially higher than other competing methods. In clinical applications, it is desirable to reduce the number of marker genes and be able to perform predictions based on a smaller set of markers. Using ten marker genes, average classification accuracy using SBGG was higher than SVM and SBGDE and slightly lower (0.5%) than random forest. It is important to note that SBGG performs substantially better in correctly identifying premetastatic (PIN, PCA) stages of prostate cancer which can prove extremely useful for diagnostics and therapeutics in clinical settings. SBGG substantially outperforms the other 3 methods in correctly identifying pre-metastatic stages of prostate cancer regardless of the number of marker genes utilized for prediction purposes.
As seen in Figure 3, SVM performance is lower than the other three methods. In multi-class classification with k categories, "ksvm" uses one-against-one approach in which binary classifiers are trained. The appropriate class is found by a voting scheme. The class that gets maximum votes is the winning class. In this paper, we declared a winning class when votes exceeded 50%, which is quite stringent. After closer examination, we found that in some cases SVM identified the correct class, but the number of votes was below the 50% threshold. This result indicates that SVM is less sensitive than the other methods.
Importantly, SBGG identified more biologically relevant gene sets in addition to showing better classification performance (Table 4). This result indicates that by having heavier tails in the prior distributions, SBGG is able to identify weaker gene expression changes that have more functional relevance to the phenotype of interest. Thus, we posit that SBGG may be a better approach to simultaneously identify marker genes for classifications as well as gaining insights into the molecular mechanisms of the phenotype under investigation.
It is important to note that the classification accuracy of all three models were compared using a selected set of 398 genes which were obtained based on p-value of a single gene analysis using an ordinal regression model. Hence, this may bias the initial gene selection process. It is possible that some biologically relevant genes to the prostate cancer progression might have been missed by this analysis due to low signal. One way to perform an initial gene selection could be to consider gene pathway information as described previously by others . Our future plan is to evaluate SBGG performance using pathway driven feature selection methods while considering more complex covariance matrix structure which takes into account gene-gene interactions. Also, we plan to incorporate literature information into the prior distributions in order to design literature informed priors that would potentially enable us to obtain machine learning models with high classification accuracy which provide a very enriched set of markers with high biological relevance to the phenotype under study.
This work and its publication was supported by the Billl & Melinda Gates Foundation and University of Memphis Center for Translational Informatics.
This article has been published as part of BMC Bioinformatics Volume 16 Supplement 13, 2015: Proceedings of the 12th Annual MCBIOS Conference. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/16/S13.
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