Bayesian analysis of transformation latent variable models with multivariate censored data

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin–Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data.

We propose a novel Bayesian model framework for discrete ordinal and count data based on conditional transformations of the responses. The conditional transformation function is estimated from the data in conjunction with an a priori chosen reference distribution. For count responses, the resulting transformation model is novel in the sense that it is a Bayesian fully parametric yet distribution-free approach that can additionally account for excess zeros with additive transformation function specifications. For ordinal categoric responses, our cumulative link transformation model allows the inclusion of linear and non-linear covariate effects that can additionally be made category-specific, resulting in (non-)proportional odds or hazards models and more, depending on the choice of the reference distribution. Inference is conducted by a generic modular Markov chain Monte Carlo algorithm where multivariate Gaussian priors enforce specific properties such as smoothness on the functional effects. To illustrate the versatility of Bayesian discrete conditional transformation models, applications to counts of patent citations in the presence of excess zeros and on treating forest health categories in a discrete partial proportional odds model are presented.

This study develops a joint modeling approach that incorporates latent traits into causal mediation analysis with multiple mediators and a survival outcome. A linear structural equation model is used to characterize the latent mediators with several highly correlated observable surrogates and depicts the relationships among multiple parallel or causally ordered mediators and the exposure. A proportional hazards model is used to derive the path-specific causal effects on the scale of hazard ratio under the counterfactual framework with a set of sequential ignorability assumptions. A Bayesian approach with Markov chain Monte Carlo algorithm is developed to perform efficient estimation of the causal effects. Posterior propriety theory is established for the proportional hazards model with latent variables. Empirical performance of the proposed method is verified through simulation studies. The proposed model is then applied to a study on the Alzheimer’s Disease Neuroimaging Initiative dataset to investigate the causal effects of APOE- allele on the disease progression, either directly or through potential mediators, such as hippocampus atrophy, ventricle expansion, and cognitive impairment.

In this paper, we aim to develop a semiparametric transformation model. Nonparametric transformation functions are modeled with Bayesian P-splines. The transformed variables can be fitted to a general nonlinear mixed model, including linear or nonlinear regression models, mixed effect models, factor analysis models, and other latent variable models as special cases. Markov chain Monte Carlo algorithms are implemented to estimate transformation functions and unknown quantities in the model. The performance of the developed methodology is demonstrated with a simulation study. Its application to a real study on polydrug use is presented.

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.

Left-censored linear regression models are quite popular models and have been deeply considered in the last three decades. In this paper, we consider a completely Bayesian approach for making a new Markov chain Monte Carlo (MCMC) algorithm with tractable full posteriors. Simulated consequences and real data analyses depict that the new Markov chain Monte Carlo algorithm has excellent mixing property and carry out very well than the present methods based on prediction accuracy.

<p>Joint models for longitudinal and survival data have been widely discussed in the literature. This thesis proposes a joint model using a stereotype model for the longitudinal ordinal responses and a Cox proportional hazards model for survival time. Our current joint model has a new feature since no literature has examined the joint model under the stereotype model. The stereotype model can improve the fit by adding extra score parameters, but it still has the advantage of requiring only a single parameter to describe the effect of a predictor on the item response levels. We give an example to model longitudinal ordinal data and survival data for patients being followed up after treatments. The main focus is on modeling both the quality of life data and the survival data simultaneously with a goal of understanding the association between the two processes over time. These two models are linked through a latent variable that characterizes the quality of life of an individual and is assumed to underlie the hazard rate. In other words, the latent variable serves as a shared variable in the joint model. We present the joint model in two different aspects: one based on a Bayesian approach and the other one a semiparametric approach using the EM algorithm. For the Bayesian approach, the latent variable is treated as a continuous variable and is assumed to have a multivariate normal distribution. The partial survival likelihood function is used in the survival component of the Bayesian joint model, while the full likelihood function is considered in the semiparametric joint model. In the latter approach the baseline hazard is assumed to be a step function and has no parametric form. The latent variable in the semiparametric joint model is then treated as a discrete variable. We illustrate our methodologies by analyzing data from the Staccato study, a randomized trial to compare two treatment methods, for Human Immunodeficiency Virus (HIV) infection of Thai patients on Highly Active Antiretroviral Therapy (HAART), in which the quality of life was assessed with a HIV Medical Outcome Study (MOS-HIV) questionnaire. Furthermore, we extend the study further to the case of multiple failure types in the survival component. Thus, the extension of the joint model consists of the stereotype model and the competing risks model. The Bayesian method is employed to estimate all unknown parameters in this extended joint model. The results we obtained are consistent for both the Bayesian joint model and the semiparametric joint model. Both models show that patients who had a better quality of life were associated with a lower hazard of HIV progression. Patients on continuous treatment also had a lower hazard of HIV progression compared with patients on CD4-guided interruption treatment.</p>

In marketing applications, it is common that some key covariates in a regression model, such as marketing mix variables or consumer profiles, are subject to missingness. The convenient method that excludes the consumers with missingness in any covariate can result in a substantial loss of efficiency and may lead to strong selection bias in the estimation of consumer preferences and sensitivities. To solve these problems, we propose a new Bayesian distribution-free approach, which can ensure that no customer is left behind in the analysis as a result of missing covariates. In this way, all customers are being considered in devising managerial policies. The proposed approach allows for flexible modeling of a joint distribution of multidimensional interrelated covariates that can contain both continuous and discrete variables. At the same time, it minimizes the impact of distributional assumptions involved in covariate modeling because the method does not require researchers to specify parametric distributions for covariates and can automatically generate suitable distributions for missing covariates. We have developed an efficient Markov chain Monte Carlo algorithm for inference. Besides robustness and flexibility, the proposed approach reduces modeling and computational efforts associated with missing covariates and therefore makes the missing covariate problems easier to handle. We evaluate the performance of the proposed method using extensive simulation studies. We then illustrate the method in two real data examples in which missing covariates occur: a mixed multinomial logit discrete-choice model in a ketchup data set and a hierarchical probit purchase incidence model in a retail store data set. These analyses demonstrate that the proposed method overcomes several important limitations of existing approaches for solving missing covariate problems and offers opportunities to make better managerial decisions with the current available marketing databases. Although our applications focus on consumer-level data, the proposed method is general and can be applied to other marketing applications where other types of marketing players are the units of analysis.

We propose a Bayesian nonparametric matrix clustering approach to analyze the latent heterogeneity structure in the shot selection data collected from professional basketball players in the National Basketball Association (NBA). The proposed method adopts a mixture of finite mixtures framework and fully uses the spatial information via a mixture of matrix normal distribution representation. We propose an efficient Markov chain Monte Carlo algorithm for posterior sampling that allows simultaneous inference on both the number of clusters and the cluster configurations. We also establish large-sample convergence properties for the posterior distribution. The compelling empirical performance of the proposed method is demonstrated via simulation studies and an application to shot chart data from selected players in the NBAs 2017–2018 regular season. Supplementary materials for this article are available online.

Inherited mutations of the BRCA1 and BRCA2 genes are known to confer an elevated risk of both breast and ovarian cancers. The effect of carrying such a mutation on survival after developing breast or ovarian cancer is less well understood. We investigate the relationship between BRCA1 and BRCA2 carrier status and survival after breast cancer. We obtained data from the Cancer and Steroid Hormone Study, a large population-based study including more than 4,000 breast cancer cases. Patient data include extensive information about breast and ovarian cancer in relatives. We obtained follow-up information about patients via record linkage with the Surveillance, Epidemiology, and End Results registry, with maximum follow-up of 15 years. In the absence of genetic testing for each individual, presence or absence of mutation at a breast cancer susceptibility gene is captured by a pair of binary latent variables whose marginal probability depends on the patient's family history of breast and ovarian cancer. We estimate the effect of genotype on survival using a Cox proportional hazards model, treating genetic status as a latent variable and controlling for stage at diagnosis, histology, whether radiation treatment was administered, the individual's smoking history, body mass, race, and age at diagnosis. Inference is accomplished using a Markov chain Monte Carlo algorithm to draw a sample from the posterior distribution of model parameters accounting for sampling error, uncertainty in the genotype of study participants, and uncertainty in estimates of genetic parameters. An analysis that does not discriminate between BRCA1 and BRCA2 estimates the genetic effect on survival after breast cancer for women carrying a mutation at either site. We find evidence that survival of nonirradiated mutation carriers is better than that of noncarriers; we estimate the probability of improved survival to be .990. As a byproduct of this analysis, we estimate that a study based on 901 women with known mutation status would yield estimates of genetic effect on survival with comparable uncertainty to that obtained in the combined-gene analysis. An analysis assessing the separate effects of the two genes indicates that carriers of either mutation may have an improved prognosis. We estimate the probability of improved survival among nonirradiated BRCA1 carriers to be .844, and that among nonirradiated BRCA2 carriers to be .924. However, substantial uncertainty remains about the magnitude of the BRCA2 effect.

Bayesian analysis relies heavily on the Markov chain Monte Carlo (MCMC) algorithm to obtain random samples from posterior distributions. In this study, we compare the performance of MCMC stopping rules and provide a guideline for determining the termination point of the MCMC algorithm in latent variable models. In simulation studies, we examine the performance of four different MCMC stopping rules: potential scale reduction factor (PSRF), fixed‐width stopping rule, Geweke's diagnostic, and effective sample size. Specifically, we evaluate these stopping rules in the context of the DINA model and the bifactor item response theory model, two commonly used latent variable models in educational and psychological measurement. Our simulation study findings suggest that single‐chain approaches outperform multiple‐chain approaches in terms of item parameter accuracy. However, when it comes to person parameter estimates, the effect of stopping rules diminishes. We caution against relying solely on the univariate PSRF, which is the most popular method, as it may terminate the algorithm prematurely and produce biased item parameter estimates if the cut‐off value is not chosen carefully. Our research offers guidance to practitioners on choosing suitable stopping rules to improve the precision of the MCMC algorithm in models involving latent variables.

A correct detection of areas with excess of pollution relies first on accurate predictions of pollutant concentrations, a task that is usually complicated by skewed histograms and the presence of censored data. The unified skew-Gaussian (SUG) random field proposed by Zareifard and Jafari Khaledi [19] offers a more flexible class of sampling spatial models to account for skewness. In this paper, we adopt a Bayesian framework to perform prediction for the SUG model in the presence of censored data. Owing to the presence of many latent variables with strongly dependent components in the model, we encounter convergence issues when using Monte Carlo Markov Chain algorithms. To overcome this obstacle, we use a computationally efficient inverse Bayes formulas sampling procedure to obtain approximately independent samples from the posterior distribution of latent variables. Then they are applied to update parameters in a Gibbs sampler scheme. This hybrid algorithm provides effective samples, resulting in some computational advantages and precise predictions. The proposed approach is illustrated with a simulation study and applied to a spatial data set which contains right censored data.

Multivariate current status data, consist of indicators of whether each of several events occur by the time of a single examination. Our interest focuses on inferences about the joint distribution of the event times. Conventional methods for analysis of multiple event-time data cannot be used because all of the event times are censored and censoring may be informative. Within a given subject, we account for correlated event times through a subject-specific latent variable, conditional upon which the various events are assumed to occur independently. We also assume that each event contributes independently to the hazard of censoring. Nonparametric step functions are used to characterize the baseline distributions of the different event times and of the examination times. Covariate and subject-specific effects are incorporated through generalized linear models. A Markov chain Monte Carlo algorithm is described for estimation of the posterior distributions of the unknowns. The methods are illustrated through application to multiple tumor site data from an animal carcinogenicity study.

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.

ABSTRACTThis study proposes a heterogeneous mediation analysis for survival data that accommodates multiple mediators and sparsity of the predictors. We introduce a joint modeling approach that links the mediation regression and proportional hazards models through Bayesian additive regression trees with shared typologies. The shared tree component is motivated by the fact that confounders and effect modifiers on the causal pathways linked by different mediators often overlap. A sparsity‐inducing prior is incorporated to capture the most relevant confounders and effect modifiers on different causal pathways. The individual‐specific interventional direct and indirect effects are derived on the scale of the logarithm of hazards and survival function. A Bayesian approach with an efficient Markov chain Monte Carlo algorithm is developed to estimate the conditional interventional effects through the Monte Carlo implementation of the mediation formula. Simulation studies are conducted to verify the empirical performance of the proposed method. An application to the ACTG175 study further demonstrates the method's utility in causal discovery and heterogeneity quantification.