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Censored data arise frequently in diverse applications in which observations to be measured may be subject to some upper and lower detection limits due to the restriction of experimental apparatus such that they are not exactly quantifiable. Mixtures of factor analyzers with censored data (MFAC) have been recently proposed for model-based density estimation and clustering of high-dimensional data in the presence of censored observations. In this paper, we consider an extended version of MFAC by considering regression equations to describe the relationship between covariates and multiply censored dependent variables. Two analytically feasible EM-type algorithms are developed for computing maximum likelihood estimates of model parameters with closed-form expressions. Moreover, we provide an information-based method to compute asymptotic standard errors of mixing proportions and regression coefficients. The utility and performance of our proposed methodology are illustrated through a simulation study and two real data examples.

Mixtures of factor analyzers (MFA) provide a promising tool for modeling and clustering high-dimensional data that contain an overwhelmingly large number of attributes measured on individuals arisen from a heterogeneous population. Due to the restriction of experimental apparatus, measurements can be limited to some lower and/or upper detection bounds and thus the data are possibly censored. In this paper, we extend the MFA to accommodate censored data, and the new model is called the MFA with censoring (MFAC). A computationally feasible alternating expectation conditional maximization (AECM) algorithm is developed to carry out maximum likelihood estimation of the MFAC model. Practical issues related to model-based clustering and recovery of censored data are also discussed. Simulation studies are conducted to examine the effect of censoring in classification, estimation and cluster validation. We also present an application of the proposed approach to two real data examples in which a certain number of left-censored observations are present.

In some large clinical studies, it may be impractical to perform the physical examination to every subject at his/her last monitoring time in order to diagnose the occurrence of the event of interest. This gives rise to survival data with missing censoring indicators where the probability of missing may depend on time of last monitoring and some covariates. We present a fully Bayesian semi-parametric method for such survival data to estimate regression parameters of the proportional hazards model of Cox (1972). Theoretical investigation and simulation studies show that our method performs better than competing methods. We apply the proposed method to analyze the survival data with missing censoring indicators from the Orofacial Pain: Prospective Evaluation and Risk Assessment (OPPERA) study.

Longitudinal HIV-1 RNA viral load measures are often subject to censoring due to upper and lower detection limits depending on the quantification assays. A complication arises when these continuous measures present a heavy-tailed behavior because inference can be seriously affected by the misspecification of their parametric distribution. For such data structures, we propose a robust nonlinear censored regression model based on the scale mixtures of normal distributions. By taking into account the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is considered. A stochastic approximation of the EM algorithm is developed to obtain the maximum likelihood estimates of the model parameters. The main advantage of this new procedure os to allow estimating the parameters of interest and evaluating the log-likelihood function easily and quickly. Furthermore, the standard errors of the fixed effects and predictions of unobservable values of the response can be obtained as a byproduct. The practical utility of the proposed method is exemplified using both simulated and real data.

Mixtures of common t factor analyzers (MCtFA) have been shown its effectiveness in robustifying mixtures of common factor analyzers (MCFA) when handling model-based clustering of the high-dimensional data with heavy tails. However, the MCtFA model may still suffer from a lack of robustness against observations whose distributions are highly asymmetric. This paper presents a further robust extension of the MCFA and MCtFA models, called the mixture of common restricted skew-t factor analyzers (MCrstFA), by assuming a restricted multivariate skew-t distribution for the common factors. The MCrstFA model can be used to accommodate severely non-normal (skewed and leptokurtic) random phenomena while preserving its parsimony in factor-analytic representation and performing graphical visualization in low-dimensional plots. A computationally feasible expectation conditional maximization either algorithm is developed to carry out maximum likelihood estimation. The numbers of factors and mixture components are simultaneously determined based on common likelihood penalized criteria. The usefulness of our proposed model is illustrated with simulated and real datasets, and experimental results signify its superiority over some existing competitors.

Longitudinal data are frequently analyzed using normal mixed effects models. Moreover, the traditional estimation methods are based on mean regression, which leads to non-robust parameter estimation under non-normal error distribution. However, at least in principle, quantile regression (QR) is more robust in the presence of outliers/influential observations and misspecification of the error distributions when compared to the conventional mean regression approach. In this context, this paper develops a likelihood-based approach for estimating QR models with correlated continuous longitudinal data using the asymmetric Laplace distribution. Our approach relies on the stochastic approximation of the EM algorithm (SAEM algorithm), obtaining maximum likelihood estimates of the fixed effects and variance components in the case of nonlinear mixed effects (NLME) models. We evaluate the finite sample performance of the SAEM algorithm and asymptotic properties of the ML estimates through simulation experiments. Moreover, two real life datasets are used to illustrate our proposed method using the $$\texttt {qrNLMM}$$ package from $$\texttt {R}$$.

The issue of model-based clustering of longitudinal data has attracted increasing attention in past two decades. Finite mixtures of Student’s- t linear mixed-effects (FM-tLME) models have been considered for implementing this task especially when data contain extreme observations. This paper presents an extended finite mixtures of Student’s- t linear mixed-effects (EFM-tLME) model, where the categorical component labels are assumed to be influenced by the observed covariates. As compared with the naive methods assuming the mixing proportions to be fixed but unknown, the proposed EFM-tLME model exploits a logistic function to link the relationship between the prior classification probabilities and the covariates of interest. To carry out maximum likelihood estimation, an alternating expectation conditional maximization (AECM) algorithm is developed under several model reduction schemes. The technique for extracting the information-based standard errors of parameter estimates is also investigated. The proposed method is illustrated using simulation experiments and real data from an AIDS clinical study.

In biomedical studies and clinical trials, repeated measures are often subject to some upper and/or lower limits of detection. Hence, the responses are either left or right censored. A complication arises when more than one series of responses is repeatedly collected on each subject at irregular intervals over a period of time and the data exhibit tails heavier than the normal distribution. The multivariate censored linear mixed effect (MLMEC) model is a frequently used tool for a joint analysis of more than one series of longitudinal data. In this context, we develop a robust generalization of the MLMEC based on the scale mixtures of normal distributions. To take into account the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is considered. For this complex longitudinal structure, we propose an exact estimation procedure to obtain the maximum-likelihood estimates of the fixed effects and variance components using a stochastic approximation of the EM algorithm. This approach allows us to estimate the parameters of interest easily and quickly as well as to obtain the standard errors of the fixed effects, the predictions of unobservable values of the responses, and the log-likelihood function as a byproduct. The proposed method is applied to analyze a set of AIDS data and is examined via a simulation study.

Despite technological advances in efficiency enhancement of quantification assays, biomedical studies on HIV RNA collect viral load responses that are often subject to detection limits. Moreover, some related covariates such as CD4 cell count may be often measured with errors. Censored non-linear mixed-effects models are routinely used to analyze this type of data and are based on normality assumptions for the between-subject and within-subject random terms. However, derived inference may not be robust when the underlying normality assumptions are questionable, especially in presence of skewness and heavy tails. In this article, we address these issues simultaneously under a Bayesian paradigm through joint modeling of the response and covariate processes using an attractive class of skew-normal independent densities. The methodology is illustrated using a case study on longitudinal HIV viral loads. Diagnostics for outlier detection is immediate from the MCMC output. Both simulation and real data analysis reveal the advantage of the proposed models in providing robust inference under non-normality situations commonly encountered in HIV/AIDS or other clinical studies. Supplementary materials accompanying this paper appear on-line.

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.

In this chapter, we study a partially linear model with autoregressive beta distributed errors [6] from the Bayesian point of view. Our proposal also provides a useful method to determine the optimal order of the autoregressive processes through an adaptive procedure using the conditional predictive ordinate (CPO) statistic [9].

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.

Nonlinear mixed-effects (NLME) models are popular in many longitudinal studies, including those on human immunodeficiency virus (HIV) viral dynamics, pharmacokinetic analysis, and studies of growth and decay analysis. Generally, the normality of the random effects is a common assumption in NLME models but it can sometimes be unrealistic, suppressing important features of among-subjects variation. In this context, the use of normal/independent distributions arises as a tool for robust modeling of NLME models. These distributions fall in a class of symmetric heavy-tailed distributions that includes the normal distribution, the generalized Student-t, Student-t, slash and the contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of normal distributions in these types of models. The aim of this paper is the estimation of NLME models considering normal/independent distributions for the error term and random effects, under the Bayesian paradigm. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selections criteria is also developed. These analyses are computationally possible due to an important result that approximates the likelihood function of a NLME model with normal/independent distributions for a simple normal/independent distribution with specified parameters. An example of the new method is presented through simulation and application to a real dataset of AIDS/HIV infected patients that was initially analyzed using a normal NLME model.

Azzalini’s skew-normal distribution is obtained through a conditional reduction of a multivariate normal distribution parameterized with a correlation matrix. It seems natural that when the parameterization of that multivariate normal distribution is complexified, more flexible skew-normal distributions could be obtained. In this note this specification strategy, previously explored by Azzalini [Scand. J. Stat. 33 (2006) 561–574] among many other authors, is formally analyzed through an identification analysis.

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

The half-normal distribution is one of the widely used probability distribution for non-negative data modeling, specifically, to describe the lifetime process under fatigue. In this paper, we introduce a new type of non-negative distribution that extends the half-normal distribution. We refer to this new distribution as the epsilon half-normal distribution. We provide mathematical properties of this new distribution. In particular, we derive the stochastic representation, explicit formulas for the n-th moment, the asymmetry and kurtosis coefficients and the moment generating function. We also discuss some inferential aspects related to the maximum likelihood estimation. We illustrate the flexibility of this type of distribution with an application to a real dataset of stress-rupture.

In this article, we study some results related to a specific class of distributions, called skew-curved-symmetric family of distributions that depends on a parameter controlling the skewness and kurtosis at the same time. Special elements of this family which are studied include symmetric and well-known asymmetric distributions. General results are given for the score function and the observed information matrix. It is shown that the observed information matrix is always singular for some special cases. We illustrate the flexibility of this class of distributions with an application to a real dataset on characteristics of Australian athletes.

We introduce a class of shape mixtures of skewed distributions and study some of its main properties. We discuss a Bayesian interpretation and some invariance results of the proposed class. We develop a Bayesian analysis of the skew-normal, skew-generalized-normal, skew-normal-t and skew-t-normal linear re- gression models under some special prior specications for the model parameters. In particular, we show that the full posterior of the skew-normal regression model parameters is proper under an arbitrary proper prior for the shape parameter and noninformative prior for the other parameters. We implement a convenient hierar- chical representation in order to obtain the corresponding posterior analysis. We illustrate our approach with an application to a real dataset on characteristics of Australian male athletes.