Idle thoughts of a ‘well‐calibrated’ Bayesian in clinical drug development

The use of Bayesian approaches in the regulated world of pharmaceutical drug development has not been without its difficulties or its critics. The recent Food and Drug Administration regulatory guidance on the use of Bayesian approaches in device submissions has mandated an investigation into the operating characteristics of Bayesian approaches and has suggested how to make adjustments in order that the proposed approaches are in a sense calibrated. In this paper, I present examples of frequentist calibration of Bayesian procedures and argue that we need not necessarily aim for perfect calibration but should be allowed to use procedures, which are well‐calibrated, a position supported by the guidance. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian inference and prediction of the Rayleigh distribution based on ordered ranked set sampling

Based on ordered ranked set sample, Bayesian estimation of the model parameter as well as prediction of the unobserved data from Rayleigh distribution are studied. The Bayes estimates of the parameter involved are obtained using both squared error and asymmetric loss functions. The Bayesian prediction approach is considered for predicting the unobserved lifetimes based on a two-sample prediction problem. A real life dataset and simulation study are used to illustrate our procedures.     

AN EVALUATION OF NON-ITERATIVE METHODS FOR ESTIMATING THE LINEAR-BY-LINEAR PARAMETER OF ORDINAL LOG-LINEAR MODELS

Parameter estimation for association and log-linear models is an important aspect of the analysis of cross-classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear-by-linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non-iterative procedures for estimating the linear-by-linear parameter for ordinal log-linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non-iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered.     

An empirical comparison of EM,SEM and MCMC performance for problematic Gaussian mixture likelihoods

We compare EM, SEM, and MCMC algorithms to estimate the parameters of the Gaussian mixture model. We focus on problems in estimation arising from the likelihood function having a sharp ridge or saddle points. We use both synthetic and empirical data with those features. The comparison includes Bayesian approaches with different prior specifications and various procedures to deal with label switching. Although the solutions provided by these stochastic algorithms are more often degenerate, we conclude that SEM and MCMC may display faster convergence and improve the ability to locate the global maximum of the likelihood function.     

Estimating the total number of distinct species using quadrat sampling and under-dependence structure

To estimate the total number of distinct species in a given region, Bayesian methods along with quadrat sampling procedures have been used by several authors. A key underlying assumption relies on the independence among the species. In this note, we analyse these estimates allowing a generalized binomial dependence between species.     

Estimation of group means using Bayesian generalized linear mixed models

Generalized linear mixed models (GLMM) are commonly used to model the treatment effect over time while controlling for important clinical covariates. Standard software procedures often provide estimates of the outcome based on the mean of the covariates; however, these estimates will be biased for the true group means in the GLMM. Implementing GLMM in the frequentist framework can lead to issues of convergence. A simulation study demonstrating the use of fully Bayesian GLMM for providing unbiased estimates of group means is shown. These models are very straightforward to implement and can be used for a broad variety of outcomes (eg, binary, categorical, and count data) that arise in clinical trials. We demonstrate the proposed method on a data set from a clinical trial in diabetes.     

Inference and prediction for pareto progressively censored data

Progressively censored data from a classical Pareto distribution are to be used to make inferences about its shape and precision parameters and the reliability function. An approximation form due to Tierney and Kadane (1986) is used for obtaining the Bayes estimates. Bayesian prediction of further observations from this distribution is also considered. When the Bayesian approach is concerned, conjugate priors for either the one or the two parameters cases are considered. To illustrate the given procedures, a numerical example and a simulation study are given.     

Bayesian ikference procedures derived via the concept of relative surprise

We consider the problem of deriving Bayesian inference procedures via the concept of relative surprise. The mathematical concept of surprise has been developed by I.J. Good in a long sequence of papers. We make a modification to this development that permits the avoidance of a serious defect; namely, the change of variable problem. We apply relative surprise to the development of estimation, hypothesis testing and model checking procedures. Important advantages of the relative surprise approach to inference include the lack of dependence on a particular loss function and complete freedom to the statistician in the choice of prior for hypothesis testing problems. Links are established with common Bayesian inference procedures such as highest posterior density regions, modal estimates and Bayes factors. From a practical perspective new inference procedures arise that possess good properties.     

Marginal maximum likelihood estimation methods for the tuning parameters of ridge,power ridge,and generalized ridge regression

This study introduces fast marginal maximum likelihood (MML) algorithms for estimating the tuning (shrinkage) parameter(s) of the ridge and power ridge regression models, and an automatic plug-in MML estimator for the generalized ridge regression model, in a Bayesian framework. These methods are applicable to multicollinear or singular covariate design matrices, including matrices where the number of covariates exceeds the sample size. According to analyses of many real and simulated datasets, these MML-based ridge methods tend to compare favorably to other tuning parameter selection methods, in terms of computation speed, prediction accuracy, and ability to detect relevant covariates.     

The E-Bayesian and hierarchical Bayesian estimations for the proportional reversed hazard rate model based on record values

In this paper, E-Bayesian and hierarchical Bayesian estimations of the shape parameter, when the underlying distribution belongs to the proportional reversed hazard rate model, are considered. Maximum likelihood, Bayesian and E-Bayesian estimates of the unknown parameter and reliability function are obtained based on record values. The Bayesian estimates are derived based on squared error and linear–exponential loss functions. It is pointed out that some previously obtained order relations of E-Bayesian estimates are inadequate and these results are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations is obtained under the same loss functions. The comparison of the derived estimates is carried out by using Monte Carlo simulations. A real data set is analysed for an illustration of the findings.     

Renewal theory and the sequential design of experiments with two states of nature

In the sequential design of experiments in experimentel is sequentially performing experiments to help him make an inference about the true state of nature. Using results from renewal twenty at derive approximations for the operations characteristics and average sample numbers for this problem when there are two states of nature.

A critical problem in the sequential design of experiments is finding a good procedure. We investigate a Bayesian formulation of this problem and use our approximations to approximate the Bayes risk. Minimization of this approximate Bayes risk over procedures is discussed as a method of finding a good procedure, but difficulties are encountered due to the discrete time character of the sequential process. To avoid these difficulties, we consider minimization of an approximation related to a diffusion process. This leads to a simple rule for the sequential selection of experiments.     

Bayesian approach to epsilon-skew-normal family

The estimation problem of epsilon-skew-normal (ESN) distribution parameters is considered within Bayesian approaches. This family of distributions contains the normal distribution, can be used for analyzing the asymmetric and near-normal data. Bayesian estimates under informative and non informative Jeffreys prior distributions are obtained and performances of ESN family and these estimates are shown via a simulation study. A real data set is also used to illustrate the ideas.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

Markov Chain Monte Carlo methods for estimating surgery duration

Developing prediction bounds for surgery duration is difficult due to the large number of distinct procedures. The variety of procedures at a multi-speciality surgery suite means that even with several years of historical data a large fraction of surgical cases will have little or no historical data for use in predicting case duration. Bayesian methods can be used to combine historical data with expert judgement to provide estimates to overcome this, but eliciting expert opinion for a probability distribution can be difficult. We combine expert judgement, expert classification of procedures by complexity category and historical data in a Markov Chain Monte Carlo model and test it against one year of actual surgery cases at a multi-speciality surgical suite.     

Biases and Uncertainty in Climate Projections

Abstract. We study statistical procedures to quantify uncertainty in multivariate climate projections based on several deterministic climate models. We introduce two different assumptions – called constant bias and constant relation respectively – for extrapolating the substantial additive and multiplicative biases present during the control period to the scenario period. There are also strong indications that the biases in the scenario period are different from the extrapolations from the control period. Including such changes in the statistical models leads to an identifiability problem that we solve in a frequentist analysis using a zero sum side condition and in a Bayesian analysis using informative priors. The Bayesian analysis provides estimates of the uncertainty in the parameter estimates and takes this uncertainty into account for the predictive distributions. We illustrate the method by analysing projections of seasonal temperature and precipitation in the Alpine region from five regional climate models in the PRUDENCE project.     

A practical sampling approach for a Bayesian mixture model with unknown number of components

Recently, mixture distribution becomes more and more popular in many scientific fields. Statistical computation and analysis of mixture models, however, are extremely complex due to the large number of parameters involved. Both EM algorithms for likelihood inference and MCMC procedures for Bayesian analysis have various difficulties in dealing with mixtures with unknown number of components. In this paper, we propose a direct sampling approach to the computation of Bayesian finite mixture models with varying number of components. This approach requires only the knowledge of the density function up to a multiplicative constant. It is easy to implement, numerically efficient and very practical in real applications. A simulation study shows that it performs quite satisfactorily on relatively high dimensional distributions. A well-known genetic data set is used to demonstrate the simplicity of this method and its power for the computation of high dimensional Bayesian mixture models.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

E-Bayesian estimation for the proportional hazard rate model based on record values

In this article, a new parameter estimation method, named E-Bayesian method, is considered to obtain the estimates of the unknown parameter and reliability function based on record values. The maximum likelihood, Bayesian, E-Bayesian, and hierarchical Bayesian estimates of the unknown parameter and reliability function are obtained when the underlying distribution belongs to the proportional hazard rate model. The Bayesian estimates are obtained based on squared error and linear-exponential loss functions. The previously obtained some relations for the E-Bayesian estimates are improved. The relationship between E-Bayesian and hierarchical Bayesian estimations are obtained under the same loss functions. The comparison of the derived estimates are carried out by using Monte Carlo simulations. Real data are analyzed for an illustration of the findings.     

Choosing shrinkage estimators for regression problems

A Bayesian formulation of the canonical form of the standard regression model is used to compare various Stein-type estimators and the ridge estimator of regression coefficients, A particular (“constant prior”) Stein-type estimator having the same pattern of shrinkage as the ridge estimator is recommended for use.