Log Pseudo Marginal Likelihood Research Articles

Analysis of recurrent event data with spatial random effects using a Bayesian approach.

Recurrent event data, which represent the occurrence of repeated incidences, are common in observational studies. Furthermore, collecting possible spatial correlations in health and environmental data is likely to provide more information for risk prediction. This article proposes a comprehensive proportional intensity model considering spatial random effects for recurrent event data using a Bayesian approach. The spatial information for areal data (where the spatial location is known up to a geographic unit such as a county) and georeferenced data (where the location is exactly observed) is examined. A traditional constant baseline intensity function, as well as a flexible piecewise constant baseline intensity function, are both under consideration. To estimate the parameters, a Markov chain Monte Carlo method with the Metropolis-Hastings algorithm and the adaptive Metropolis algorithm are applied. To assess the performance of model fitting, the deviance information criterion and log pseudo marginal likelihood are proposed. Overall, simulation studies demonstrate that the proposed model is significantly better than models that do not consider spatial effects if spatial correlations exist. Finally, our approach is implemented using a dataset related to the recurrence of cardiovascular diseases, which incorporates spatial information.

Patient- and Areal-Level Risk Factors Associated With Lung Cancer Mortality in Victoria, Australia: A Bayesian Spatial Survival Analysis.

In Australia, lung cancer is the leading cause of cancer-related deaths. In Victoria, the mortality risk is assumed to vary across Local Government Areas (LGAs) due to variations in socioeconomic advantage, remoteness, and healthcare accessibility. Thus, we applied Bayesian spatial survival models to examine the geographic variation in lung cancer survival in Victoria. Data on lung cancer cases were extracted from the Victorian Lung Cancer Registry (VLCR). To account for spatial dependence and risk factors of survival in lung cancer patients, we employed a Bayesian spatial survival model. Conditional Autoregressive (CAR) prior was assigned to model the spatial dependence. Deviance Information Criterion (DIC), Watanabe Akaike Information Criterion (WAIC), and Log Pseudo Marginal Likelihood (LPML) were used for model comparison. In the final best-fitted model, the Adjusted Hazard Ratio (AHR) with the 95% Credible Interval (CrI) was reported. The outcome variable was the survival status of lung cancer patients, defined as whether they survived or died during the follow-up period (death was our interest). Our study revealed substantial variations in lung cancer mortality in Victoria. Poor Eastern Cooperative Oncology Group (ECOG) performance status, diagnosed at a regional hospital, Small Cell Lung Cancer (SCLC), advanced age, and advanced clinical stage were associated with a higher risk of mortality, whereas being female, presented at Multidisciplinary Team (MDT) meeting, and diagnosed at a metropolitan private hospital were significantly associated with a lower risk of mortality. Identifying geographical disparities in lung cancer survival may help shape healthcare policy to implement more targeted and effective lung cancer care services.

Bayesian modeling of clustered competing risks survival times with spatial random effects

n some studies, survival data are arranged spatially such as geographical regions. Incorporating spatial associationin these data not only can increase the accuracy and efficiency of the parameter estimation, but it also investigatesthe spatial patterns of survivorship. In this paper, we considered a Bayesian hierarchical survival model in thesetting of competing risks for the spatially clustered HIV/AIDS data. In this model, a Weibull Parametric distributionwith the spatial random effects in the form of county-failure type-level was used. A multivariate intrinsic conditionalautoregressive (MCAR) distribution was employed to model the areal spatial random effects. Comparison amongcompeting models was performed by the deviance information criterion and log pseudo-marginal likelihood. Weillustrated the gains of our model through the simulation studies and application to the HIV/AIDS data.

A Bayesian small area model with order restrictions for contingency tables

Estimating the probabilities of possible outcomes for small areas can be difficult, due to a lack of available data from national surveys. One of the statistical techniques for small area estimation is using multinomial Dirichlet models to borrow information among small areas. We study Bayesian diagnostics for multinomial counts from small areas. Within each area, the probabilities of counts for possible outcomes are ordered (e.g., unimodal ordering). Specifically we consider Bayesian diagnostics for a multinomial Dirichlet model with order restriction which shares a common effect among areas. The log pseudo marginal likelihood (LPML) is a well-known Bayesian criterion for comparing models. Since the order restriction significantly increases the difficulty, we develop an algorithm to compute LPML. We use a special-designed importance function to increase the efficiency of Monte Carlo integration, thereby gaining a higher precision for estimation of LPML. The proposed methodology is applied to a case study of body mass index (BMI) and a simulation study to test different scenarios.

The Effect of Time-dependent Prognostic Factors on Survival of Non-Small Cell Lung Cancer using Bayesian Extended Cox Model

Background: Lung cancer is one of the most common cancers around the world. The aim of this study was to use Extended Cox Model (ECM) with Bayesian approach to survey the behavior of potential time-varying prognostic factors of Non-small cell lung cancer.
 Materials and Methods: Survival status of all 190 patients diagnosed with Non-Small Cell lung cancer referring to hospitals in Yazd were recorded from 2009 to 2013 by phone call. We fitted conventional Cox proportional hazards (Cox PH) as well as Bayesian ECM. Inference for estimated risk ratios was based on 90% credible intervals. Log pseudo marginal likelihood criteria (LMPL) was used for model comparison. Statistical computations were based on R language.
 Results: In this study, 190 patients with non-small cell lung cancer were followed, of whom 160 died because of the disease (84.2%). Median of survival time was 8 ± 0.076 month. After fitting the Cox PH Model, it was determined that the PH assumption was not satisfied for the type of treatment, the disease stage, and pathology status variables (p <0.001). LPML for Cox PH and Bayesian ECM was -431.593 and -401.01, respectively. Estimated hazard ratio curves based on Bayesian ECM showed that the risk ratio for these variables exhibited significant time varying behavior on hazard of lung cancer through follow up time.
 Conclusion: Based on LMPL, Bayesian ECM was found to have a better fit than Cox PH Model which declares, results from Cox PH should be interpreted with care. Especially, from beginning of the study to about 20 month after, very high risk ratio was estimated for variables whose PH was not satisfying for them.

Bayesian regularization for flexible baseline hazard functions in Cox survival models.

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi-parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B-spline function. For those "semi-parametric" proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo-marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the "semi-parametric" baseline hazard specification, the B-splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of therisk.

Modeling of spatio-temporally clustered survival HIV/AIDS data in the presence of competing risks setting

In some applications, clustered survival data are arranged spatio-temporally such as geographical regions over multiple time periods. Incorporating spatio-temporal variation in these data not only can improve the accuracy and efficiency of parameter estimation, but it also investigates spatial pattern of survivorship over the study period for identifying high-risk areas. Competing risks in survival data concern a situation where there is more than one cause of failure, but only the occurrence of the first one is observable. In this paper, we considered several Bayesian hierarchical survival models in the setting of competing risks for the spatio-temporally clustered HIV/AIDS data. An intrinsic conditional autoregressive (ICAR) distribution and a multivariate intrinsic conditional autoregressive (MICAR) distribution were employed to model random effect terms. The comparison between competing models was performed using the deviance information criterion and log pseudo-marginal likelihood. We illustrated the gains of final model through simulation study and application to the HIV/AIDS data.

SpBayesSurv: Fitting Bayesian Spatial Survival Models Using R

Spatial survival analysis has received a great deal of attention over the last 20 years due to the important role that geographical information can play in predicting survival. This paper provides an introduction to a set of programs for implementing some Bayesian spatial survival models in R using the package spBayesSurv. The function survregbayes includes the three most commonly-used semiparametric models: proportional hazards, proportional odds, and accelerated failure time. All manner of censored survival times are simultaneously accommodated including uncensored, interval censored, current-status, left and right censored, and mixtures of these. Left-truncated data are also accommodated. Time-dependent covariates are allowed under the piecewise constant assumption. Both georeferenced and areally observed spatial locations are handled via frailties. Model fit is assessed with conditional Cox-Snell residual plots, and model choice is carried out via the log pseudo marginal likelihood, the deviance information criterion and the Watanabe-Akaike information criterion. The accelerated failure time frailty model with a covariate-dependent baseline is included in the function frailtyGAFT. In addition, the package also provides two marginal survival models: proportional hazards and linear dependent Dirichlet process mixture, where the spatial dependence is modeled via spatial copulas. Note that the package can also handle non-spatial data using non-spatial versions of aforementioned models.

Exploring the modeling and site-ranking performance of Bayesian spatiotemporal crash frequency models with mixture components

The current study introduces the flexible approach of mixture components to model the spatiotemporal interaction for ranking of hazardous sites and compares the model performance with the conventional methods. In case of predictive accuracy based on in-sample errors (posterior deviance), the Mixture-5 demonstrated superior performance in majority of the cases, indicating the advantage of mixture approach to accurately predict crash counts. LPML (log pseudo marginal likelihood) was also calculated as a cross-validation measure based on out-of-sample errors and this criterion also established the dominance of Mixture-5, further reinforcing the superiority of the mixture approach from different perspectives.The site ranking evaluation results demonstrated the advantages of adopting the mixture approach. In terms of total rank difference (TRD) results, there were several discrepancies between the two approaches, suggesting that two approaches designate unsafe sites differently. Another site ranking criterion, site consistency test (SCT), was employed to explore the difference in identification of unsafe sites based on two datasets: estimated crash count (traditional) and the spatial variability across time. The advantage of mixture models to act as a complimentary approach for site ranking was revealed by the spatial variability SCT results. The method consistency test (MCT) results also indicate the advantages of mixture models over the Base one. These findings suggested that mixture approach may prove helpful in the network screening step of safety management process to identify sites which may turn unsafe in the future and escape the detection from traditional methods.

A Bayesian transition model for missing longitudinal binary outcomes and an application to a smoking cessation study.

Smoking cessation intervention studies often produce data on smoking status at discrete follow-up assessments, often with missing data in different amounts at each assessment. Smoking status in these studies is a dynamic process with individuals transitioning from smoking to abstinent, as well as abstinent to smoking, at different times during the intervention. Directly assessing transitions provides an opportunity to answer important questions like 'Does the proposed intervention help smokers remain abstinent or quit smoking more effectively than other interventions?' In this article, we model changes in smoking status and examine how interventions and other covariates affect the transitions. We propose a Bayesian approach for fitting the transition model to the observed data and impute missing outcomes based on a logistic model, which accounts for both missing at random (MAR) and missing not at random (MNAR) mechanisms. The proposed Bayesian approach treats missing data as additional unknown quantities and samples them from their posterior distributions. The performance of the proposed method is investigated through simulation studies and illustrated by data from a randomized controlled trial of smoking cessation interventions. Finally, posterior predictive checking and log pseudo marginal likelihood (LPML) are used to assess model assumptions and perform model comparisons, respectively.

Bayesian hierarchical modeling of substate area estimates from the Medicare CAHPS survey.

Each year, surveys are conducted to assess the quality of care for Medicare beneficiaries, using instruments from the Consumer Assessment of Healthcare Providers and Systems (CAHPS®) program. Currently, survey measures presented for Fee-for-Service beneficiaries are either pooled at the state level or unpooled for smaller substate areas nested within the state; the choice in each state is based on statistical tests of measure heterogeneity across areas within state. We fit spatial-temporal Bayesian random-effects models using a flexible parameterization to estimate mean scores for each of the domains formed by 94 areas in 32 states measured over 5 years. A Bayesian hat matrix provides a heuristic interpretation of the way the model combines information for estimates in these domains. The model can be used to choose between reporting of state- or substate-level direct estimates in each state, or as a source of alternative small-area estimates superior to either direct estimate. We compare several candidate models using log pseudomarginal likelihood and posterior predictive checks. Results from the best-performing model for 8 measures surveyed from 2012 to 2016 show substantial reductions in mean squared error (MSE) over direct estimates.

A Unified Framework for Fitting Bayesian Semiparametric Models to Arbitrarily Censored Survival Data, Including Spatially Referenced Data

ABSTRACTA comprehensive, unified approach to modeling arbitrarily censored spatial survival data is presented for the three most commonly used semiparametric models: proportional hazards, proportional odds, and accelerated failure time. Unlike many other approaches, all manner of censored survival times are simultaneously accommodated including uncensored, interval censored, current-status, left and right censored, and mixtures of these. Left-truncated data are also accommodated leading to models for time-dependent covariates. Both georeferenced (location exactly observed) and areally observed (location known up to a geographic unit such as a county) spatial locations are handled; formal variable selection makes model selection especially easy. Model fit is assessed with conditional Cox–Snell residual plots, and model choice is carried out via log pseudo marginal likelihood (LPML) and deviance information criterion (DIC). Baseline survival is modeled with a novel transformed Bernstein polynomial prior. All models are fit via a new function which calls efficient compiled C++ in the R package spBayesSurv. The methodology is broadly illustrated with simulations and real data applications. An important finding is that proportional odds and accelerated failure time models often fit significantly better than the commonly used proportional hazards model. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Bayesian spatiotemporal crash frequency models with mixture components for space-time interactions

The traffic safety research has developed spatiotemporal models to explore the variations in the spatial pattern of crash risk over time. Many studies observed notable benefits associated with the inclusion of spatial and temporal correlation and their interactions. However, the safety literature lacks sufficient research for the comparison of different temporal treatments and their interaction with spatial component. This study developed four spatiotemporal models with varying complexity due to the different temporal treatments such as (I) linear time trend; (II) quadratic time trend; (III) Autoregressive-1 (AR-1); and (IV) time adjacency. Moreover, the study introduced a flexible two-component mixture for the space-time interaction which allows greater flexibility compared to the traditional linear space-time interaction. The mixture component allows the accommodation of global space-time interaction as well as the departures from the overall spatial and temporal risk patterns. This study performed a comprehensive assessment of mixture models based on the diverse criteria pertaining to goodness-of-fit, cross-validation and evaluation based on in-sample data for predictive accuracy of crash estimates.The assessment of model performance in terms of goodness-of-fit clearly established the superiority of the time-adjacency specification which was evidently more complex due to the addition of information borrowed from neighboring years, but this addition of parameters allowed significant advantage at posterior deviance which subsequently benefited overall fit to crash data. The Base models were also developed to study the comparison between the proposed mixture and traditional space-time components for each temporal model. The mixture models consistently outperformed the corresponding Base models due to the advantages of much lower deviance.For cross-validation comparison of predictive accuracy, linear time trend model was adjudged the best as it recorded the highest value of log pseudo marginal likelihood (LPML). Four other evaluation criteria were considered for typical validation using the same data for model development. Under each criterion, observed crash counts were compared with three types of data containing Bayesian estimated, normal predicted, and model replicated ones. The linear model again performed the best in most scenarios except one case of using model replicated data and two cases involving prediction without including random effects. These phenomena indicated the mediocre performance of linear trend when random effects were excluded for evaluation. This might be due to the flexible mixture space-time interaction which can efficiently absorb the residual variability escaping from the predictable part of the model. The comparison of Base and mixture models in terms of prediction accuracy further bolstered the superiority of the mixture models as the mixture ones generated more precise estimated crash counts across all four models, suggesting that the advantages associated with mixture component at model fit were transferable to prediction accuracy. Finally, the residual analysis demonstrated the consistently superior performance of random effect models which validates the importance of incorporating the correlation structures to account for unobserved heterogeneity.

Bayesian Approach to Personalized Benefit-Risk Assessment

ABSTRACTBenefit-risk assessment is critical in evaluating the effectiveness of a new drug before and after the approval. Some benefit-risk measures depend on the probabilities of benefit-risk categories in which the subject-level benefit and risk outcomes are characterized. The existing benefit-risk methods for analyzing the categorical data depend only on the frequencies of mutually exclusive and collectively exhaustive categories that the subjects fall in, and thus ignore the subject-level differences. We propose a Bayesian method for analyzing the subject-level data with multiple visits. A generalized linear model is used to model the subject-level response probability, with respect to a “reference” category, assuming a logit model with subject-level category effects and multiple visit effects. The random longitudinal visit effects are modeled by a multivariate normal distribution with zero means and first-order autoregressive structured variance-covariance matrices. In the proposed Bayesian setup, a Dirichlet process is used as a prior for the subject-level category effect to catch the similarity among the subject responses. We develop an efficient Markov chain Monte Carlo algorithm for implementing the proposed method, and illustrate the estimation of individual benefit-risk profiles through simulation. The performance of the proposed model fit is evaluated using two model selection approaches, namely, the deviance information criterion (DIC) and the log-pseudo marginal likelihood (LPML). We analyze a clinical trial data using the proposed method to assess the subject-level or personalized benefit-risk in each arm, and to evaluate the aggregated benefit-risk difference between the treatments at different visits.

Hierarchical Models for Semicompeting Risks Data With Application to Quality of End-of-Life Care for Pancreatic Cancer

ABSTRACTReadmission following discharge from an initial hospitalization is a key marker of quality of healthcare in the United States. For the most part, readmission has been studied among patients with “acute” health conditions, such as pneumonia and heart failure, with analyses based on a logistic-Normal generalized linear mixed model. Naïve application of this model to the study of readmission among patients with “advanced” health conditions such as pancreatic cancer, however, is problematic because it ignores death as a competing risk. A more appropriate analysis is to imbed such a study within the semicompeting risks framework. To our knowledge, however, no comprehensive statistical methods have been developed for cluster-correlated semicompeting risks data. To resolve this gap in the literature we propose a novel hierarchical modeling framework for the analysis of cluster-correlated semicompeting risks data that permits parametric or nonparametric specifications for a range of components giving analysts substantial flexibility as they consider their own analyses. Estimation and inference is performed within the Bayesian paradigm since it facilitates the straightforward characterization of (posterior) uncertainty for all model parameters, including hospital-specific random effects. Model comparison and choice is performed via the deviance information criterion and the log-pseudo marginal likelihood statistic, both of which are based on a partially marginalized likelihood. An efficient computational scheme, based on the Metropolis-Hastings-Green algorithm, is developed and had been implemented in the R package SemiCompRisks. A comprehensive simulation study shows that the proposed framework performs very well in a range of data scenarios, and outperforms competitor analysis strategies. The proposed framework is motivated by and illustrated with an ongoing study of the risk of readmission among Medicare beneficiaries diagnosed with pancreatic cancer. Using data on n = 5298 patients at J=112 hospitals in the six New England states between 2000–2009, key scientific questions we consider include the role of patient-level risk factors on the risk of readmission and the extent of variation in risk across hospitals not explained by differences in patient case-mix. Supplementary materials for this article are available online.

Posterior Predictive Bayesian Phylogenetic Model Selection

We present two distinctly different posterior predictive approaches to Bayesian phylogenetic model selection and illustrate these methods using examples from green algal protein-coding cpDNA sequences and flowering plant rDNA sequences. The Gelfand-Ghosh (GG) approach allows dissection of an overall measure of model fit into components due to posterior predictive variance (GGp) and goodness-of-fit (GGg), which distinguishes this method from the posterior predictive P-value approach. The conditional predictive ordinate (CPO) method provides a site-specific measure of model fit useful for exploratory analyses and can be combined over sites yielding the log pseudomarginal likelihood (LPML) which is useful as an overall measure of model fit. CPO provides a useful cross-validation approach that is computationally efficient, requiring only a sample from the posterior distribution (no additional simulation is required). Both GG and CPO add new perspectives to Bayesian phylogenetic model selection based on the predictive abilities of models and complement the perspective provided by the marginal likelihood (including Bayes Factor comparisons) based solely on the fit of competing models to observed data.

A semiparametric Bayesian approach for mark-recapture estimation

This paper describes a semiparametric Bayesian approach for modelling mark-recapture data. A main assumption in modelling mark-recapture data is that survival probabilities are homogeneous. We relax this assumption by modelling survival probabilities as a function of two parameters which explain variations due to unknown biological and environmental reasons. The heterogeneity in travel times and survival probabilities is accounted using the Dirichlet process. The Dirichlet process also provides a clustering mechanism which is often suitable for mark-recapture data where groups of animals can be thought of as arising from the same cohort. The approach is highlighted using actual data arising from thed Pacific Ocean Shelf Tracking (POST) project. Log-pseudo marginal likelihood (LPML) model selection procedure indicates that the proposed model performs better over conventional alternative methods.

Meta-analysis using Dirichlet process

This article develops a Bayesian approach for meta-analysis using the Dirichlet process. The key aspect of the Dirichlet process in meta-analysis is the ability to assess evidence of statistical heterogeneity or variation in the underlying effects across study while relaxing the distributional assumptions. We assume that the study effects are generated from a Dirichlet process. Under a Dirichlet process model, the study effects parameters have support on a discrete space and enable borrowing of information across studies while facilitating clustering among studies. We illustrate the proposed method by applying it to a dataset on the Program for International Student Assessment on 30 countries. Results from the data analysis, simulation studies, and the log pseudo-marginal likelihood model selection procedure indicate that the Dirichlet process model performs better than conventional alternative methods.

Spatially Dependent Polya Tree Modeling for Survival Data

With the proliferation of spatially oriented time-to-event data, spatial modeling in the survival context has received increased recent attention. A traditional way to capture a spatial pattern is to introduce frailty terms in the linear predictor of a semiparametric model, such as proportional hazards or accelerated failure time. We propose a new methodology to capture the spatial pattern by assuming a prior based on a mixture of spatially dependent Polya trees for the baseline survival in the proportional hazards model. Thanks to modern Markov chain Monte Carlo (MCMC) methods, this approach remains computationally feasible in a fully hierarchical Bayesian framework. We compare the spatially dependent mixture of Polya trees (MPT) approach to the traditional spatial frailty approach, and illustrate the usefulness of this method with an analysis of Iowan breast cancer survival data from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. Our method provides better goodness of fit over the traditional alternatives as measured by log pseudo marginal likelihood (LPML), the deviance information criterion (DIC), and full sample score (FSS) statistics.