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

Size: px
Start display at page:

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

Transcription

1 Regression: Part II

2 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

3 Assumptions of Linear Model Homoskedasticity No error in X variables Error in Y variables is measurement error Normally distributed error Observations are independent No missing data

4 Heteroskedasticity

5 Solutions 1) Transform the data 1) Pro: No additional parameters 2) Cons: No longer modeling the original data, likelihood & process model have different meaning, backtransformation non-trivial (Jensen's Inequality) 2) Model the variance 1) Pro: working with original data and model, no tranf. 2) Con: additional process model and parameters (and priors)

6 Heteroskedasticity y~n 1 2 x, 1 2 x 2 X Y Data Model β, α Process Model Β 0,V β A 0,V α Parameter Model

7 Example: Linear varying SD y~n 1 2 x, 1 2 x 2 Likelihood (R) LnL = function(theta,x,y){ beta = theta[1:2] alpha = theta[3:4] -sum(dnorm(y,beta[1]+beta[2]*x,alpha[1]+alpha[2]*x,log=true)) } Bayes (WinBUGS) model{ for(i in 1:2) { beta[i] ~ dnorm(0,0.001)} ## priors for(i in 1:2) { alpha[i] ~ dlnorm(0,0.001)} for(i in 1:n){ prec[i] <- 1/pow(alpha[1] + alpha[2]*x[i],2) mu[i] <- beta[1]+beta[2]*x[i] y[i] ~ dnorm(mu[i],prec[i]) } }

8 Likelihood

9 Bayes

10 Additional thoughts on modeling variance Need not be linear Can model in terms of sd, variance, or precision Can vary with treatments/factors or categorical variables e.g. can relax the ANOVA assumptions of equal variance among treatments

11 Assumptions of Linear Model Homoskedasticity No error in X variables Error in Y variables is measurement error Normally distributed error Observations are independent No missing data

12 Errors in Variables Regression model assumes all the error is in the Y Often know there is non-negligable error in the measurement of X

13 Errors in Variables = x 1 2 y~n, 2 x o ~N x, 2 ~N B V 0, B 2 ~IG s s 1, 2 2 ~IG t t 1, 2 x~n X V 0, X Process model Data model for y Data model for x Prior for beta Prior for sigma Prior for tau Prior for X

14 Errors in Variables y~n X, 2 x o ~N x, 2 X (o) Y Data Model X β, σ Process Model X 0,V x Β 0,V β s 1,s 2 Parameter Model

15 Full Posterior p, 2, 2, X y, X o N y x, N x o x, 2 N B V 0, B IG 2 s s IG 2 t t 1, 2 1, 2 Conditionals N x X 0, V X p N y x, 2 N B V 0 1 0, B p 2 N y x, 2 IG 2 s s 0 1 1, 2 p 2 N x o x, 2 IG 2 t t 1, 2 p X N x o x, 2 N x X V 0, X

16 model { ## priors for(i in 1:2) { beta[i] ~ dnorm(0,0.001)} sigma ~ dgamma(0.1,0.1) tau ~ dgamma(0.1,0.1) for(i in 1:n) { xt[i] ~ dunif(0,10)} } for(i in 1:n){ x[i] ~ dnorm(xt[i],tau) mu[i] <- beta[1]+beta[2]*x[i] y[i] ~ dnorm(mu[i],sigma) }

17

18

19

20 Conceptually within the MCMC Update the regression model given the current values of X Update the observation error in X based on the difference between the current and observed values of X Update the values of X based on the observed values of X and the regression model Overall, integrate over the possible values of X

21 Additional Thoughts on EIV x o ~g x Errors in X's need not be Normal Errors need not be additive Can account for known biases x o ~N 0 1 x, 2

22 Additional Thoughts on EIV x o ~g x Errors in X's need not be Normal Errors need not be additive Can account for known biases x o ~N 0 1 x, 2 Observed data can be a different type (proxy) Very useful to have informative priors

23 Latent Variables Variables that are not directly observed Values are inferred from model Parameter model: prior on value Data and Process models provide constraint p X N y 0 1 x, 2 N x o x, 2 N x X 0, V X MCMC integrates over (by sampling) the values the unobserved variable could take on Contribute to uncertainty in parameters, model Ignoring this variability can lead to falsely overconfident conclusions

24 Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables Error in Y variables is measurement error Normally distributed error Observations are independent No missing data

25 Missing data models y~n X, 2 Let's assume a standard multiple regression model (homoskedastic, no error in X) If some of the y's are missing Can just predict the distribution of those values using the model PI What if some of the X's are missing The observed y is more likely to have come from some values of X than others

26 Less Likely More Likely

27 Missing Data =X y~n, 2 ~N B V 0, B 2 ~IG s s 1, 2 x ~N X V mis 0, X Process model Data model for y Prior for beta Prior for sigma Prior for missing X p x mis N X, 2 N x X 0, V X

28 Missing Data Model y~n X, 2 X Y Data Model X mis β, σ Process Model X 0,V x Β 0,V β s 1,s 2 Parameter Model

29 Conceptually within the MCMC Update the regression model based on ALL the rows of data conditioned on the current values of the missing data Update the missing data based on the current regression model and the values that all other covariates take on Overall, integrate over the uncertainty in missing X's Model uncertainty increases, but less so than if whole rows of data was dropped (partial info.)

30 ASSUMPTION!! Missing data models assume that the data is missing at random If data is missing SYSTEMATICALLY it can not be estimated

31 BUGS example: Simple Regression model{ ## priors for(i in 1:2) { beta[i] ~ dnorm(0,0.001)} sigma ~ dgamma(0.1,0.1) for(i in mis) { x[i] ~ dunif(0,10)} } for(i in 1:n){ mu[i] <- beta[1]+beta[2]*x[i] y[i] ~ dnorm(mu[i],sigma) } Vector giving indices of missing values X Y NA 7.5

32 Example

33 Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables No missing data Missing data model Normally distributed error Error in Y variables is measurement error Observations are independent

34 Generalized Linear Models Retains linear function Allows for alternate PDFs to be used in likelihood However, with many non-normal PDFs the range of the model parameters does not allow a linear function to be used safely Pois(λ): λ > 0 Binom(n,θ) 0 < θ < 1 Typically a link function is used to relate linear model to PDF

35 Distribution Normal Exponential Gamma Poisson Binomial Multinomial Link Functions Canonical Link Functions Link Name Link Function Mean Function Identity Xb = µ µ = Xb Inverse Log Logit Xb = µ -1 µ = (Xb) -1 Xb = ln(µ) µ = exp(xb) Xb=ln = exp Xb 1 1 exp Xb Can use most any function as a link function but may only be valid over a restricted range Many are technically nonlinear functions

36 Logit Xb=ln 1 Interpretation: Log of the ODDS RATIO logit(0.5) = 0.0

37 Logistic Regression Common model for the analysis of boolean data (0/1, True/False, Present/Absent) Assumes a Bernoulli likelihood Bern(θ) = Binom(1,θ) Likelihood specification y~bern logit =X Data Model Process Model Bayesian ~N B 0, V B Parameter Model

38 Logistic Regression y~binom 1,logit 1 X X Y Data Model β Process Model Β 0,V β Parameter Model

39

40

41 Logistic Regression in R Option 1 built in function glm(y ~ x, family = binomial(link= logit )) Option 2 homebrew lnl = function(beta){ -dbinom(y,1,ilogit(beta[0] + beta[1]*x),log=t) }

42

43 Call: glm(formula = y ~ x, family = binomial()) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-15 *** x < 2e-16 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 249 degrees of freedom Residual deviance: on 248 degrees of freedom AIC:

44

45 Alternative link functions probit Normal CDF cauchit - Cauchy CDF log -- =exp X cloglog - Complimentary log-log Asymmetric, often used for high or low probabilities =1 exp exp X If you code yourself, any function that projects from Real to (0,1)

46

47 Coming next... GLM Bayesian Logistic Poisson Regression Multinomial Continuing our exploration of relaxing the assumptions of linear models

Generalized Linear Models

Generalized Linear Models Generalized Linear Models Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables No missing data Missing data model Normally distributed error Error in

More information

Generalized Linear Models

Generalized Linear Models Generalized Linear Models Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables No missing data Missing data model Normally distributed error GLM Error

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

Bayesian Graphical Models

Bayesian Graphical Models Graphical Models and Inference, Lecture 16, Michaelmas Term 2009 December 4, 2009 Parameter θ, data X = x, likelihood L(θ x) p(x θ). Express knowledge about θ through prior distribution π on θ. Inference

More information

Generalized linear models

Generalized linear models Generalized linear models Douglas Bates November 01, 2010 Contents 1 Definition 1 2 Links 2 3 Estimating parameters 5 4 Example 6 5 Model building 8 6 Conclusions 8 7 Summary 9 1 Generalized Linear Models

More information

Generalized Linear Models. stat 557 Heike Hofmann

Generalized Linear Models. stat 557 Heike Hofmann Generalized Linear Models stat 557 Heike Hofmann Outline Intro to GLM Exponential Family Likelihood Equations GLM for Binomial Response Generalized Linear Models Three components: random, systematic, link

More information

Tento projekt je spolufinancován Evropským sociálním fondem a Státním rozpočtem ČR InoBio CZ.1.07/2.2.00/

Tento projekt je spolufinancován Evropským sociálním fondem a Státním rozpočtem ČR InoBio CZ.1.07/2.2.00/ Tento projekt je spolufinancován Evropským sociálním fondem a Státním rozpočtem ČR InoBio CZ.1.07/2.2.00/28.0018 Statistical Analysis in Ecology using R Linear Models/GLM Ing. Daniel Volařík, Ph.D. 13.

More information

LOGISTIC REGRESSION Joseph M. Hilbe

LOGISTIC REGRESSION Joseph M. Hilbe LOGISTIC REGRESSION Joseph M. Hilbe Arizona State University Logistic regression is the most common method used to model binary response data. When the response is binary, it typically takes the form of

More information

Lecture 19. Spatial GLM + Point Reference Spatial Data. Colin Rundel 04/03/2017

Lecture 19. Spatial GLM + Point Reference Spatial Data. Colin Rundel 04/03/2017 Lecture 19 Spatial GLM + Point Reference Spatial Data Colin Rundel 04/03/2017 1 Spatial GLM Models 2 Scottish Lip Cancer Data Observed Expected 60 N 59 N 58 N 57 N 56 N value 80 60 40 20 0 55 N 8 W 6 W

More information

Logistic Regression - problem 6.14

Logistic Regression - problem 6.14 Logistic Regression - problem 6.14 Let x 1, x 2,, x m be given values of an input variable x and let Y 1,, Y m be independent binomial random variables whose distributions depend on the corresponding values

More information

Statistics in Environmental Research (BUC Workshop Series) II Problem sheet - WinBUGS - SOLUTIONS

Statistics in Environmental Research (BUC Workshop Series) II Problem sheet - WinBUGS - SOLUTIONS Statistics in Environmental Research (BUC Workshop Series) II Problem sheet - WinBUGS - SOLUTIONS 1. (a) The posterior mean estimate of α is 14.27, and the posterior mean for the standard deviation of

More information

7/28/15. Review Homework. Overview. Lecture 6: Logistic Regression Analysis

7/28/15. Review Homework. Overview. Lecture 6: Logistic Regression Analysis Lecture 6: Logistic Regression Analysis Christopher S. Hollenbeak, PhD Jane R. Schubart, PhD The Outcomes Research Toolbox Review Homework 2 Overview Logistic regression model conceptually Logistic regression

More information

9 Generalized Linear Models

9 Generalized Linear Models 9 Generalized Linear Models The Generalized Linear Model (GLM) is a model which has been built to include a wide range of different models you already know, e.g. ANOVA and multiple linear regression models

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

Logistic Regression. James H. Steiger. Department of Psychology and Human Development Vanderbilt University

Logistic Regression. James H. Steiger. Department of Psychology and Human Development Vanderbilt University Logistic Regression James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) Logistic Regression 1 / 38 Logistic Regression 1 Introduction

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

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

Regression models. Generalized linear models in R. Normal regression models are not always appropriate. Generalized linear models. Examples.

Regression models. Generalized linear models in R. Normal regression models are not always appropriate. Generalized linear models. Examples. Regression models Generalized linear models in R Dr Peter K Dunn http://www.usq.edu.au Department of Mathematics and Computing University of Southern Queensland ASC, July 00 The usual linear regression

More information

A Generalized Linear Model for Binomial Response Data. Copyright c 2017 Dan Nettleton (Iowa State University) Statistics / 46

A Generalized Linear Model for Binomial Response Data. Copyright c 2017 Dan Nettleton (Iowa State University) Statistics / 46 A Generalized Linear Model for Binomial Response Data Copyright c 2017 Dan Nettleton (Iowa State University) Statistics 510 1 / 46 Now suppose that instead of a Bernoulli response, we have a binomial response

More information

Administration. Homework 1 on web page, due Feb 11 NSERC summer undergraduate award applications due Feb 5 Some helpful books

Administration. Homework 1 on web page, due Feb 11 NSERC summer undergraduate award applications due Feb 5 Some helpful books STA 44/04 Jan 6, 00 / 5 Administration Homework on web page, due Feb NSERC summer undergraduate award applications due Feb 5 Some helpful books STA 44/04 Jan 6, 00... administration / 5 STA 44/04 Jan 6,

More information

Metropolis-Hastings Algorithm

Metropolis-Hastings Algorithm Strength of the Gibbs sampler Metropolis-Hastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to

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

Nonlinear Models. and. Hierarchical Nonlinear Models

Nonlinear Models. and. Hierarchical Nonlinear Models Nonlinear Models and Hierarchical Nonlinear Models Start Simple Progressively Add Complexity Tree Allometries Diameter vs Height with a hierarchical species effect Three response variables: Ht, crown depth,

More information

Two Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 26 May :00 16:00

Two Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 26 May :00 16:00 Two Hours MATH38052 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER GENERALISED LINEAR MODELS 26 May 2016 14:00 16:00 Answer ALL TWO questions in Section

More information

Logistic Regressions. Stat 430

Logistic Regressions. Stat 430 Logistic Regressions Stat 430 Final Project Final Project is, again, team based You will decide on a project - only constraint is: you are supposed to use techniques for a solution that are related to

More information

WinBUGS : part 2. Bruno Boulanger Jonathan Jaeger Astrid Jullion Philippe Lambert. Gabriele, living with rheumatoid arthritis

WinBUGS : part 2. Bruno Boulanger Jonathan Jaeger Astrid Jullion Philippe Lambert. Gabriele, living with rheumatoid arthritis WinBUGS : part 2 Bruno Boulanger Jonathan Jaeger Astrid Jullion Philippe Lambert Gabriele, living with rheumatoid arthritis Agenda 2! Hierarchical model: linear regression example! R2WinBUGS Linear Regression

More information

You can specify the response in the form of a single variable or in the form of a ratio of two variables denoted events/trials.

You can specify the response in the form of a single variable or in the form of a ratio of two variables denoted events/trials. The GENMOD Procedure MODEL Statement MODEL response = < effects > < /options > ; MODEL events/trials = < effects > < /options > ; You can specify the response in the form of a single variable or in the

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

Exercise 5.4 Solution

Exercise 5.4 Solution Exercise 5.4 Solution Niels Richard Hansen University of Copenhagen May 7, 2010 1 5.4(a) > leukemia

More information

Generalised linear models. Response variable can take a number of different formats

Generalised linear models. Response variable can take a number of different formats Generalised linear models Response variable can take a number of different formats Structure Limitations of linear models and GLM theory GLM for count data GLM for presence \ absence data GLM for proportion

More information

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

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

More information

Bayesian Inference for Regression Parameters

Bayesian Inference for Regression Parameters Bayesian Inference for Regression Parameters 1 Bayesian inference for simple linear regression parameters follows the usual pattern for all Bayesian analyses: 1. Form a prior distribution over all unknown

More information

Bayesian Model Diagnostics and Checking

Bayesian Model Diagnostics and Checking Earvin Balderama Quantitative Ecology Lab Department of Forestry and Environmental Resources North Carolina State University April 12, 2013 1 / 34 Introduction MCMCMC 2 / 34 Introduction MCMCMC Steps in

More information

Lecture 3. G. Cowan. Lecture 3 page 1. Lectures on Statistical Data Analysis

Lecture 3. G. Cowan. Lecture 3 page 1. Lectures on Statistical Data Analysis Lecture 3 1 Probability (90 min.) Definition, Bayes theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests (90 min.) general concepts, test statistics,

More information

DIC: Deviance Information Criterion

DIC: Deviance Information Criterion (((( Welcome Page Latest News DIC: Deviance Information Criterion Contact us/bugs list WinBUGS New WinBUGS examples FAQs DIC GeoBUGS DIC (Deviance Information Criterion) is a Bayesian method for model

More information

Log-linear Models for Contingency Tables

Log-linear Models for Contingency Tables Log-linear Models for Contingency Tables Statistics 149 Spring 2006 Copyright 2006 by Mark E. Irwin Log-linear Models for Two-way Contingency Tables Example: Business Administration Majors and Gender A

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

Generalized linear models for binary data. A better graphical exploratory data analysis. The simple linear logistic regression model

Generalized linear models for binary data. A better graphical exploratory data analysis. The simple linear logistic regression model Stat 3302 (Spring 2017) Peter F. Craigmile Simple linear logistic regression (part 1) [Dobson and Barnett, 2008, Sections 7.1 7.3] Generalized linear models for binary data Beetles dose-response example

More information

Weakness of Beta priors (or conjugate priors in general) They can only represent a limited range of prior beliefs. For example... There are no bimodal beta distributions (except when the modes are at 0

More information

Normal distribution We have a random sample from N(m, υ). The sample mean is Ȳ and the corrected sum of squares is S yy. After some simplification,

Normal distribution We have a random sample from N(m, υ). The sample mean is Ȳ and the corrected sum of squares is S yy. After some simplification, Likelihood Let P (D H) be the probability an experiment produces data D, given hypothesis H. Usually H is regarded as fixed and D variable. Before the experiment, the data D are unknown, and the probability

More information

SCHOOL OF MATHEMATICS AND STATISTICS. Linear and Generalised Linear Models

SCHOOL OF MATHEMATICS AND STATISTICS. Linear and Generalised Linear Models SCHOOL OF MATHEMATICS AND STATISTICS Linear and Generalised Linear Models Autumn Semester 2017 18 2 hours Attempt all the questions. The allocation of marks is shown in brackets. RESTRICTED OPEN BOOK EXAMINATION

More information

R Hints for Chapter 10

R Hints for Chapter 10 R Hints for Chapter 10 The multiple logistic regression model assumes that the success probability p for a binomial random variable depends on independent variables or design variables x 1, x 2,, x k.

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models Generalized Linear Models - part III Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs.

More information

Robust Bayesian Regression

Robust Bayesian Regression Readings: Hoff Chapter 9, West JRSSB 1984, Fúquene, Pérez & Pericchi 2015 Duke University November 17, 2016 Body Fat Data: Intervals w/ All Data Response % Body Fat and Predictor Waist Circumference 95%

More information

Introduction to the Generalized Linear Model: Logistic regression and Poisson regression

Introduction to the Generalized Linear Model: Logistic regression and Poisson regression Introduction to the Generalized Linear Model: Logistic regression and Poisson regression Statistical modelling: Theory and practice Gilles Guillot gigu@dtu.dk November 4, 2013 Gilles Guillot (gigu@dtu.dk)

More information

Modeling Overdispersion

Modeling Overdispersion James H. Steiger Department of Psychology and Human Development Vanderbilt University Regression Modeling, 2009 1 Introduction 2 Introduction In this lecture we discuss the problem of overdispersion in

More information

Classification. Chapter Introduction. 6.2 The Bayes classifier

Classification. Chapter Introduction. 6.2 The Bayes classifier Chapter 6 Classification 6.1 Introduction Often encountered in applications is the situation where the response variable Y takes values in a finite set of labels. For example, the response Y could encode

More information

Weakness of Beta priors (or conjugate priors in general) They can only represent a limited range of prior beliefs. For example... There are no bimodal beta distributions (except when the modes are at 0

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

Logistic & Tobit Regression

Logistic & Tobit Regression Logistic & Tobit Regression Different Types of Regression Binary Regression (D) Logistic transformation + e P( y x) = 1 + e! " x! + " x " P( y x) % ln$ ' = ( + ) x # 1! P( y x) & logit of P(y x){ P(y

More information

Random Independent Variables

Random Independent Variables Random Independent Variables Example: e 1, e 2, e 3 ~ Poisson(5) X 1 = e 1 + e 3 X 2 = e 2 +e 3 X 3 ~ N(10,10) ind. of e 1 e 2 e 3 Y = β 0 + β 1 X 1 + β 2 X 2 +β 3 X 3 + ε ε ~ N(0,σ 2 ) β 0 = 0 β 1 = 1

More information

UNIVERSITY OF TORONTO. Faculty of Arts and Science APRIL 2010 EXAMINATIONS STA 303 H1S / STA 1002 HS. Duration - 3 hours. Aids Allowed: Calculator

UNIVERSITY OF TORONTO. Faculty of Arts and Science APRIL 2010 EXAMINATIONS STA 303 H1S / STA 1002 HS. Duration - 3 hours. Aids Allowed: Calculator UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL 2010 EXAMINATIONS STA 303 H1S / STA 1002 HS Duration - 3 hours Aids Allowed: Calculator LAST NAME: FIRST NAME: STUDENT NUMBER: There are 27 pages

More information

Generalized Linear Models

Generalized Linear Models Generalized Linear Models 1/37 The Kelp Data FRONDS 0 20 40 60 20 40 60 80 100 HLD_DIAM FRONDS are a count variable, cannot be < 0 2/37 Nonlinear Fits! FRONDS 0 20 40 60 log NLS 20 40 60 80 100 HLD_DIAM

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

Clinical Trials. Olli Saarela. September 18, Dalla Lana School of Public Health University of Toronto.

Clinical Trials. Olli Saarela. September 18, Dalla Lana School of Public Health University of Toronto. Introduction to Dalla Lana School of Public Health University of Toronto olli.saarela@utoronto.ca September 18, 2014 38-1 : a review 38-2 Evidence Ideal: to advance the knowledge-base of clinical medicine,

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

Multivariate Statistics in Ecology and Quantitative Genetics Summary

Multivariate Statistics in Ecology and Quantitative Genetics Summary Multivariate Statistics in Ecology and Quantitative Genetics Summary Dirk Metzler & Martin Hutzenthaler http://evol.bio.lmu.de/_statgen 5. August 2011 Contents Linear Models Generalized Linear Models Mixed-effects

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

Lecture 6. Prior distributions

Lecture 6. Prior distributions Summary Lecture 6. Prior distributions 1. Introduction 2. Bivariate conjugate: normal 3. Non-informative / reference priors Jeffreys priors Location parameters Proportions Counts and rates Scale parameters

More information

Age 55 (x = 1) Age < 55 (x = 0)

Age 55 (x = 1) Age < 55 (x = 0) Logistic Regression with a Single Dichotomous Predictor EXAMPLE: Consider the data in the file CHDcsv Instead of examining the relationship between the continuous variable age and the presence or absence

More information

Generalized Models: Part 1

Generalized Models: Part 1 Generalized Models: Part 1 Topics: Introduction to generalized models Introduction to maximum likelihood estimation Models for binary outcomes Models for proportion outcomes Models for categorical outcomes

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

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

Logistic Regression 21/05

Logistic Regression 21/05 Logistic Regression 21/05 Recall that we are trying to solve a classification problem in which features x i can be continuous or discrete (coded as 0/1) and the response y is discrete (0/1). Logistic regression

More information

BMI 541/699 Lecture 22

BMI 541/699 Lecture 22 BMI 541/699 Lecture 22 Where we are: 1. Introduction and Experimental Design 2. Exploratory Data Analysis 3. Probability 4. T-based methods for continous variables 5. Power and sample size for t-based

More information

1. Hypothesis testing through analysis of deviance. 3. Model & variable selection - stepwise aproaches

1. Hypothesis testing through analysis of deviance. 3. Model & variable selection - stepwise aproaches Sta 216, Lecture 4 Last Time: Logistic regression example, existence/uniqueness of MLEs Today s Class: 1. Hypothesis testing through analysis of deviance 2. Standard errors & confidence intervals 3. Model

More information

MSH3 Generalized linear model

MSH3 Generalized linear model Contents MSH3 Generalized linear model 5 Logit Models for Binary Data 173 5.1 The Bernoulli and binomial distributions......... 173 5.1.1 Mean, variance and higher order moments.... 173 5.1.2 Normal limit....................

More information

Statistics 203 Introduction to Regression Models and ANOVA Practice Exam

Statistics 203 Introduction to Regression Models and ANOVA Practice Exam Statistics 203 Introduction to Regression Models and ANOVA Practice Exam Prof. J. Taylor You may use your 4 single-sided pages of notes This exam is 7 pages long. There are 4 questions, first 3 worth 10

More information

Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square

Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square Likelihood Ratio Test in High-Dimensional Logistic Regression Is Asymptotically a Rescaled Chi-Square Yuxin Chen Electrical Engineering, Princeton University Coauthors Pragya Sur Stanford Statistics Emmanuel

More information

Logit Regression and Quantities of Interest

Logit Regression and Quantities of Interest Logit Regression and Quantities of Interest Stephen Pettigrew March 5, 2014 Stephen Pettigrew Logit Regression and Quantities of Interest March 5, 2014 1 / 59 Outline 1 Logistics 2 Generalized Linear Models

More information

STAT Lecture 11: Bayesian Regression

STAT Lecture 11: Bayesian Regression STAT 491 - Lecture 11: Bayesian Regression Generalized Linear Models Generalized linear models (GLMs) are a class of techniques that include linear regression, logistic regression, and Poisson regression.

More information

Class Notes: Week 8. Probit versus Logit Link Functions and Count Data

Class Notes: Week 8. Probit versus Logit Link Functions and Count Data Ronald Heck Class Notes: Week 8 1 Class Notes: Week 8 Probit versus Logit Link Functions and Count Data This week we ll take up a couple of issues. The first is working with a probit link function. While

More information

Logit Regression and Quantities of Interest

Logit Regression and Quantities of Interest Logit Regression and Quantities of Interest Stephen Pettigrew March 4, 2015 Stephen Pettigrew Logit Regression and Quantities of Interest March 4, 2015 1 / 57 Outline 1 Logistics 2 Generalized Linear Models

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models Generalized Linear Models - part II Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs.

More information

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

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

Introduction to Generalized Models

Introduction to Generalized Models Introduction to Generalized Models Today s topics: The big picture of generalized models Review of maximum likelihood estimation Models for binary outcomes Models for proportion outcomes Models for categorical

More information

BUGS Bayesian inference Using Gibbs Sampling

BUGS Bayesian inference Using Gibbs Sampling BUGS Bayesian inference Using Gibbs Sampling Glen DePalma Department of Statistics May 30, 2013 www.stat.purdue.edu/~gdepalma 1 / 20 Bayesian Philosophy I [Pearl] turned Bayesian in 1971, as soon as I

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

STA 450/4000 S: January

STA 450/4000 S: January STA 450/4000 S: January 6 005 Notes Friday tutorial on R programming reminder office hours on - F; -4 R The book Modern Applied Statistics with S by Venables and Ripley is very useful. Make sure you have

More information

On the Inference of the Logistic Regression Model

On the Inference of the Logistic Regression Model On the Inference of the Logistic Regression Model 1. Model ln =(; ), i.e. = representing false. The linear form of (;) is entertained, i.e. ((;)) ((;)), where ==1 ;, with 1 representing true, 0 ;= 1+ +

More information

Variability within multi-component systems. Bayesian inference in probabilistic risk assessment The current state of the art

Variability within multi-component systems. Bayesian inference in probabilistic risk assessment The current state of the art PhD seminar series Probabilistics in Engineering : g Bayesian networks and Bayesian hierarchical analysis in engeering g Conducted by Prof. Dr. Maes, Prof. Dr. Faber and Dr. Nishijima Variability within

More information

CS 361: Probability & Statistics

CS 361: Probability & Statistics October 17, 2017 CS 361: Probability & Statistics Inference Maximum likelihood: drawbacks A couple of things might trip up max likelihood estimation: 1) Finding the maximum of some functions can be quite

More information

STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis. 1. Indicate whether each of the following is true (T) or false (F).

STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis. 1. Indicate whether each of the following is true (T) or false (F). STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis 1. Indicate whether each of the following is true (T) or false (F). (a) T In 2 2 tables, statistical independence is equivalent to a population

More information

Bayesian inference for sample surveys. Roderick Little Module 2: Bayesian models for simple random samples

Bayesian inference for sample surveys. Roderick Little Module 2: Bayesian models for simple random samples Bayesian inference for sample surveys Roderick Little Module : Bayesian models for simple random samples Superpopulation Modeling: Estimating parameters Various principles: least squares, method of moments,

More information

Nonlinear Models. What do you do when you don t have a line? What do you do when you don t have a line? A Quadratic Adventure

Nonlinear Models. What do you do when you don t have a line? What do you do when you don t have a line? A Quadratic Adventure What do you do when you don t have a line? Nonlinear Models Spores 0e+00 2e+06 4e+06 6e+06 8e+06 30 40 50 60 70 longevity What do you do when you don t have a line? A Quadratic Adventure 1. If nonlinear

More information

NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION. ST3241 Categorical Data Analysis. (Semester II: ) April/May, 2011 Time Allowed : 2 Hours

NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION. ST3241 Categorical Data Analysis. (Semester II: ) April/May, 2011 Time Allowed : 2 Hours NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION Categorical Data Analysis (Semester II: 2010 2011) April/May, 2011 Time Allowed : 2 Hours Matriculation No: Seat No: Grade Table Question 1 2 3 4 5 6 Full marks

More information

R Output for Linear Models using functions lm(), gls() & glm()

R Output for Linear Models using functions lm(), gls() & glm() LM 04 lm(), gls() &glm() 1 R Output for Linear Models using functions lm(), gls() & glm() Different kinds of output related to linear models can be obtained in R using function lm() {stats} in the base

More information

Probability Distributions Columns (a) through (d)

Probability Distributions Columns (a) through (d) Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)

More information

STA 303 H1S / 1002 HS Winter 2011 Test March 7, ab 1cde 2abcde 2fghij 3

STA 303 H1S / 1002 HS Winter 2011 Test March 7, ab 1cde 2abcde 2fghij 3 STA 303 H1S / 1002 HS Winter 2011 Test March 7, 2011 LAST NAME: FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 303 STA 1002 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator. Some formulae

More information

Stat 5102 Final Exam May 14, 2015

Stat 5102 Final Exam May 14, 2015 Stat 5102 Final Exam May 14, 2015 Name Student ID The exam is closed book and closed notes. You may use three 8 1 11 2 sheets of paper with formulas, etc. You may also use the handouts on brand name distributions

More information

TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1

TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1 TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1 1.1 The Probability Model...1 1.2 Finite Discrete Models with Equally Likely Outcomes...5 1.2.1 Tree Diagrams...6 1.2.2 The Multiplication Principle...8

More information

Hands on cusp package tutorial

Hands on cusp package tutorial Hands on cusp package tutorial Raoul P. P. P. Grasman July 29, 2015 1 Introduction The cusp package provides routines for fitting a cusp catastrophe model as suggested by (Cobb, 1978). The full documentation

More information

ssh tap sas913, sas https://www.statlab.umd.edu/sasdoc/sashtml/onldoc.htm

ssh tap sas913, sas https://www.statlab.umd.edu/sasdoc/sashtml/onldoc.htm Kedem, STAT 430 SAS Examples: Logistic Regression ==================================== ssh abc@glue.umd.edu, tap sas913, sas https://www.statlab.umd.edu/sasdoc/sashtml/onldoc.htm a. Logistic regression.

More information

Statistical Methods in HYDROLOGY CHARLES T. HAAN. The Iowa State University Press / Ames

Statistical Methods in HYDROLOGY CHARLES T. HAAN. The Iowa State University Press / Ames Statistical Methods in HYDROLOGY CHARLES T. HAAN The Iowa State University Press / Ames Univariate BASIC Table of Contents PREFACE xiii ACKNOWLEDGEMENTS xv 1 INTRODUCTION 1 2 PROBABILITY AND PROBABILITY

More information

Package HGLMMM for Hierarchical Generalized Linear Models

Package HGLMMM for Hierarchical Generalized Linear Models Package HGLMMM for Hierarchical Generalized Linear Models Marek Molas Emmanuel Lesaffre Erasmus MC Erasmus Universiteit - Rotterdam The Netherlands ERASMUSMC - Biostatistics 20-04-2010 1 / 52 Outline General

More information

Bayesian Methods for Machine Learning

Bayesian Methods for Machine Learning Bayesian Methods for Machine Learning CS 584: Big Data Analytics Material adapted from Radford Neal s tutorial (http://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf), Zoubin Ghahramni (http://hunch.net/~coms-4771/zoubin_ghahramani_bayesian_learning.pdf),

More information

NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION (SOLUTIONS) ST3241 Categorical Data Analysis. (Semester II: )

NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION (SOLUTIONS) ST3241 Categorical Data Analysis. (Semester II: ) NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION (SOLUTIONS) Categorical Data Analysis (Semester II: 2010 2011) April/May, 2011 Time Allowed : 2 Hours Matriculation No: Seat No: Grade Table Question 1 2 3

More information

Now consider the case where E(Y) = µ = Xβ and V (Y) = σ 2 G, where G is diagonal, but unknown.

Now consider the case where E(Y) = µ = Xβ and V (Y) = σ 2 G, where G is diagonal, but unknown. Weighting We have seen that if E(Y) = Xβ and V (Y) = σ 2 G, where G is known, the model can be rewritten as a linear model. This is known as generalized least squares or, if G is diagonal, with trace(g)

More information

STAC51: Categorical data Analysis

STAC51: Categorical data Analysis STAC51: Categorical data Analysis Mahinda Samarakoon April 6, 2016 Mahinda Samarakoon STAC51: Categorical data Analysis 1 / 25 Table of contents 1 Building and applying logistic regression models (Chap

More information