Elicitation of Subjective Probability Distributions

To incorporate expert opinion into a Bayesian analysis, it must be quantified as a prior distribution through an elicitation process that asks the expert meaningful questions whose answers determine this distribution. The aim of this thesis is to fill some gaps in the available techniques for eliciting prior distributions for Generalized Linear Models (GLMs) and multinomial models. A general method for quantifying opinion about GLMs was developed in Garthwaite and Al-Awadhi (2006). They model the relationship between each continuous predictor and the dependant variable as a piecewise-linear function with a regression coefficient at each of its dividing points. However, coefficients were assumed a priori independent if associated with different predictors. We relax this simplifying assumption and propose three new methods for eliciting positive-definite variance-covariance matrices of a multivariate normal prior distribution. In addition, we extend the method of Garthwaite and Dickey (1988) for eliciting an inverse chi-squared conjugate prior for the error variance in normal linear models. We also propose a novel method for eliciting a lognormal prior distribution for the scale parameter of a gamma GLM. For multinomial models, novel methods are proposed that quantify expert opinion about a conjugate Dirichlet distribution and, additionally, about three more general and flexible prior distributions. First, an elicitation method is proposed for the generalized Dirichlet distribution that was introduced by Connor and Mosimann (1969). Second, a method is developed for eliciting the Gaussian copula as a multivariate distribution with marginal beta priors. Third, a further novel method is constructed that quantifies expert opinion about the most flexible alternate prior, the logistic normal distribution (Aitchison, 1986). This third method is extended to the case of multinomial models with explanatory covariates. All proposed methods in this thesis are designed to be used with interactive Prior Elicitation Graphical Software (PEGS) that is freely available at http://statistics.open.ac.uk/elicitation

Modified confidence intervals for the Mahalanobis distance

Reiser (2001) proposes a method of forming confidence interval for a Mahalanobis distance that yields intervals which have exactly the nominal coverage, but sometimes the interval is View the MathML source (0,0). We consider the case where Mahalanobis distance quantifies the difference between an individual and a population mean, and suggest a modification that avoids implausible intervals

BIMAM—a tool for imputing variables missing across datasets using a Bayesian imputation and analysis model

Motivation Combination of multiple datasets is routine in modern epidemiology. However, studies may have measured different sets of variables; this is often inefficiently dealt with by excluding studies or dropping variables. Multilevel multiple imputation methods to impute these ‘systematically’ missing data (as opposed to ‘sporadically’ missing data within a study) are available, but problems may arise when many random effects are needed to allow for heterogeneity across studies. We show that the Bayesian IMputation and Analysis Model (BIMAM) implemented in our tool works well in this situation. General features BIMAM performs imputation and analysis simultaneously. It imputes both binary and continuous systematically and sporadically missing data, and analyses binary and continuous outcomes. BIMAM is a user-friendly, freely available tool that does not require knowledge of Bayesian methods. BIMAM is an R Shiny application. It is downloadable to a local machine and it automatically installs the required freely available packages (R packages, including R2MultiBUGS and MultiBUGS). Availability BIMAM is available at [www.alecstudy.org/bimam]

On point estimation of the abnormality of a Mahalanobis index

Mahalanobis distance may be used as a measure of the disparity between an individual’s profile of scores and the average profile of a population of controls. The degree to which the individual’s profile is unusual can then be equated to the proportion of the population who would have a larger Mahalanobis distance than the individual. Several estimators of this proportion are examined. These include plug-in maximum likelihood estimators, medians, the posterior mean from a Bayesian probability matching prior, an estimator derived from a Taylor expansion, and two forms of polynomial approximation, one based on Bernstein polynomial and one on a quadrature method. Simulations show that some estimators, including the commonly-used plug-in maximum likelihood estimators, can have substantial bias for small or moderate sample sizes. The polynomial approximations yield estimators that have low bias, with the quadrature method marginally to be preferred over Bernstein polynomials. However, the polynomial estimators sometimes yield infeasible estimates that are outside the 0–1 range. While none of the estimators are perfectly unbiased, the median estimators match their definition; in simulations their estimates of the proportion have a median error close to zero. The standard median estimator can give unrealistically small estimates (including 0) and an adjustment is proposed that ensures estimates are always credible. This latter estimator has much to recommend it when unbiasedness is not of paramount importance, while the quadrature method is recommended when bias is the dominant issue

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java

Prevalence and Population Attributable Risk for Chronic Airflow Obstruction in a Large Multinational Study

Rationale: The Global Burden of Disease programme identified smoking, and ambient and household air pollution as the main drivers of death and disability from Chronic Obstructive Pulmonary Disease (COPD).Objective: To estimate the attributable risk of chronic airflow obstruction (CAO), a quantifiable characteristic of COPD, due to several risk factors.Methods: The Burden of Obstructive Lung Disease study is a cross-sectional study of adults, aged≥40, in a globally distributed sample of 41 urban and rural sites. Based on data from 28,459 participants, we estimated the prevalence of CAO, defined as a post-bronchodilator one-second forced expiratory volume to forced vital capacity ratio Measurements and Main Results: Mean prevalence of CAO was 11.2% in men and 8.6% in women. Mean PAR for smoking was 5.1% in men and 2.2% in women. The next most influential risk factors were poor education levels, working in a dusty job for ≥10 years, low body mass index (BMI), and a history of tuberculosis. The risk of CAO attributable to the different risk factors varied across sites.Conclusions: While smoking remains the most important risk factor for CAO, in some areas poor education, low BMI and passive smoking are of greater importance. Dusty occupations and tuberculosis are important risk factors at some sites

On eliciting expert opinion in generalized linear models

Suitable elicitation methods play a key role in Bayesian analysis of generalized linear models (GLMs) by obtaining and including expert knowledge as a prior distribution for the model parameters. Some elicitation methods for GLMs available in the literature focus mainly on logistic regression. A more general elicitation method of quantifying opinion about any GLM was developed in Garthwaite and Al-Awadhi (2006). The relationship between each continuous predictor and the dependant variable was modeled as a piecewise-linear function and each of its dividing points is accompanied with a regression coefficient. However, a simplifying assumption was made regarding independence between these coefficients, in the sense that regression coefficients were a priori independent if associated with different predictors. In this current research we relax the independence assumption between coefficients of different variables. In this case the variance-covariance matrix of the prior distribution is no longer block-diagonal. A method of elicitation for this more complex case is given and it is shown that the resulting covariance matrix is positive-definite. The method was designed to be used with the aid of interactive graphical software. It has been used in practical case studies to quantify the opinions of ecologists and medical doctors (Al-Awadhi and Garthwaite (2006); Garthwaite, Chilcott, Jenkinson, and Tappenden (2008)). The software is being revised and extended further in this research to handle the case of GLM with correlated pairs of covariates

Eliciting prior distributions for extra parameters in some generalized linear models

To elicit an informative prior distribution for a normal linear model or a gamma generalized linear model (GLM), expert opinion must be quantified about both the regression coefficients and the extra parameters of these models. The latter task has attracted comparatively little attention. In this article, we introduce two elicitation methods that aim to complete the prior structure of the normal and gamma GLMs. First, we develop a method of assessing a conjugate prior distribution for the error variance in normal linear models. The method quantifies an expert's opinions through assessments of a median and conditional medians. Second, we propose a novel method for eliciting a lognormal prior distribution for the scale parameter of gamma GLMs. Given the mean value of a gamma distributed response variable, the method is based on conditional quartile assessments. It can also be used to quantify an expert's opinion about the prior distribution for the shape parameter of any gamma random variable, if the mean of the distribution has been elicited or is assumed to be known. In the context of GLMs, the mean value is determined by the regression coefficients. Interactive graphics is the medium through which assessments for the two proposed methods are elicited. Examples illustrating use of the methods are given. Computer programs that implement both methods are available

Eliciting Dirichlet and Connor–Mosimann prior distributions for multinomial models

This paper addresses the task of eliciting an informative prior distribution for multinomial models. We first introduce a method of eliciting univariate beta distributions for the probability of each category, conditional on the probabilities of other categories. Two different forms of multivariate prior are derived from the elicited beta distributions. First, we determine the hyperparameters of a Dirichlet distribution by reconciling the assessed parameters of the univariate beta conditional distributions. Although the Dirichlet distribution is the standard conjugate prior distribution for multinomial models, it is not flexible enough to represent a broad range of prior information. Second, we use the beta distributions to determine the parameters of a Connor–Mosimann distribution, which is a generalization of a Dirichlet distribution and is also a conjugate prior for multinomial models. It has a larger number of parameters than the standard Dirichlet distribution and hence a more flexible structure. The elicitation methods are designed to be used with the aid of interactive graphical user-friendly software

Two-Term Edgeworth Expansions for the Classes of U- and V-statistics

Much effort has been devoted to deriving Edgeworth expansions for various classes of statistics that are asymptotically normally distributed, with derivations tailored to the individual structure of each class. Expansions with smaller error rates are needed for more accurate statistical inference. Two such Edgeworth expansions are derived analytically in this paper. One is a two-term expansion for the standardized U-statistic of order m, m ⩾ 3, with an error rate o(n− 1). The other is an expansion with the same error rate for the distribution of the standardized V-statistic of the same order. In deriving the Edgeworth expansion, we made use of the close connection between the V- and U-statistics, which permits to first derive the needed expansion for the related U-statistic, then extend it to the V-statistic, taking into consideration the estimation of all difference terms between the two statistics