Bayesian Data Integration and Variable Selection for Pan-Cancer Survival Prediction using Protein Expression Data

Summary:

Accurate prognostic prediction using molecular information is a challenging area of research which is essential to develop precision medicine. In this paper, we develop translational models to identify major actionable proteins that are associated with clinical outcomes like the survival time of the patients. There are considerable statistical and computational challenges due to the large dimension of the problems. Furthermore, the data are available for different tumor types hence data integration for various tumors is desirable. Having censored survival outcomes escalates one more level of complexity in the inferential procedure. We develop Bayesian hierarchical survival models which accommodate all these challenges aforementioned here. We use hierarchical Bayesian accelerated failure time (AFT) model for the survival regression. Furthermore, we assume sparse horseshoe prior distribution for the regression coefficients to identify the major proteomic drivers. We allow to borrow strength across tumor groups by introducing a correlation structure among the prior distributions. The proposed methods have been used to analyze data from the recently curated “The Cancer Proteome Atlas” (TCPA) which contains RPPA based high quality protein expression data as well as detailed clinical annotation including survival times. Our simulation and the TCPA data analysis illustrate the efficacy of the proposed integrative model which links different tumors with the correlated prior structures.

1. Introduction

Accurate prognostic prediction using molecular profiles is an essential ingredient to develop precision medicine. Molecular profiling data emerging from high-throughput technologies can be used to search for new biomarkers and to develop accurate prognostic tools and predictive models. In this paper, we used proteomics data to develop similar translational models for identifying major actionable proteins that are associated with clinical outcomes like the survival time of the patients.

Direct analysis of high dimensional proteomics data has received widespread attention because it represents a powerful approach to understand the pathophysiology and therapy of cancer, which cannot be achieved by analyses solely driven by genomics or transcriptomics (Li et al., 2013; Akbani et al., 2014). Baladandayuthapani et al. (2014) noted that many proteins are regulated by post-translational modifications, such as phosphorylation or cleavage events, that are not detected by the analysis of DNA or RNA. Moreover, several studies have also demonstrated marked discordance between mRNA and protein expression levels, particularly for genes in kinase signaling and cell cycle regulation pathways (Shankavaram et al., 2007). It has been demonstrated recently, in both cancer cell lines and tumors, that different genetic mutations in the same signaling pathway can result in significant differences in the quantitative activation levels of downstream pathway effectors (Park et al., 2010). These observations support the suggestion that direct measurements are essential to measure protein activation. To fulfill this thrust a new protein expression data visualization tool has been generated for 31 types of tumors using reverse-phase protein arrays (RPPAs) and have been uploaded in a user-friendly data portal, The Cancer Proteome Atlas (Li et al., 2017).

The primary source of the data motivating our methodological work comes from this TCPA. The current data covers more than 7500 tumor samples and signaling pathways in cancer such as P13K, MAPK, and mTOR. The 31 cancer types include Bladder, Breast, Colon, Brain, Head and Neck, Kidney, Lung (adenocarcinoma and squamous cell carcinoma), Rectum, Ovarian, and Uterine cancers. Moreover, TCPA presents the tumor cell lines: 439 samples in four cell line sets, including both baseline and drug-treated cell lines. Figure 1 in Li et al. (2013) shows a detailed schematic of the TCPA data portal. The mRNA expression and DNA (copy number) were matched over large cohorts of well-characterized TCGA patient tumors. In addition, there is detailed clinical annotation that includes survival times and clinical subtype/stage information on the tumor samples. To the best of our knowledge, this represents the largest collection of cancer functional proteomics data with parallel genomic, transcriptomic and clinical data currently available. For a detailed description of TCPA data portal we refer the readers to Li et al. (2013) and references therein.

The information available in TCPA are collected and presented for each tumor type which is likely to be different and independent across tumor types. A schematic diagram of TCPA data is provided in Table 1. To accomplish the scientific goal of enhancing the statistical power of the inference procedure integration of such data has been shown extremely useful; we cite Hamid et al. (2009) for a comprehensive review. For instance, Daemen et al. (2009) used kernel based approaches to integrate the genomics data.

Table 1.

TCPA Data Structure

Tumor Groups	Survival time of Subjects	Measurements of Protein Expressions
Tumor 1	t11	x111, x121, … , x1p
	t21	x211, x221, … , x2p
	…	…
	tn1 1	xn1 11, xn1 21, … , xn1p
Tumor 2	t12	x112, x122, … , x1p
	t22	x212, x222, … , x2p
	…	…
	tn2 2	xn2 12, xn222, … , xn2p
Tumor k	t1k	x11k, x12k, … , x1p
	t2k	x21k, x22k, … , x2p
	…	…
	tnkk	xnk1k, xnk2k, … , xnkp

The availability of detailed matched proteomics data on hundreds of tumors collated by TCPA provides a major opportunity to develop an integrated picture of commonalities, differences and emergent themes across tumor lineages. In this aim, we propose to develop methods to integrate Pan-Cancer data across tumor lineages. The overreaching goal of this Pan-Cancer effort is to provide a birds eye view of the functional proteome encompassing multiple tumor lineages which may help to suggest potential targets that are applicable to disease subsets or across diseases. The Pan-Cancer analysis was launched by TCGA which involves integration across tumor types and organs of origin to gain better analytical breadth (Weinstein et al., 2013). Our interest in this paper is to develop a Pan-Cancer model which will particularly link different tumor groups.

Each tumor group in the TCPA data portal consists of at least 189 proteins. Therefore, we must deal with high dimensional statistical analysis after merging the data. Because of the easy interpretation, one seeks for a parsimonious model selection strategy which can be achieved via several regularization techniques existing in the literature. For instance, there exist different penalized regressions, lasso (Tibshirani, 1996) and extensions of lasso including adaptive lasso (Zou, 2006), and many others. The Bayesian variable selection literature is also rich; Stochastic Search Variable Selection (SSVS) (George and McCulloch, 1993), variable selection with shrinking and diffusing priors (Narisetty et al., 2014) to name a few. The prior specification is critical in any Bayesian analysis, if not essential. Hence, to achieve shrinkage and sparsity via the prior formulation is generally of interest. Toward this end, Carvalho et al. (2010) proposed shrinkage estimation via the Horseshoe prior. Other propositions include Dirichlet-Laplace prior of Bhattacharya et al. (2015). However, the applications of these shrinkage priors in the setting of censored data are very limited till now. To our knowledge, Peltola et al. (2014) provided a comparison study using different shrinkage priors and concluded in the favor of Horseshoe prior in terms of the predictive abilities of the models. They considered a parametric Weibull model on the survival time.

In the settings of survival regressions, the straightforward implementation of these variable selection techniques may not be tenable particularly due to the censored observations which requires additional sampling methods from the censored space. For example, Lee and Mallick (2004) assumed a linear model on the scale parameter of the Weibull survival regression or assumed a Gamma process on the Cox proportional hazard model (Cox, 1972), while Zhang et al. (2018) proposed a Dirichlet process prior on the Accelerated Failure Rate (AFT) model (Miller, 1976). In addition, following data augmentation approach of Tanner and Wong (1984) the censored observations can be imputed (Bonato et al., 2011). On the other hand, the survival versions of penalized regularizations have also been discussed; important references include lasso (Tibshirani, 1997) and ridge regression (Li and Luan, 2002) in the context of Cox model. When developing the AFT model is of particular interest then there exist several proposal in the literature, for example, Huang et al. (2006); Cai et al. (2009) derived some regularized versions including lasso and Huang and Ma (2010) discussed bridge regression, and (Wang and Song, 2011) developed adaptive lasso for AFT models. Furthermore, Khan and Shaw (2016, 2017) developed a class of adaptive elastic net techniques and synthesized techniques for variable selection in AFT models. However, they did not consider Bayesian settings in their research.

In this paper we develop a Bayesian hierarchical AFT model. Unlike Cox proportional hazard model in which the covariates act multiplicatively on the hazard function, the AFT model considers the additive effect of the covariates on the log of survival responses which results in an intuitive linear regression interpretation (Wei, 1992). In addition, when dealing with high dimensional proteomics data it has been reported to have the poor mixing in the MCMC chain in fitting a Cox model (Sha et al., 2006). In Cox proportional hazard models the regression parameters cannot be integrated out due to unavailability of any conjugate priors thus requires complex MCMC procedures. In the contrary, in our Bayesian log normal AFT model, as we show in this article, most of the conditional distributions are available and thereby straightforward Gibbs sampling can be employed to update the chain.

Although the variable selection techniques described above have been useful to identify important features in genomics data, they have not been developed to incorporate the data integration procedures. In the contrary, the existing integration methods are not well examined for high dimensional Pan-Cancer settings for a single platform. In particular, in the presence of censored data we observe a lack of a unified method which can be employed to deal with all these issues. Toward this end, we propose a Bayesian hierarchical model which fits a log-normal AFT model in each group. Additionally, to achieve integration across human tumor groups we model the mean parameter of the prior distributions on the coefficients to borrow strength across groups. To accomplish sparsity via the shrinkage priors we place Horseshoe priors (or variants) on the coefficients. The full methodology has been combined and implemented in R package hsaft. The resulting estimates are shown to be efficient compared to the existing regression methods run for each tumor group separately. At the end, selection of the most important proteins for the TCPA data is done using predictive survival curve. In particular, we use Brier score (Brier, 1950) to asses the predictive performance of our fitted model.

A conceptually related field is multi-task survival analysis which is very recently being introduced in Li et al. (2016); Wang et al. (2017); Liu et al. (2018). Among these Wang et al. (2017) propose a unified framework to integrate the multiple survival models for multiple tumor groups and provide a set of output models for each groups. In the contrary we propose a single model which accommodates all the tumor groups simultaneously. Furthermore, they worked with the Cox proportional hazard model while we employ the Bayesian analysis on parametric AFT model.

The remainder of this article is organized as follows. In Section 2, we discuss the Bayesian log-normal AFT model, the prior formulation for borrowing strength, the sampling strategies from the posterior distribution space of the parameters, and the consistency property of the parameter estimator. In Section 3 we describe the survival prediction and the variable selection strategy from the MCMC samples. To show that the proposed method is superior than the individual analysis, we present simulation examples in Section 4. Finally, we illustrate the application of our proposed technique in TCPA data in Section 5 followed by a brief discussion in Section 6 to conclude this article.

2. Model

From Table 1 it can be noted that the RPPA based protein expression data in TCPA is expressed for several tumors; for instance, there are three types of kidney tumors, two types of lung cancers and so on. The objective is to identify the major proteins common across the tumor groups which explains the survival of the subjects. Essentially, our goal is to fit a Pan-Cancer model across multiple types of cancer. We wish to identify if there is any special characteristic which is common across these cancers. That way, we also learn about the presence of any important cancer specific characteristic. Furthermore by borrowing information among different cancers, we have better power to identify important cancer specific features.

Toward this goal, we consider r groups of cancers (or tumors) present in the data. Suppose that the interest is to make inference on the survival times of subjects of each cancer from p proteins which are same across all tumor groups. We use the Accelerated Failure Time (AFT) model which regresses the survival time on the covariates. The AFT model is given by, $\log (t_{ik})={\sum }_{j=1}^p{x}_{ijk}{\beta }_{jk}+{ϵ}_{ik}$, i = 1, … , n_i, j = 1, … , p, k = 1, … , r, where i denotes the patient, j denotes the protein and k denotes the type of cancer. Likewise, t_ik is the survival time of i-th patient who has the k-th cancer, x_ijk is the corresponding pth protein expression in the TCPA data, β = (β₁₁, … , β_pr) is the vector of regression coefficients, and ϵ is the error vector. Assumption of ϵ ~ N(0, σ²I) gives raise to the log normal AFT model, whereas, one could assume other distributions such as t distribution (Kleinbaum and Klein, 2006).

Letting c_ik to be the censoring time, the observed time may be denoted by $t_{ik}^{*}=\min (t_{ik},c_{ik})$; the corresponding observed censored indicator is δ_ik = I{t_ik ⩽ c_ik}, I{.} being the censoring indicator. Since the response is right censored, we follow the data augmentation approach of Tanner and Wong (1984) to impute the censored data w_ik (see also Bonato et al. 2011), $w_{ik}=\log t_{ik}^{*}$, if t_ik is event time; and $w_{ik}>\log t_{ik}^{*}$, if t_ik is right censored.

2.1. Shrinkage Prior

Due to the presence of large number of proteins, we will carry out a variable selection procedure in this AFT model to identify the important ones. We consider the shrinkage priors on the coefficients. In the shrinkage framework, a scale-mixture representation of the global local priors allows parameters to be updated in blocks via a fairly automatic Gibbs sampler (Bhattacharya et al., 2016) which makes it convenient for large scale problems. Here, we adopt Horseshoe prior (Carvalho et al., 2010; Peltola et al., 2014). The hierarchical Horseshoe representation for the AFT model is:

$$\log t_{ik}|{\beta }_{jk},{\sigma }^2\sim N\left({\sum }_{j=1}^p{x}_{ijk}{\beta }_{jk},{\sigma }^2\right){\beta }_{jk}|{\lambda }_{jk},T,{\sigma }^2\sim N(b_{Pj},{\lambda }_{jk}^2{T}^2{\sigma }^2),b_{Pj}\sim N(0,{\sigma }_P^2){\lambda }_{jk}\sim C^{+}(0,1),T\sim C^{+}(0,1),{\sigma }^2\sim \pi ({\sigma }^2)=1/{\sigma }^2$$ (1)

where, C⁺(0,1) is the truncated Cauchy density given by f (x) = 1/π(1 + x²), x > 0. The conditional distribution and posterior computations are discussed in the Web Appendix A.

2.2. Borrowing Strength

In the above formulation, a common τ allows borrowing strength across different cancer groups and proteins, while λ_jk’s provide protein and cancer-specific deviations. Additionally, the different tumor groups do not have same number of observations, hence straightforward regression for each tumor may not reveal the true picture of dependencies between the response and predictors which will be further illustrated in Sections 4 and 5. In order to resolve this issue we analyze the whole data together, that is, we integrate the data in our regression analysis. There are many advantages in doing so. For instance, it aids to identify non-obvious relationships existing in the data (Jansen et al. 2002). Moreover, the interpretation of the result will be simultaneous and hence more easily understandable. In addition, individual regression for each tumor group may suffer from lack of sufficient number of observations which we overcome through this type of data integration. The integration of the data can be done in several ways and Bayesian hierarchical models are particularly suitable for this purpose.

In order to take advantage of Bayesian hierarchical models, a convenient way is to borrow strength via the prior elicitation and carrying out joint estimation of the parameters across tumor groups. Ibrahim et al. (2002) specified a class of hierarchical priors on the regression coefficients in such a way that the correlation among covariates (proteins in our case) can be captured and strength can be borrowed across covariates. However, in our scenario, we wish to borrow strength across tumor groups, which can be accomplished by specifying a mean parameter for the coefficients in equation (1).

The hyper-prior elicitation in equation (1) helps us to create correlation among tumors for a given protein. The correlation between k-th group and k′-th group can be derived (Web Appendix G) as follows:

$$\mathrm{Corr}({\beta }_{jk},{\beta }_{j{k}^{\prime }})=\frac{{\sigma }_P^2}{{({\lambda }_{jk}^2{T}^2{\sigma }^2+{\sigma }_P^2)}^{\frac{1}{2}}{({\lambda }_{j{k}^{\prime }}^2{T}^2{\sigma }^2+{\sigma }_P^2)}^{\frac{1}{2}}}.$$ (2)

We note that, the correlation formulation depends on the hyperparameters σ_P, λ, τ, and σ. Since, τ and σ are global and are not group specific, they appear not to control borrow strength across tumor groups. On the other hand, ${\sigma }_P^2$ is present both in the numerator and the denominator of the right hand side in equation (2), so increasing value of ${\sigma }_P^2$ precludes to increase the value of correlation among proteins for different tumor groups and this is further illustrated in Web Appendix C. Nonetheless, it is the hyperparameter λ which helps us to borrow strength among tumors. Lower the value of ${\lambda }_{jk}^2$ higher will be the induced correlation. Note that, since ${\lambda }_{jk}^2$ also serves as shrinkage factor for the corresponding j-th protein for the k-th group, lower value of ${\lambda }_{jk}^2$ will lead to declare the protein as nonsignificant. Hence, although counter intuitive, maximum borrowing strength among tumor groups actually happens via the non-significant proteins. This phenomenon will be observed via the posterior analysis of the correlations in TCPA data in Section 5.

Note that, within each group an individual independent analysis can be carried out simply by setting the b_Pj = 0 in (1) which will be referred as “local” method in this paper.

3. Posterior Inference from MCMC Output

In the Bayesian settings the inference is based on the posterior samples of the MCMC chains. The survival curve prediction is outlined in the Web Appendix B. Other inferences proceed as follows.

To assess the variable selection performance in our simulation experiments we consider the true positive rate (TPR) or sensitivity which is defined by the proportion of truly identified variables by an analysis which were originally present in the model. On the other hand, false positive rate (FPR) is given by the proportion of falsely identified variables which were not present originally. Plotting the TPR against FPR for varying cut-offs provides the so-called ratio operating characteristic (ROC) curve. It may be noted that, under a perfect fit TPR should be close to 1 and FPR should be close to 0. This follows that, the greater the area under the curve, the better is the fit.

The strategies described in the previous discussion provide convincing ways to compare different methodologies in simulation studies. Unfortunately, in TCPA data, we are unaware of the true proteins which are responsible for explaining the survival of the patients, rather our job is exactly to find these to communicate with our clinical collaborators further. Due to the absence of a comprehensive path to do the variable selection, we propose a rank based approach via a goodness of fit measure. To achive this we use Integrated Brier Score (IBS) (Hothorn et al., 2006; Bonato et al., 2011) (Definition is provided in Web Appendix J).

We propose the following protein selection scheme based on the IBS. The proteins are ordered according to their absolute value of the posterior means of the regression parameter estimates. Now a subset of top most significant proteins can be used to fit a AFT model and corresponding IBS can be calculated. Like a forward selection criteria, if we keep increasing number of proteins in the model per their order, the IBS will keep decreasing. Then the selection of ordered proteins proceeds until a desired value of IBS. A practical demonstration is provided in Section 5. We note that, changing the threshold of IBS is equivalent to changing the threshold value to compute the TPR and FPR when we know the true model. In this way, depending on the threshold, we can recover the significant proteins which is common for more than one cancer group if they are included using the IBS threshold. For instance, in the example of tumors in female body (see Web Appendix F) the protein DIRAS3 becomes significant for at least two cancer types namely Breast cancer and Ovarian cancer.

4. Simulation Examples

In this section, we discuss simulation scenarios to illustrate the methodologies discussed above. For the simulation, we consider 5 groups each having 80 proteins. The sample sizes of the groups are set to 40, 50, 70, 100, and 120 respectively, so that, the first three groups have p > n layouts. Furthermore, the total number of parameters (400) is more than the total number of observations (380). The covariates are generated from standard normal distribution independently. For the response generation, the σ² is set to 1. The censoring distribution is assumed follow a Gamma distribution. In this way, the censoring rate can be varied generally by changing the shape and scale parameters accordingly. We simulate 100 datasets in this manner.

We consider two primary settings. In the first, which we refer to as Example 1, we generate β in the following manner. We assume the first protein is significant for all groups. Then the second protein is significant for the first group only. Similarly, the third, fourth, fifth, and sixth proteins are significant for the second, third, fourth, and fifth groups only respectively. Then we keep replicating this generation of β for every other 10 variables. In order to make a protein significant we set corresponding β = 1. So, for each block of 10 proteins the data generating matrix for β will look like:

$$\left[\begin{array}{cccccccccc}1& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 1& 0& 0& 0& 0\end{array}\right]$$

after which we assign a random positive or negative sign on each element.

We fit 5 different models in the generated data. Independent AFT regression with Horseshoe prior is fitted for each group. This is referred as “local”. Then our proposed model discussed in Section 2.2 is fitted, which can be referred as “group-corr”. As indicated in Web Appendix H, we also fit the model which borrows strength across groups and across proteins simultaneously (“all-corr”). In addition to these Bayesian methods we apply two penalized cox regression – lasso (Tibshirani 1997, Simon et al. 2011) and adaptive lasso – “alasso”, (Li et al., 2015) for each group separately.

In the setting of this example, we declare a variable as significant when the absolute estimated β is more than a given threshold value. Once the variables are identified using a given threshold, we can compute the TPR and FPR. For 100 simulations of this experiment, the mean of these 100 TPR’s and FPR’s can be calculated. Now varying the threshold from a range between 0 and 1, will produce a series of mean TPR’s and FPR’s which can be used to plot ROC curve, given in Figure 1, for each method. One can note that the area under the ROC curves for the borrowing strength structure are greater than the that due to other methods which indicates that whatever the threshold is chosen, the variable selection performance of our proposed methodology remains superior. The area under the ROC curve (AUC) results are given in Table 2, which confirm our findings. For the Bayesian analyses, the posterior means have been calculated using 10000 samples after 5000 burnin. From the ROC plots and the numerical results, it is evident that, correlation structures are actually helping to estimate the true parameters.

Figure 1.

The ROC curves for different methods, from left to right subplots are for varying censoring rates – around 35%, 50%, and 76% respectively. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

Table 2.

Area under the ROC curves plotted in Figure 1

Censoring Rate	35%	48%	76%
local	0.659	0.646	0.590
group-corr	0.695	0.689	0.648
all-corr	0.694	0.689	0.648
lasso	0.415	0.406	0.452
alasso	0.632	0.621	0.406

In Example 2, we generate the simulated data under a similar set up as in Example 1, however we consider a correlation structure among groups. In what follows, we produce the design matrix in a manner such that the rows of two different tumor groups are highly correlated with correlation 0.8. We plot the ROC curves in Figure 2 and report the AUC values for this ROC’s in Table 3. Table 3 exhibits consistent performances of all methods as in the previous example.

Figure 2.

The ROC curves for different methods, from left to right subplots are for varying censoring rates – around 35%, 56%, and 76% respectively, in the presence of correlation. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

Table 3.

Area under the ROC curves plotted in Figure 2, correlated scenario.

Censoring Rate	35%	56%	76%
local	0.661	0.649	0.597
group-corr	0.712	0.709	0.660
all-corr	0.712	0.708	0.655
lasso	0.377	0.444	0.460
alasso	0.557	0.388	0.418

5. TCPA Protein Expression Data

We apply the methodologies described above in two types of TCPA data – Kideny tumors and tumors in female bodies. While the first is described below, the second application is discussed in the Web Appendix F.

There are three types of kidney cancers viz. KICH (Kidney Chromophobe), KIRC (Kidney renal clear cell carcinoma), and KIRP (Kidney renal papillary cell carcinoma). According to Linehan and Ricketts (2013), all types of kidney cancer are different, making it even more important to characterize each one. In 2017, it is estimated that there will be 63,990 new cases of kidney cancer and 14,400 deaths as a result of this disease (American Cancer Society 2017 report). Chromophobe (KICH) kidney cancer accounts for five percent of these cancer cases. On the other hand, Renal Cell Carcinoma is the most common type of kidney cancer which are broadly classified into KIRC and KIRP.

The protein data for these tumors has 63, 469, and 215 samples respectively with 189 proteins which follows that KICH group has more number of proteins than the number of observations whereas the KIRP consists of nearly same number of samples and proteins. In addition, approximately 75.9% samples are censored and Web Figure S1 presents group-wise observed Kaplan-Meir plots.

For finding the significant protein selections associated with the survival cancer outcome, this data is used to apply our methodologies developed in this paper. In Web Figure S3 we plot the posterior estimates of the coefficients for a single MCMC chain. We insert the posterior summary for the group-corr method along with summary for the regressions run within each tumor group (local). All proteins have been standardized before the analyses. For the statistical inference 20000 posterior samples are collected after discarding 10000 burnin samples.

5.1. Protein Selection

One of the key goals of the clinical researchers is to predict the survival times of the individuals which can be accomplished by computing the posterior predictive survival curve. Additionally, as mentioned in Section 3, we aim to select important proteins using this predictive summary. The mean IBS (defined in Section 3) obtained from 4 MCMC chains for group-corr method is 0.168 while the mean IBS for the local method is 0.201 which implies that the group-corr method has better predictive ability. It should be mentioned here that IBS due to all-corr method is same (0.168) as that due to group-corr method. So, for simplicity, we restrict ourselves considering the group-corr method only for the following discussion.

In particular, parsimonious model selection is desirable in the presence of many features and hence sparsity is assumed in these kinds of analyses because of the easy interpretation. Toward this end, we identify top 8 most significant proteins from the posterior estimates of group-corr method and use these proteins to run a simple AFT model with log normal assumption. This gives us the IBS as 0.172 which is very close to the IBS of original fitting with all proteins. So, one can conclude in the favor of these 8 proteins explaining maximum variation present in the data. Again, to confirm our outcome, we run similar AFT model 10 times with randomly selected 8 proteins which results the IBS as 0.331 which is more than the IBS 0.172 due to group-corr. Since models with lower IBS is preferred, the variable selection technique using the correlation can be argued as a reliable technique. The selected proteins are listed in Table 4. The effects of these proteins on the cancers have been well studied in the literature. For example, Advani et al. (2015) showed how medication to CRAF_pS338 improves the treatment; Duckworth et al. (2016) concluded that overexpression of GAB2 promotes tumor growth; similarly, SF2 has been established as a critical pathway for human cancer cell survival, dissemination, and resistance to drug therapy (Wang et al., 2014). PCADHERIN, FOXO3A_pS318S321, and DIRAS3 are the top 3 proteins recovered by all 4 chains. Not surprisingly, these are well known for kidney tumor growth and invasion (Blaschke et al., 2002; Ni et al., 2014; Chen et al., 2016).

Table 4.

Top 8 proteins selected for kidney cancers

Chain 1	Chain 2	Chain 3	Chain 4
PCADHERIN	PCADHERIN	PCADHERIN	PCADHERIN
FOXO3A_pS318S321	DIRAS3	FOXO3A_pS318S321	DIRAS3
DIRAS3	FOXO3A_pS318S321	DIRAS3	FOXO3A_pS318S321
SF2	RAD51	RAD51	RAD51
RAD51	SF2	GAB2	SF2
GAB2	GAB2	SF2	GAB2
HER3_pY1298	BETACATENIN	HER3_pY1298	CRAF_pS338
CRAF_pS338	HER3_pY1298	BAK	HER3_pY1298

6. Conclusion

In this paper, we have proposed a Bayesian variable selection technique which accommodates both high-dimensional shrinkage and integration of the censored data. We have only considered log-normal AFT model set up. Nonetheless, the extension to any other distribution is immediate. Furthermore, use of the latent variable technique in may also make us possible to extend this method for high dimensional Bayesian Cox regression. Another future research topic will be to explore borrowing strength from multiple factors efficiently. For instance, Kling et al. (2015) considered the incorporation of sample sizes with the hope of eliminating the sample size effects of different groups from the final inference. One could follow their suggestion to incorporate similar factor terms in the prior elicitation.

Instead of using an improper prior, a vague Inverse Gamma prior on σ² may be more suitable. Another recommendation could be to integrate out parameters in obtaining the marginal distributions of w_ik which will be a student’s t distribution by suitably choosing the shape and rate parameters of Inverse Gamma prior on σ² (Sha et al., 2006). Other suggestions include placing a hyper-prior on ${\sigma }_P^2$. For example, Ibrahim et al. (2002) argued for specifying an Inverse Gamma prior would help to borrow strength more. On the other hand, Polson and Scott (2012) recommended a Half-Cauchy prior. Nevertheless, in our setup, we did not see much improvement in the estimations using these priors.

The future research is due to explore the impact of different sample sizes of the tumor groups. Moreover, the proposed method has been applied solely on two real-world datasets, and thus the effectiveness of the method should be thoroughly tested in the future on further applications. Nonetheless, our contribution toward the overreaching goal of extracting broad information out of the RPPA based data bank TCPA will help to gain better understanding of Pan-Cancer models.