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Infinite mixtureofexperts model for sparse survival regression with application to breast cancer
BMC Bioinformatics volume 11, Article number: S8 (2010)
Abstract
Background
We present an infinite mixtureofexperts model to find an unknown number of subgroups within a given patient cohort based on survival analysis. The effect of patient features on survival is modeled using the Cox’s proportionality hazards model which yields a nonstandard regression component. The model is able to find key explanatory factors (chosen from main effects and higherorder interactions) for each subgroup by enforcing sparsity on the regression coefficients via the Bayesian GroupLasso.
Results
Simulated examples justify the need of such an elaborate framework for identifying subgroups along with their key characteristics versus other simpler models. When applied to a breastcancer dataset consisting of survival times and protein expression levels of patients, it results in identifying two distinct subgroups with different survival patterns (lowrisk and highrisk) along with the respective sets of compound markers.
Conclusions
The unified framework presented here, combining elements of cluster and feature detection for survival analysis, is clearly a powerful tool for analyzing survival patterns within a patient group. The model also demonstrates the feasibility of analyzing complex interactions which can contribute to definition of novel prognostic compound markers.
Background
Survival Analysis is a branch of statistics dealing with the analysis of timetofailure data and is applicable to a variety of domains like biology, engineering, economics etc. More generally, it is the analysis of timetoevent data where an event could signify death, failure etc. Particularly in the context of disease studies, it is a powerful tool for understanding the effect of patient features on survival patterns within a group. A parametric approach to such an analysis involves the estimation of parameters of a probability density function which models time. The model is further extended by considering the effect of covariates (X) on time via a regression component. Cox’s proportionality hazards model, as explained in [1], is a popular model for modeling such an effect:
(1)
where h_{0}(t) is the baseline hazard function (chance of instant death given survival till time t), x is the vector of covariates and β is a vector of regression coefficients. In this paper, we focus on covariates which are categorical in nature, since it is a frequently encountered case in biological applications.
In the past, such models have been extended to a mixture model (mixture of survival experts) in order to find subgroups in data with respect to survival time along with measuring the effect of covariates within each subgroup. In this context, (Rosen and Tanner) [2] define a finite mixtureofexperts (MOE) model by maximizing the partial likelihood for the regression coefficients and by using some heuristics to resolve the number of experts in the model. A more recent attempt at this analysis, which was carried out by [3], uses a maximum likelihood approach to infer the parameters of the model and the Akaike information criterion (AIC) to determine the number of mixture components. A Bayesian version of the mixture model has been investigated by [4], which analyzes the model with respect to time but does not capture the effect of covariates. On the other hand, the work by [5] performs variable selection based on the covariates but ignores the clustering aspect of the modeling. Similarly, [6] defines an infinite mixture model but does not include a mixture of experts, hence assuming all the covariates to be generated from the same distribution and also assumes a common shape parameter for the Weibull distribution.
In this paper, we unify the various important elements of this analysis into a Bayesian infinite mixtureofexperts (MOE) framework to model survival time, while capturing the effect of covariates and also dealing with an unknown number of mixing components. The number of experts are inferred using a Dirichlet process prior on the mixing proportions, which overcomes the issue of deciding the number of mixture components beforehand [7]. The regression component, introduced via the proportionality hazards model, is nonstandard since the Weibull distribution is not part of the exponential family of distributions due to the lack of fixedlength sufficient statistics. Another novel feature of this framework is the addition of sparsity constraints to the regression coefficients β in order to determine the key explanatory factors (covariates) for each mixture component. Since the covariates are discrete in nature, each variable is transformed to a group of dummy variables and sparsity is achieved by applying a Bayesian version of the GroupLasso (as described in [8] and [9]) which is based on a sparse constraint for grouped coefficients [10]. We demonstrate the ability of the model to recover the right sparsity pattern with simulated examples. In a related work, [11] show sparsistency (sparse pattern consistency) of the lasso in the limit of large observations. The following sections describe all the components of this unified framework with some results on a breastcancer dataset.
Methods
In this section, we explain the overall model in an incremental way starting first with a regression model for survival analysis and then attaching a clustering model to it. This also highlights the incremental nature of the algorithm presented for inference.
Bayesian survival regression
To begin with, we focus on defining a single cluster model. For survival analysis, we model the distribution of a random variable T (representing time) over the interval [0, ∞). Further, a standard survival function is defined based on the cumulative distribution over T as follows:
(2)
which models the probability of an individual surviving up to time t_{0}. The hazard function h(t), the instantaneous rate of failure at time t, is defined as follows:
(3)
For modeling purposes, our choice of distribution for modeling time is the Weibull distribution which is flexible in terms of being able to model a variety of survival functions and hazard rates. Apart from flexibility, it is also the only distribution which captures both the accelerated time model and the proportionality hazards model, see [12] for details. The Weibull distribution is defined as follows:
(4)
where α_{ w } and λ_{ w } are the shape and scale parameters, respectively. Based on the above definition and assuming rightcensored data (see [1] for details), the likelihood can be written as:
(5)
where N is the number of observations, δ_{ i } = 0 when the i^{th} observation is censored and 1 otherwise. Further, to model the effect of covariates x on the distribution over time, we apply Cox’s proportional hazards model. Under this model, the covariates are assumed to have a multiplicative effect on the hazard function:
h(tx) = h_{0}(t) exp(f(x, β)), (6)
where h_{0}(t) is the baseline hazard function, x is the vector of covariates and β is a vector of regression coefficients. In our model, we assume the function f to be a linear predictor i.e. f(x, β) = η = x^{T}β. We also consider higherorder interactions (firstorder  pairs of features, and secondorder  triplets of features etc.) instead of modeling just the main effects (individual features). Further flexibility is added to the linear predictor by adding a random effect in the following manner:
η = x^{t}β + ∈, where ∈ ~ N(0,σ^{2}). (7)
The likelihood is modified as follows to include the covariate effect:
(8)
We note that although most parts of the model described so far resemble an enhancement of a generalized linear model (GLM) (see [13]) called a randomintercept model, it is not strictly a GLM since the Weibull distribution lacks fixedlength sufficient statistics and is not considered, in a strict sense, to be part of the exponential family of distributions unless the shape parameter is known. Although the Weibull distribution lacks fixedlength sufficient statistics, for the two parameters (α_{ w }, λ _{ w }), it is still possible to define a joint conjugate prior ([14]), as is explained in the subsection on priors eq. (10). In order to provide a full Bayesian treatment of the model, we define suitable conjugate priors for the other parameters of the model, namely σ and β.
Contrast coding
In biological applications, it is very common to encounter categorical data. When the x_{ i } ’s are categorical variables, a suitable coding procedure is applied to the variables (see standard textbooks like [15]) in order to obtain the design matrix for inference. Apart from single variables (interactions of order zero), the design matrix also consists of higherorder 1st order (pairwise interactions) and 2nd order (triplet interactions). An example of a two variable (with three categories) observation matrix with a firstorder interaction transformed using dummy coding is shown in Fig. 1 (top). A default dummy coding procedure leads to overparametrization (redundancy in the number of columns) and this effect becomes profound with greater number of levels and higherorder interactions. Also in many biological applications, the categorical variables have a natural ordering in the values that they take, for example  intensity values. Based on these requirements, we use polynomial contrast codes since they are suited for ordered categorical variables and avoid overparametrization by representing a Klevel variable with K−1 columns (see Fig. 1 (bottom)). This results in representing each categorical variable as a group of contrastcoded variables. Hence, to create the full design matrix, first the levels are contrastcoded (using a standard R function) which gives us the codes for respective levels (see Fig. 1 (bottomright)) and then each observation is recoded (for main effects and higherorder interactions) using these codes as reference.
Priors
One of the major requirements of the model is to find the key explanatory factors from data. To achieve this goal, we need to apply sparsity constraints on the regression coefficients β to identify the key interactions. As described, the coding procedure gives rise to groups of contrastcoded variables. This transformation of data leads to the task of inferring sparsity on a group level, i.e. on grouped dummy variables, where each group represents a single variable in the original formulation.
Hence, for parameter β, we apply the general prior defined in [9] to a special case for Bayesian GroupLasso (as defined in [8] for a Poisson model), which is suitable for sparse inference in grouped variables for the model that we have defined. The sparse prior is motivated by the classical Grouplasso which can be recovered in the logspace based on defining the prior as a product of Multivariate Laplacians. Although a direct representation of the prior exists, in order to make the posterior analysis feasible (to obtain standard conditional posteriors), we redefine the prior as a twolevel hierarchical model, by introducing latent variables λ_{ g }.For the Bayesian Group Lasso, the hierarchical prior over the regression coefficients is defined as follows:
(9)
where G is the number of groups, p_{ g } is the size of group g, ρ and σ^{2} play the role of the Lagrange parameter in classical GroupLasso and each β_{ g } is a scaled mixture of MultivariateGaussians. Based on (9), we can derive the marginal pdf of β_{ g } analytically as a product of Multivariate Laplacians (for details, see [8]).
A full Bayesian treatment of the model is achieved by introducing a prior on σ^{2}, based on a standard conjugate joint prior (see [16]), described as a product of a Normal distribution of β given σ and an inversechi square distribution of , and a conjugate Gamma prior on ρ. Although the Weibull distribution lacks fixedlength sufficient statistics, for the remaining two parameters (α_{ w } , λ_{ w }), it is possible to define a joint conjugate prior, as explained in [14]:
(10)
where a,b,c > 0 and d allows us to deal with the lack of fixedlength sufficient statistics.
The full model with all the variables is described in Figure 2.
Posteriors
In practice, sampling from the posterior distribution will not be possible directly, hence we propose to use a Gibbs sampling strategy for stochastic integration. The sampling process further enables this procedure to be incorporated very naturally as another step in the clustering algorithm discussed in the next section. Additionally, for the lasso model, the BlockedGibbs sampler has been shown to be geometrically ergodic in [17]. Hence the convergence of the Gibbs sampler is expected to be very rapid. Multiplying the priors with the likelihood and rearranging the relevant terms yields the full conditional posteriors, which are needed in the Gibbs sampler for carrying out the stochastic integrations. The posterior for σ, β, ρ and are exactly as defined in [8]. The conditional posterior of η_{ i } is difficult to sample from since it is not of standard form. However, since the conditional posterior is logconcave, we propose the use of Laplace approximation, similar to that in [18], which approximates the conditional posterior to a Normal distribution and simplifies sampling considerably. Although alternatives exist in the form of adaptiverejection sampling, the Laplace approximation gives results that are indistinguishable while speeding up computations considerably.
For the Weibull parameters α_{ w } and λ _{ w }, sampling based on their individual posteriors conditioned on each other is avoided, since this results in a slow mixing of the Markov chain due to a high correlation between samples from the two conditionals. To overcome this issue, the conditional posterior of (α _{ w }, λ _{ w }) is split up into the conditional of λ _{ w } given α_{ w } which results in an InverseGamma distribution,
(11)
where y is the number of deaths (number of data points for which δ_{ i } = 1) and the marginal of α_{ w } which is derived based on the work in [14]:
(12)
where P_{ y } is the product of t_{ i } ’s for which δ_{ i } = 1 and (●) represents all the unknown parameters. This marginal results in a nonstandard distribution, and sampling is done via a discretized version of the same.
Infinite mixture of survival experts
Finite mixture of experts. The previous section described the inference procedure when the data is assumed to be generated from one global group. We further enhance this idea by removing this assumption and model data which is potentially generated from multiple (and known number of) subgroups/clusters in data. In order to model the clustering in terms of the combined effects of features x and survival time t, we use an MOE model as defined in [19] (see Figure 3: Left panel). It consists of a fixed number of experts, each expert explaining the distribution of time for a particular region in the covariate space. Hence the t based clusters or mixing components, represented by experts, are probability distributions conditioned on the covariates x. The distribution of t can be written based on a standard mixture model conditioned on x as:
(13)
where (●) represents all the unknown parameters and c_{ j } ’s are the mixture components. The first term in eq. (13) is the gate function which decides which j^{th} expert is best suited for making a prediction for feature vector x. Using Bayes’ rule, we can rewrite the model in the following way in order to resemble a standard mixture model, as shown in [20]):
(14)
This representation allows us to visualize each mixture component as a joint distribution over (x, t). The distribution over x is modeled as a Normal distribution as show in Figure 2. The standard joint conjugate prior of NormalInvχ^{2} is applied to the parameters . The posterior conditionals are also of standard form and hence can be easily incorporated into the Gibbs sampling scheme introduced in the previous section. To complete the Bayesian picture, we need to apply a suitable prior to the mixing proportions c. In a finite MOE model, a Dirichlet distribution is a standard conjugate prior to the mixing proportions. All other parameters and priors, based on the modeling of (x, t), follow from the previous section.
Infinite mixture of experts. The above model was described for the case when the underlying number of clusters is fixed/known. We now add the final enhancement to our model by removing this limiting assumption as well. The model is extended to an infinite mixtureofexperts by replacing finite clusters by infinite clusters and hence replacing the Dirichlet distribution by a Dirichlet process (DP) as prior for the mixing proportions, similar to [20]. The Dirichlet process is a distribution on distributions i.e. a particular sample from a DP is also a probability distribution from which samples can be drawn. The draws from a DP are discrete hence making it a useful prior for clustering purposes. In this manner, the effective number of clusters can be inferred from data by carrying out MCMC sampling from the posterior distribution. This model extension is described in a hierarchical manner as follows (see Figure 3):
(15)
where DP denotes a dirichlet process prior with base distribution G_{0} and a concentration parameter α, c_{ i } is the latent class to which an observation (x_{ i } , t_{ i } ) belongs and ϕ _{ c } denotes the parameters which determine the distribution of class c. Further hierarchy is added to ϕ _{ c } (parameters) by adding suitable priors as defined in Section 2.
Markov Chain Monte Carlo (MCMC) sampling for Inference and Parameter Estimation. The inference of the infinitemixtureofexperts model is carried out by MCMC sampling of the posterior distribution. Although there exist nonconjugate versions of the Dirichlet process algorithms (as given in [21]) which can be applied for inference, for practical reasons, we use a truncated version of the Dirichlet process called the DirichletMultinomial allocation model [22], by specifying an upper bound on maximum number of clusters based on the prior knowledge of the particular application. It serves as a good approximation to the DP measure and results in a finitesum random probability measure which is computationally easy to deal with and easy to implement. More specifically, we carry out a BlockedGibbs sampling on a truncated Dirichlet process (see Algorithm 1 for details). After initializing all the parameters, the sampling algorithm is executed till the point of convergence. The point of convergence can be determined based on the lengthcontrol diagnosis explained in [23] or fixed to a maximum number of iterations based on studying the traceplots of the sampling process in simulations.
Results and discussion
Simulations. In order to demonstrate the effectiveness of the model, experiments were carried out on simulated data. The first experiment shows the capability of the model to correctly identify two subgroups in data along with identifying the key explanatory factors in both groups. The dataset of size 150 was generated from two equally proportioned clusters with (5, 5) and (1,1) being the shape and scale parameters for the Weibull distribution for each cluster. The features consisted of 7 variables with expansion up to 2nd order interactions (63 terms). For the first cluster, the significant factors included main effects X 1, X 3 and X 4, all first order interactions with these three variables i.e. (X 1 : X 3), (X 1 : X 4), (X 3 : X 4) and a second order interaction (X 1 : X 3 : X 4). Similarly, for the second cluster, the significant factors included main effects X 2, X 6 and X 7, all first order interactions with these three variables (i.e. (X 2 : X 6), (X 2 : X 7), (X 6 : X 7)) and a second order interaction (X 2 : X 6 : X 7).
Significance was achieved by assigning β values of (3, 3, 3, 3, 3, 3, 3) and (3, 3, 3, 3, 3, 3, 3) to the specific factors in the respective clusters and the rest of the β coefficients to zero. The covariates themselves were sampled from a Normal distribution with means (0.3, 0.3, 0.3, 0.3, 0.3, 0.3, 0.3) and (0.7, 0.7, 0.7, 0.7, 0.7, 0.7, 0.7) for each cluster respectively. The Gibbs sampling process was executed for 50,000 iterations and the burnin was observed to be very early (in the first ≈ 100 iterations). Both the clusters were detected and all the true significant factors for both clusters were identified successfully. See Figure 4 for details.
In the second experiment, we compare our mixtureofexperts model to a global single cluster model in order to justify the need for a mixture model. The training data generated in the first experiment was used again for learning the parameters of a singlecluster model. In order to compare the two models, a separate test set (of size 500) was generated additionally to evaluate the performance of both models by comparing the loglikelihood of all the test points based on the parameters learned by both models. The perpoint comparison is shown in Figure 5 which indicates the improvement achieved by using a MOE model. We also performed a standard KruskalWallis rank test which also ranks the MOE model higher than the single cluster model (see Figure 5 left panel). Apart from the quantitative evaluation, we also see in terms of identifying the significant factors (see Figure 5 right panel), that the single cluster model does poorly, both in recognizing the true factors and in terms of false positives. This can be explained based on the fact that in a single cluster model, the model has to assume a common baseline model (for both clusters). Then, in order to adjust for the real survival patterns, it can only achieve the same effect by making suitable adjustments to the regression component. In doing so, the model compromises in terms of the identification of significant factors from data. As a result, we see that the MOE model performs much better than a onecluster model, hence justifying the need for a clusterbased model.
Application to BreastCancer dataset. The dataset consists of measured intensity levels obtained from tissue microarrays of the following markers: karyopherinalpha2 (KPNA2), nuclear staining for p53, the anticytokeratin CK5/6, the fibrous structural protein CollagenVI, the interαtrypsin inhibitor ITIH5, the estrogen receptor (ER) and the human epidermal growth factor receptor HER2. From these categorical variables we constructed covariates arranged in a design matrix which includes all dummycoded interactions up to the second order.
Crossvalidation experiments were conducted for both the MOE and single cluster model which gave rise to similar trends but with unclear significance. Despite of the fact that this dataset is one of the biggest of its kind, the rather low number of samples (270 patients) remains the main challenge in these scenarios. A further difficulty is the large number of censored patients (60%), which is a common problem in long term retrospective studies.
Over a wide range of priorvalues, the Dirichlet process mixture model for selecting “survival experts” finds two large and highly stable clusters. In order to externally validate these clusters, we analyze the survival of the underlying patient populations by way of classical KaplanMeier plots, see Figure 6. It is obvious that the survival experiences of patients belonging to the two clusters differ significantly, with cluster 1 basically containing all patients who die early. In Figure 7, the interaction patterns within the two clusters are shown as lines connecting pairs or triplets of markers, where the linewidth encodes the significance in terms of posterior quantiles which do not contain zero.
The highrisk patient cluster is characterized by a global underexpression of ER and overexpression of basically all other markers, in particular KPNA2, CK5/6 and HER2. Overexpression of the latter two markers clearly identifies this cluster as a collection of basal and HER2type breastcancer patients. The occurrence of KPNA2 in the highrisk group is also in accordance with previous studies: KPNA2 is a member of the karyopherin (importin) family, which is part of the nuclear transport protein complex. KPNA2 overexpression has been shown in several gene expression signatures in breast cancer and other cancer types. KPNA2 overexpression has been previously identified as a possible prognostic marker in breast cancer [24].
The groupLasso detects several strong higherorder interactions. Interpreting these interaction terms can be a complex problem, but a close analysis of the contrast codes and the sign of the regression coefficients shows that the weak prognosis of members in this cluster is dominated by some of the combinations, details in Table 1 where ↘ means underexpression and ↗ overexpression.
The observation that highorder interaction terms seem to be even more indicative than the individual main effects is a highly interesting result of this study which may lead to the definition of novel prognostic markers for better differentiation between highrisk patients. Together with our medical partners we are currently testing these new hypothetical compoundmarkers.
The lowrisk cluster has a clear luminaltype signature (strong ER response). Hardly any significant patterns can be identified which, however, is quite understandable by noticing that the survival curve is almost flat for these patients: in the proportional hazards model the individual covariates influence the “passage of time”, and a flat curve basically means that there is almost no intraclass variation that could be explained by individual covariate effects.
Conclusions
We have introduced a fully Bayesian survival infinite mixtureofexperts model which extends classical approaches by including feature selection for contrastcoded categorical variables. Random links and a mixtureofexperts architecture allow for both stochastic and modeldriven deviations from the underlying parametric survival model. The inherent clustering property of the final model makes it possible to identify patient groups which are homogeneous with respect to the predictive power of their covariates for the observed survival times. The builtin Bayesian feature selection mechanism reveals clusterspecific explanatory factors and interactions. Due to the Bayesian treatment within a suitably expanded model, posterior samples can be generated efficiently which makes it possible to assess the statistical significance based on a very large number of draws.
Applied to survival data from a breast cancer study, the model identified two stable patient clusters that show a clear distinction in terms of survival probability. Several strong highorder interactions between marker proteins were detected which carry more information about the survival targets as the markers themselves. Not only does this result confirm earlier studies, it also shows that the analysis of complex interactions is feasible and may lead to the definition of novel prognostic markers. We are currently conducting new experiments to test these new hypothetical compoundmarkers.
Abbreviations
 AIC:

Akaike information criterion
 MOE:

Mixture of experts
 GLM:

Generalized linear model
 MCMC:

Markov chain Monte Carlo
 DP:

Dirichlet Process
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Acknowledgements
The work was supported by a grant of the Swiss SystemsX.ch Initiative (Swiss National Science Foundation) to the project ”LiverX” (Competence Center for Systems Physiology and Metabolic Diseases). We also acknowledge financial support from the FET programme within the EU FP7, under the SIMBAD project (Contract 213250).
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Authors contributions
SR, TJF, JMB and VR have contributed toward designing the model and drafting the manuscript. PJW and ED are domain experts in pathology and molecular biology and have contributed with respect to conducting biological experiments, generating the required samples and in analyzing the results, i.e. estimating the protein expression on the immunohistochemical stained slides. All authors read and approved the final manuscript.
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Raman, S., Fuchs, T.J., Wild, P.J. et al. Infinite mixtureofexperts model for sparse survival regression with application to breast cancer. BMC Bioinformatics 11, S8 (2010). https://doi.org/10.1186/1471210511S8S8
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Keywords
 Weibull Distribution
 Lasso
 Dirichlet Process
 Markov Chain Monte Carlo Sampling
 Baseline Hazard Function