Simulation and Analysis of Bivariate Beta Distribution for Interval Response Variables Francois Mercier and Amy Racine

Total Page:16

File Type:pdf, Size:1020Kb

Simulation and Analysis of Bivariate Beta Distribution for Interval Response Variables Francois Mercier and Amy Racine Simulation and analysis of bivariate Beta distribution for interval response variables Francois Mercier and Amy Racine M&S-CIS, Novartis Pharma AG [email: [email protected]] Objective Results 1/ Data generation In certain clinical trials, the response variable is restricted into an interval (e.g. visual analogue scale VAS). The Beta distribution parameters c1, c2 and c3 can be derived from the mean and standard deviation of Y1 This response variable may be measured repeatedly during the trial. A natural distribution for such a variable is as follows (note: c4 = (c1+c2) –c3): a multivariate Beta distribution. For clinical trial simulations, it is therefore necessary to generate data from é (1-mˆ )ù c = mˆ 2 ´ Y 1 - mˆ such a distribution in order to perform inference and/or predictions. In the context of the clinical trial described earlier, depending on 1 ê Y1 2 ú Y1 ë sˆY1 û We propose hereafter to address the particular case of the bivariate Beta (bivBeta) distribution and present the the sample mean at time 1 and time 2, sample variance and æ 1 ö approach (and corresponding code) we use to generate and analyze this type of data. expected coefficient of correlation, we would obtain the following c = c ´ç -1÷ Beta parameters estimates, in the two below examples (Table 1 2 1 ç ˆ ÷ è m Y1 ø and Figure 2): c = c + c ´mˆ Methods 3 ( 1 2 ) Y 2 Ex. m =0.4, m =0.5, c = 2 a = 2.357 1 2 1 0 1 s =0.2, r=0.7 c = 3 a = 0.143 1 2 1 c = 2.5 a = 0.642 We assume a clinical trial in which a response variable Y, characterized by an interval distribution, is evaluated 3 2 Table 1: Elicitation of alpha values in c = 2.5 a = 1.857 at both early (Y1) and late stage (Y2). The clinical trial may have been designed in such a way that at the 4 3 two specific examples Ex. m =0.6, m =0.7, c = 3 a = 1.161 1 2 1 0 occasion of an interim analysis, one wants to predict Y2, based on the available Y1 data for decision making 2 s =0.2, r=0.5 c = 2 a = 0.339 1 2 1 (e.g. futility or success of the trial). c = 3.5 a = 0.839 3 2 Projection à bivBeta c = 1.5 a = 2.661 Simulation of time trends in a random variable is 4 3 frequently done within a multivariate framework where the different variables represent the state of Y1~Beta() Y2~Beta() the random variable at different point in time. FPFV Interim Final Analysis Analysis With random variables having possibly skewed interval distributions (see for instance, Figure 1), it is not recommended to use e.g.a truncated multivariate normal distribution for simulation. Rather, it is recommends to use multivariate Beta distributions. The flexible Beta distribution is widely used in life 15 sciences to describe the probability density distribution of proportions or relative frequencies of a random univariate variable. 10 Generation of univariate Beta-distributed random - variable is straightforward, but generating pairs of Figure 2: Scatterplots and densityplots of bivariate Beta distributed variables. Red line represents x=y and Frequency correlated Beta-distributed random variables is more 5 cross represents median Y1 and median Y2. complex since there is no natural multivariate extension of univariate Beta distribution (Johnson 2/ Data analysis and Kotz, 1976 [1]). To analyze the bivBeta data, a Bayesian approach was used considering the distribution of the endpoint as 0 bivariate Normal, after logittransformation. As an illustration, we simulated a simple clinical trial scenario -5 0 5 10 Following Catalani(2002)[2] it is proposed to use a involving one placebo group and one active treatment group. The simulated data corresponded to 20 Dirichlet distribution to simulate outcomes from a Score patients per group, with Beta distributed data at two occasions T1 and T2. The output of this analysis is bivBetadistribution. provided hereafter: Figure 1: Distribution of a score range from -10 to 50, as an illustration of interval response variable. 1/ Introduction of a shared random variable The marginals in a Dirichlet distribution are Beta variables Let {X1, X2, X3} ~ Dirichlet(3, a 0, a 1, a 2, a 3), with Xi = Zi/(Z1+Z2+Z3), i = 1, 2, 3 where Z i’s, are independent gamma variables Zi ~ G(shape=a j, scale=1) A popular technique for generation of correlated random variables is to introduce a shared random variable: Define Y1 = X1+X3 Y1 ~ Be(a 1+a 3, a 0+a 2) Y2 = X2+X3 Y2 ~ Be(a 2+a 3, a 0+a 1) Set g = (a +a +a +a ),then we can derive the correlation coefficient([2]) 0 1 2 3 In a second example (not reported here) we simulated a scenario where Y1 (response at T1) were complete but Y2 (response at T2) r(Y , Y ) = (-a a + a a ) / sqrt((a +a )(a +a )(a +a )(a +a ))’ 1 2 1 2 0 3 1 3 0 2 2 3 0 1 were incomplete. This scenario could, for instance, correspond to the situation observed at an interim point during a clinical trial. 2/ Elicitation of the alphas Using the same data set as described above but replacing 50% of Sampling data from a bivariatedensity with beta marginals, with the original data at T2 by missing values, we could fit the same parameters, c , c and c , c , and r, positive correlation coefficient, 1 2 3 4 model and obtain similar parameter estimates (but more variability in the meanT2 estimations) [Results available on demand]. Set c1 = a 1+a 3 c2 = a 0+a 2 c3 = a 2+a 3 and c4 = a 0+a 1 = c1+c2-c3 Conclusion which implies c1+c2>c3 and r = (-a 1a 2+a 0a 3)/sqrt(c1c2c3(c1+c2-c3))’ We present a method to generate bivariate Beta distributed data, where: Assuming r>0, we solve for {a , a , a , a }, as functions of {c , c , c , c , r} 0 1 2 3 1 2 3 4 1. Beta variables are derived from Dirichlet distribution, and we obtain: 2. Dirichlet distribution parameters are obtained in case of correlated variables. a = r* sqrt(c c c (c +c -c ))+ c c / (c +c ) 3 1 2 3 1 2 3 1 3 1 2 Although it was not possible to fit models for bivariate Beta variables directly (in WinBUGS), we could fit models considering these variables as bivariateNormal, after appropriate transformation. it follows, a = c -a 1 1 3 References a 2 = c3 -a 3 a 0 = c2 -c3+a 3 [1] Johnson, N.L., and Kotz, S. (1976) Distributions in statistics: Continuous MultivariateDistributions. Wiley, New York. [2] Catalani, M. (2002) Sample from a couple of positively correlated variates. http://arxiv.org/abs/math/0209090v1 PAGE (Population Approach Group Europe) Acknowledgements: Marseille (F), 16-18 June 2008 - Billy Amzal - Marina Savelieva.
Recommended publications
  • The Exponential Family 1 Definition
    The Exponential Family David M. Blei Columbia University November 9, 2016 The exponential family is a class of densities (Brown, 1986). It encompasses many familiar forms of likelihoods, such as the Gaussian, Poisson, multinomial, and Bernoulli. It also encompasses their conjugate priors, such as the Gamma, Dirichlet, and beta. 1 Definition A probability density in the exponential family has this form p.x / h.x/ exp >t.x/ a./ ; (1) j D f g where is the natural parameter; t.x/ are sufficient statistics; h.x/ is the “base measure;” a./ is the log normalizer. Examples of exponential family distributions include Gaussian, gamma, Poisson, Bernoulli, multinomial, Markov models. Examples of distributions that are not in this family include student-t, mixtures, and hidden Markov models. (We are considering these families as distributions of data. The latent variables are implicitly marginalized out.) The statistic t.x/ is called sufficient because the probability as a function of only depends on x through t.x/. The exponential family has fundamental connections to the world of graphical models (Wainwright and Jordan, 2008). For our purposes, we’ll use exponential 1 families as components in directed graphical models, e.g., in the mixtures of Gaussians. The log normalizer ensures that the density integrates to 1, Z a./ log h.x/ exp >t.x/ d.x/ (2) D f g This is the negative logarithm of the normalizing constant. The function h.x/ can be a source of confusion. One way to interpret h.x/ is the (unnormalized) distribution of x when 0. It might involve statistics of x that D are not in t.x/, i.e., that do not vary with the natural parameter.
    [Show full text]
  • A Skew Extension of the T-Distribution, with Applications
    J. R. Statist. Soc. B (2003) 65, Part 1, pp. 159–174 A skew extension of the t-distribution, with applications M. C. Jones The Open University, Milton Keynes, UK and M. J. Faddy University of Birmingham, UK [Received March 2000. Final revision July 2002] Summary. A tractable skew t-distribution on the real line is proposed.This includes as a special case the symmetric t-distribution, and otherwise provides skew extensions thereof.The distribu- tion is potentially useful both for modelling data and in robustness studies. Properties of the new distribution are presented. Likelihood inference for the parameters of this skew t-distribution is developed. Application is made to two data modelling examples. Keywords: Beta distribution; Likelihood inference; Robustness; Skewness; Student’s t-distribution 1. Introduction Student’s t-distribution occurs frequently in statistics. Its usual derivation and use is as the sam- pling distribution of certain test statistics under normality, but increasingly the t-distribution is being used in both frequentist and Bayesian statistics as a heavy-tailed alternative to the nor- mal distribution when robustness to possible outliers is a concern. See Lange et al. (1989) and Gelman et al. (1995) and references therein. It will often be useful to consider a further alternative to the normal or t-distribution which is both heavy tailed and skew. To this end, we propose a family of distributions which includes the symmetric t-distributions as special cases, and also includes extensions of the t-distribution, still taking values on the whole real line, with non-zero skewness. Let a>0 and b>0be parameters.
    [Show full text]
  • On a Problem Connected with Beta and Gamma Distributions by R
    ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i) 1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0, (1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0<w<l, Ío T(a)r(a) 1 for w > 1. Now we can state the converse problem as follows : Let X and Y be two independently and identically distributed random variables having a common distribution function Fix). Suppose that W = Xj{X + Y) has a Beta distribution of the form (1.2). Then the question is whether £(x) is necessarily a Gamma distribution of the form (1.1). This problem was posed by Mauldon in [9]. He also showed that the converse problem is not true in general and constructed an example of a non-Gamma distribution with this property using the solution of an integral equation which was studied by Goodspeed in [2]. In the present paper we carry out a systematic investigation of this problem. In §2, we derive some general properties possessed by this class of distribution laws Fix).
    [Show full text]
  • A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess
    S S symmetry Article A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess Guillermo Martínez-Flórez 1, Víctor Leiva 2,* , Emilio Gómez-Déniz 3 and Carolina Marchant 4 1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 14014, Colombia; [email protected] 2 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile 3 Facultad de Economía, Empresa y Turismo, Universidad de Las Palmas de Gran Canaria and TIDES Institute, 35001 Canarias, Spain; [email protected] 4 Facultad de Ciencias Básicas, Universidad Católica del Maule, 3466706 Talca, Chile; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 June 2020; Accepted: 19 August 2020; Published: 1 September 2020 Abstract: In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Keywords: beta distribution; centered skew-normal distribution; maximum-likelihood methods; Monte Carlo simulations; proportions; R software; rates; zero/one inflated data 1.
    [Show full text]
  • Lecture 2 — September 24 2.1 Recap 2.2 Exponential Families
    STATS 300A: Theory of Statistics Fall 2015 Lecture 2 | September 24 Lecturer: Lester Mackey Scribe: Stephen Bates and Andy Tsao 2.1 Recap Last time, we set out on a quest to develop optimal inference procedures and, along the way, encountered an important pair of assertions: not all data is relevant, and irrelevant data can only increase risk and hence impair performance. This led us to introduce a notion of lossless data compression (sufficiency): T is sufficient for P with X ∼ Pθ 2 P if X j T (X) is independent of θ. How far can we take this idea? At what point does compression impair performance? These are questions of optimal data reduction. While we will develop general answers to these questions in this lecture and the next, we can often say much more in the context of specific modeling choices. With this in mind, let's consider an especially important class of models known as the exponential family models. 2.2 Exponential Families Definition 1. The model fPθ : θ 2 Ωg forms an s-dimensional exponential family if each Pθ has density of the form: s ! X p(x; θ) = exp ηi(θ)Ti(x) − B(θ) h(x) i=1 • ηi(θ) 2 R are called the natural parameters. • Ti(x) 2 R are its sufficient statistics, which follows from NFFC. • B(θ) is the log-partition function because it is the logarithm of a normalization factor: s ! ! Z X B(θ) = log exp ηi(θ)Ti(x) h(x)dµ(x) 2 R i=1 • h(x) 2 R: base measure.
    [Show full text]
  • Distributions (3) © 2008 Winton 2 VIII
    © 2008 Winton 1 Distributions (3) © onntiW8 020 2 VIII. Lognormal Distribution • Data points t are said to be lognormally distributed if the natural logarithms, ln(t), of these points are normally distributed with mean μ ion and standard deviatσ – If the normal distribution is sampled to get points rsample, then the points ersample constitute sample values from the lognormal distribution • The pdf or the lognormal distribution is given byf - ln(x)()μ 2 - 1 2 f(x)= e ⋅ 2 σ >(x 0) σx 2 π because ∞ - ln(x)()μ 2 - ∞ -() t μ-2 1 1 2 1 2 ∫ e⋅ 2eσ dx = ∫ dt ⋅ 2σ (where= t ln(x)) 0 xσ2 π 0σ2 π is the pdf for the normal distribution © 2008 Winton 3 Mean and Variance for Lognormal • It can be shown that in terms of μ and σ σ2 μ + E(X)= 2 e 2⋅μ σ + 2 σ2 Var(X)= e( ⋅ ) e - 1 • If the mean E and variance V for the lognormal distribution are given, then the corresponding μ and σ2 for the normal distribution are given by – σ2 = log(1+V/E2) – μ = log(E) - σ2/2 • The lognormal distribution has been used in reliability models for time until failure and for stock price distributions – The shape is similar to that of the Gamma distribution and the Weibull distribution for the case α > 2, but the peak is less towards 0 © 2008 Winton 4 Graph of Lognormal pdf f(x) 0.5 0.45 μ=4, σ=1 0.4 0.35 μ=4, σ=2 0.3 μ=4, σ=3 0.25 μ=4, σ=4 0.2 0.15 0.1 0.05 0 x 0510 μ and σ are the lognormal’s mean and std dev, not those of the associated normal © onntiW8 020 5 IX.
    [Show full text]
  • Theoretical Properties of the Weighted Feller-Pareto Distributions." Asian Journal of Mathematics and Applications, 2014: 1-12
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Georgia Southern University: Digital Commons@Georgia Southern Georgia Southern University Digital Commons@Georgia Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of 2014 Theoretical Properties of the Weighted Feller- Pareto Distributions Oluseyi Odubote Georgia Southern University Broderick O. Oluyede Georgia Southern University, [email protected] Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/math-sci-facpubs Part of the Mathematics Commons Recommended Citation Odubote, Oluseyi, Broderick O. Oluyede. 2014. "Theoretical Properties of the Weighted Feller-Pareto Distributions." Asian Journal of Mathematics and Applications, 2014: 1-12. source: http://scienceasia.asia/index.php/ama/article/view/173/ https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/307 This article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 2014, Article ID ama0173, 12 pages ISSN 2307-7743 http://scienceasia.asia THEORETICAL PROPERTIES OF THE WEIGHTED FELLER-PARETO AND RELATED DISTRIBUTIONS OLUSEYI ODUBOTE AND BRODERICK O. OLUYEDE Abstract. In this paper, for the first time, a new six-parameter class of distributions called weighted Feller-Pareto (WFP) and related family of distributions is proposed. This new class of distributions contains several other Pareto-type distributions such as length-biased (LB) Pareto, weighted Pareto (WP I, II, III, and IV), and Pareto (P I, II, III, and IV) distributions as special cases.
    [Show full text]
  • Field Guide to Continuous Probability Distributions
    Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations.
    [Show full text]
  • Bayesian Analysis 1 Introduction
    Bayesian analysis Class Notes Manuel Arellano March 8, 2016 1 Introduction Bayesian methods have traditionally had limited influence in empirical economics, but they have become increasingly important with the popularization of computer-intensive stochastic simulation algorithms in the 1990s. This is particularly so in macroeconomics, where applications of Bayesian inference include vector autoregressions (VARs) and dynamic stochastic general equilibrium (DSGE) models. Bayesian approaches are also attractive in models with many parameters, such as panel data models with individual heterogeneity and flexible nonlinear regression models. Examples include discrete choice models of consumer demand in the fields of industrial organization and marketing. An empirical study uses data to learn about quantities of interest (parameters). A likelihood function or some of its features specify the information in the data about those quantities. Such specification typically involves the use of a priori information in the form of parametric or functional restrictions. In the Bayesian approach to inference, one not only assigns a probability measure to the sample space but also to the parameter space. Specifying a probability distribution over potential parameter values is the conventional way of modelling uncertainty in decision-making, and offers a systematic way of incorporating uncertain prior information into statistical procedures. Outline The following section introduces the Bayesian way of combining a prior distribution with the likelihood of the data to generate point and interval estimates. This is followed by some comments on the specification of prior distributions. Next we turn to discuss asymptotic approximations; the main result is that in regular cases there is a large-sample equivalence between Bayesian probability statements and frequentist confidence statements.
    [Show full text]
  • On Families of Generalized Pareto Distributions: Properties and Applications
    Journal of Data Science 377-396 , DOI: 10.6339/JDS.201804_16(2).0008 On Families of Generalized Pareto Distributions: Properties and Applications D. Hamed1 and F. Famoye2 and C. Lee2 1 Department of Mathematics, Winthrop University, Rock Hill, SC, USA; 2Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA In this paper, we introduce some new families of generalized Pareto distributions using the T-R{Y} framework. These families of distributions are named T-Pareto{Y} families, and they arise from the quantile functions of exponential, log-logistic, logistic, extreme value, Cauchy and Weibull distributions. The shapes of these T-Pareto families can be unimodal or bimodal, skewed to the left or skewed to the right with heavy tail. Some general properties of the T-Pareto{Y} family are investigated and these include the moments, modes, mean deviations from the mean and from the median, and Shannon entropy. Several new generalized Pareto distributions are also discussed. Four real data sets from engineering, biomedical and social science are analyzed to demonstrate the flexibility and usefulness of the T-Pareto{Y} families of distributions. Key words: Shannon entropy; quantile function; moment; T-X family. 1. Introduction The Pareto distribution is named after the well-known Italian-born Swiss sociologist and economist Vilfredo Pareto (1848-1923). Pareto [1] defined Pareto’s Law, which can be stated as N Axa , where N represents the number of persons having income x in a population. Pareto distribution is commonly used in modelling heavy tailed distributions, including but not limited to income, insurance and city size populations.
    [Show full text]
  • Probability Density Function of Non-Reactive Solute Concentration in Heterogeneous Porous Formations ⁎ Alberto Bellin A, , Daniele Tonina B,C
    Journal of Contaminant Hydrology 94 (2007) 109–125 www.elsevier.com/locate/jconhyd Probability density function of non-reactive solute concentration in heterogeneous porous formations ⁎ Alberto Bellin a, , Daniele Tonina b,c a Dipartimento di Ingegneria Civile e Ambientale, Università di Trento, via Mesiano 77, I-38050 Trento, Italy b Earth and Planetary Science Department, University of California, Berkeley, USA c USFS, Rocky Mountain Research Station 322 East Front Street, Suite 401 Boise, ID, 83702, USA Received 20 June 2006; received in revised form 15 May 2007; accepted 17 May 2007 Available online 12 June 2007 Abstract Available models of solute transport in heterogeneous formations lack in providing complete characterization of the predicted concentration. This is a serious drawback especially in risk analysis where confidence intervals and probability of exceeding threshold values are required. Our contribution to fill this gap of knowledge is a probability distribution model for the local concentration of conservative tracers migrating in heterogeneous aquifers. Our model accounts for dilution, mechanical mixing within the sampling volume and spreading due to formation heterogeneity. It is developed by modeling local concentration dynamics with an Ito Stochastic Differential Equation (SDE) that under the hypothesis of statistical stationarity leads to the Beta probability distribution function (pdf) for the solute concentration. This model shows large flexibility in capturing the smoothing effect of the sampling volume and the associated reduction of the probability of exceeding large concentrations. Furthermore, it is fully characterized by the first two moments of the solute concentration, and these are the same pieces of information required for standard geostatistical techniques employing Normal or Log-Normal distributions.
    [Show full text]
  • A Multivariate Beta-Binomial Model Which Allows to Conduct Bayesian Inference Regarding Θ Or Transformations Thereof
    SIMULTANEOUS INFERENCE FOR MULTIPLE PROPORTIONS:AMULTIVARIATE BETA-BINOMIAL MODEL Max Westphal∗ Institute for Statistics University of Bremen Bremen, Germany March 20, 2020 ABSTRACT Statistical inference in high-dimensional settings is challenging when standard unregular- ized methods are employed. In this work, we focus on the case of multiple correlated proportions for which we develop a Bayesian inference framework. For this purpose, we construct an m-dimensional Beta distribution from a 2m-dimensional Dirichlet distribution, building on work by Olkin and Trikalinos(2015). This readily leads to a multivariate Beta- binomial model for which simple update rules from the common Dirichlet-multinomial model can be adopted. From the frequentist perspective, this approach amounts to adding pseudo-observations to the data and allows a joint shrinkage estimation of mean vector and covariance matrix. For higher dimensions (m > 10), the extensive model based on 2m parameters starts to become numerically infeasible. To counter this problem, we utilize a reduced parametrisation which has only 1 + m(m + 1)=2 parameters describing first and second order moments. A copula model can then be used to approximate the (posterior) multivariate Beta distribution. A natural inference goal is the construction of multivariate credible regions. The properties of different credible regions are assessed in a simulation arXiv:1911.00098v4 [stat.ME] 20 Mar 2020 study in the context of investigating the accuracy of multiple binary classifiers. It is shown that the extensive and copula approach lead to a (Bayes) coverage probability very close to the target level. In this regard, they outperform credible regions based on a normal approxi- mation of the posterior distribution, in particular for small sample sizes.
    [Show full text]