Latent Variable Models for Binary Data. Suppose that for a given vector of explanatory variables x, the latent

Size: px
Start display at page:

Download "Latent Variable Models for Binary Data. Suppose that for a given vector of explanatory variables x, the latent"

Transcription

1 Latent Variable Models for Binary Data Suppose that for a given vector of explanatory variables x, the latent variable, U, has a continuous cumulative distribution function F (u; x) and that the binary response Y = 1 is recorded if and only if U > 0: θ = Pr(Y = 1 x) = 1 F (0; x). Since U is not directly observed there is no loss of generality in taking the critical (i.e., cutoff point) to be 0. In addition, we can take the standard deviation of U (or some other measure of dispersion) to be 1, without loss of generality. 1

2 Probit Models For example, if U N(x β, 1) it follows that θ i = Pr(Y = 1 x i ) = Φ(x iβ), where Φ( ) is the cumulative normal distribution function Φ(t) = (2π) 1/2 t exp( 1 2 z2 ) dz. The relation is linearized by the inverse normal transformation Φ 1 (θ) = x iβ = p j=1 x ij β j. 2

3 We have regarded the cutoff value of U as fixed and the mean of U to be changing with x. Alternatively, one could assume that the distribution of U is fixed and allow the critical value to vary with x (e.g., dose) In toxicology studies where dose is the explanatory variable it makes sense to let V denote the minimum level of dose needed to produce a response (i.e., tolerance) 3

4 Under the second formulation, y i = 1 if x i β > v i It follows that Pr(Y = 1 x i ) = Pr(V x iβ). Note that the shape of the dose-response curve is determined by the distribution function of V If V N(0, 1), then Pr(Y = 1 x i ) = Φ(x iβ), and it follows that the U and V formulations are equivalent The U formulation is more common 4

5 Latent Utilities & Choice Models Suppose that Fred is choosing between 2 brands of a product (say, Ben & Jerry s or Haigen Daz) Fred has a utility for Ben & Jerry s (denoted by Z i1 ) and a utility for Haigen Daz (denotes by Z i2 ) Letting the difference in utilities be represented by the normal linear model, we have U i = Z i1 Z i2 = x iβ + ɛ i, where ɛ i N(0, 1). If Fred has a higher utility for Ben & Jerry s, then Z i1 > Z i2, U i > 0, and Fred will choose Ben & Jerry s (Y i = 1) 5

6 This latent utility formulation is again equivalent to a probit model for the binary response. The generalization to a multinomial response is straightforward by introducing k latent utilities instead of 2, and letting an individual s response (i.e., choice) correspond to the category with the maximum utility Although the probit model is preferred in bioassay and social sciences applications, the logistic model is preferred in the biomedical sciences Of course, the choice of distribution function for U (and hence the choice of link in the binary response GLM) should be motivated by model fit. 6

7 Logistic Regression The normal form is only one possibility for the distribution of U. Another is the logistic distribution with location x iβ and unit scale. The logistic distribution has cumulative distribution function so that F (u) = exp(u x iβ) 1 + exp(u x iβ), F (0; x i ) = 1/{1 + exp(x iβ)}, It follows that Pr(Y = 1 x i ) = Pr(U > 0 x i ) = 1 F (0; x i ) = 1/{1+exp( x iβ)}. To linearize this relation, we take the logit transformation of both sides, log{θ i /(1 θ i )} = x iβ. 7

8 Homework Exercise: For x i = (1, x) and β 2 > 0, reformulate the logistic regression in terms of a threshold model (i.e., the V formulation of the probit model described above). Derive the probability density function (pdf) obtained by differentiating Pr(Y = 1 x i ) with respect to x. Reparameterize in terms of τ = 1/β 2 and µ = β 1 /β 2. Plot this pdf for µ = 0 and πτ/ 3 = 1 along with the N(0,1) pdf in S-PLUS. Which density has the fatter tails? Is the pdf for x in the logistic case in the exponential family? 8

9 Some Generalizations of the Logistic Model The logistic regression model assumes a restricted dose-response shape and it is possible generalize the model to relax the restriction Aranda-Ordaz (1981) proposed two families of linearizing transformation, which can easily be inverted and which span a range of forms. The first, which is restricted to symmetric cases (i.e., invariant to interchanging success & failure) is 2 θ ν (1 θ) ν ν θ ν + (1 θ). ν In the limit as ν 0, this is logistic and for ν = 1 this is linear The second family has log[{(1 θ) ν 1}/ν], which reduces to the extreme value model when ν = 0 and the logistic when ν = 1. 9

10 When there is doubt about the transformation, a formal approach is to use one or the other of the above transformations and to fit the resulting model for a range of possible values for ν A profile likelihood can be obtained for ν by plotting the maximized likelihood against ν (Frequentist) Potentially, one could choose a standard form, such as the logistic, if the corresponding value of ν falls with the 95% profile likelihood confidence region. Alternatively, we could choose a prior density for ν and implement a Bayesian approach. 10

11 Data Augmentation Algorithms for Probit Models (Albert & Chib, 1993, JASA, ) Now, suppose that p i = Pr(y i = 1 x i, β) = Φ(x iβ), where Φ( ) is the N(0,1) cdf. As discussed previously, this probit regression model is equivalent to: y i = 1(z i > 0), z i N(x iβ, 1), where z 1,..., z n are independent latent variables. Note that if the z i are known and a multivariate normal prior is chosen for β, the posterior distribution is multivariate normal. 11

12 The z i are unknown latent variables, which we introduce for computational convenience and which have no impact on the model interpretation. By introducing the z i s, we are augmenting the observed data y = (y 1,..., y n ) with latent data z = (z 1,..., z n ). The joint posterior density of the unobservables β and z is π(β, z y) π(β) n i=1 {1(z i > 0) 1(y i = 1) + 1(z i 0) 1(y i = 0)}N(z i ; x iβ, 1), which is the prior for β times the prior for z given β times the likelihood for y given z and β. 12

13 Note that, integrating out the latent data, we have π(β y) = π(β, z y) dz π(β) n i=1 = π(β) n i=1 = π(β) n i=1 { 0 N(z i; x iβ, 1) dz } 1(y i =0) { i N(z 0 i ; x } iβ, 1) dz i Φ( x iβ) 1(y i=0) Φ(x iβ) 1(y i=1), {1 Φ(x iβ)} 1(y i=0) Φ(x iβ) 1(y i=1), and computation could proceed for this binary-response glm using Gibbs sampling with ARS (e.g., in WinBUGS) This procedure updates the parameters one at a time and requires programming of ARS. 13

14 Alternative: Data Augmentation Gibbs sampler 1. Choose initial values for β and prior density, β N(β 0, Σ 0 ). 2. Impute the latent data by sampling from the full conditional distribution, π(z i β, y) {1(z i > 0) 1(y i = 1) + 1(z i 0) 1(y i = 0)}N(z i ; x iβ, 1) d = truncated N(x iβ, 1), for z i > 0 if y i = 1, and z i 0 if y i = Update β (jointly!) by sampling from the full conditional, π(β z, y) d = N( β, Σ β ), where Σ β = (Σ X X) 1 is the posterior covariance conditional on z and β = Σ β (Σ 1 0 β 0 + X z). 4. Repeat steps 2-3 until apparent convergence and calculate posterior summaries for β based on a large number of additional iterates. 14

15 Some Comments If you like probit models, the Albert and Chib algorithm is extremely useful being very easy to program and efficient relative to ARS. Probit models have the disadvantage that the regression coefficients cannot be expressed as a simple analytic function of the probability of response. However, by approximating the logistic model using a scale mixture of normals, one can modify the Albert and Chib approach for logistic regression (O Brien and Dunson, 2003, ISDS Discussion Paper 03-08). Underlying normal models are not limited to univariate binary data - generalizations extremely useful. 15

16 Extending GLMs for Correlated Data GLMs assume that the observations y 1,..., y n are independent draws from an exponential family distribution However, in many applications, there may be dependency in the outcome data For example, in longitudinal studies, repeated observations are collected for each study subject 16

17 Longitudinal Studies For subject i (i = 1,..., n), outcome data consist of an n i 1 vector of measurements at follow-up times t i,1,..., t i,ni. Instead of a single measurement, y i, for subject i, we have a vector of measurements, y = (y i1,..., y i,ni ). Since different measurements for a subject may be correlated, the standard GLM is not appropriate 17

18 Possibilities for Repeated Measures Data 1. Conditional Model: Allow the linear predictor to differ for the different study subjects, η ij = x ijβ + b i, where x ij does not include an intercept and b i is a subject-specific parameter (i.e., subject is a blocking factor in ANOVA jargon). 2. Marginal Model: Specify a marginal model for the population averaged response, and construct a variance estimator which takes into account the correlation structure (e.g., Generalized Estimating Equations, Liang and Zeger, 1986) 3. Mixed Model: Assume the regression coefficients for a subject are drawn from a population distribution, and estimate both the population and individual-specific parameters (Laird and Ware, 1982). 18

Standard Errors & Confidence Intervals. N(0, I( β) 1 ), I( β) = [ 2 l(β, φ; y) β i β β= β j

Standard Errors & Confidence Intervals. N(0, I( β) 1 ), I( β) = [ 2 l(β, φ; y) β i β β= β j Standard Errors & Confidence Intervals β β asy N(0, I( β) 1 ), where I( β) = [ 2 l(β, φ; y) ] β i β β= β j We can obtain asymptotic 100(1 α)% confidence intervals for β j using: β j ± Z 1 α/2 se( β j )

More information

Bayesian Inference in GLMs. Frequentists typically base inferences on MLEs, asymptotic confidence

Bayesian Inference in GLMs. Frequentists typically base inferences on MLEs, asymptotic confidence Bayesian Inference in GLMs Frequentists typically base inferences on MLEs, asymptotic confidence limits, and log-likelihood ratio tests Bayesians base inferences on the posterior distribution of the unknowns

More information

STA 216, GLM, Lecture 16. October 29, 2007

STA 216, GLM, Lecture 16. October 29, 2007 STA 216, GLM, Lecture 16 October 29, 2007 Efficient Posterior Computation in Factor Models Underlying Normal Models Generalized Latent Trait Models Formulation Genetic Epidemiology Illustration Structural

More information

Bayesian Multivariate Logistic Regression

Bayesian Multivariate Logistic Regression Bayesian Multivariate Logistic Regression Sean M. O Brien and David B. Dunson Biostatistics Branch National Institute of Environmental Health Sciences Research Triangle Park, NC 1 Goals Brief review of

More information

Bayesian Hypothesis Testing in GLMs: One-Sided and Ordered Alternatives. 1(w i = h + 1)β h + ɛ i,

Bayesian Hypothesis Testing in GLMs: One-Sided and Ordered Alternatives. 1(w i = h + 1)β h + ɛ i, Bayesian Hypothesis Testing in GLMs: One-Sided and Ordered Alternatives Often interest may focus on comparing a null hypothesis of no difference between groups to an ordered restricted alternative. For

More information

Gibbs Sampling in Latent Variable Models #1

Gibbs Sampling in Latent Variable Models #1 Gibbs Sampling in Latent Variable Models #1 Econ 690 Purdue University Outline 1 Data augmentation 2 Probit Model Probit Application A Panel Probit Panel Probit 3 The Tobit Model Example: Female Labor

More information

Bayes methods for categorical data. April 25, 2017

Bayes methods for categorical data. April 25, 2017 Bayes methods for categorical data April 25, 2017 Motivation for joint probability models Increasing interest in high-dimensional data in broad applications Focus may be on prediction, variable selection,

More information

STA216: Generalized Linear Models. Lecture 1. Review and Introduction

STA216: Generalized Linear Models. Lecture 1. Review and Introduction STA216: Generalized Linear Models Lecture 1. Review and Introduction Let y 1,..., y n denote n independent observations on a response Treat y i as a realization of a random variable Y i In the general

More information

Econometrics Lecture 5: Limited Dependent Variable Models: Logit and Probit

Econometrics Lecture 5: Limited Dependent Variable Models: Logit and Probit Econometrics Lecture 5: Limited Dependent Variable Models: Logit and Probit R. G. Pierse 1 Introduction In lecture 5 of last semester s course, we looked at the reasons for including dichotomous variables

More information

Lecture 16: Mixtures of Generalized Linear Models

Lecture 16: Mixtures of Generalized Linear Models Lecture 16: Mixtures of Generalized Linear Models October 26, 2006 Setting Outline Often, a single GLM may be insufficiently flexible to characterize the data Setting Often, a single GLM may be insufficiently

More information

Bayesian Linear Regression

Bayesian Linear Regression Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective

More information

Generalized Linear Models. Last time: Background & motivation for moving beyond linear

Generalized Linear Models. Last time: Background & motivation for moving beyond linear Generalized Linear Models Last time: Background & motivation for moving beyond linear regression - non-normal/non-linear cases, binary, categorical data Today s class: 1. Examples of count and ordered

More information

A Fully Nonparametric Modeling Approach to. BNP Binary Regression

A Fully Nonparametric Modeling Approach to. BNP Binary Regression A Fully Nonparametric Modeling Approach to Binary Regression Maria Department of Applied Mathematics and Statistics University of California, Santa Cruz SBIES, April 27-28, 2012 Outline 1 2 3 Simulation

More information

The linear model is the most fundamental of all serious statistical models encompassing:

The linear model is the most fundamental of all serious statistical models encompassing: Linear Regression Models: A Bayesian perspective Ingredients of a linear model include an n 1 response vector y = (y 1,..., y n ) T and an n p design matrix (e.g. including regressors) X = [x 1,..., x

More information

Stat 535 C - Statistical Computing & Monte Carlo Methods. Lecture 15-7th March Arnaud Doucet

Stat 535 C - Statistical Computing & Monte Carlo Methods. Lecture 15-7th March Arnaud Doucet Stat 535 C - Statistical Computing & Monte Carlo Methods Lecture 15-7th March 2006 Arnaud Doucet Email: arnaud@cs.ubc.ca 1 1.1 Outline Mixture and composition of kernels. Hybrid algorithms. Examples Overview

More information

Stat 542: Item Response Theory Modeling Using The Extended Rank Likelihood

Stat 542: Item Response Theory Modeling Using The Extended Rank Likelihood Stat 542: Item Response Theory Modeling Using The Extended Rank Likelihood Jonathan Gruhl March 18, 2010 1 Introduction Researchers commonly apply item response theory (IRT) models to binary and ordinal

More information

MULTILEVEL IMPUTATION 1

MULTILEVEL IMPUTATION 1 MULTILEVEL IMPUTATION 1 Supplement B: MCMC Sampling Steps and Distributions for Two-Level Imputation This document gives technical details of the full conditional distributions used to draw regression

More information

Gibbs Sampling in Linear Models #2

Gibbs Sampling in Linear Models #2 Gibbs Sampling in Linear Models #2 Econ 690 Purdue University Outline 1 Linear Regression Model with a Changepoint Example with Temperature Data 2 The Seemingly Unrelated Regressions Model 3 Gibbs sampling

More information

A Bayesian Mixture Model with Application to Typhoon Rainfall Predictions in Taipei, Taiwan 1

A Bayesian Mixture Model with Application to Typhoon Rainfall Predictions in Taipei, Taiwan 1 Int. J. Contemp. Math. Sci., Vol. 2, 2007, no. 13, 639-648 A Bayesian Mixture Model with Application to Typhoon Rainfall Predictions in Taipei, Taiwan 1 Tsai-Hung Fan Graduate Institute of Statistics National

More information

Motivation Scale Mixutres of Normals Finite Gaussian Mixtures Skew-Normal Models. Mixture Models. Econ 690. Purdue University

Motivation Scale Mixutres of Normals Finite Gaussian Mixtures Skew-Normal Models. Mixture Models. Econ 690. Purdue University Econ 690 Purdue University In virtually all of the previous lectures, our models have made use of normality assumptions. From a computational point of view, the reason for this assumption is clear: combined

More information

Partial factor modeling: predictor-dependent shrinkage for linear regression

Partial factor modeling: predictor-dependent shrinkage for linear regression modeling: predictor-dependent shrinkage for linear Richard Hahn, Carlos Carvalho and Sayan Mukherjee JASA 2013 Review by Esther Salazar Duke University December, 2013 Factor framework The factor framework

More information

STA 216: GENERALIZED LINEAR MODELS. Lecture 1. Review and Introduction. Much of statistics is based on the assumption that random

STA 216: GENERALIZED LINEAR MODELS. Lecture 1. Review and Introduction. Much of statistics is based on the assumption that random STA 216: GENERALIZED LINEAR MODELS Lecture 1. Review and Introduction Much of statistics is based on the assumption that random variables are continuous & normally distributed. Normal linear regression

More information

Stat 5101 Lecture Notes

Stat 5101 Lecture Notes Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random

More information

Wageningen Summer School in Econometrics. The Bayesian Approach in Theory and Practice

Wageningen Summer School in Econometrics. The Bayesian Approach in Theory and Practice Wageningen Summer School in Econometrics The Bayesian Approach in Theory and Practice September 2008 Slides for Lecture on Qualitative and Limited Dependent Variable Models Gary Koop, University of Strathclyde

More information

Generalized Linear Models for Non-Normal Data

Generalized Linear Models for Non-Normal Data Generalized Linear Models for Non-Normal Data Today s Class: 3 parts of a generalized model Models for binary outcomes Complications for generalized multivariate or multilevel models SPLH 861: Lecture

More information

November 2002 STA Random Effects Selection in Linear Mixed Models

November 2002 STA Random Effects Selection in Linear Mixed Models November 2002 STA216 1 Random Effects Selection in Linear Mixed Models November 2002 STA216 2 Introduction It is common practice in many applications to collect multiple measurements on a subject. Linear

More information

Bayesian linear regression

Bayesian linear regression Bayesian linear regression Linear regression is the basis of most statistical modeling. The model is Y i = X T i β + ε i, where Y i is the continuous response X i = (X i1,..., X ip ) T is the corresponding

More information

Bayesian Inference in the Multivariate Probit Model

Bayesian Inference in the Multivariate Probit Model Bayesian Inference in the Multivariate Probit Model Estimation of the Correlation Matrix by Aline Tabet A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science

More information

Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes 1

Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes 1 Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, Discrete Changes 1 JunXuJ.ScottLong Indiana University 2005-02-03 1 General Formula The delta method is a general

More information

Bayesian Linear Models

Bayesian Linear Models Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics, School of Public

More information

Stat 579: Generalized Linear Models and Extensions

Stat 579: Generalized Linear Models and Extensions Stat 579: Generalized Linear Models and Extensions Linear Mixed Models for Longitudinal Data Yan Lu April, 2018, week 15 1 / 38 Data structure t1 t2 tn i 1st subject y 11 y 12 y 1n1 Experimental 2nd subject

More information

Bayesian Linear Models

Bayesian Linear Models Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department

More information

Linear Regression Models P8111

Linear Regression Models P8111 Linear Regression Models P8111 Lecture 25 Jeff Goldsmith April 26, 2016 1 of 37 Today s Lecture Logistic regression / GLMs Model framework Interpretation Estimation 2 of 37 Linear regression Course started

More information

Model Selection in GLMs. (should be able to implement frequentist GLM analyses!) Today: standard frequentist methods for model selection

Model Selection in GLMs. (should be able to implement frequentist GLM analyses!) Today: standard frequentist methods for model selection Model Selection in GLMs Last class: estimability/identifiability, analysis of deviance, standard errors & confidence intervals (should be able to implement frequentist GLM analyses!) Today: standard frequentist

More information

Default Priors and Effcient Posterior Computation in Bayesian

Default Priors and Effcient Posterior Computation in Bayesian Default Priors and Effcient Posterior Computation in Bayesian Factor Analysis January 16, 2010 Presented by Eric Wang, Duke University Background and Motivation A Brief Review of Parameter Expansion Literature

More information

Bayesian non-parametric model to longitudinally predict churn

Bayesian non-parametric model to longitudinally predict churn Bayesian non-parametric model to longitudinally predict churn Bruno Scarpa Università di Padova Conference of European Statistics Stakeholders Methodologists, Producers and Users of European Statistics

More information

BAYESIAN ANALYSIS OF BINARY REGRESSION USING SYMMETRIC AND ASYMMETRIC LINKS

BAYESIAN ANALYSIS OF BINARY REGRESSION USING SYMMETRIC AND ASYMMETRIC LINKS Sankhyā : The Indian Journal of Statistics 2000, Volume 62, Series B, Pt. 3, pp. 372 387 BAYESIAN ANALYSIS OF BINARY REGRESSION USING SYMMETRIC AND ASYMMETRIC LINKS By SANJIB BASU Northern Illinis University,

More information

Single-level Models for Binary Responses

Single-level Models for Binary Responses Single-level Models for Binary Responses Distribution of Binary Data y i response for individual i (i = 1,..., n), coded 0 or 1 Denote by r the number in the sample with y = 1 Mean and variance E(y) =

More information

Online Appendix to: Marijuana on Main Street? Estimating Demand in Markets with Limited Access

Online Appendix to: Marijuana on Main Street? Estimating Demand in Markets with Limited Access Online Appendix to: Marijuana on Main Street? Estating Demand in Markets with Lited Access By Liana Jacobi and Michelle Sovinsky This appendix provides details on the estation methodology for various speci

More information

Gibbs Sampling for the Probit Regression Model with Gaussian Markov Random Field Latent Variables

Gibbs Sampling for the Probit Regression Model with Gaussian Markov Random Field Latent Variables Gibbs Sampling for the Probit Regression Model with Gaussian Markov Random Field Latent Variables Mohammad Emtiyaz Khan Department of Computer Science University of British Columbia May 8, 27 Abstract

More information

ECON 594: Lecture #6

ECON 594: Lecture #6 ECON 594: Lecture #6 Thomas Lemieux Vancouver School of Economics, UBC May 2018 1 Limited dependent variables: introduction Up to now, we have been implicitly assuming that the dependent variable, y, was

More information

Factor Analytic Models of Clustered Multivariate Data with Informative Censoring (refer to Dunson and Perreault, 2001, Biometrics 57, )

Factor Analytic Models of Clustered Multivariate Data with Informative Censoring (refer to Dunson and Perreault, 2001, Biometrics 57, ) Factor Analytic Models of Clustered Multivariate Data with Informative Censoring (refer to Dunson and Perreault, 2001, Biometrics 57, 302-308) Consider data in which multiple outcomes are collected for

More information

Bayesian Linear Models

Bayesian Linear Models Bayesian Linear Models Sudipto Banerjee September 03 05, 2017 Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles Linear Regression Linear regression is,

More information

A general mixed model approach for spatio-temporal regression data

A general mixed model approach for spatio-temporal regression data A general mixed model approach for spatio-temporal regression data Thomas Kneib, Ludwig Fahrmeir & Stefan Lang Department of Statistics, Ludwig-Maximilians-University Munich 1. Spatio-temporal regression

More information

Gibbs Sampling in Endogenous Variables Models

Gibbs Sampling in Endogenous Variables Models Gibbs Sampling in Endogenous Variables Models Econ 690 Purdue University Outline 1 Motivation 2 Identification Issues 3 Posterior Simulation #1 4 Posterior Simulation #2 Motivation In this lecture we take

More information

Hierarchical Modeling for Univariate Spatial Data

Hierarchical Modeling for Univariate Spatial Data Hierarchical Modeling for Univariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Spatial Domain 2 Geography 890 Spatial Domain This

More information

Generalized Linear Models 1

Generalized Linear Models 1 Generalized Linear Models 1 STA 2101/442: Fall 2012 1 See last slide for copyright information. 1 / 24 Suggested Reading: Davison s Statistical models Exponential families of distributions Sec. 5.2 Chapter

More information

I. Multinomial Logit Suppose we only have individual specific covariates. Then we can model the response probability as

I. Multinomial Logit Suppose we only have individual specific covariates. Then we can model the response probability as Econ 513, USC, Fall 2005 Lecture 15 Discrete Response Models: Multinomial, Conditional and Nested Logit Models Here we focus again on models for discrete choice with more than two outcomes We assume that

More information

Modeling Binary Outcomes: Logit and Probit Models

Modeling Binary Outcomes: Logit and Probit Models Modeling Binary Outcomes: Logit and Probit Models Eric Zivot December 5, 2009 Motivating Example: Women s labor force participation y i = 1 if married woman is in labor force = 0 otherwise x i k 1 = observed

More information

Marginal Specifications and a Gaussian Copula Estimation

Marginal Specifications and a Gaussian Copula Estimation Marginal Specifications and a Gaussian Copula Estimation Kazim Azam Abstract Multivariate analysis involving random variables of different type like count, continuous or mixture of both is frequently required

More information

12 Modelling Binomial Response Data

12 Modelling Binomial Response Data c 2005, Anthony C. Brooms Statistical Modelling and Data Analysis 12 Modelling Binomial Response Data 12.1 Examples of Binary Response Data Binary response data arise when an observation on an individual

More information

Analysing geoadditive regression data: a mixed model approach

Analysing geoadditive regression data: a mixed model approach Analysing geoadditive regression data: a mixed model approach Institut für Statistik, Ludwig-Maximilians-Universität München Joint work with Ludwig Fahrmeir & Stefan Lang 25.11.2005 Spatio-temporal regression

More information

Bayesian Analysis of Latent Variable Models using Mplus

Bayesian Analysis of Latent Variable Models using Mplus Bayesian Analysis of Latent Variable Models using Mplus Tihomir Asparouhov and Bengt Muthén Version 2 June 29, 2010 1 1 Introduction In this paper we describe some of the modeling possibilities that are

More information

Lecture 13: More on Binary Data

Lecture 13: More on Binary Data Lecture 1: More on Binary Data Link functions for Binomial models Link η = g(π) π = g 1 (η) identity π η logarithmic log π e η logistic log ( π 1 π probit Φ 1 (π) Φ(η) log-log log( log π) exp( e η ) complementary

More information

Generalized Linear Models

Generalized Linear Models Generalized Linear Models Advanced Methods for Data Analysis (36-402/36-608 Spring 2014 1 Generalized linear models 1.1 Introduction: two regressions So far we ve seen two canonical settings for regression.

More information

Generalized Linear Models. Kurt Hornik

Generalized Linear Models. Kurt Hornik Generalized Linear Models Kurt Hornik Motivation Assuming normality, the linear model y = Xβ + e has y = β + ε, ε N(0, σ 2 ) such that y N(μ, σ 2 ), E(y ) = μ = β. Various generalizations, including general

More information

Lecture 8: The Metropolis-Hastings Algorithm

Lecture 8: The Metropolis-Hastings Algorithm 30.10.2008 What we have seen last time: Gibbs sampler Key idea: Generate a Markov chain by updating the component of (X 1,..., X p ) in turn by drawing from the full conditionals: X (t) j Two drawbacks:

More information

An Introduction to Bayesian Linear Regression

An Introduction to Bayesian Linear Regression An Introduction to Bayesian Linear Regression APPM 5720: Bayesian Computation Fall 2018 A SIMPLE LINEAR MODEL Suppose that we observe explanatory variables x 1, x 2,..., x n and dependent variables y 1,

More information

Lecture 5: Spatial probit models. James P. LeSage University of Toledo Department of Economics Toledo, OH

Lecture 5: Spatial probit models. James P. LeSage University of Toledo Department of Economics Toledo, OH Lecture 5: Spatial probit models James P. LeSage University of Toledo Department of Economics Toledo, OH 43606 jlesage@spatial-econometrics.com March 2004 1 A Bayesian spatial probit model with individual

More information

The Logit Model: Estimation, Testing and Interpretation

The Logit Model: Estimation, Testing and Interpretation The Logit Model: Estimation, Testing and Interpretation Herman J. Bierens October 25, 2008 1 Introduction to maximum likelihood estimation 1.1 The likelihood function Consider a random sample Y 1,...,

More information

Part 8: GLMs and Hierarchical LMs and GLMs

Part 8: GLMs and Hierarchical LMs and GLMs Part 8: GLMs and Hierarchical LMs and GLMs 1 Example: Song sparrow reproductive success Arcese et al., (1992) provide data on a sample from a population of 52 female song sparrows studied over the course

More information

Linear Regression. Data Model. β, σ 2. Process Model. ,V β. ,s 2. s 1. Parameter Model

Linear Regression. Data Model. β, σ 2. Process Model. ,V β. ,s 2. s 1. Parameter Model Regression: Part II Linear Regression y~n X, 2 X Y Data Model β, σ 2 Process Model Β 0,V β s 1,s 2 Parameter Model Assumptions of Linear Model Homoskedasticity No error in X variables Error in Y variables

More information

Goals. PSCI6000 Maximum Likelihood Estimation Multiple Response Model 1. Multinomial Dependent Variable. Random Utility Model

Goals. PSCI6000 Maximum Likelihood Estimation Multiple Response Model 1. Multinomial Dependent Variable. Random Utility Model Goals PSCI6000 Maximum Likelihood Estimation Multiple Response Model 1 Tetsuya Matsubayashi University of North Texas November 2, 2010 Random utility model Multinomial logit model Conditional logit model

More information

Discussion of Missing Data Methods in Longitudinal Studies: A Review by Ibrahim and Molenberghs

Discussion of Missing Data Methods in Longitudinal Studies: A Review by Ibrahim and Molenberghs Discussion of Missing Data Methods in Longitudinal Studies: A Review by Ibrahim and Molenberghs Michael J. Daniels and Chenguang Wang Jan. 18, 2009 First, we would like to thank Joe and Geert for a carefully

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate

More information

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline. Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,

More information

σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =

σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) = Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,

More information

Generalized Linear Models Introduction

Generalized Linear Models Introduction Generalized Linear Models Introduction Statistics 135 Autumn 2005 Copyright c 2005 by Mark E. Irwin Generalized Linear Models For many problems, standard linear regression approaches don t work. Sometimes,

More information

Contents. Part I: Fundamentals of Bayesian Inference 1

Contents. Part I: Fundamentals of Bayesian Inference 1 Contents Preface xiii Part I: Fundamentals of Bayesian Inference 1 1 Probability and inference 3 1.1 The three steps of Bayesian data analysis 3 1.2 General notation for statistical inference 4 1.3 Bayesian

More information

Fall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.

Fall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A. 1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n

More information

Logistic Regression. Seungjin Choi

Logistic Regression. Seungjin Choi Logistic Regression Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/

More information

Nonparametric Bayesian modeling for dynamic ordinal regression relationships

Nonparametric Bayesian modeling for dynamic ordinal regression relationships Nonparametric Bayesian modeling for dynamic ordinal regression relationships Athanasios Kottas Department of Applied Mathematics and Statistics, University of California, Santa Cruz Joint work with Maria

More information

Generalized Linear Models

Generalized Linear Models York SPIDA John Fox Notes Generalized Linear Models Copyright 2010 by John Fox Generalized Linear Models 1 1. Topics I The structure of generalized linear models I Poisson and other generalized linear

More information

Generalized logit models for nominal multinomial responses. Local odds ratios

Generalized logit models for nominal multinomial responses. Local odds ratios Generalized logit models for nominal multinomial responses Categorical Data Analysis, Summer 2015 1/17 Local odds ratios Y 1 2 3 4 1 π 11 π 12 π 13 π 14 π 1+ X 2 π 21 π 22 π 23 π 24 π 2+ 3 π 31 π 32 π

More information

CTDL-Positive Stable Frailty Model

CTDL-Positive Stable Frailty Model CTDL-Positive Stable Frailty Model M. Blagojevic 1, G. MacKenzie 2 1 Department of Mathematics, Keele University, Staffordshire ST5 5BG,UK and 2 Centre of Biostatistics, University of Limerick, Ireland

More information

Measurement error as missing data: the case of epidemiologic assays. Roderick J. Little

Measurement error as missing data: the case of epidemiologic assays. Roderick J. Little Measurement error as missing data: the case of epidemiologic assays Roderick J. Little Outline Discuss two related calibration topics where classical methods are deficient (A) Limit of quantification methods

More information

Bayesian GLMs and Metropolis-Hastings Algorithm

Bayesian GLMs and Metropolis-Hastings Algorithm Bayesian GLMs and Metropolis-Hastings Algorithm We have seen that with conjugate or semi-conjugate prior distributions the Gibbs sampler can be used to sample from the posterior distribution. In situations,

More information

Linear, Generalized Linear, and Mixed-Effects Models in R. Linear and Generalized Linear Models in R Topics

Linear, Generalized Linear, and Mixed-Effects Models in R. Linear and Generalized Linear Models in R Topics Linear, Generalized Linear, and Mixed-Effects Models in R John Fox McMaster University ICPSR 2018 John Fox (McMaster University) Statistical Models in R ICPSR 2018 1 / 19 Linear and Generalized Linear

More information

Linear Regression With Special Variables

Linear Regression With Special Variables Linear Regression With Special Variables Junhui Qian December 21, 2014 Outline Standardized Scores Quadratic Terms Interaction Terms Binary Explanatory Variables Binary Choice Models Standardized Scores:

More information

STAT 518 Intro Student Presentation

STAT 518 Intro Student Presentation STAT 518 Intro Student Presentation Wen Wei Loh April 11, 2013 Title of paper Radford M. Neal [1999] Bayesian Statistics, 6: 475-501, 1999 What the paper is about Regression and Classification Flexible

More information

Default Priors and Efficient Posterior Computation in Bayesian Factor Analysis

Default Priors and Efficient Posterior Computation in Bayesian Factor Analysis Default Priors and Efficient Posterior Computation in Bayesian Factor Analysis Joyee Ghosh Institute of Statistics and Decision Sciences, Duke University Box 90251, Durham, NC 27708 joyee@stat.duke.edu

More information

Supplementary Material for Analysis of Job Satisfaction: The Case of Japanese Private Companies

Supplementary Material for Analysis of Job Satisfaction: The Case of Japanese Private Companies Supplementary Material for Analysis of Job Satisfaction: The Case of Japanese Private Companies S1. Sampling Algorithms We assume that z i NX i β, Σ), i =1,,n, 1) where Σ is an m m positive definite covariance

More information

The Multilevel Logit Model for Binary Dependent Variables Marco R. Steenbergen

The Multilevel Logit Model for Binary Dependent Variables Marco R. Steenbergen The Multilevel Logit Model for Binary Dependent Variables Marco R. Steenbergen January 23-24, 2012 Page 1 Part I The Single Level Logit Model: A Review Motivating Example Imagine we are interested in voting

More information

GEE for Longitudinal Data - Chapter 8

GEE for Longitudinal Data - Chapter 8 GEE for Longitudinal Data - Chapter 8 GEE: generalized estimating equations (Liang & Zeger, 1986; Zeger & Liang, 1986) extension of GLM to longitudinal data analysis using quasi-likelihood estimation method

More information

Bayesian Inference. Chapter 4: Regression and Hierarchical Models

Bayesian Inference. Chapter 4: Regression and Hierarchical Models Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Ausín and Mike Wiper Department of Statistics Universidad Carlos III de Madrid Advanced Statistics and Data Mining Summer School

More information

Conjugate Analysis for the Linear Model

Conjugate Analysis for the Linear Model Conjugate Analysis for the Linear Model If we have good prior knowledge that can help us specify priors for β and σ 2, we can use conjugate priors. Following the procedure in Christensen, Johnson, Branscum,

More information

Data-analysis and Retrieval Ordinal Classification

Data-analysis and Retrieval Ordinal Classification Data-analysis and Retrieval Ordinal Classification Ad Feelders Universiteit Utrecht Data-analysis and Retrieval 1 / 30 Strongly disagree Ordinal Classification 1 2 3 4 5 0% (0) 10.5% (2) 21.1% (4) 42.1%

More information

STAT5044: Regression and Anova

STAT5044: Regression and Anova STAT5044: Regression and Anova Inyoung Kim 1 / 18 Outline 1 Logistic regression for Binary data 2 Poisson regression for Count data 2 / 18 GLM Let Y denote a binary response variable. Each observation

More information

36-720: The Rasch Model

36-720: The Rasch Model 36-720: The Rasch Model Brian Junker October 15, 2007 Multivariate Binary Response Data Rasch Model Rasch Marginal Likelihood as a GLMM Rasch Marginal Likelihood as a Log-Linear Model Example For more

More information

Bayesian Inference on Joint Mixture Models for Survival-Longitudinal Data with Multiple Features. Yangxin Huang

Bayesian Inference on Joint Mixture Models for Survival-Longitudinal Data with Multiple Features. Yangxin Huang Bayesian Inference on Joint Mixture Models for Survival-Longitudinal Data with Multiple Features Yangxin Huang Department of Epidemiology and Biostatistics, COPH, USF, Tampa, FL yhuang@health.usf.edu January

More information

Weighted Least Squares I

Weighted Least Squares I Weighted Least Squares I for i = 1, 2,..., n we have, see [1, Bradley], data: Y i x i i.n.i.d f(y i θ i ), where θ i = E(Y i x i ) co-variates: x i = (x i1, x i2,..., x ip ) T let X n p be the matrix of

More information

Bayesian Inference. Chapter 9. Linear models and regression

Bayesian Inference. Chapter 9. Linear models and regression Bayesian Inference Chapter 9. Linear models and regression M. Concepcion Ausin Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering

More information

Hierarchical Modelling for Univariate Spatial Data

Hierarchical Modelling for Univariate Spatial Data Hierarchical Modelling for Univariate Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department

More information

Hybrid Censoring; An Introduction 2

Hybrid Censoring; An Introduction 2 Hybrid Censoring; An Introduction 2 Debasis Kundu Department of Mathematics & Statistics Indian Institute of Technology Kanpur 23-rd November, 2010 2 This is a joint work with N. Balakrishnan Debasis Kundu

More information

PQL Estimation Biases in Generalized Linear Mixed Models

PQL Estimation Biases in Generalized Linear Mixed Models PQL Estimation Biases in Generalized Linear Mixed Models Woncheol Jang Johan Lim March 18, 2006 Abstract The penalized quasi-likelihood (PQL) approach is the most common estimation procedure for the generalized

More information

Research Division Federal Reserve Bank of St. Louis Working Paper Series

Research Division Federal Reserve Bank of St. Louis Working Paper Series Research Division Federal Reserve Bank of St Louis Working Paper Series Kalman Filtering with Truncated Normal State Variables for Bayesian Estimation of Macroeconomic Models Michael Dueker Working Paper

More information

ST3241 Categorical Data Analysis I Generalized Linear Models. Introduction and Some Examples

ST3241 Categorical Data Analysis I Generalized Linear Models. Introduction and Some Examples ST3241 Categorical Data Analysis I Generalized Linear Models Introduction and Some Examples 1 Introduction We have discussed methods for analyzing associations in two-way and three-way tables. Now we will

More information

Econometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland

Econometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2018 Part III Limited Dependent Variable Models As of Jan 30, 2017 1 Background 2 Binary Dependent Variable The Linear Probability

More information

Comparison of multiple imputation methods for systematically and sporadically missing multilevel data

Comparison of multiple imputation methods for systematically and sporadically missing multilevel data Comparison of multiple imputation methods for systematically and sporadically missing multilevel data V. Audigier, I. White, S. Jolani, T. Debray, M. Quartagno, J. Carpenter, S. van Buuren, M. Resche-Rigon

More information

Statistical Estimation

Statistical Estimation Statistical Estimation Use data and a model. The plug-in estimators are based on the simple principle of applying the defining functional to the ECDF. Other methods of estimation: minimize residuals from

More information