Generalized Linear Models
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Economic Choices
ECONOMIC CHOICES Daniel McFadden* This Nobel lecture discusses the microeconometric analysis of choice behavior of consumers who face discrete economic alternatives. Before the 1960's, economists used consumer theory mostly as a logical tool, to explore conceptually the properties of alternative market organizations and economic policies. When the theory was applied empirically, it was to market-level or national-accounts-level data. In these applications, the theory was usually developed in terms of a representative agent, with market-level behavior given by the representative agent’s behavior writ large. When observations deviated from those implied by the representative agent theory, these differences were swept into an additive disturbance and attributed to data measurement errors, rather than to unobserved factors within or across individual agents. In statistical language, traditional consumer theory placed structural restrictions on mean behavior, but the distribution of responses about their mean was not tied to the theory. In the 1960's, rapidly increasing availability of survey data on individual behavior, and the advent of digital computers that could analyze these data, focused attention on the variations in demand across individuals. It became important to explain and model these variations as part of consumer theory, rather than as ad hoc disturbances. This was particularly obvious for discrete choices, such as transportation mode or occupation. The solution to this problem has led to the tools we have today for microeconometric analysis of choice behavior. I will first give a brief history of the development of this subject, and place my own contributions in context. After that, I will discuss in some detail more recent developments in the economic theory of choice, and modifications to this theory that are being forced by experimental evidence from cognitive psychology. -
An Introduction to Poisson Regression Russ Lavery, K&L Consulting Services, King of Prussia, PA, U.S.A
NESUG 2010 Statistics and Analysis An Animated Guide: An Introduction To Poisson Regression Russ Lavery, K&L Consulting Services, King of Prussia, PA, U.S.A. ABSTRACT: This paper will be a brief introduction to Poisson regression (theory, steps to be followed, complications and interpretation) via a worked example. It is hoped that this will increase motivation towards learning this useful statistical technique. INTRODUCTION: Poisson regression is available in SAS through the GENMOD procedure (generalized modeling). It is appropriate when: 1) the process that generates the conditional Y distributions would, theoretically, be expected to be a Poisson random process and 2) when there is no evidence of overdispersion and 3) when the mean of the marginal distribution is less than ten (preferably less than five and ideally close to one). THE POISSON DISTRIBUTION: The Poison distribution is a discrete Percent of observations where the random variable X is expected distribution and is appropriate for to have the value x, given that the Poisson distribution has a mean modeling counts of observations. of λ= P(X=x, λ ) = (e - λ * λ X) / X! Counts are observed cases, like the 0.4 count of measles cases in cities. You λ can simply model counts if all data 0.35 were collected in the same measuring 0.3 unit (e.g. the same number of days or 0.3 0.5 0.8 same number of square feet). 0.25 λ 1 0.2 3 You can use the Poisson Distribution = 5 for modeling rates (rates are counts 0.15 20 per unit) if the units of collection were 8 different. -
A Generalized Linear Model for Principal Component Analysis of Binary Data
A Generalized Linear Model for Principal Component Analysis of Binary Data Andrew I. Schein Lawrence K. Saul Lyle H. Ungar Department of Computer and Information Science University of Pennsylvania Moore School Building 200 South 33rd Street Philadelphia, PA 19104-6389 {ais,lsaul,ungar}@cis.upenn.edu Abstract they are not generally appropriate for other data types. Recently, Collins et al.[5] derived generalized criteria We investigate a generalized linear model for for dimensionality reduction by appealing to proper- dimensionality reduction of binary data. The ties of distributions in the exponential family. In their model is related to principal component anal- framework, the conventional PCA of real-valued data ysis (PCA) in the same way that logistic re- emerges naturally from assuming a Gaussian distribu- gression is related to linear regression. Thus tion over a set of observations, while generalized ver- we refer to the model as logistic PCA. In this sions of PCA for binary and nonnegative data emerge paper, we derive an alternating least squares respectively by substituting the Bernoulli and Pois- method to estimate the basis vectors and gen- son distributions for the Gaussian. For binary data, eralized linear coefficients of the logistic PCA the generalized model's relationship to PCA is anal- model. The resulting updates have a simple ogous to the relationship between logistic and linear closed form and are guaranteed at each iter- regression[12]. In particular, the model exploits the ation to improve the model's likelihood. We log-odds as the natural parameter of the Bernoulli dis- evaluate the performance of logistic PCA|as tribution and the logistic function as its canonical link. -
Generalized Linear Models (Glms)
San Jos´eState University Math 261A: Regression Theory & Methods Generalized Linear Models (GLMs) Dr. Guangliang Chen This lecture is based on the following textbook sections: • Chapter 13: 13.1 – 13.3 Outline of this presentation: • What is a GLM? • Logistic regression • Poisson regression Generalized Linear Models (GLMs) What is a GLM? In ordinary linear regression, we assume that the response is a linear function of the regressors plus Gaussian noise: 0 2 y = β0 + β1x1 + ··· + βkxk + ∼ N(x β, σ ) | {z } |{z} linear form x0β N(0,σ2) noise The model can be reformulate in terms of • distribution of the response: y | x ∼ N(µ, σ2), and • dependence of the mean on the predictors: µ = E(y | x) = x0β Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University3/24 Generalized Linear Models (GLMs) beta=(1,2) 5 4 3 β0 + β1x b y 2 y 1 0 −1 0.0 0.2 0.4 0.6 0.8 1.0 x x Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University4/24 Generalized Linear Models (GLMs) Generalized linear models (GLM) extend linear regression by allowing the response variable to have • a general distribution (with mean µ = E(y | x)) and • a mean that depends on the predictors through a link function g: That is, g(µ) = β0x or equivalently, µ = g−1(β0x) Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University5/24 Generalized Linear Models (GLMs) In GLM, the response is typically assumed to have a distribution in the exponential family, which is a large class of probability distributions that have pdfs of the form f(x | θ) = a(x)b(θ) exp(c(θ) · T (x)), including • Normal - ordinary linear regression • Bernoulli - Logistic regression, modeling binary data • Binomial - Multinomial logistic regression, modeling general cate- gorical data • Poisson - Poisson regression, modeling count data • Exponential, Gamma - survival analysis Dr. -
Logistic Regression, Dependencies, Non-Linear Data and Model Reduction
COMP6237 – Logistic Regression, Dependencies, Non-linear Data and Model Reduction Markus Brede [email protected] Lecture slides available here: http://users.ecs.soton.ac.uk/mb8/stats/datamining.html (Thanks to Jason Noble and Cosma Shalizi whose lecture materials I used to prepare) COMP6237: Logistic Regression ● Outline: – Introduction – Basic ideas of logistic regression – Logistic regression using R – Some underlying maths and MLE – The multinomial case – How to deal with non-linear data ● Model reduction and AIC – How to deal with dependent data – Summary – Problems Introduction ● Previous lecture: Linear regression – tried to predict a continuous variable from variation in another continuous variable (E.g. basketball ability from height) ● Here: Logistic regression – Try to predict results of a binary (or categorical) outcome variable Y from a predictor variable X – This is a classification problem: classify X as belonging to one of two classes – Occurs quite often in science … e.g. medical trials (will a patient live or die dependent on medication?) Dependent variable Y Predictor Variables X The Oscars Example ● A fictional data set that looks at what it takes for a movie to win an Oscar ● Outcome variable: Oscar win, yes or no? ● Predictor variables: – Box office takings in millions of dollars – Budget in millions of dollars – Country of origin: US, UK, Europe, India, other – Critical reception (scores 0 … 100) – Length of film in minutes – This (fictitious) data set is available here: https://www.southampton.ac.uk/~mb1a10/stats/filmData.txt Predicting Oscar Success ● Let's start simple and look at only one of the predictor variables ● Do big box office takings make Oscar success more likely? ● Could use same techniques as below to look at budget size, film length, etc. -
Bayesian Inference Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Aus´ınand Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 1 / 35 Objective AFM Smith Dennis Lindley We analyze the Bayesian approach to fitting normal and generalized linear models and introduce the Bayesian hierarchical modeling approach. Also, we study the modeling and forecasting of time series. Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 2 / 35 Contents 1 Normal linear models 1.1. ANOVA model 1.2. Simple linear regression model 2 Generalized linear models 3 Hierarchical models 4 Dynamic models Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 3 / 35 Normal linear models A normal linear model is of the following form: y = Xθ + ; 0 where y = (y1;:::; yn) is the observed data, X is a known n × k matrix, called 0 the design matrix, θ = (θ1; : : : ; θk ) is the parameter set and follows a multivariate normal distribution. Usually, it is assumed that: 1 ∼ N 0 ; I : k φ k A simple example of normal linear model is the simple linear regression model T 1 1 ::: 1 where X = and θ = (α; β)T . x1 x2 ::: xn Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 4 / 35 Normal linear models Consider a normal linear model, y = Xθ + . A conjugate prior distribution is a normal-gamma distribution: -
Small-Sample Properties of Tests for Heteroscedasticity in the Conditional Logit Model
HEDG Working Paper 06/04 Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole May 2006 ISSN 1751-1976 york.ac.uk/res/herc/hedgwp Small-sample properties of tests for heteroscedasticity in the conditional logit model Arne Risa Hole National Primary Care Research and Development Centre Centre for Health Economics University of York May 16, 2006 Abstract This paper compares the small-sample properties of several asymp- totically equivalent tests for heteroscedasticity in the conditional logit model. While no test outperforms the others in all of the experiments conducted, the likelihood ratio test and a particular variety of the Wald test are found to have good properties in moderate samples as well as being relatively powerful. Keywords: conditional logit, heteroscedasticity JEL classi…cation: C25 National Primary Care Research and Development Centre, Centre for Health Eco- nomics, Alcuin ’A’Block, University of York, York YO10 5DD, UK. Tel.: +44 1904 321404; fax: +44 1904 321402. E-mail address: [email protected]. NPCRDC receives funding from the Department of Health. The views expressed are not necessarily those of the funders. 1 1 Introduction In most applications of the conditional logit model the error term is assumed to be homoscedastic. Recently, however, there has been a growing interest in testing the homoscedasticity assumption in applied work (Hensher et al., 1999; DeShazo and Fermo, 2002). This is partly due to the well-known result that heteroscedasticity causes the coe¢ cient estimates in discrete choice models to be inconsistent (Yatchew and Griliches, 1985), but also re‡ects a behavioural interest in factors in‡uencing the variance of the latent variables in the model (Louviere, 2001; Louviere et al., 2002). -
Estimation of Discrete Choice Models Including Socio-Economic Explanatory Variables, Diskussionsbeiträge - Serie II, No
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Ronning, Gerd Working Paper Estimation of discrete choice models including socio- economic explanatory variables Diskussionsbeiträge - Serie II, No. 42 Provided in Cooperation with: Department of Economics, University of Konstanz Suggested Citation: Ronning, Gerd (1987) : Estimation of discrete choice models including socio-economic explanatory variables, Diskussionsbeiträge - Serie II, No. 42, Universität Konstanz, Sonderforschungsbereich 178 - Internationalisierung der Wirtschaft, Konstanz This Version is available at: http://hdl.handle.net/10419/101753 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
Ologit — Ordered Logistic Regression
Title stata.com ologit — Ordered logistic regression Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description ologit fits ordered logit models of ordinal variable depvar on the independent variables indepvars. The actual values taken on by the dependent variable are irrelevant, except that larger values are assumed to correspond to “higher” outcomes. Quick start Ordinal logit model of y on x1 and categorical variables a and b ologit y x1 i.a i.b As above, and include interaction between a and b and report results as odds ratios ologit y x1 a##b, or With bootstrap standard errors ologit y x1 i.a i.b, vce(bootstrap) Analysis restricted to cases where catvar = 0 using svyset data with replicate weights svy bootstrap, subpop(if catvar==0): ologit y x1 i.a i.b Menu Statistics > Ordinal outcomes > Ordered logistic regression 1 2 ologit — Ordered logistic regression Syntax ologit depvar indepvars if in weight , options options Description Model offset(varname) include varname in model with coefficient constrained to 1 constraints(constraints) apply specified linear constraints SE/Robust vce(vcetype) vcetype may be oim, robust, cluster clustvar, bootstrap, or jackknife Reporting level(#) set confidence level; default is level(95) or report odds ratios nocnsreport do not display constraints display options control columns and column formats, row spacing, line width, display of omitted variables and base and empty cells, and factor-variable labeling Maximization maximize options control the maximization process; seldom used collinear keep collinear variables coeflegend display legend instead of statistics indepvars may contain factor variables; see [U] 11.4.3 Factor variables. -
Multinomial Logistic Regression
26-Osborne (Best)-45409.qxd 10/9/2007 5:52 PM Page 390 26 MULTINOMIAL LOGISTIC REGRESSION CAROLYN J. ANDERSON LESLIE RUTKOWSKI hapter 24 presented logistic regression Many of the concepts used in binary logistic models for dichotomous response vari- regression, such as the interpretation of parame- C ables; however, many discrete response ters in terms of odds ratios and modeling prob- variables have three or more categories (e.g., abilities, carry over to multicategory logistic political view, candidate voted for in an elec- regression models; however, two major modifica- tion, preferred mode of transportation, or tions are needed to deal with multiple categories response options on survey items). Multicate- of the response variable. One difference is that gory response variables are found in a wide with three or more levels of the response variable, range of experiments and studies in a variety of there are multiple ways to dichotomize the different fields. A detailed example presented in response variable. If J equals the number of cate- this chapter uses data from 600 students from gories of the response variable, then J(J – 1)/2 dif- the High School and Beyond study (Tatsuoka & ferent ways exist to dichotomize the categories. Lohnes, 1988) to look at the differences among In the High School and Beyond study, the three high school students who attended academic, program types can be dichotomized into pairs general, or vocational programs. The students’ of programs (i.e., academic and general, voca- socioeconomic status (ordinal), achievement tional and general, and academic and vocational). test scores (numerical), and type of school How the response variable is dichotomized (nominal) are all examined as possible explana- depends, in part, on the nature of the variable. -
Logit and Ordered Logit Regression (Ver
Getting Started in Logit and Ordered Logit Regression (ver. 3.1 beta) Oscar Torres-Reyna Data Consultant [email protected] http://dss.princeton.edu/training/ PU/DSS/OTR Logit model • Use logit models whenever your dependent variable is binary (also called dummy) which takes values 0 or 1. • Logit regression is a nonlinear regression model that forces the output (predicted values) to be either 0 or 1. • Logit models estimate the probability of your dependent variable to be 1 (Y=1). This is the probability that some event happens. PU/DSS/OTR Logit odelm From Stock & Watson, key concept 9.3. The logit model is: Pr(YXXXFXX 1 | 1= , 2 ,...=k β ) +0 β ( 1 +2 β 1 +βKKX 2 + ... ) 1 Pr(YXXX 1= | 1 , 2k = ,... ) 1−+(eβ0 + βXX 1 1 + β 2 2 + ...βKKX + ) 1 Pr(YXXX 1= | 1 , 2= ,... ) k ⎛ 1 ⎞ 1+ ⎜ ⎟ (⎝ eβ+0 βXX 1 1 + β 2 2 + ...βKK +X ⎠ ) Logit nd probita models are basically the same, the difference is in the distribution: • Logit – Cumulative standard logistic distribution (F) • Probit – Cumulative standard normal distribution (Φ) Both models provide similar results. PU/DSS/OTR It tests whether the combined effect, of all the variables in the model, is different from zero. If, for example, < 0.05 then the model have some relevant explanatory power, which does not mean it is well specified or at all correct. Logit: predicted probabilities After running the model: logit y_bin x1 x2 x3 x4 x5 x6 x7 Type predict y_bin_hat /*These are the predicted probabilities of Y=1 */ Here are the estimations for the first five cases, type: 1 x2 x3 x4 x5 x6 x7 y_bin_hatbrowse y_bin x Predicted probabilities To estimate the probability of Y=1 for the first row, replace the values of X into the logit regression equation.