Volume 11 Supplement 8
Infinite mixture-of-experts model for sparse survival regression with application to breast cancer
© Raman et al; licensee BioMed Central Ltd. 2010
Published: 26 October 2010
We present an infinite mixture-of-experts model to find an unknown number of sub-groups 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 non-standard regression component. The model is able to find key explanatory factors (chosen from main effects and higher-order interactions) for each sub-group by enforcing sparsity on the regression coefficients via the Bayesian Group-Lasso.
Simulated examples justify the need of such an elaborate framework for identifying sub-groups along with their key characteristics versus other simpler models. When applied to a breast-cancer dataset consisting of survival times and protein expression levels of patients, it results in identifying two distinct sub-groups with different survival patterns (low-risk and high-risk) along with the respective sets of compound markers.
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.
where h0(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 sub-groups in data with respect to survival time along with measuring the effect of covariates within each sub-group. In this context, (Rosen and Tanner)  define a finite mixture-of-experts (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 , 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 , which analyzes the model with respect to time but does not capture the effect of covariates. On the other hand, the work by  performs variable selection based on the covariates but ignores the clustering aspect of the modeling. Similarly,  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 mixture-of-experts (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 . The regression component, introduced via the proportionality hazards model, is non-standard since the Weibull distribution is not part of the exponential family of distributions due to the lack of fixed-length 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 Group-Lasso (as described in  and ) which is based on a sparse constraint for grouped coefficients . We demonstrate the ability of the model to recover the right sparsity pattern with simulated examples. In a related work,  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 breast-cancer dataset.
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
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(t|x) = h0(t) exp(f(x, β)), (6)
where h0(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 higher-order interactions (first-order - pairs of features, and second-order - 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,σ2). (7)
We note that although most parts of the model described so far resemble an enhancement of a generalized linear model (GLM) (see ) called a random-intercept model, it is not strictly a GLM since the Weibull distribution lacks fixed-length 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 fixed-length sufficient statistics, for the two parameters (α w , λ w ), it is still possible to define a joint conjugate prior (), 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.
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 contrast-coded 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.
where G is the number of groups, p g is the size of group g, ρ and σ2 play the role of the Lagrange parameter in classical Group-Lasso and each β g is a scaled mixture of Multivariate-Gaussians. Based on (9), we can derive the marginal pdf of β g analytically as a product of Multivariate Laplacians (for details, see ).
where a,b,c > 0 and d allows us to deal with the lack of fixed-length sufficient statistics.
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 Blocked-Gibbs sampler has been shown to be geometrically ergodic in . 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 . The conditional posterior of η i is difficult to sample from since it is not of standard form. However, since the conditional posterior is log-concave, we propose the use of Laplace approximation, similar to that in , which approximates the conditional posterior to a Normal distribution and simplifies sampling considerably. Although alternatives exist in the form of adaptive-rejection sampling, the Laplace approximation gives results that are indistinguishable while speeding up computations considerably.
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 non-standard distribution, and sampling is done via a discretized version of the same.
Infinite mixture of survival experts
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 Normal-Inv-χ2 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.
Algorithm 1 Blocked Gibbs Sampling for a Truncated Dirichlet process
Input: N observations D = (x i , t i ).
Initialize: c i = random cluster assignments and parameters .
Draw from the posterior of the joint distribution p(π, Φ*, c) by drawing from the conditionals.
while NotCoverged do
Sample Φ* | π, c, D - This is carried out individually for each parameter in the model conditioned on the rest.
Sample c | Φ*, π, D - For i = 1,…, N, draw values , c i = 1,…, M.
Sample π | Φ*, c, D - The mixture proportions are drawn based on the posterior P(π|α)P(c|π).
Markov Chain Monte Carlo (MCMC) sampling for Inference and Parameter Estimation. The inference of the infinite-mixture-of-experts model is carried out by MCMC sampling of the posterior distribution. Although there exist non-conjugate versions of the Dirichlet process algorithms (as given in ) which can be applied for inference, for practical reasons, we use a truncated version of the Dirichlet process called the Dirichlet-Multinomial allocation model , 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 finite-sum random probability measure which is computationally easy to deal with and easy to implement. More specifically, we carry out a Blocked-Gibbs 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 length-control diagnosis explained in  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 sub-groups 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).
Application to Breast-Cancer dataset. The dataset consists of measured intensity levels obtained from tissue microarrays of the following markers: karyopherin-alpha-2 (KPNA2), nuclear staining for p53, the anti-cytokeratin CK5/6, the fibrous structural protein Collagen-VI, the inter-α-trypsin 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 dummy-coded interactions up to the second order.
Cross-validation 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.
The high-risk 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 HER2-type breast-cancer patients. The occurrence of KPNA2 in the high-risk 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 .
Interpretation of interaction terms
The observation that high-order 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 high-risk patients. Together with our medical partners we are currently testing these new hypothetical compound-markers.
The low-risk cluster has a clear luminal-type 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 intra-class variation that could be explained by individual covariate effects.
We have introduced a fully Bayesian survival infinite mixture-of-experts model which extends classical approaches by including feature selection for contrast-coded categorical variables. Random links and a mixture-of-experts architecture allow for both stochastic and model-driven 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 built-in Bayesian feature selection mechanism reveals cluster-specific 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 high-order 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 compound-markers.
List of abbreviations
Akaike information criterion
Mixture of experts
Generalized linear model
Markov chain Monte Carlo
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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