Parametric and Nonparametric Bayesian Models for Ecological Inference in 2 × 2 Tables∗
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Parametric Models
An Introduction to Event History Analysis Oxford Spring School June 18-20, 2007 Day Two: Regression Models for Survival Data Parametric Models We’ll spend the morning introducing regression-like models for survival data, starting with fully parametric (distribution-based) models. These tend to be very widely used in social sciences, although they receive almost no use outside of that (e.g., in biostatistics). A General Framework (That Is, Some Math) Parametric models are continuous-time models, in that they assume a continuous parametric distribution for the probability of failure over time. A general parametric duration model takes as its starting point the hazard: f(t) h(t) = (1) S(t) As we discussed yesterday, the density is the probability of the event (at T ) occurring within some differentiable time-span: Pr(t ≤ T < t + ∆t) f(t) = lim . (2) ∆t→0 ∆t The survival function is equal to one minus the CDF of the density: S(t) = Pr(T ≥ t) Z t = 1 − f(t) dt 0 = 1 − F (t). (3) This means that another way of thinking of the hazard in continuous time is as a conditional limit: Pr(t ≤ T < t + ∆t|T ≥ t) h(t) = lim (4) ∆t→0 ∆t i.e., the conditional probability of the event as ∆t gets arbitrarily small. 1 A General Parametric Likelihood For a set of observations indexed by i, we can distinguish between those which are censored and those which aren’t... • Uncensored observations (Ci = 1) tell us both about the hazard of the event, and the survival of individuals prior to that event. -
Case Study Applications of Statistics in Institutional Research
/-- ; / \ \ CASE STUDY APPLICATIONS OF STATISTICS IN INSTITUTIONAL RESEARCH By MARY ANN COUGHLIN and MARIAN PAGAN( Case Study Applications of Statistics in Institutional Research by Mary Ann Coughlin and Marian Pagano Number Ten Resources in Institutional Research A JOINTPUBLICA TION OF THE ASSOCIATION FOR INSTITUTIONAL RESEARCH AND THE NORTHEASTASSO CIATION FOR INSTITUTIONAL REASEARCH © 1997 Association for Institutional Research 114 Stone Building Florida State University Ta llahassee, Florida 32306-3038 All Rights Reserved No portion of this book may be reproduced by any process, stored in a retrieval system, or transmitted in any form, or by any means, without the express written permission of the publisher. Printed in the United States To order additional copies, contact: AIR 114 Stone Building Florida State University Tallahassee FL 32306-3038 Tel: 904/644-4470 Fax: 904/644-8824 E-Mail: [email protected] Home Page: www.fsu.edul-airlhome.htm ISBN 1-882393-06-6 Table of Contents Acknowledgments •••••••••••••••••.•••••••••.....••••••••••••••••.••. Introduction .•••••••••••..•.•••••...•••••••.....••••.•••...••••••••• 1 Chapter 1: Basic Concepts ..•••••••...••••••...••••••••..••••••....... 3 Characteristics of Variables and Levels of Measurement ...................3 Descriptive Statistics ...............................................8 Probability, Sampling Theory and the Normal Distribution ................ 16 Chapter 2: Comparing Group Means: Are There Real Differences Between Av erage Faculty Salaries Across Departments? •...••••••••.••.•••••• -
Robust Estimation on a Parametric Model Via Testing 3 Estimators
Bernoulli 22(3), 2016, 1617–1670 DOI: 10.3150/15-BEJ706 Robust estimation on a parametric model via testing MATHIEU SART 1Universit´ede Lyon, Universit´eJean Monnet, CNRS UMR 5208 and Institut Camille Jordan, Maison de l’Universit´e, 10 rue Tr´efilerie, CS 82301, 42023 Saint-Etienne Cedex 2, France. E-mail: [email protected] We are interested in the problem of robust parametric estimation of a density from n i.i.d. observations. By using a practice-oriented procedure based on robust tests, we build an estimator for which we establish non-asymptotic risk bounds with respect to the Hellinger distance under mild assumptions on the parametric model. We show that the estimator is robust even for models for which the maximum likelihood method is bound to fail. A numerical simulation illustrates its robustness properties. When the model is true and regular enough, we prove that the estimator is very close to the maximum likelihood one, at least when the number of observations n is large. In particular, it inherits its efficiency. Simulations show that these two estimators are almost equal with large probability, even for small values of n when the model is regular enough and contains the true density. Keywords: parametric estimation; robust estimation; robust tests; T-estimator 1. Introduction We consider n independent and identically distributed random variables X1,...,Xn de- fined on an abstract probability space (Ω, , P) with values in the measure space (X, ,µ). E F We suppose that the distribution of Xi admits a density s with respect to µ and aim at estimating s by using a parametric approach. -
Regression Assumptions in Clinical Psychology Research Practice—A Systematic Review of Common Misconceptions
Regression assumptions in clinical psychology research practice—a systematic review of common misconceptions Anja F. Ernst and Casper J. Albers Heymans Institute for Psychological Research, University of Groningen, Groningen, The Netherlands ABSTRACT Misconceptions about the assumptions behind the standard linear regression model are widespread and dangerous. These lead to using linear regression when inappropriate, and to employing alternative procedures with less statistical power when unnecessary. Our systematic literature review investigated employment and reporting of assumption checks in twelve clinical psychology journals. Findings indicate that normality of the variables themselves, rather than of the errors, was wrongfully held for a necessary assumption in 4% of papers that use regression. Furthermore, 92% of all papers using linear regression were unclear about their assumption checks, violating APA- recommendations. This paper appeals for a heightened awareness for and increased transparency in the reporting of statistical assumption checking. Subjects Psychiatry and Psychology, Statistics Keywords Linear regression, Statistical assumptions, Literature review, Misconceptions about normality INTRODUCTION One of the most frequently employed models to express the influence of several predictors Submitted 18 November 2016 on a continuous outcome variable is the linear regression model: Accepted 17 April 2017 Published 16 May 2017 Yi D β0 Cβ1X1i Cβ2X2i C···CβpXpi C"i: Corresponding author Casper J. Albers, [email protected] This equation predicts the value of a case Yi with values Xji on the independent variables Academic editor Xj (j D 1;:::;p). The standard regression model takes Xj to be measured without error Shane Mueller (cf. Montgomery, Peck & Vining, 2012, p. 71). The various βj slopes are each a measure of Additional Information and association between the respective independent variable Xj and the dependent variable Y. -
A PARTIALLY PARAMETRIC MODEL 1. Introduction a Notable
A PARTIALLY PARAMETRIC MODEL DANIEL J. HENDERSON AND CHRISTOPHER F. PARMETER Abstract. In this paper we propose a model which includes both a known (potentially) nonlinear parametric component and an unknown nonparametric component. This approach is feasible given that we estimate the finite sample parameter vector and the bandwidths simultaneously. We show that our objective function is asymptotically equivalent to the individual objective criteria for the parametric parameter vector and the nonparametric function. In the special case where the parametric component is linear in parameters, our single-step method is asymptotically equivalent to the two-step partially linear model esti- mator in Robinson (1988). Monte Carlo simulations support the asymptotic developments and show impressive finite sample performance. We apply our method to the case of a partially constant elasticity of substitution production function for an unbalanced sample of 134 countries from 1955-2011 and find that the parametric parameters are relatively stable for different nonparametric control variables in the full sample. However, we find substan- tial parameter heterogeneity between developed and developing countries which results in important differences when estimating the elasticity of substitution. 1. Introduction A notable development in applied economic research over the last twenty years is the use of the partially linear regression model (Robinson 1988) to study a variety of phenomena. The enthusiasm for the partially linear model (PLM) is not confined to -
Options for Development of Parametric Probability Distributions for Exposure Factors
EPA/600/R-00/058 July 2000 www.epa.gov/ncea Options for Development of Parametric Probability Distributions for Exposure Factors National Center for Environmental Assessment-Washington Office Office of Research and Development U.S. Environmental Protection Agency Washington, DC 1 Introduction The EPA Exposure Factors Handbook (EFH) was published in August 1997 by the National Center for Environmental Assessment of the Office of Research and Development (EPA/600/P-95/Fa, Fb, and Fc) (U.S. EPA, 1997a). Users of the Handbook have commented on the need to fit distributions to the data in the Handbook to assist them when applying probabilistic methods to exposure assessments. This document summarizes a system of procedures to fit distributions to selected data from the EFH. It is nearly impossible to provide a single distribution that would serve all purposes. It is the responsibility of the assessor to determine if the data used to derive the distributions presented in this report are representative of the population to be assessed. The system is based on EPA’s Guiding Principles for Monte Carlo Analysis (U.S. EPA, 1997b). Three factors—drinking water, population mobility, and inhalation rates—are used as test cases. A plan for fitting distributions to other factors is currently under development. EFH data summaries are taken from many different journal publications, technical reports, and databases. Only EFH tabulated data summaries were analyzed, and no attempt was made to obtain raw data from investigators. Since a variety of summaries are found in the EFH, it is somewhat of a challenge to define a comprehensive data analysis strategy that will cover all cases. -
Exponential Families and Theoretical Inference
EXPONENTIAL FAMILIES AND THEORETICAL INFERENCE Bent Jørgensen Rodrigo Labouriau August, 2012 ii Contents Preface vii Preface to the Portuguese edition ix 1 Exponential families 1 1.1 Definitions . 1 1.2 Analytical properties of the Laplace transform . 11 1.3 Estimation in regular exponential families . 14 1.4 Marginal and conditional distributions . 17 1.5 Parametrizations . 20 1.6 The multivariate normal distribution . 22 1.7 Asymptotic theory . 23 1.7.1 Estimation . 25 1.7.2 Hypothesis testing . 30 1.8 Problems . 36 2 Sufficiency and ancillarity 47 2.1 Sufficiency . 47 2.1.1 Three lemmas . 48 2.1.2 Definitions . 49 2.1.3 The case of equivalent measures . 50 2.1.4 The general case . 53 2.1.5 Completeness . 56 2.1.6 A result on separable σ-algebras . 59 2.1.7 Sufficiency of the likelihood function . 60 2.1.8 Sufficiency and exponential families . 62 2.2 Ancillarity . 63 2.2.1 Definitions . 63 2.2.2 Basu's Theorem . 65 2.3 First-order ancillarity . 67 2.3.1 Examples . 67 2.3.2 Main results . 69 iii iv CONTENTS 2.4 Problems . 71 3 Inferential separation 77 3.1 Introduction . 77 3.1.1 S-ancillarity . 81 3.1.2 The nonformation principle . 83 3.1.3 Discussion . 86 3.2 S-nonformation . 91 3.2.1 Definition . 91 3.2.2 S-nonformation in exponential families . 96 3.3 G-nonformation . 99 3.3.1 Transformation models . 99 3.3.2 Definition of G-nonformation . 103 3.3.3 Cox's proportional risks model . -
A Causation Coefficient and Taxonomy of Correlation/Causation Relationships
A causation coefficient and taxonomy of correlation/causation relationships Joshua Brulé∗ Abstract This paper introduces a causation coefficient which is defined in terms of probabilistic causal models. This coefficient is suggested as the natural causal analogue of the Pearson correlation coefficient and permits comparing causation and correlation to each other in a simple, yet rigorous manner. Together, these coefficients provide a natural way to classify the possible correlation/causation relationships that can occur in practice and examples of each relationship are provided. In addition, the typical relationship between correlation and causation is analyzed to provide insight into why correlation and causation are often conflated. Finally, example calculations of the causation coefficient are shown on a real data set. Introduction The maxim, “Correlation is not causation”, is an important warning to analysts, but provides very little information about what causation is and how it relates to correlation. This has prompted other attempts at summarizing the relationship. For example, Tufte [1] suggests either, “Observed covariation is necessary but not sufficient for causality”, which is demonstrably false arXiv:1708.05069v1 [stat.ME] 5 Aug 2017 or, “Correlation is not causation but it is a hint”, which is correct, but still underspecified. In what sense is correlation a ‘hint’ to causation? Correlation is well understood and precisely defined. Generally speaking, correlation is any statistical relationship involving dependence, i.e. the ran- dom variables are not independent. More specifically, correlation can refer ∗Department of Computer Science, University of Maryland, College Park. [email protected] 1 to a descriptive statistic that summarizes the nature of the dependence. -
A Primer on the Exponential Family of Distributions
A Primer on the Exponential Family of Distributions David R. Clark, FCAS, MAAA, and Charles A. Thayer 117 A PRIMER ON THE EXPONENTIAL FAMILY OF DISTRIBUTIONS David R. Clark and Charles A. Thayer 2004 Call Paper Program on Generalized Linear Models Abstract Generahzed Linear Model (GLM) theory represents a significant advance beyond linear regression theor,], specifically in expanding the choice of probability distributions from the Normal to the Natural Exponential Famdy. This Primer is intended for GLM users seeking a hand)' reference on the model's d]smbutional assumptions. The Exponential Faintly of D,smbutions is introduced, with an emphasis on variance structures that may be suitable for aggregate loss models m property casualty insurance. 118 A PRIMER ON THE EXPONENTIAL FAMILY OF DISTRIBUTIONS INTRODUCTION Generalized Linear Model (GLM) theory is a signtficant advance beyond linear regression theory. A major part of this advance comes from allowmg a broader famdy of distributions to be used for the error term, rather than just the Normal (Gausstan) distributton as required m hnear regression. More specifically, GLM allows the user to select a distribution from the Exponentzal Family, which gives much greater flexibility in specifying the vanance structure of the variable being forecast (the "'response variable"). For insurance apphcations, this is a big step towards more realistic modeling of loss distributions, while preserving the advantages of regresston theory such as the ability to calculate standard errors for estimated parameters. The Exponentml family also includes several d~screte distributions that are attractive candtdates for modehng clatm counts and other events, but such models will not be considered here The purpose of this Primer is to give the practicmg actuary a basic introduction to the Exponential Family of distributions, so that GLM models can be designed to best approximate the behavior of the insurance phenomenon. -
2 May 2021 Parametric Bootstrapping in a Generalized Extreme Value
Parametric bootstrapping in a generalized extreme value regression model for binary response DIOP Aba∗ E-mail: [email protected] Equipe de Recherche en Statistique et Mod`eles Al´eatoires Universit´eAlioune Diop, Bambey, S´en´egal DEME El Hadji E-mail: [email protected] Laboratoire d’Etude et de Recherche en Statistique et D´eveloppement Universit´eGaston Berger, Saint-Louis, S´en´egal Abstract Generalized extreme value (GEV) regression is often more adapted when we in- vestigate a relationship between a binary response variable Y which represents a rare event and potentiel predictors X. In particular, we use the quantile function of the GEV distribution as link function. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, test of hypothesis) to sample es- timates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an estimator by measuring those properties when sampling from an approximating distribution. In this paper, we fitted the generalized extreme value regression model, then we performed parametric bootstrap method for testing hupthesis, estimating confidence interval of parameters for generalized extreme value regression model and a real data application. arXiv:2105.00489v1 [stat.ME] 2 May 2021 Keywords: generalized extreme value, parametric bootstrap, confidence interval, test of hypothesis, Dengue. ∗Corresponding author 1 1 Introduction Classical methods of statistical inference do not provide correct answers to all the concrete problems the user may have. They are only valid under some specific application conditions (e.g. normal distribution of populations, independence of samples etc.). -
The Quest for Statistical Significance: Ignorance, Bias and Malpractice of Research Practitioners Joshua Abah
The Quest for Statistical Significance: Ignorance, Bias and Malpractice of Research Practitioners Joshua Abah To cite this version: Joshua Abah. The Quest for Statistical Significance: Ignorance, Bias and Malpractice of Research Practitioners. International Journal of Research & Review (www.ijrrjournal.com), 2018, 5 (3), pp.112- 129. hal-01758493 HAL Id: hal-01758493 https://hal.archives-ouvertes.fr/hal-01758493 Submitted on 4 Apr 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - ShareAlike| 4.0 International License International Journal of Research and Review www.gkpublication.in E-ISSN: 2349-9788; P-ISSN: 2454-2237 Review Article The Quest for Statistical Significance: Ignorance, Bias and Malpractice of Research Practitioners Joshua Abah Abah Department of Science Education University of Agriculture, Makurdi, Nigeria ABSTRACT There is a growing body of evidence on the prevalence of ignorance, biases and malpractice among researchers which questions the authenticity, validity and integrity of the knowledge been propagated in professional circles. The push for academic relevance and career advancement have driven some research practitioners into committing gross misconduct in the form of innocent ignorance, sloppiness, malicious intent and outright fraud. -
Parametric Versus Semi and Nonparametric Regression Models
Parametric versus Semi and Nonparametric Regression Models Hamdy F. F. Mahmoud1;2 1 Department of Statistics, Virginia Tech, Blacksburg VA, USA. 2 Department of Statistics, Mathematics and Insurance, Assiut University, Egypt. Abstract Three types of regression models researchers need to be familiar with and know the requirements of each: parametric, semiparametric and nonparametric regression models. The type of modeling used is based on how much information are available about the form of the relationship between response variable and explanatory variables, and the random error distribution. In this article, differences between models, common methods of estimation, robust estimation, and applications are introduced. The R code for all the graphs and analyses presented here, in this article, is available in the Appendix. Keywords: Parametric, semiparametric, nonparametric, single index model, robust estimation, kernel regression, spline smoothing. 1 Introduction The aim of this paper is to answer many questions researchers may have when they fit their data, such as what are the differences between parametric, semi/nonparamertic models? which one should be used to model a real data set? what estimation method should be used?, and which type of modeling is better? These questions and others are addressed by examples. arXiv:1906.10221v1 [stat.ME] 24 Jun 2019 Assume that a researcher collected data about Y, a response variable, and explanatory variables, X = (x1; x2; : : : ; xk). The model that describes the relationship between Y and X can be written as Y = f(X; β) + , (1) where β is a vector of k parameters, is an error term whose distribution may or may not be normal, f(·) is some function that describe the relationship between Y and X.