Bayesian clustering and product partition models

Summary. We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies—originally devised for nonparametric Bayesian problems—to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.     

Least squares estimation of regression parameters in mixed effects models with unmeasured covariates

We consider mixed effects models for longitudinal, repeated measures or clustered data. Unmeasured or omitted covariates in such models may be correlated with the included covanates, and create model violations when not taken into account. Previous research and experience with longitudinal data sets suggest a general form of model which should be considered when omitted covariates are likely, such as in observational studies. We derive the marginal model between the response variable and included covariates, and consider model fitting using the ordinary and weighted least squares methods, which require simple non-iterative computation and no assumptions on the distribution of random covariates or error terms, Asymptotic properties of the least squares estimators are also discussed. The results shed light on the structure of least squares estimators in mixed effects models, and provide large sample procedures for statistical inference and prediction based on the marginal model. We present an example of the relationship between fluid intake and output in very low birth weight infants, where the model is found to have the assumed structure.     

Spherical regression models with general covariates and anisotropic errors

Statistics and Computing - Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example...     

Bayesian clustering for row effects models

We deal with two-way contingency tables having ordered column categories. We use a row effects model wherein each interaction term is assumed to have a multiplicative form involving a row effect parameter and a fixed column score. We propose a methodology to cluster row effects in order to simplify the interaction structure and to enhance the interpretation of the model. Our method uses a product partition model with a suitable specification of the cohesion function, so that we can carry out our analysis on a collection of models of varying dimensions using a straightforward MCMC sampler. The methodology is illustrated with reference to simulated and real data sets.     

A semiparametric extended hazard regression model with time-dependent covariates

We introduce a general class of semiparametric hazard regression models, called extended hazard (EH) models, that are designed to accommodate various survival schemes with time-dependent covariates. The EH model contains both the Cox model and the accelerated failure time (AFT) model as its subclasses so that we can use this nested structure to perform model selection between the Cox model and the AFT model. A class of estimating equations using counting process and martingale techniques is developed to estimate the regression parameters of the proposed model. The performance of the estimating procedure and the impact of model misspecification are assessed through simulation studies. Two data examples, Stanford heart transplant data and Mediterranean fruit flies, egg-laying data, are used to demonstrate the usefulness of the EH model.     

A Cox-type regression model with change-points in the covariates

We consider a Cox-type regression model with change-points in the covariates. A change-point specifies the unknown threshold at which the influence of a covariate shifts smoothly, i.e., the regression parameter may change over the range of a covariate and the underlying regression function is continuous but not differentiable. The model can be used to describe change-points in different covariates but also to model more than one change-point in a single covariate. Estimates of the change-points and of the regression parameters are derived and their properties are investigated. It is shown that not only the estimates of the regression parameters are [Formula: see text] -consistent but also the estimates of the change-points in contrast to the conjecture of other authors. Asymptotic normality is shown by using results developed for M-estimators. At the end of this paper we apply our model to an actuarial dataset, the PBC dataset of Fleming and Harrington (Counting processes and survival analysis, 1991) and to a dataset of electric motors.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.     

Cox regression for current status data with mismeasured covariates

Covariate measurement error problems have been extensively studied in the context of right‐censored data but less so for current status data. Motivated by the zebrafish basal cell carcinoma (BCC) study, where the occurrence time of BCC was only known to lie before or after a sacrifice time and where the covariate (Sonic hedgehog expression) was measured with error, the authors describe a semiparametric maximum likelihood method for analyzing current status data with mismeasured covariates under the proportional hazards model. They show that the estimator of the regression coefficient is asymptotically normal and efficient and that the profile likelihood ratio test is asymptotically Chi‐squared. They also provide an easily implemented algorithm for computing the estimators. They evaluate their method through simulation studies, and illustrate it with a real data example. The Canadian Journal of Statistics 39: 73–88; 2011 © 2011 Statistical Society of Canada     

Additive hazards regression of current status data with auxiliary covariates

This paper discusses the regression analysis of current status failure time data arising from the additive hazards model with auxiliary covariates. As often occurs in practice, it is impossible or impractical to measure the exact magnitude of covariates for all subjects in a study. To compensate the missing information, some auxiliary covariates are utilized instead. We propose two easy-to-implement procedures for estimation of regression parameters by making use of auxiliary information. The asymptotic properties of the resulting estimators are established and extensive numerical studies indicate that both procedures work well in practice.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

Exponential tilt models for two-group comparison with censored data

We study application of the Exponential Tilt Model (ETM) to compare survival distributions in two groups. The ETM assumes a parametric form for the density ratio of the two distributions. It accommodates a broad array of parametric models such as the log-normal and gamma models and can be sufficiently flexible to allow for crossing hazard and crossing survival functions. We develop a nonparametric likelihood approach to estimate ETM parameters in the presence of censoring and establish related asymptotic results. We compare the ETM to the Proportional Hazards Model (PHM) in simulation studies. When the proportional hazards assumption is not satisfied but the ETM assumption is, the ETM has better power for testing the hypothesis of no difference between the two groups. And, importantly, when the ETM relation is not satisfied but the PHM assumption is, the ETM can still have power reasonably close to that of the PHM. Application of the ETM is illustrated by a gastrointestinal tumor study.     

Joint regression modeling for missing categorical covariates in generalized linear models

Missing covariates data is a common issue in generalized linear models (GLMs). A model-based procedure arising from properly specifying joint models for both the partially observed covariates and the corresponding missing indicator variables represents a sound and flexible methodology, which lends itself to maximum likelihood estimation as the likelihood function is available in computable form. In this paper, a novel model-based methodology is proposed for the regression analysis of GLMs when the partially observed covariates are categorical. Pair-copula constructions are used as graphical tools in order to facilitate the specification of the high-dimensional probability distributions of the underlying missingness components. The model parameters are estimated by maximizing the weighted log-likelihood function by using an EM algorithm. In order to compare the performance of the proposed methodology with other well-established approaches, which include complete-cases and multiple imputation, several simulation experiments of Binomial, Poisson and Normal regressions are carried out under both missing at random and non-missing at random mechanisms scenarios. The methods are illustrated by modeling data from a stage III melanoma clinical trial. The results show that the methodology is rather robust and flexible, representing a competitive alternative to traditional techniques.     

Adaptive estimation for varying coefficient models with nonstationary covariates

In this paper, the adaptive estimation for varying coefficient models proposed by Chen, Wang, and Yao (2015 Chen, Y., Q. Wang, and W. Yao. 2015. Adaptive estimation for varying coefficient models. Journal of Multivariate Analysis 137:17–31.[Crossref], [Web of Science ®] , [Google Scholar]) is extended to allowing for nonstationary covariates. The asymptotic properties of the estimator are obtained, showing different convergence rates for the integrated covariates and stationary covariates. The nonparametric estimator of the functional coefficient with integrated covariates has a faster convergence rate than the estimator with stationary covariates, and its asymptotic distribution is mixed normal. Moreover, the adaptive estimation is more efficient than the least square estimation for non normal errors. A simulation study is conducted to illustrate our theoretical results.     

Weighted quantile regression with nonelliptically structured covariates

Although quantile regression estimators are robust against low leverage observations with atypically large responses (Koenker & Bassett 1978), they can be seriously affected by a few points that deviate from the majority of the sample covariates. This problem can be alleviated by downweighting observations with high leverage. Unfortunately, when the covariates are not elliptically distributed, Mahalanobis distances may not be able to correctly identify atypical points. In this paper the authors discuss the use of weights based on a new leverage measure constructed using Rosenblatt's multivariate transformation which is able to reflect nonelliptical structures in the covariate space. The resulting weighted estimators are consistent, asymptotically normal, and have a bounded influence function. In addition, the authors also discuss a selection criterion for choosing the downweighting scheme. They illustrate their approach with child growth data from Finland. Finally, their simulation studies suggest that this methodology has good finite‐sample properties.     

Using the EM algorithm for Bayesian variable selection in logistic regression models with related covariates

We develop a Bayesian variable selection method for logistic regression models that can simultaneously accommodate qualitative covariates and interaction terms under various heredity constraints. We use expectation-maximization variable selection (EMVS) with a deterministic annealing variant as the platform for our method, due to its proven flexibility and efficiency. We propose a variance adjustment of the priors for the coefficients of qualitative covariates, which controls false-positive rates, and a flexible parameterization for interaction terms, which accommodates user-specified heredity constraints. This method can handle all pairwise interaction terms as well as a subset of specific interactions. Using simulation, we show that this method selects associated covariates better than the grouped LASSO and the LASSO with heredity constraints in various exploratory research scenarios encountered in epidemiological studies. We apply our method to identify genetic and non-genetic risk factors associated with smoking experimentation in a cohort of Mexican-heritage adolescents.     

Random coefficients regression models. a review 1

A review is given of random regression coefficients models. The emphasis is put on the problem of estimating the mean regression coefficients and the covariance matrix of the coefficients. Prediction of the individual random coefficients is not discussed. The main purpose of the review is to point to the practical aspects of the models and the problem of statistical inference in finite samples. Some problems for future research are indicated.     

Nonparametric estimation of regression models with mixed discrete and continuous covariates by the K-nn method

In this article we consider the problem of estimating a nonparametric conditional mean function with mixed discrete and continuous covariates by the nonparametric k-nearest-neighbor (k-nn) method. We derive the asymptotic normality result of the proposed estimator and use Monte Carlo simulations to demonstrate its finite sample performance. We also provide an illustrative empirical example of our method.     

Cox regression of clustered event times with covariates missing not at random

Motivated by a recent tuberculosis (TB) study, this paper is concerned with covariates missing not at random (MNAR) and models the potential intracluster correlation by a frailty. We consider the regression analysis of right‐censored event times from clustered subjects under a Cox proportional hazards frailty model and present the semiparametric maximum likelihood estimator (SPMLE) of the model parameters. An easy‐to‐implement pseudo‐SPMLE is then proposed to accommodate more realistic situations using readily available supplementary information on the missing covariates. Algorithms are provided to compute the estimators and their consistent variance estimators. We demonstrate that both the SPMLE and the pseudo‐SPMLE are consistent and asymptotically normal by the arguments based on the theory of modern empirical processes. The proposed approach is examined numerically via simulation and illustrated with an analysis of the motivating TB study data.