The Prediction of Random Effects in Generalized Linear Mixed Model
|
|
- Clarissa Curtis
- 5 years ago
- Views:
Transcription
1 The Prediction of Random Effects in Generalized Linear Mixed Model Wenxue Ge Department of Mathematics and Statistics University of Victoria, Canada Nov 20, of Victoria
2 Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
3 Outline Background GLM and GLMM 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
4 GLM and GLMM Exponential Family f(y;θ) = s(y) t(θ) exp[a(y)b(θ)] f(y;θ) = exp[ a(y)b(θ) + c(θ) + d(y) ] -where s(y) = exp[ d(y) ]; t(θ) = exp[ c(θ) ] Canonical if a(y) = y; b(θ) is natural parameter
5 GLM and GLMM Generalized Linear Model - Y 1,...,Y N from the exponential family 1. Y i has the canonical form f(y i ;θ i ) = exp [ y i b i (θ i ) + c i (θ i ) + d i (y i )] 2. Y i s same distribution form - Link function g(µ i ) = x T i β Monotone, Differentiable
6 GLM and GLMM Generalized Linear Mixed Model - Random Effects Incorporating correlation, broader inference - General Model
7 Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
8 , or BLUP - "Best": miminum MSE (not within all predictors) - Model Var [ ] u = e [ ] G 0 σ 2 0 R E(u) = 0, E(e) = 0
9 Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
10 Henderson s Justification, u N q (0,Gσ 2 ) f (y, u) = g(y u)h(u) (1) = (2πσ 2 ) 1 2 n 1 2 q (det G 0 ) exp{ 0 R 2σ 2 [u T G 1 u + (y Xβ Z u) T R 1 (y Xβ Z u)]} Bayesian Derivation, u N q (0,Gσ 2 ) - β: uniform, improper prior - Posterior density is proportional to (1)
11 Minimize u T G 1 u + (y Xβ Z u) T R 1 (y Xβ Z u) Mixed model equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3) - Eliminate u û = (Z T R 1 Z + G 1 ) 1 Z T R 1 (y X ˆβ) (4) gives X T R 1 X ˆβ X T R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 X ˆβ = X T R 1 y X T R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 y
12 Simplify X T (R 1 R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 )X ˆβ = X T (R 1 R 1 Z (Z T R 1 Z + G 1 ) 1 Z T R 1 )y and (R+ZGZ T ) 1 = R 1 R 1 Z (Z T R 1 Z +G 1 ) 1 Z T R 1 gives X T (R + ZGZ T ) 1 X ˆβ = X T (R + ZGZ T ) 1 y (5) Solution (Note:û = (Z T R 1 Z +G 1 ) 1 Z T R 1 (y X ˆβ)) ˆβ = (X T (R + ZGZ T ) 1 X) 1 X T (R + ZGZ T ) 1 y (6) û = (Z T R 1 Z + G 1 ) 1 {[Z T R 1 Z T R 1 X] [X T (R + ZGZ T ) 1 X] 1 X T (R + ZGZ T ) 1 }y (7)
13 Outline Background 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
14 BLUP(b T β + c T u) - b T and c T are given - Seek λ : λ T y unbiase for b T β + c T u and minimize var(λ T y (b T β + c T u)) + 2m T (X T λ X T b) = λ T var(y)λ + c T var(u)c-2λ T cov(u, y T ) +2 m T (X T λ X T b) 2m Lagrange multipliers Yields λ T soly as BLUP(b T β + c T u) = b T ˆβ + c T (Z T R 1 Z + G 1 ) 1 Z T R 1 (y X ˆβ)
15 Harville and Robinson - More intuitive - Linear Combination: b T β + c T u - Any Linear Unbiased Estimator: b T ˆβ + c T û + a T y - Where X T a = 0 b T and c T are given
16 Robinson Continued E[yy T ] = Xββ T X T + ZGZ T σ 2 + Rσ 2 E[uy T ] = GZ T σ 2 E[(ˆβ β)y T a] =[X T (R + ZGZ T ) 1 X] 1 X T (ZGZ T + R) 1 E[yy T ]a βe[y T ]a =ββ T X T a + [X T (R + ZGZ T ) 1 X] 1 X T σ 2 a ββ T X T a =0 Same steps E[(û u)y T a] = 0
17 "Best" E{[b T (ˆβ β) + c T (û u)a T y] [b T (ˆβ β) + c T (û u)a T y] T } = E{[b T (ˆβ β) + c T (û u)][b T (ˆβ β) + c T (û u)] T } + E[a T yy T a] + E{[b T (ˆβ β) + c T (û u)]y T a} + E{a T y[b T (ˆβ β) + c T (û u)]} = E{[b T (ˆβ β) + c T (û u)][b T (ˆβ β) + c T (û u)] T } + E[a T yy T a]
18 Outline Background First Lactation Yields 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
19 First Lactation Yields Model: y 9x1 = X 9x3 β 3x1 + Z 9x4 u 4x1 + e 9x1 y : Yield β : Herd u : Sire
20 First Lactation Yields Mixed model Equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3) R 9x9 = I 9x9 G 4x4 = 0.1I 4x4
21 First Lactation Yields Mixed model Equations X T R 1 X ˆβ + X T R 1 Z û = X T R 1 y (2) Z T R 1 X ˆβ + (Z T R 1 Z + G 1 )û = Z T R 1 y (3)
22 First Lactation Yields Solutions
23 Outline Background Expansions 1 Background GLM and GLMM 2 3 First Lactation Yields 4 Expansions
24 Expansions Why have random effects Expansion - Goldberger s Derivition Not require normal - Restricted Maximum Likelihood Variance parameters
25 Expansions
26 Expansions
MIXED MODELS THE GENERAL MIXED MODEL
MIXED MODELS This chapter introduces best linear unbiased prediction (BLUP), a general method for predicting random effects, while Chapter 27 is concerned with the estimation of variances by restricted
More informationChapter 5 Prediction of Random Variables
Chapter 5 Prediction of Random Variables C R Henderson 1984 - Guelph We have discussed estimation of β, regarded as fixed Now we shall consider a rather different problem, prediction of random variables,
More informationMixed-Models. version 30 October 2011
Mixed-Models version 30 October 2011 Mixed models Mixed models estimate a vector! of fixed effects and one (or more) vectors u of random effects Both fixed and random effects models always include a vector
More informationMixed-Model Estimation of genetic variances. Bruce Walsh lecture notes Uppsala EQG 2012 course version 28 Jan 2012
Mixed-Model Estimation of genetic variances Bruce Walsh lecture notes Uppsala EQG 01 course version 8 Jan 01 Estimation of Var(A) and Breeding Values in General Pedigrees The above designs (ANOVA, P-O
More information1 Mixed effect models and longitudinal data analysis
1 Mixed effect models and longitudinal data analysis Mixed effects models provide a flexible approach to any situation where data have a grouping structure which introduces some kind of correlation between
More informationSTA216: 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 informationGeneralized linear models
Generalized linear models Søren Højsgaard Department of Mathematical Sciences Aalborg University, Denmark October 29, 202 Contents Densities for generalized linear models. Mean and variance...............................
More informationGeneralized 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 informationLinear Methods for Prediction
Chapter 5 Linear Methods for Prediction 5.1 Introduction We now revisit the classification problem and focus on linear methods. Since our prediction Ĝ(x) will always take values in the discrete set G we
More informationMS&E 226: Small Data. Lecture 11: Maximum likelihood (v2) Ramesh Johari
MS&E 226: Small Data Lecture 11: Maximum likelihood (v2) Ramesh Johari ramesh.johari@stanford.edu 1 / 18 The likelihood function 2 / 18 Estimating the parameter This lecture develops the methodology behind
More informationOutline of GLMs. Definitions
Outline of GLMs Definitions This is a short outline of GLM details, adapted from the book Nonparametric Regression and Generalized Linear Models, by Green and Silverman. The responses Y i have density
More information21. Best Linear Unbiased Prediction (BLUP) of Random Effects in the Normal Linear Mixed Effects Model
21. Best Linear Unbiased Prediction (BLUP) of Random Effects in the Normal Linear Mixed Effects Model Copyright c 2018 (Iowa State University) 21. Statistics 510 1 / 26 C. R. Henderson Born April 1, 1911,
More informationSB1a Applied Statistics Lectures 9-10
SB1a Applied Statistics Lectures 9-10 Dr Geoff Nicholls Week 5 MT15 - Natural or canonical) exponential families - Generalised Linear Models for data - Fitting GLM s to data MLE s Iteratively Re-weighted
More informationIntroduction to Generalized Linear Models
Introduction to Generalized Linear Models Edps/Psych/Soc 589 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2018 Outline Introduction (motivation
More informationLecture 9 Multi-Trait Models, Binary and Count Traits
Lecture 9 Multi-Trait Models, Binary and Count Traits Guilherme J. M. Rosa University of Wisconsin-Madison Mixed Models in Quantitative Genetics SISG, Seattle 18 0 September 018 OUTLINE Multiple-trait
More informationVarious types of likelihood
Various types of likelihood 1. likelihood, marginal likelihood, conditional likelihood, profile likelihood, adjusted profile likelihood 2. semi-parametric likelihood, partial likelihood 3. empirical likelihood,
More informationVariational Bayes and Variational Message Passing
Variational Bayes and Variational Message Passing Mohammad Emtiyaz Khan CS,UBC Variational Bayes and Variational Message Passing p.1/16 Variational Inference Find a tractable distribution Q(H) that closely
More informationGeneralized 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 informationThe Simple Regression Model. Part II. The Simple Regression Model
Part II The Simple Regression Model As of Sep 22, 2015 Definition 1 The Simple Regression Model Definition Estimation of the model, OLS OLS Statistics Algebraic properties Goodness-of-Fit, the R-square
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7
MA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7 1 Random Vectors Let a 0 and y be n 1 vectors, and let A be an n n matrix. Here, a 0 and A are non-random, whereas y is
More informationIntroduction to Estimation Methods for Time Series models Lecture 2
Introduction to Estimation Methods for Time Series models Lecture 2 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 2 SNS Pisa 1 / 21 Estimators:
More informationLinear Mixed Models. One-way layout REML. Likelihood. Another perspective. Relationship to classical ideas. Drawbacks.
Linear Mixed Models One-way layout Y = Xβ + Zb + ɛ where X and Z are specified design matrices, β is a vector of fixed effect coefficients, b and ɛ are random, mean zero, Gaussian if needed. Usually think
More informationIntroduction 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 informationStatistics 203: Introduction to Regression and Analysis of Variance Penalized models
Statistics 203: Introduction to Regression and Analysis of Variance Penalized models Jonathan Taylor - p. 1/15 Today s class Bias-Variance tradeoff. Penalized regression. Cross-validation. - p. 2/15 Bias-variance
More informationApplied Linear Statistical Methods
Applied Linear Statistical Methods (short lecturenotes) Prof. Rozenn Dahyot School of Computer Science and Statistics Trinity College Dublin Ireland www.scss.tcd.ie/rozenn.dahyot Hilary Term 2016 1. Introduction
More informationChapter 3 Best Linear Unbiased Estimation
Chapter 3 Best Linear Unbiased Estimation C R Henderson 1984 - Guelph In Chapter 2 we discussed linear unbiased estimation of k β, having determined that it is estimable Let the estimate be a y, and if
More informationContextual Effects in Modeling for Small Domains
University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Engineering and Information Sciences 2011 Contextual Effects in
More informationGauss Markov & Predictive Distributions
Gauss Markov & Predictive Distributions Merlise Clyde STA721 Linear Models Duke University September 14, 2017 Outline Topics Gauss-Markov Theorem Estimability and Prediction Readings: Christensen Chapter
More informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More informationLinear Methods for Prediction
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationLogistic regression. 11 Nov Logistic regression (EPFL) Applied Statistics 11 Nov / 20
Logistic regression 11 Nov 2010 Logistic regression (EPFL) Applied Statistics 11 Nov 2010 1 / 20 Modeling overview Want to capture important features of the relationship between a (set of) variable(s)
More informationSTA 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 informationLecture 5: BLUP (Best Linear Unbiased Predictors) of genetic values. Bruce Walsh lecture notes Tucson Winter Institute 9-11 Jan 2013
Lecture 5: BLUP (Best Linear Unbiased Predictors) of genetic values Bruce Walsh lecture notes Tucson Winter Institute 9-11 Jan 013 1 Estimation of Var(A) and Breeding Values in General Pedigrees The classic
More informationVARIANCE COMPONENT ESTIMATION & BEST LINEAR UNBIASED PREDICTION (BLUP)
VARIANCE COMPONENT ESTIMATION & BEST LINEAR UNBIASED PREDICTION (BLUP) V.K. Bhatia I.A.S.R.I., Library Avenue, New Delhi- 11 0012 vkbhatia@iasri.res.in Introduction Variance components are commonly used
More informationThe consequences of misspecifying the random effects distribution when fitting generalized linear mixed models
The consequences of misspecifying the random effects distribution when fitting generalized linear mixed models John M. Neuhaus Charles E. McCulloch Division of Biostatistics University of California, San
More informationPart IB Statistics. Theorems with proof. Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua. Lent 2015
Part IB Statistics Theorems with proof Based on lectures by D. Spiegelhalter Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationChapter 11 MIVQUE of Variances and Covariances
Chapter 11 MIVQUE of Variances and Covariances C R Henderson 1984 - Guelph The methods described in Chapter 10 for estimation of variances are quadratic, translation invariant, and unbiased For the balanced
More informationStat 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 informationChapter 12 REML and ML Estimation
Chapter 12 REML and ML Estimation C. R. Henderson 1984 - Guelph 1 Iterative MIVQUE The restricted maximum likelihood estimator (REML) of Patterson and Thompson (1971) can be obtained by iterating on MIVQUE,
More informationSCHOOL 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 informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 47 Outline 1 Regression 2 Simple Linear regression 3 Basic concepts in regression 4 How to estimate unknown parameters 5 Properties of Least Squares Estimators:
More informationOutline for today. Computation of the likelihood function for GLMMs. Likelihood for generalized linear mixed model
Outline for today Computation of the likelihood function for GLMMs asmus Waagepetersen Department of Mathematics Aalborg University Denmark Computation of likelihood function. Gaussian quadrature 2. Monte
More informationThe 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 informationApproximating models. Nancy Reid, University of Toronto. Oxford, February 6.
Approximating models Nancy Reid, University of Toronto Oxford, February 6 www.utstat.utoronto.reid/research 1 1. Context Likelihood based inference model f(y; θ), log likelihood function l(θ; y) y = (y
More informationParameter Estimation
Parameter Estimation Consider a sample of observations on a random variable Y. his generates random variables: (y 1, y 2,, y ). A random sample is a sample (y 1, y 2,, y ) where the random variables y
More informationStats 579 Intermediate Bayesian Modeling. Assignment # 2 Solutions
Stats 579 Intermediate Bayesian Modeling Assignment # 2 Solutions 1. Let w Gy) with y a vector having density fy θ) and G having a differentiable inverse function. Find the density of w in general and
More informationParametric Bootstrap Methods for Bias Correction in Linear Mixed Models
CIRJE-F-801 Parametric Bootstrap Methods for Bias Correction in Linear Mixed Models Tatsuya Kubokawa University of Tokyo Bui Nagashima Graduate School of Economics, University of Tokyo April 2011 CIRJE
More informationMLES & Multivariate Normal Theory
Merlise Clyde September 6, 2016 Outline Expectations of Quadratic Forms Distribution Linear Transformations Distribution of estimates under normality Properties of MLE s Recap Ŷ = ˆµ is an unbiased estimate
More informationStat260: Bayesian Modeling and Inference Lecture Date: March 10, 2010
Stat60: Bayesian Modelin and Inference Lecture Date: March 10, 010 Bayes Factors, -priors, and Model Selection for Reression Lecturer: Michael I. Jordan Scribe: Tamara Broderick The readin for this lecture
More informationPractical Econometrics. for. Finance and Economics. (Econometrics 2)
Practical Econometrics for Finance and Economics (Econometrics 2) Seppo Pynnönen and Bernd Pape Department of Mathematics and Statistics, University of Vaasa 1. Introduction 1.1 Econometrics Econometrics
More informationReview of Econometrics
Review of Econometrics Zheng Tian June 5th, 2017 1 The Essence of the OLS Estimation Multiple regression model involves the models as follows Y i = β 0 + β 1 X 1i + β 2 X 2i + + β k X ki + u i, i = 1,...,
More informationLecture Notes. Introduction
5/3/016 Lecture Notes R. Rekaya June 1-10, 016 Introduction Variance components play major role in animal breeding and genetic (estimation of BVs) It has been an active area of research since early 1950
More information3. Linear Regression With a Single Regressor
3. Linear Regression With a Single Regressor Econometrics: (I) Application of statistical methods in empirical research Testing economic theory with real-world data (data analysis) 56 Econometrics: (II)
More informationREGRESSION WITH SPATIALLY MISALIGNED DATA. Lisa Madsen Oregon State University David Ruppert Cornell University
REGRESSION ITH SPATIALL MISALIGNED DATA Lisa Madsen Oregon State University David Ruppert Cornell University SPATIALL MISALIGNED DATA 10 X X X X X X X X 5 X X X X X 0 X 0 5 10 OUTLINE 1. Introduction 2.
More informationSTAT5044: Regression and Anova
STAT5044: Regression and Anova Inyoung Kim 1 / 15 Outline 1 Fitting GLMs 2 / 15 Fitting GLMS We study how to find the maxlimum likelihood estimator ˆβ of GLM parameters The likelihood equaions are usually
More informationTopic 25 - One-Way Random Effects Models. Outline. Random Effects vs Fixed Effects. Data for One-way Random Effects Model. One-way Random effects
Topic 5 - One-Way Random Effects Models One-way Random effects Outline Model Variance component estimation - Fall 013 Confidence intervals Topic 5 Random Effects vs Fixed Effects Consider factor with numerous
More information36-720: Linear Mixed Models
36-720: Linear Mixed Models Brian Junker October 8, 2007 Review: Linear Mixed Models (LMM s) Bayesian Analogues Facilities in R Computational Notes Predictors and Residuals Examples [Related to Christensen
More informationStatistical Models. ref: chapter 1 of Bates, D and D. Watts (1988) Nonlinear Regression Analysis and its Applications, Wiley. Dave Campbell 2009
Statistical Models ref: chapter 1 of Bates, D and D. Watts (1988) Nonlinear Regression Analysis and its Applications, Wiley Dave Campbell 2009 Today linear regression in terms of the response surface geometry
More informationStatistics: Learning models from data
DS-GA 1002 Lecture notes 5 October 19, 2015 Statistics: Learning models from data Learning models from data that are assumed to be generated probabilistically from a certain unknown distribution is a crucial
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR
More informationMixed models in R using the lme4 package Part 7: Generalized linear mixed models
Mixed models in R using the lme4 package Part 7: Generalized linear mixed models Douglas Bates University of Wisconsin - Madison and R Development Core Team University of
More information(I AL BL 2 )z t = (I CL)ζ t, where
ECO 513 Fall 2011 MIDTERM EXAM The exam lasts 90 minutes. Answer all three questions. (1 Consider this model: x t = 1.2x t 1.62x t 2 +.2y t 1.23y t 2 + ε t.7ε t 1.9ν t 1 (1 [ εt y t = 1.4y t 1.62y t 2
More informationAdvanced Quantitative Methods: maximum likelihood
Advanced Quantitative Methods: Maximum Likelihood University College Dublin March 23, 2011 1 Introduction 2 3 4 5 Outline Introduction 1 Introduction 2 3 4 5 Preliminaries Introduction Ordinary least squares
More informationHomework 1: Solutions
Homework 1: Solutions Statistics 413 Fall 2017 Data Analysis: Note: All data analysis results are provided by Michael Rodgers 1. Baseball Data: (a) What are the most important features for predicting players
More informationRestricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model
Restricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model Xiuming Zhang zhangxiuming@u.nus.edu A*STAR-NUS Clinical Imaging Research Center October, 015 Summary This report derives
More information557: MATHEMATICAL STATISTICS II BIAS AND VARIANCE
557: MATHEMATICAL STATISTICS II BIAS AND VARIANCE An estimator, T (X), of θ can be evaluated via its statistical properties. Typically, two aspects are considered: Expectation Variance either in terms
More informationBrief Review on Estimation Theory
Brief Review on Estimation Theory K. Abed-Meraim ENST PARIS, Signal and Image Processing Dept. abed@tsi.enst.fr This presentation is essentially based on the course BASTA by E. Moulines Brief review on
More informationGeneralized 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 informationPhysics 403. Segev BenZvi. Parameter Estimation, Correlations, and Error Bars. Department of Physics and Astronomy University of Rochester
Physics 403 Parameter Estimation, Correlations, and Error Bars Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Review of Last Class Best Estimates and Reliability
More informationOutline. Mixed models in R using the lme4 package Part 5: Generalized linear mixed models. Parts of LMMs carried over to GLMMs
Outline Mixed models in R using the lme4 package Part 5: Generalized linear mixed models Douglas Bates University of Wisconsin - Madison and R Development Core Team UseR!2009,
More informationRegression. Oscar García
Regression Oscar García Regression methods are fundamental in Forest Mensuration For a more concise and general presentation, we shall first review some matrix concepts 1 Matrices An order n m matrix is
More informationNow 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 informationSolutions for Econometrics I Homework No.3
Solutions for Econometrics I Homework No3 due 6-3-15 Feldkircher, Forstner, Ghoddusi, Pichler, Reiss, Yan, Zeugner April 7, 6 Exercise 31 We have the following model: y T N 1 X T N Nk β Nk 1 + u T N 1
More informationTHE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS We begin with a relatively simple special case. Suppose y ijk = µ + τ i + u ij + e ijk, (i = 1,..., t; j = 1,..., n; k = 1,..., m) β =
More informationPeter Hoff Linear and multilinear models April 3, GLS for multivariate regression 5. 3 Covariance estimation for the GLM 8
Contents 1 Linear model 1 2 GLS for multivariate regression 5 3 Covariance estimation for the GLM 8 4 Testing the GLH 11 A reference for some of this material can be found somewhere. 1 Linear model Recall
More informationAdvanced Quantitative Methods: maximum likelihood
Advanced Quantitative Methods: Maximum Likelihood University College Dublin 4 March 2014 1 2 3 4 5 6 Outline 1 2 3 4 5 6 of straight lines y = 1 2 x + 2 dy dx = 1 2 of curves y = x 2 4x + 5 of curves y
More informationWU Weiterbildung. Linear Mixed Models
Linear Mixed Effects Models WU Weiterbildung SLIDE 1 Outline 1 Estimation: ML vs. REML 2 Special Models On Two Levels Mixed ANOVA Or Random ANOVA Random Intercept Model Random Coefficients Model Intercept-and-Slopes-as-Outcomes
More informationMultiple Regression Analysis
Multiple Regression Analysis y = 0 + 1 x 1 + x +... k x k + u 6. Heteroskedasticity What is Heteroskedasticity?! Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,
More informationTopic 12 Overview of Estimation
Topic 12 Overview of Estimation Classical Statistics 1 / 9 Outline Introduction Parameter Estimation Classical Statistics Densities and Likelihoods 2 / 9 Introduction In the simplest possible terms, the
More informationProblem 1 (20) Log-normal. f(x) Cauchy
ORF 245. Rigollet Date: 11/21/2008 Problem 1 (20) f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 4 2 0 2 4 Normal (with mean -1) 4 2 0 2 4 Negative-exponential x x f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.5
More informationHierarchical Generalized Linear Model Approach For Estimating Of Working Population In Kepulauan Riau Province
IOP Conference Series: Earth and Environmental Science PAPER OPEN ACCESS Hierarchical Generalized Linear Model Approach For Estimating Of Working Population In Kepulauan Riau Province To cite this article:
More informationVariations. ECE 6540, Lecture 10 Maximum Likelihood Estimation
Variations ECE 6540, Lecture 10 Last Time BLUE (Best Linear Unbiased Estimator) Formulation Advantages Disadvantages 2 The BLUE A simplification Assume the estimator is a linear system For a single parameter
More informationBayesian 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 informationChapter 5: Generalized Linear Models
w w w. I C A 0 1 4. o r g Chapter 5: Generalized Linear Models b Curtis Gar Dean, FCAS, MAAA, CFA Ball State Universit: Center for Actuarial Science and Risk Management M Interest in Predictive Modeling
More informationBayesian 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 informationThe Gauss-Markov Model. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 61
The Gauss-Markov Model Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 61 Recall that Cov(u, v) = E((u E(u))(v E(v))) = E(uv) E(u)E(v) Var(u) = Cov(u, u) = E(u E(u)) 2 = E(u 2
More informationSTAT763: Applied Regression Analysis. Multiple linear regression. 4.4 Hypothesis testing
STAT763: Applied Regression Analysis Multiple linear regression 4.4 Hypothesis testing Chunsheng Ma E-mail: cma@math.wichita.edu 4.4.1 Significance of regression Null hypothesis (Test whether all β j =
More informationGeneralized 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 informationThe Statistical Property of Ordinary Least Squares
The Statistical Property of Ordinary Least Squares The linear equation, on which we apply the OLS is y t = X t β + u t Then, as we have derived, the OLS estimator is ˆβ = [ X T X] 1 X T y Then, substituting
More informationLikelihood-Based Methods
Likelihood-Based Methods Handbook of Spatial Statistics, Chapter 4 Susheela Singh September 22, 2016 OVERVIEW INTRODUCTION MAXIMUM LIKELIHOOD ESTIMATION (ML) RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION (REML)
More informationLecture 2. (See Exercise 7.22, 7.23, 7.24 in Casella & Berger)
8 HENRIK HULT Lecture 2 3. Some common distributions in classical and Bayesian statistics 3.1. Conjugate prior distributions. In the Bayesian setting it is important to compute posterior distributions.
More informationGeneralized Estimating Equations
Outline Review of Generalized Linear Models (GLM) Generalized Linear Model Exponential Family Components of GLM MLE for GLM, Iterative Weighted Least Squares Measuring Goodness of Fit - Deviance and Pearson
More informationThe loss function and estimating equations
Chapter 6 he loss function and estimating equations 6 Loss functions Up until now our main focus has been on parameter estimating via the maximum likelihood However, the negative maximum likelihood is
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationSummer School in Statistics for Astronomers V June 1 - June 6, Regression. Mosuk Chow Statistics Department Penn State University.
Summer School in Statistics for Astronomers V June 1 - June 6, 2009 Regression Mosuk Chow Statistics Department Penn State University. Adapted from notes prepared by RL Karandikar Mean and variance Recall
More informationABHELSINKI UNIVERSITY OF TECHNOLOGY
Cross-Validation, Information Criteria, Expected Utilities and the Effective Number of Parameters Aki Vehtari and Jouko Lampinen Laboratory of Computational Engineering Introduction Expected utility -
More informationarxiv: v1 [math.st] 22 Dec 2018
Optimal Designs for Prediction in Two Treatment Groups Rom Coefficient Regression Models Maryna Prus Otto-von-Guericke University Magdeburg, Institute for Mathematical Stochastics, PF 4, D-396 Magdeburg,
More informationSignal Processing - Lecture 7
1 Introduction Signal Processing - Lecture 7 Fitting a function to a set of data gathered in time sequence can be viewed as signal processing or learning, and is an important topic in information theory.
More informationBayesian 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 informationST3241 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