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16
Adaptive Lasso for Cox’s Proportional Hazards Model
 Biometrika
, 2007
"... We investigate the variable selection problem for Cox’s proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalised log partial likelihood with the adaptivelyweig ..."
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Cited by 20 (4 self)
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We investigate the variable selection problem for Cox’s proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalised log partial likelihood with the adaptivelyweighted L1 penalty on regression coefficients, providing what we call the adaptive Lasso estimator. The method incorporates different penalties for different coefficients: unimportant variables receive larger penalties than important ones, so that important variables tend to be retained in the selection process, whereas unimportant variables are more likely to be dropped. Theoretical properties, such as consistency and rate of convergence of the estimator, are studied. We also show that, with proper choice of regularisation parameters, the proposed estimator has the oracle properties. The convex optimisation nature of the method leads to an efficient computation algorithm. Both simulated and real examples show that the method performs competitively.
Estimating Ratios of Normalizing Constants for Densities with Different Dimensions
 STATISTICA SINICA
, 1997
"... In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Mont ..."
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Cited by 16 (2 self)
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In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong 1996), the path sampling method (Gelman and Meng 1994), and the ratio importance sampling method (Chen and Shao 1994) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative meansquare errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.
Computational Methods for Multiplicative Intensity Models using Weighted Gamma . . .
 PROCESSES: PROPORTIONAL HAZARDS, MARKED POINT PROCESSES AND PANEL COUNT DATA
, 2004
"... We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share ..."
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Cited by 16 (4 self)
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We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of Bayesian hierarchical models using the Dirichlet process. Using this fact, along with an approximation for the weighted gamma process, we show that with some care, one can adapt efficient algorithms used for the Dirichlet process to this setting. We discuss blocked Gibbs sampling procedures and Pólya urn Gibbs samplers. We illustrate our methods with applications to proportional hazard models, Poisson spatial regression models, recurrent events, and panel count data.
Bayesian Inferences in the Cox Model for Order Restricted Hypotheses
"... ... This article proposes a Bayesian approach for addressing hypotheses of this type. Ve reparameterize the Cox model in terms of a cumulative product of parameters having conjugate prior densities, consisting of mixtures of point masses at one and truncated garnma densities. Due to the structure of ..."
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Cited by 7 (4 self)
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... This article proposes a Bayesian approach for addressing hypotheses of this type. Ve reparameterize the Cox model in terms of a cumulative product of parameters having conjugate prior densities, consisting of mixtures of point masses at one and truncated garnma densities. Due to the structure of the model, posterior computation can proceed via a simple and efficient Gibbs sampling algorithm. Posterior probabilities for the global null hypothesis and subhypotheses coinparing the hazards fbr specific groups eau be calcnlated directly from the output of a single Gibbs ('.hain. The approach allows for level sets across which a predictor has no effect. Generalizations to multiple predictors are described, and the method is applied to a study of emergency medical treatment for stroke.
AdaptiveLASSO for Cox’s proportional hazards model
 Biometrika
, 2007
"... We investigate the variable selection problem for Cox’s proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalized log partial likelihood with the adaptivelyweig ..."
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Cited by 7 (2 self)
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We investigate the variable selection problem for Cox’s proportional hazards model, and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience. The new method is based on a penalized log partial likelihood with the adaptivelyweighted L1 penalty on regression coefficients, and is named adaptiveLASSO (ALASSO) estimator. Instead of applying the same penalty to all the coefficients as other shrinkage methods, the ALASSO advocates different penalties for different coefficients: unimportant variables receive larger penalties than important variables. In this way, important variables can be protectively preserved in the model selection process, while unimportant ones are shrunk more towards zero and thus more likely to be dropped from the model. We study the consistency and rate of convergence of the proposed estimator. Further, with proper choice of regularization parameters, we have shown that the ALASSO perform as well as the oracle procedure in variable selection; namely, it works as well as if the correct submodel were known. Another advantage of the ALASSO is its convex optimization form and convenience in implementation. Simulated and real examples show that the ALASSO estimator compares favorably with the LASSO.
A Bayes method for a monotone hazard rate via Spaths
 Ann. Statist
, 2006
"... A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of Spaths. A closed and tract ..."
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Cited by 5 (1 self)
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A class of random hazard rates, that is defined as a mixture of an indicator kernel convoluted with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of Spaths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over Spaths. The path characterization or the estimator is proved to be a RaoBlackwellization of an existing partition characterization or partitionsum estimator. This accentuates the importance of Spath in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that Spath characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.
Approaches for Semiparametric Bayesian Regression
 Computational Approach for Full Nonparametric Bayesian Inference under Dirichlet Process Mixture Models,&quot; Journal of Computational and Graphical Statistics
, 1997
"... Developing regression relationships is a primary inferential activity. We consider such relationships in the context of hierarchical models incorporating linear structure at each stage. Modern statistical work encourages less presumptive, i.e., nonparametric specifications for at least a portion of ..."
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Cited by 4 (2 self)
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Developing regression relationships is a primary inferential activity. We consider such relationships in the context of hierarchical models incorporating linear structure at each stage. Modern statistical work encourages less presumptive, i.e., nonparametric specifications for at least a portion of the modeling. That is, we seek to enrich the class of standard parametric hierarchical models by wandering nonparametrically near (in some sense) the standard class but retaining the linear structure. This enterprise falls within what is referred to as semiparametric modeling. We focus here on nonparametric modeling of monotone functions associated with the model. Such monotone functions arise, for example, as the stochastic mechanism itself using the cumulative distribution function, as the link function in a generalized linear model, as the cumulative hazard function in survival analysis models, and as the calibration function in errorsinvariables models. Nonparametric approaches for mod...
Bayesian model selection and averaging in additive and proportional hazards models
, 2004
"... Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selecti ..."
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Cited by 2 (1 self)
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Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additivemultiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to submodels corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure nonnegative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.
A bayes method for a bathtub failure rate via two spaths
 National University of Singapore
, 2007
"... A class of semiparametric hazard/failure rates with a bathtub shape is of interest. It does not only provide a great deal of flexibility over existing parametric methods in the modeling aspect but also results in a closed and tractable Bayes estimator for the bathtubshaped failure rate (BFR). Such ..."
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Cited by 1 (0 self)
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A class of semiparametric hazard/failure rates with a bathtub shape is of interest. It does not only provide a great deal of flexibility over existing parametric methods in the modeling aspect but also results in a closed and tractable Bayes estimator for the bathtubshaped failure rate (BFR). Such an estimator is derived to be a finite sum over two Spaths due to an explicit posterior analysis in terms of two (conditionally independent) Spaths. These, newly discovered, explicit results can be proved to be a RaoBlackwellization of counterpart results in terms of partitions that are readily available by a specialization of James (2005)’s work. We develop both iterative and noniterative computational procedures based on existing efficient Monte Carlo methods for sampling one single Spath. Numerical simulations are given to demonstrate the practicality and the effectiveness of our methodology. Last but not least, two applications of the proposed method are discussed, of which one is about a Bayesian test for failure rates and the other is related to modeling with covariates. 1