Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

A Framework for Random-Effects ROC Analysis: Biases with the Bootstrap and Other Variance Estimators

In this article, we analyze the three-way bootstrap estimate of the variance of the reader-averaged nonparametric area under the receiver operating characteristic (ROC) curve. The setting for this work is medical imaging, and the experimental design involves sampling from three distributions: a set of normal and diseased cases (patients), and a set of readers (doctors). The experiment we consider is fully crossed in that each reader reads each case. A reading generates a score that indicates the reader's level of suspicion that the patient is diseased. The distribution of scores for the normal patients is compared to the distribution of scores for the diseased patients via an ROC curve, and the area under the ROC curve (AUC) summarizes the reader's diagnostic ability to separate the normal patients from the diseased ones. We find that the bootstrap estimate of the variance of the reader-averaged AUC is biased, and we represent this bias in terms of moments of success outcomes. This representation helps unify and improve several current methods for multi-reader multi-case (MRMC) ROC analysis.     

A factor model approach for the joint segmentation with between‐series correlation

We consider the detection of changes in the mean of a set of time series. The breakpoints are allowed to be series specific, and the series are assumed to be correlated. The correlation between the series is supposed to be constant along time but is allowed to take an arbitrary form. We show that such a dependence structure can be encoded in a factor model. Thanks to this representation, the inference of the breakpoints can be achieved via dynamic programming, which remains one the most efficient algorithms. We propose a model selection procedure to determine both the number of breakpoints and the number of factors. This proposed method is implemented in the FASeg R package, which is available on the CRAN. We demonstrate the performances of our procedure through simulation experiments and present an application to geodesic data.     

A lack-of-fit test for parametric zero-inflated Poisson models

Count data often contain many zeros. In parametric regression analysis of zero-inflated count data, the effect of a covariate of interest is typically modelled via a linear predictor. This approach imposes a restrictive, and potentially questionable, functional form on the relation between the independent and dependent variables. To address the noted restrictions, a flexible parametric procedure is employed to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. The semiparametric zero-inflated Poisson regression model is fitted by maximizing the likelihood function through an expectation–maximization algorithm. The smooth estimate of the functional form of the covariate effect can enhance modelling flexibility. Within this modelling framework, a log-likelihood ratio test is used to assess the adequacy of the covariate function. Simulation results show that the proposed test has excellent power in detecting the lack of fit of a linear predictor. A real-life data set is used to illustrate the practicality of the methodology.     

Analyzing clustered count data with a cluster-specific random effect zero-inflated Conway–Maxwell–Poisson distribution

Count data analysis techniques have been developed in biological and medical research areas. In particular, zero-inflated versions of parametric count distributions have been used to model excessive zeros that are often present in these assays. The most common count distributions for analyzing such data are Poisson and negative binomial. However, a Poisson distribution can only handle equidispersed data and a negative binomial distribution can only cope with overdispersion. However, a Conway–Maxwell–Poisson (CMP) distribution [4] can handle a wide range of dispersion. We show, with an illustrative data set on next-generation sequencing of maize hybrids, that both underdispersion and overdispersion can be present in genomic data. Furthermore, the maize data set consists of clustered observations and, therefore, we develop inference procedures for a zero-inflated CMP regression that incorporates a cluster-specific random effect term. Unlike the Gaussian models, the underlying likelihood is computationally challenging. We use a numerical approximation via a Gaussian quadrature to circumvent this issue. A test for checking zero-inflation has also been developed in our setting. Finite sample properties of our estimators and test have been investigated by extensive simulations. Finally, the statistical methodology has been applied to analyze the maize data mentioned before.     

Identifying Nonlinear Relationships in Regression using the ACE Algorithm

This paper introduces an alternating conditional expectation (ACE) algorithm: a non-parametric approach for estimating the transformations that lead to the maximal multiple correlation of a response and a set of independent variables in regression and correlation analysis. These transformations can give the data analyst insight into the relationships between these variables so that this can be best described and non-linear relationships uncovered. Using the Bayesian information criterion (BIC), we show how to find the best closed-form approximations for the optimal ACE transformations. By means of ACE and BIC, the model fit can be considerably improved compared with the conventional linear model as demonstrated in the two simulated and two real datasets in this paper.     

Local influence analysis for regression models with scale mixtures of skew-normal distributions

The robust estimation and the local influence analysis for linear regression models with scale mixtures of multivariate skew-normal distributions have been developed in this article. The main virtue of considering the linear regression model under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows an easy implementation of inference. Inspired by the expectation maximization algorithm, we have developed a local influence analysis based on the conditional expectation of the complete-data log-likelihood function, which is a measurement invariant under reparametrizations. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and with Cook's well-known approach it can be very difficult to obtain measures of the local influence. Some useful perturbation schemes are discussed. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, a real data set has been analyzed, illustrating the usefulness of the proposed methodology.     

Quantile Regression via the EM Algorithm

The three-parameter asymmetric Laplace distribution (ALD) has received increasing attention in the field of quantile regression due to an important feature between its location and asymmetric parameters. On the basis of the representation of the ALD as a normal-variance–mean mixture with an exponential mixing distribution, this article develops EM and generalized EM algorithms, respectively, for computing regression quantiles of linear and nonlinear regression models. It is interesting to show that the proposed EM algorithm and the MM (Majorization–Minimization) algorithm for quantile regressions are really the same in terms of computation, since the updating formula of them are the same. This provides a good example that connects the EM and MM algorithms. Simulation studies show that the EM algorithm can successfully recover the true parameters in quantile regressions.     

A random cluster algorithm for some continuous Markov random fields

Based on a random cluster representation, the Swendsen–Wang algorithm for the Ising and Potts distributions is extended to a class of continuous Markov random fields. The algorithm can be described briefly as follows. A given configuration is decomposed into clusters. Probabilities for flipping the values of the random variables in each cluster are calculated. According to these probabilities, values of all the random variables in each cluster will be either updated or kept unchanged and this is done independently across the clusters. A new configuration is then obtained. We will show through a simulation study that, like the Swendsen–Wang algorithm in the case of Ising and Potts distributions, the cluster algorithm here also outperforms the Gibbs sampler in beating the critical slowing down for some strongly correlated Markov random fields.     

Bayesian inference under progressive type-I interval censoring

Bayesian estimation for population parameter under progressive type-I interval censoring is studied via Markov Chain Monte Carlo (MCMC) simulation. Two competitive statistical models, generalized exponential and Weibull distributions for modeling a real data set containing 112 patients with plasma cell myeloma, are studied for illustration. In model selection, a novel Bayesian procedure which involves a mixture model is proposed. Then the mix proportion is estimated through MCMC and used as the model selection criterion.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

A density function connected with a non-negative self-decomposable random variable

The innovation random variable for a non-negative self-decomposable random variable can have a compound Poisson distribution. In this case, we provide the density function for the compounded variable. When it does not have a compound Poisson representation, there is a straightforward and easily available compound Poisson approximation for which the density function of the compounded variable is also available. These results can be used in the simulation of Ornstein–Uhlenbeck type processes with given marginal distributions. Previously, simulation of such processes used the inverse of the corresponding tail Lévy measure. We show this approach corresponds to the use of an inverse cdf method of a certain distribution. With knowledge of this distribution and hence density function, the sampling procedure is open to direct sampling methods.     

Pricing dynamic fund protections for a hyperexponential jump diffusion process

This article deals with the valuation of dynamic fund protections (DFPs) under a jump diffusion model, where the jump size follows a hyperexponential distribution. The closed-form solution of the value of DFP is obtained in terms of Laplace transform. A numerical example is provided to show that the explicit solution is easy to implement by using the Gaver–Stehfest algorithm. Effects of key parameters are analyzed at last. The valuation method developed in this work can be used in pricing various variable annuities and path-dependent financial products.     

A model with mixed binary responses and censored observations

In this paper, we study the maximum likelihood estimation of a model with mixed binary responses and censored observations. The model is very general and includes the Tobit model and the binary choice model as special cases. We show that, by using additional binary choice observations, our method is more efficient than the traditional Tobit model. Two iterative procedures are proposed to compute the maximum likelihood estimator (MLE) for the model based on the EM algorithm (Dempster et al, 1977) and the Newton-Raphson method. The uniqueness of the MLE is proved. The simulation results show that the inconsistency and inefficiency can be significant when the Tobit method is applied to the present mixed model. The experiment results also suggest that the EM algorithm is much faster than the Newton-Raphson method for the present mixed model. The method also allows one to combine two data sets, the smaller data set with more detailed observations and the larger data set with less detailed binary choice observations in order to improve the efficiency of estimation. This may entail substantial savings when one conducts surveys.     

The new unified representation of multivariate skewed distributions

There are so many proposals in construction skewed distributions, and it is worth finding an overall class which covers all of these proposals. We introduce a new unified representation of multivariate skewed distributions. We will show that this new unified multivariate form of skewed distributions includes all of the continuous multivariate skewed distributions in the literature. This new unified representation is based on the multivariate probability integral transformation and can be decomposed into one factor that is original multivariate symmetric probability density function (pdf) f on ? ^k and skewed factor defined by a pdf p on [0, 1] ^k . This decomposition leads us to prove some useful properties of this new unified form. Stochastic representations and basic properties of this new form are also investigated in this article. Our work is motivated by considering the different skewing mechanisms which lead to different skewed distributions and show that all of these common-used distributions can be viewed as a new unified form.     

Bayesian inference for joint location and scale nonlinear models with skew-normal errors

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the dataset under consideration involves asymmetric outcomes. In this article, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis for joint location and scale nonlinear models with skew-normal errors, which relax the normality assumption and include the normal one as a special case. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of MCMC methods to simulate samples from the joint posterior distribution. Finally, simulation studies and a real example are used to illustrate the proposed methodology.     

The sinh-normal/independent nonlinear regression model

The normal/independent family of distributions is an attractive class of symmetric heavy-tailed density functions. They have a nice hierarchical representation to make inferences easily. We propose the Sinh-normal/independent distribution which extends the Sinh-normal (SN) distribution [23]. We discuss some of its properties and propose the Sinh-normal/independent nonlinear regression model based on a similar setup of Lemonte and Cordeiro [18], who applied the Birnbaum–Saunders distribution. We develop an EM-algorithm for maximum likelihood estimation of the model parameters. In order to examine the robustness of this flexible class against outlying observations, we perform a simulation study and analyze a real data set to illustrate the usefulness of the new model.     

Inference about the Regression Parameters Using Median-Ranked Set Sampling

The ranked set samples and median ranked set samples in particular have been used extensively in the literature due to many reasons. In some situations, the experimenter may not be able to quantify or measure the response variable due to the high cost of data collection, however it may be easier to rank the subject of interest. The purpose of this article is to study the asymptotic distribution of the parameter estimators of the simple linear regression model. We show that these estimators using median ranked set sampling scheme converge in distribution to the normal distribution under weak conditions. Moreover, we derive large sample confidence intervals for the regression parameters as well as a large sample prediction interval for new observation. Also, we study the properties of these estimators for small sample setup and conduct a simulation study to investigate the behavior of the distributions of the proposed estimators.     

Mixed Graphical Models with Missing Data and the Partial Imputation EM Algorithm

In this paper we discuss graphical models for mixed types of continuous and discrete variables with incomplete data. We use a set of hyperedges to represent an observed data pattern. A hyperedge is a set of variables observed for a group of individuals. In a mixed graph with two types of vertices and two types of edges, dots and circles represent discrete and continuous variables respectively. A normal graph represents a graphical model and a hypergraph represents an observed data pattern. In terms of the mixed graph, we discuss decomposition of mixed graphical models with incomplete data, and we present a partial imputation method which can be used in the EM algorithm and the Gibbs sampler to speed their convergence. For a given mixed graphical model and an observed data pattern, we try to decompose a large graph into several small ones so that the original likelihood can be factored into a product of likelihoods with distinct parameters for small graphs. For the case that a graph cannot be decomposed due to its observed data pattern, we can impute missing data partially so that the graph can be decomposed.     

A nonlinear mixed-effects model for multivariate longitudinal data with partially observed outcomes with application to HIV disease dynamics

The measurable multiple bio-markers for a disease are used as indicators for studying the response variable of interest in order to monitor and model disease progression. However, it is common for subjects to drop out of the studies prematurely resulting in unbalanced data and hence complicating the inferences involving such data. In this paper we consider a case where data are unbalanced among subjects and also within a subject because for some reason only a subset of the multiple outcomes of the response variable are observed at any one occasion. We propose a nonlinear mixed-effects model for the multivariate response variable data and derive a joint likelihood function that takes into account the partial dropout of the outcomes of the response variable. We further show how the methodology can be used in the estimation of the parameters that characterise HIV disease dynamics. An approximation technique of the parameters is also given and illustrated using a routine observational HIV dataset.