Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

Modeling the correlation structure of data that have multiple levels of association

Some modem approaches for the analysis of non-normally distributed and correlated data, including Liang and Zeger's ( 1986 ) method of generalized estimating equations (GEE), model the pattern of association among outcomes by assuming a structure for their correlation matrix. A number of relatively simple patterned correlation matrices are available for measurements with one level of correlation. However, modeling the correlation structure of data with multiple levels, or causes, of association is not as straightforward; this note discusses some of the difficulties and discusses a simple class of correlation models that may prove useful in this endeavor.     

Multivariate Dispersion Models Generated From Gaussian Copula

In this paper a class of multivariate dispersion models generated from the multivariate Gaussian copula is presented. Being a multivariate extension of Jørgensen's (1987a) dispersion models, this class of multivariate models is parametrized by marginal position, dispersion and dependence parameters, producing a large variety of multivariate discrete and continuous models including the multivariate normal as a special case. Properties of the multivariate distributions are investigated, some of which are similar to those of the multivariate normal distribution, which makes these models potentially useful for the analysis of correlated non-normal data in a way analogous to that of multivariate normal data. As an example, we illustrate an application of the models to the regression analysis of longitudinal data, and establish an asymptotic relationship between the likelihood equation and the generalized estimating equation of Liang & Zeger (1986).     

On approximate likelihood inference in a poisson mixed model

A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance σ² of a Poisson mixed model. In the first step, it is proposed to construct a small σ²-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and σ². In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so-called Stein-type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.     

Bayesian semiparametric reproductive dispersion mixed models for non-normal longitudinal data: estimation and case influence analysis

Semiparametric reproductive dispersion mixed model (SPRDMM) is a natural extension of the reproductive dispersion model and the semiparametric mixed model. In this paper, we relax the normality assumption of random effects in SPRDMM and use a truncated and centred Dirichlet process prior to specify random effects, and present the Bayesian P-spline to approximate the smoothing unknown function. A hybrid algorithm combining the block Gibbs sampler and the Metropolis–Hastings algorithm is implemented to sample observations from the posterior distribution. Also, we develop Bayesian case deletion influence measure for SPRDMM based on the φ-divergence and present those computationally feasible formulas. Several simulation studies and a real example are presented to illustrate the proposed methodologies.     

Fitting and testing the Marshall–Olkin extended Weibull model with randomly censored data

The random censorship model (RCM) is commonly used in biomedical science for modeling life distributions. The popular non-parametric Kaplan–Meier estimator and some semiparametric models such as Cox proportional hazard models are extensively discussed in the literature. In this paper, we propose to fit the RCM with the assumption that the actual life distribution and the censoring distribution have a proportional odds relationship. The parametric model is defined using Marshall–Olkin's extended Weibull distribution. We utilize the maximum-likelihood procedure to estimate model parameters, the survival distribution, the mean residual life function, and the hazard rate as well. The proportional odds assumption is also justified by the newly proposed bootstrap Komogorov–Smirnov type goodness-of-fit test. A simulation study on the MLE of model parameters and the median survival time is carried out to assess the finite sample performance of the model. Finally, we implement the proposed model on two real-life data sets.     

面板数据的自适应Lasso分位回归方法研究

如何在对参数进行估计的同时自动选择重要解释变量,一直是面板数据分位回归模型中讨论的热点问题之一。通过构造一种含多重随机效应的贝叶斯分层分位回归模型,在假定固定效应系数先验服从一种新的条件Laplace分布的基础上,给出了模型参数估计的Gibbs抽样算法。考虑到不同重要程度的解释变量权重系数压缩程度应该不同,所构造的先验信息具有自适应性的特点,能够准确地对模型中重要解释变量进行自动选取,且设计的切片Gibbs抽样算法能够快速有效地解决模型中各个参数的后验均值估计问题。模拟结果显示,新方法在参数估计精确度和变量选择准确度上均优于现有文献的常用方法。通过对中国各地区多个宏观经济指标的面板数据进行建模分析,演示了新方法估计参数与挑选变量的能力。     

Longitudinal Data Estimating Equations for Dispersion Models

Liang & Zeger's generalized estimating equation approach for analysis of longitudinal data is extended to marginal distributions of dispersion model type. This includes for example the von Mises and simplex distributions, suitable for angles and proportions, respectively. Both modelling of position and joint modelling of position and dispersion is considered, and the method is applied to a set of bird orientation data.     

Bayesian shrinkage estimates for regression coefficients in m populations

For a linear regression model over m populations with separate regression coefficients but a common error variance, a Bayesian model is employed to obtain regression coefficient estimates which are shrunk toward an overall value. The formulation uses Normal priors on the coefficients and diffuse priors on the grand mean vectors, the error variance, and the between-to-error variance ratios. The posterior density of the parameters which were given diffuse priors is obtained. From this the posterior means and variances of regression coefficients and the predictive mean and variance of a future observation are obtained directly by numerical integration in the balanced case, and with the aid of series expansions in the approximately balanced case. An example is presented and worked out for the case of one predictor variable. The method is an extension of Box & Tiao's Bayesian estimation of means in the balanced one-way random effects model.     

Optimal estimation of polynomial hazard functions

The counting process formulation (Aalen, 1978) for the analysis of life time data is briefly reviewed. This formulation is used to arrive at a regression type model and a smooth estimate of the hazard function. In the regression model, the error terms are martingales and Nelson's estimator is the dependent variable. An optimal approach for estimating the parameters of the polynomial is considered, Asymptotic normality of the optimal estimate is proved and an illustrative example is given.     

The beta distribution as a latent response model for ordinal data (I): Estimation of location and dispersion parameters

Ordinal data are often modeled using a continuous latent response distribution, which is partially observed through windows of adjacent intervals defined by cutpoints. In this paper we propose the beta distribution as a model for the latent response. The beta distribution has several advantages over the other common distributions used, e.g. , normal and logistic. In particular, it enables separate modeling of location and dispersion effects which is essential in the Taguchi method of robust design. First, we study the problem of estimating the location and dispersion parameters of a single beta distribution (representing a single treatment) from ordinal data assuming known equispaced cutpoints. Two methods of estimation are compared: the maximum likelihood method and the method of moments. Two methods of treating the data are considered: in raw discrete form and in smoothed continuousized form. A large scale simulation study is carried out to compare the different methods. The mean square errors of the estimates are obtained under a variety of parameter configurations. Comparisons are made based on the ratios of the mean square errors (called the relative efficiencies). No method is universally the best, but the maximum likelihood method using continuousized data is found to perform generally well, especially for estimating the dispersion parameter. This method is also computationally much faster than the other methods and does not experience convergence difficulties in case of sparse or empty cells. Next, the problem of estimating unknown cutpoints is addressed. Here the multiple treatments setup is considered since in an actual application, cutpoints are common to all treatments, and must be estimated from all the data. A two-step iterative algorithm is proposed for estimating the location and dispersion parameters of the treatments, and the cutpoints. The proposed beta model and McCullagh's (1980) proportional odds model are compared by fitting them to two real data sets.     

Estimates for mean and dispersion effects in unreplicated factorial designs

One important goal of experimentation in quality improvement is to minimize the variability of a product or process around a target mean value. Factors which affect variances as well as factors that affect the mean can be identified using the analysis of mean and dispersion. Box and Meyer (1986b) proposed a method of model identification and maximum likelihood estimation for mean and dispersion effects from unreplicated designs. In this article, we address two problems associated with MLE’s. First, asymptotic variance of MLE's for dispersion effects which can be used to judge the significance of factors can be misleading. A possible explanation is provided; simulation results also indicate that the asymptotic, variance underestimates.     

Extending the inference function for augmented margins method to implement trivariate Clayton copula-based SUR Tobit models

Abstract

In this paper, we perform the analysis of the SUR Tobit model for three left-censored dependent variables by modeling its nonlinear dependence structure through the one-parameter Clayton copula. For unbiased parameter estimation, we propose an extension of the Inference Function for Augmented Margins (IFAM) method to the trivariate case. The interval estimation for the model parameters using resampling procedures is also discussed. We perform simulation and empirical studies, whose satisfactory results indicate the good performance of the proposed model and methods. Our procedure is illustrated using real data on consumption of food items (salad dressings, lettuce, tomato) by Americans.     

Quasi-Likelihood Approach to Bioavailability and Bioequivalence Analysis

The outcomes AUCT (area-under-curve from time zero to time t) of n individuals randomized to one of two groups TR or RT, where the group name denotes the order in which the subjects receive a test formulation (T) or a reference formulation (R), are used to assess average bioequivalence for the two formulations. The classical method is the mixed model, for example, proc mixed or proc glm with random statement in SAS can be used to analyze this type of data. This is equivalent to the marginal likelihood approach in a normal–normal model. There are some limitations for this approach. It is not appropriate if the random effect is not normally distributed. In this article, we introduce a hierarchical quasi-likelihood approach. Instead of assuming the random effect is normal, we make assumptions only about the mean and the variance function of the random effect. Our method is flexible to model the random effect. Since we can estimate the random effect for each individual, we can check the adequacy of the distribution assumption about the random effect. This method can also be used to handle high-dimensional crossover data. Simulation studies are conducted to check the finite sample performance of the method under various conditions and two real data examples are used for illustration.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

A weibull process failure mechanism for the economic design of mewma control charts

The design parameters of the multivariate exponentially weighted moving average (MEWMA) control chart may be chosen according to economic and/or statistical considerations. The economic model proposed for the design of the'MEWMA chart assumes a Markovian process failure mechanism following an exponential distribution. We'assess the sensitivity of the resulting economic design for the MEWMA to deviations from this assumption. In particular, the generalization, from an exponential to a Weibull distribution of process failure, is used to study the selection of MEWMA chart parameters given process cost and time information. We conclude that the quality of the resulting design (in terms of expected cost) is not substantially affected by mis-specification of the distribution of process failure.     

The generalized Gudermannian distribution: inference and volatility modelling

In this paper, we introduce a new distribution, called generalized Gudermannian (GG) distribution, and its skew extension for GARCH models in modelling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are obtained including probability density and cumulative distribution functions, moments, and stochastic representation. The maximum likelihood method is used to estimate unknown parameters of the proposed model and finite sample performance of maximum likelihood estimates are evaluated by means of Monte-Carlo simulation study. The real data application on Nikkei 225 index is given to demonstrate the performance of GARCH model specified under skew extension of GG innovation distribution against normal, Student's-t, skew normal and generalized error and skew generalized error distributions in terms of the accuracy of VaR forecasts. The empirical results show that the GARCH model with GG innovation distribution produces the most accurate VaR forecasts for all confidence levels.     

AN EXTENSION OF THE EM ALGORITHM FOR OPTIMIZATION OF CONSTRAINED LIKELIHOOD: AN APPLICATION IN TOXICOLOGY

Since its introduction in the mid 1970's, the EM algorithm has found a widespread popularity for solving the likelihood equations. Several investigators have used the algorithm in a variety of problems with incomplete information to obtain maximum likelihood estimates in numerous applications. The algorithm, however, becomes inappropriate when the underlying equations are subject to some constraints. Although an extension has been proposed to derive solutions when the parameters are subject to a set of linear constraints, the evaluation of likelihood equations from incomplete data when the equations are subject to a nonlinear constraint is still an open problem. Here, we consider a mixture model, a classical example of incomplete data, and discuss the problem of maximum likelihood estimation of the model parameters when the parameters have to satisfy a nonlinear constraint. An extension of the EM algorithm based on the celebrated Lagrange multiplier will be proposed to solve the equations. An application of the methodology in animal bioassay experiments for risk assessment of toxic substances will be described and data from a toxicological experiment will be used to illustrate the results.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

A generalized Q–Q plot for longitudinal data

Most biomedical research is carried out using longitudinal studies. The method of generalized estimating equations (GEEs) introduced by Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] and Zeger and Liang [Longitudinal data analysis for discrete and continuous outcomes, Biometrics 42 (1986), pp. 121–130] has become a standard method for analyzing non-normal longitudinal data. Since then, a large variety of GEEs have been proposed. However, the model diagnostic problem has not been explored intensively. Oh et al. [Modeldiagnostic plots for repeated measures data using the generalized estimating equations approach, Comput. Statist. Data Anal. 53 (2008), pp. 222–232] proposed residual plots based on the quantile–quantile (Q–Q) plots of the χ²-distribution for repeated-measures data using the GEE methodology. They considered the Pearson, Anscombe and deviance residuals. In this work, we propose to extend this graphical diagnostic using a generalized residual. A simulation study is presented as well as two examples illustrating the proposed generalized Q–Q plots.