带线性约束的多元线性回归模型参数估计

本文首先构造线性约束条件下的多元线性回归模型的样本似然函数,利用Lagrange法证明其合理性。其次,从似然函数的角度讨论线性约束条件对模型参数的影响,对由传统理论得出的参数估计作出贝叶斯与经验贝叶斯的改进。做贝叶斯改进时,将矩阵正态-Wishart分布作为模型参数和精度阵的联合共轭先验分布,结合构造的似然函数得出参数的后验分布,计算出参数的贝叶斯估计;做经验贝叶斯改进时,将样本分组,从方差的角度讨论由子样得出的参数估计对总样本的参数估计的影响,计算出经验贝叶斯估计。最后,利用Matlab软件生成的随机矩阵做模拟。结果表明,这两种改进后的参数估计均较由传统理论得出的参数估计更精确,拟合结果的误差比更小,可信度更高,在大数据的情况下,这种计算方法的速度更快。     

Statistical model and inference for pharmacokinetic data

In this paper, we consider a statistical model for the drug concentration–time profiles that are obtained in a pharmacokinetic (PK) study when the drug is orally administered. In the proposed statistical PK model, the subject-specific concentration–time curve is described by the one-compartment PK model with first-order absorption and elimination. Moreover, a multivariate generalized gamma distribution is developed for the joint distribution of the drug concentrations that are repeatedly measured from the same subject. We then construct confidence intervals for the subject–exposure parameters which provide a further insight into the individual exposure of the drug under study. The proposed statistical PK model and the associated inference are then applied to illustrate a real data set. A simulation study is also implemented to investigate the performances of the coverage probability and expected length of the proposed confidence intervals. Finally, we give conclusions and discussions on the application of the proposed procedures.     

PkStaMp Library for Constructing Optimal Population Designs for PK/PD Studies

We discuss a Matlab-based library for constructing optimal sampling schemes for pharmacokinetic (PK) and pharmacodynamic (PD) studies. The software relies on optimal design theory for nonlinear mixed effects models and, in particular, on the first-order optimization algorithm. The library includes a number of popular compartmental PK and combined PK/PD models and can be extended to include more models. An outline of inputs/outputs is provided, some algorithmic details and examples are presented, and future work is discussed.     

Optimal parametric design with applications to pharmacokinetic and pharmacodynamic trials

This paper considers optimal parametric designs, i.e. designs represented by probability measures determined by a set of parameters, for nonlinear models and illustrates their use in designs for pharmacokinetic (PK) and pharmacokinetic/pharmacodynamic (PK/PD) trials. For some practical problems, such as designs for modelling PK/PD relationship, this is often the only feasible type of design, as the design points follow a PK model and cannot be directly controlled. Even for ordinary design problems the parametric designs have some advantages over the traditional designs, which often have too few design points for model checking and may not be robust to model and parameter misspecifications. We first describe methods and algorithms to construct the parametric design for ordinary nonlinear design problems and show that the parametric designs are robust to parameter misspecification and have good power for model discrimination. Then we extend this design method to construct optimal repeated measurement designs for nonlinear mixed models. We also use this parametric design for modelling a PK/PD relationship and propose a simulation based algorithm. The application of parametric designs is illustrated with a three-parameter open one-compartment PK model for the ordinary design and repeated measurement design, and an Emax model for the phamacokinetic/pharmacodynamic trial design.     

Bayesian multiple change-point estimation for exponential distribution with truncated and censored data

This paper considers the multiple change-point estimation for exponential distribution with truncated and censored data by Gibbs sampling. After all the missing data of interest is filled in by some sampling methods such as rejection sampling method, the complete-data likelihood function is obtained. The full conditional distributions of all parameters are discussed. The means of Gibbs samples are taken as Bayesian estimations of the parameters. The implementation steps of Gibbs sampling are introduced in detail. Finally random simulation test is developed, and the results show that Bayesian estimations are fairly accurate.     

A Simulation Based Study for Comparing Tests Associated With Receiver Operating Characteristic (ROC) Curves

Receiver Operating Characteristic curves and the Area Under Curve (AUC) are widely used to evaluate the predictive accuracy of diagnostic tests. The parametric methods of estimating AUCs are well established while nonparametric methods, such as Wilcoxon's method, lack proper research. This study considered three standard error techniques, namely, Hanley and McNeil, Hanley and Tilaki, and DeLong methods. Several parameters were considered, while measuring the predictor on a binary scale. The normality and type I error rate was violated for Hanley and McNeil's method while asymptotically DeLong's method performed better. Hanley and Tilaki's Jackknife method and DeLong's method performed equally well.     

Evaluations of Bayesian and maximum likelihood methods in PK models with below‐quantification‐limit data

Pharmacokinetic (PK) data often contain concentration measurements below the quantification limit (BQL). While specific values cannot be assigned to these observations, nevertheless these observed BQL data are informative and generally known to be lower than the lower limit of quantification (LLQ). Setting BQLs as missing data violates the usual missing at random (MAR) assumption applied to the statistical methods, and therefore leads to biased or less precise parameter estimation. By definition, these data lie within the interval [0, LLQ], and can be considered as censored observations. Statistical methods that handle censored data, such as maximum likelihood and Bayesian methods, are thus useful in the modelling of such data sets. The main aim of this work was to investigate the impact of the amount of BQL observations on the bias and precision of parameter estimates in population PK models (non‐linear mixed effects models in general) under maximum likelihood method as implemented in SAS and NONMEM, and a Bayesian approach using Markov chain Monte Carlo (MCMC) as applied in WinBUGS. A second aim was to compare these different methods in dealing with BQL or censored data in a practical situation. The evaluation was illustrated by simulation based on a simple PK model, where a number of data sets were simulated from a one‐compartment first‐order elimination PK model. Several quantification limits were applied to each of the simulated data to generate data sets with certain amounts of BQL data. The average percentage of BQL ranged from 25% to 75%. Their influence on the bias and precision of all population PK model parameters such as clearance and volume distribution under each estimation approach was explored and compared. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian Estimation of Bivariate Exponential Distributions Based on Linex and Quadratic Loss Functions: A Survival Approach with Censored Samples

In this article, four bivariate exponential (BVE) distributions with subject to right censoring samples are presented. Bayesian estimates of the parameters of BVE are obtained through Linex and quadratic loss functions. Gamma prior distribution has been suggested to reforming the posterior function. The estimations and standard errors of parameters have also been obtained through simulation method. Markov chain Monte Carlo (MCMC) method is employed for the case of Block-Buse bivariate distribution because there was no closed form for estimator criteria. Simulation studies have been conducted to show that the computation parts can be implemented easily and comparing the estimated values due to two methods and with the true values as well.     

Bayesian Bandwidth Estimation in Nonparametric Time-Varying Coefficient Models

Bandwidth plays an important role in determining the performance of nonparametric estimators, such as the local constant estimator. In this article, we propose a Bayesian approach to bandwidth estimation for local constant estimators of time-varying coefficients in time series models. We establish a large sample theory for the proposed bandwidth estimator and Bayesian estimators of the unknown parameters involved in the error density. A Monte Carlo simulation study shows that (i) the proposed Bayesian estimators for bandwidth and parameters in the error density have satisfactory finite sample performance; and (ii) our proposed Bayesian approach achieves better performance in estimating the bandwidths than the normal reference rule and cross-validation. Moreover, we apply our proposed Bayesian bandwidth estimation method for the time-varying coefficient models that explain Okun’s law and the relationship between consumption growth and income growth in the U.S. For each model, we also provide calibrated parametric forms of the time-varying coefficients. Supplementary materials for this article are available online.     

Discrete Scale Models for Survival–Sacrifice Experiments

Analyses of carcinogenicity experiments involving occult (hidden) tumours are usually based on cause-of-death information or the results of many interim sacrifices. A simple compartmental model is described that does not involve the cause of death. The method of analysis requires only one interim sacrifice, in addition to the usual terminal kill, to ensure that the tumour incidence rates can be estimated. One advantage of the approach is demonstrated in the analysis of glomerulosclerosis following exposure to ionizing radiation. Although the semiparametric model involves fewer parameters, estimates of key functions derived in this analysis are similar to those obtained previously by using a nonparametric method that involves many more parameters.     

A semi‐parametric approach to fitting a nonlinear mixed PK/PD model with an effect compartment using SAS

Modelling of the relationship between concentration (PK) and response (PD) plays an important role in drug development. The modelling becomes complicated when the drug concentration and response measurements are not taken simultaneously and/or hysteresis occurs between the response and the concentration. A model‐based approach fits a joint pharmacokinetic (PK) and concentration–response (PK/PD) model, including an effect compartment if necessary, to concentration and response data. However, this approach relies on the PK data being well described by a common PK model. We propose an algorithm for a semi‐parametric approach to fitting nonlinear mixed PK/PD models including an effect compartment using linear interpolation and extrapolation for concentration data. This approach is independent of the PK model, and the algorithm can easily be implemented using SAS PROC NLMIXED. Practical issues in programming and computing are also discussed. The properties of this approach are examined using simulations. This approach is used to analyse data from a study of the PK/PD relationship between insulin and glucose levels. Copyright © 2005 John Wiley & Sons, Ltd.     

Bayesian sieve methods: approximation rates and adaptive posterior contraction rates

In the last 20 years, a lot of achievements have been made in the study of posterior contraction rates of nonparametric Bayesian methods, and plenty of them involve sieve priors, but mainly for specific models or sieves. We provide a posterior contraction theorem for general parametric sieve priors. The theorem has weaker and simpler conditions compared with the existing results, and indicates that the sieve prior is rate adaptive. We apply the general theorem to density estimations and nonparametric regression with jumps. We also provided a reversible jump MCMC (Markov Chain Monte Carlo) algorithm for the sieve prior.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

Kullback-leibler information approach to the optimum measurement point for bayesian estimation

When an appropriate parametric model and a prior distribution of its parameters are given to describe clinical time courses of a dynamic biological process, Bayesian approaches allow us to estimate the entire profiles from a few or even a single observation per subject. The goodness of the estimation depends on the measurement points at which the observations were made. The number of measurement points per subject is generally limited to one or two. The limited measurement points have to be selected carefully. This paper proposes an approach to the selection of the optimum measurement point for Bayesian estimations of clinical time courses. The selection is made among given candidates, based on the goodness of estimation evaluated by the Kullback-Leibler information. This information measures the discrepancy of an estimated time course from the true one specified by a given appropriate model. The proposed approach is applied to a pharmacokinetic analysis, which is a typical clinical example where the selection is required. The results of the present study strongly suggest that the proposed approach is applicable to pharmacokinetic data and has a wide range of clinical applications.     

A Bayesian model for cross-study differential gene expression

In this paper we define a hierarchical Bayesian model for microarray expression data collected from several studies and use it to identify genes that show differential expression between two conditions. Key features include shrinkage across both genes and studies, and flexible modeling that allows for interactions between platforms and the estimated effect, as well as concordant and discordant differential expression across studies. We evaluated the performance of our model in a comprehensive fashion, using both artificial data, and a "split-study" validation approach that provides an agnostic assessment of the model's behavior not only under the null hypothesis, but also under a realistic alternative. The simulation results from the artificial data demonstrate the advantages of the Bayesian model. The 1 - AUC values for the Bayesian model are roughly half of the corresponding values for a direct combination of t- and SAM-statistics. Furthermore, the simulations provide guidelines for when the Bayesian model is most likely to be useful. Most noticeably, in small studies the Bayesian model generally outperforms other methods when evaluated by AUC, FDR, and MDR across a range of simulation parameters, and this difference diminishes for larger sample sizes in the individual studies. The split-study validation illustrates appropriate shrinkage of the Bayesian model in the absence of platform-, sample-, and annotation-differences that otherwise complicate experimental data analyses. Finally, we fit our model to four breast cancer studies employing different technologies (cDNA and Affymetrix) to estimate differential expression in estrogen receptor positive tumors versus negative ones. Software and data for reproducing our analysis are publicly available.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures

We propose a Bayesian nonparametric instrumental variable approach under additive separability that allows us to correct for endogeneity bias in regression models where the covariate effects enter with unknown functional form. Bias correction relies on a simultaneous equations specification with flexible modeling of the joint error distribution implemented via a Dirichlet process mixture prior. Both the structural and instrumental variable equation are specified in terms of additive predictors comprising penalized splines for nonlinear effects of continuous covariates. Inference is fully Bayesian, employing efficient Markov chain Monte Carlo simulation techniques. The resulting posterior samples do not only provide us with point estimates, but allow us to construct simultaneous credible bands for the nonparametric effects, including data-driven smoothing parameter selection. In addition, improved robustness properties are achieved due to the flexible error distribution specification. Both these features are challenging in the classical framework, making the Bayesian one advantageous. In simulations, we investigate small sample properties and an investigation of the effect of class size on student performance in Israel provides an illustration of the proposed approach which is implemented in an R package bayesIV. Supplementary materials for this article are available online.