STA 216, GLM, Lecture 16. October 29, 2007
|
|
- Jasmine Baker
- 5 years ago
- Views:
Transcription
1 STA 216, GLM, Lecture 16 October 29, 2007
2 Efficient Posterior Computation in Factor Models Underlying Normal Models Generalized Latent Trait Models Formulation Genetic Epidemiology Illustration Structural Equation Models
3 How can we do efficient computation? Efficient posterior computation in factor analysis models is very challenging Typical Gibbs sampler can be subject to extreme slow-mixing Centering does not provide complete solution - can only center one measurement for each latent variable & eliminates conjugacy unless prior non-exchangeable What to do?
4 Parameter Expansion (PX) Originally proposed as a method for speeding up convergence of EM algorithm Redundant parameters are carefully introduced to allow faster convergence of EM & better mixing Gibbs samplers The idea is to induce the redundant parameters in such a way as to avoid changing the target distribution in the MCMC algorithm Hence, the posterior is not changed, but one reduces autocorrelation.
5 PX in Hierarchical Models It is very difficult to obtain a PX-accelerated Gibbs sampler in general cases Hard to avoid changing the target distribution Gelman (2005, Bayesian Analysis) proposes to use PX to induce a better prior, while also speeding up mixing in the setting of variance component models Ghosh & Dunson (2007) extend to factor analysis models
6 Homework Exercise Propose a PX Gibbs sampler for the sperm concentration latent factor regression model from lecture 15 Simulate data under the model and compare the PX Gibbs sampler to a typical Gibbs sampler without PX Due - next Friday
7 What if our data are categorical? The above models assume that the different elements of y i are continuous and normally distributed In most settings in which factor analysis models are used, at least some of the elements of y i are instead ordered categorical It is appealing to have a general framework for modeling of correlated measurements having different scales (continuous, binary, ordinal)
8 Underlying Normal Models To solve this problem, we can considering the following modification to the measurement model: y ij = g j (y ij; τ j ), j = 1,..., p y i = µ + Λη i + ɛ i, ɛ i N 3 (0, Σ), Here, y i are the observed variables & yi are normal latent variables underlying y i g j ( ; τ j ) = link function possibly involving threshold parameters τ j
9 Link functions in underlying normal models For continuous items (i.e., y ij is continuous), g j is chosen as the identity link For binary items (y ij {0, 1}), we choose a threshold link y ij = 1(y ij > 0) For ordered categorical items (y ij {1,..., c j }), we generalize the binary case to let y ij = c j l=1 = l 1(τ j,l 1 < y ij τ jl ), with τ j0 =, τ j1 = 0, τ j,cj =.
10 Some Comments Note that we are using an underlying multivariate normal model to characterize dependence in observations having a variety of scales In factor analysis, the dependence is induced through shared dependence on the latent factors Posterior computation is straightforward using a data augmentation Gibbs sampler, which imputes the y ij from their truncated normal full conditional distributions. Other sampling steps proceed as if yij were observed data
11 Generalized Latent Trait Models Underlying normal specification induces normal linear models on the continuous items & probit-type models on categorical items Structure is restrictive - may prefer to use a different GLM for each item, while allowing dependence Replace underlying normal measurement model with generalized latent trait model (GLTM): η ij = µ j + λ jξ i, where η ij =linear predictor in GLM for outcome type j, ξ i = (ξ i1,..., ξ ir ) =vector of latent traits
12 Comments on GLTMs GLTMs allow modeling also with count outcomes & for more flexible models for the individual items (e.g., logistic, complementary log-log, etc instead of just probit) Important - latent traits impact both dependence in the different elements of y i & lack of fit in the individual item GLMs Harder to fit such models routinely, though adaptive rejection sampling & other tricks possible
13 Dangers of GLTMs Dual role of latent variable component in accommodating dependence & lack of fit individual item links creates problems in interpretation Consider the case in which y ij is a 0/1 indicator of a disease, with i indexing family of j indexing individual within a family Following model commonly used to assess within-family dependence in probability of disease logit { Pr(y ij = 1 x i, β, ξ i ) } = x iβ + ξ i, ξ i N(0, ψ)
14 Application to Genetic Epidemiology Studies ξ i = difference in the log-odds of disease for family i relative to the population average Such differences among families are commonly attributable to genetic effects The estimated value of ψ is used to infer the magnitude of the genetic component Diseases having small ψ will exhibit limited within-family dependence & hence should have a small genetic component
15 Genetic Epidemiology Example (continued) Anything wrong with this interpretation? It has been shown that one can identify the fixed effects, β, & the random effects variance, ψ even if data are only available for a single individual per family. How can this be?? Random effect included to allow within-family dependence with a single individual per family no need for a random effect?
16 Punch Line We have identifiability even with a single individual per family because induced link function is no longer logistic In particular, we have a logistic-normal link function: Pr(y i = 1 x i, β, ψ) = g ψ (x iβ) ( = {1 + exp( x iβ + ξ i )} 1 (2πψ) 1/2 exp 1 ) 2ψ ξ2 i dξ i, Shape of the link function varies as ψ varies, so we can estimate β, ψ even with a single subject per family
17 Some Further Comments Is ψ interpretable as a genetic heterogeneity in this case? What if we have a few families with multiple individuals, and many with a single individual? Answer: ψ measures both lack-of-fit in the logistic link & heterogeneity among families. To obtain reliable inferences on genetic heterogeneity, you should use a flexible link function
18 Some General Comments about GLTMs Used simple random intercept genetic epi example as illustration Need to worry about these issues just as much in more complex settings involving multivariate outcomes having different scales Normal & underlying normal more robust to such issues, since one does not change the link in marginalizing out latent variables To allow additional flexibility, one can work within underlying normal family - e.g., using scale mixtures of normals.
19 Structural Equation Models (Bollen, 1989) When interest focuses on modeling of relationships among latent traits, factor analysis needs to be extended Structural Equation Models (SEMs) provide a broader framework Specified in two components: (1) measurement model relating observed variables to latent traits; (2) structural model characterizing joint distribution of latent traits
20 Structural Equation Models (Bollen, 1989) The measured data consist of a vector of response variables, y i = (y i1,..., y ip ), & a vector of predictor variables, x i = (x i1,..., x iq ). Measurement model: y i = µ y + Λ y η i + ɛ y i, ɛy i N p(0, Σ y ) x i = µ x + Λ x ξ i + ɛ x i, ɛ x i N p (0, Σ x ) Just two separate factor analysis models of y i and x i components
21 Linear Structural Relations (LISREL) model LISREL model: η i = Bη i + Γξ i + δ i, ξ i N s (0, I s ), δ i N r (0, I r ). Describes joint distribution and association among latent variables B has zeros along diagonals - putting η i standard notation in LISREL model (typically, B has upper triangular elements = 0) Γ=often parameters of primary interest - characterize association among latent predictor and response variables
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 informationDefault 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 informationLecture 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 informationBayes 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 informationBayesian 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 informationBayesian 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 informationGeneralized 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 informationDefault 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 informationA 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 informationContents. 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 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 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 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 informationPart 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 informationStandard 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 informationComputationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models
Computationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models Tihomir Asparouhov 1, Bengt Muthen 2 Muthen & Muthen 1 UCLA 2 Abstract Multilevel analysis often leads to modeling
More informationBayesian 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 informationPartial 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 informationFactor 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 informationStat 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 informationMULTILEVEL 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 informationThe 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 informationDynamic Generalized Linear Models
Dynamic Generalized Linear Models Jesse Windle Oct. 24, 2012 Contents 1 Introduction 1 2 Binary Data (Static Case) 2 3 Data Augmentation (de-marginalization) by 4 examples 3 3.1 Example 1: CDF method.............................
More informationUsing 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 informationNovember 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 informationLecture 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 informationBayesian 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 informationNonparametric Bayes tensor factorizations for big data
Nonparametric Bayes tensor factorizations for big data David Dunson Department of Statistical Science, Duke University Funded from NIH R01-ES017240, R01-ES017436 & DARPA N66001-09-C-2082 Motivation Conditional
More informationGibbs 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 informationGibbs 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 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 informationBayes Model Selection with Path Sampling: Factor Models
with Path Sampling: Factor Models Ritabrata Dutta and Jayanta K Ghosh Purdue University 07/02/11 Factor Models in Applications Factor Models in Applications Factor Models Factor Models and Factor analysis
More informationBayesian 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 informationBayesian 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 informationCTDL-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 informationBayesian methods for latent trait modeling of longitudinal data
Bayesian methods for latent trait modeling of longitudinal data DAVID B. DUNSON Biostatistics Branch, National Institute of Environmental Health Sciences MD A3-03, P.O. Box 12233 Research Triangle Park,
More informationAnders Skrondal. Norwegian Institute of Public Health London School of Hygiene and Tropical Medicine. Based on joint work with Sophia Rabe-Hesketh
Constructing Latent Variable Models using Composite Links Anders Skrondal Norwegian Institute of Public Health London School of Hygiene and Tropical Medicine Based on joint work with Sophia Rabe-Hesketh
More informationScaling up Bayesian Inference
Scaling up Bayesian Inference David Dunson Departments of Statistical Science, Mathematics & ECE, Duke University May 1, 2017 Outline Motivation & background EP-MCMC amcmc Discussion Motivation & background
More informationStat 451 Lecture Notes Markov Chain Monte Carlo. Ryan Martin UIC
Stat 451 Lecture Notes 07 12 Markov Chain Monte Carlo Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapters 8 9 in Givens & Hoeting, Chapters 25 27 in Lange 2 Updated: April 4, 2016 1 / 42 Outline
More informationSingle-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 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 informationRichard N. Jones, Sc.D. HSPH Kresge G2 October 5, 2011
Harvard Catalyst Biostatistical Seminar Neuropsychological Proles in Alzheimer's Disease and Cerebral Infarction: A Longitudinal MIMIC Model An Overview of Structural Equation Modeling using Mplus Richard
More informationChapter 1. Modeling Basics
Chapter 1. Modeling Basics What is a model? Model equation and probability distribution Types of model effects Writing models in matrix form Summary 1 What is a statistical model? A model is a mathematical
More informationModel Assumptions; Predicting Heterogeneity of Variance
Model Assumptions; Predicting Heterogeneity of Variance Today s topics: Model assumptions Normality Constant variance Predicting heterogeneity of variance CLP 945: Lecture 6 1 Checking for Violations of
More informationNonparametric 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 informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 3 Linear
More information36-720: 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 informationApplication of Latent Class with Random Effects Models to Longitudinal Data. Ken Beath Macquarie University
Application of Latent Class with Random Effects Models to Longitudinal Data Ken Beath Macquarie University Introduction Latent trajectory is a method of classifying subjects based on longitudinal data
More informationComparing IRT with Other Models
Comparing IRT with Other Models Lecture #14 ICPSR Item Response Theory Workshop Lecture #14: 1of 45 Lecture Overview The final set of slides will describe a parallel between IRT and another commonly used
More informationNonparametric Bayes Modeling
Nonparametric Bayes Modeling Lecture 6: Advanced Applications of DPMs David Dunson Department of Statistical Science, Duke University Tuesday February 2, 2010 Motivation Functional data analysis Variable
More informationCentering Predictor and Mediator Variables in Multilevel and Time-Series Models
Centering Predictor and Mediator Variables in Multilevel and Time-Series Models Tihomir Asparouhov and Bengt Muthén Part 2 May 7, 2018 Tihomir Asparouhov and Bengt Muthén Part 2 Muthén & Muthén 1/ 42 Overview
More informationUsing Bayesian Priors for More Flexible Latent Class Analysis
Using Bayesian Priors for More Flexible Latent Class Analysis Tihomir Asparouhov Bengt Muthén Abstract Latent class analysis is based on the assumption that within each class the observed class indicator
More informationIntroduction to Bayesian Statistics and Markov Chain Monte Carlo Estimation. EPSY 905: Multivariate Analysis Spring 2016 Lecture #10: April 6, 2016
Introduction to Bayesian Statistics and Markov Chain Monte Carlo Estimation EPSY 905: Multivariate Analysis Spring 2016 Lecture #10: April 6, 2016 EPSY 905: Intro to Bayesian and MCMC Today s Class An
More informationMotivation 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 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 informationWageningen 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 informationBayesian Nonparametric Modeling for Multivariate Ordinal Regression
Bayesian Nonparametric Modeling for Multivariate Ordinal Regression arxiv:1408.1027v3 [stat.me] 20 Sep 2016 Maria DeYoreo Department of Statistical Science, Duke University and Athanasios Kottas Department
More informationGeneralized Linear Models I
Statistics 203: Introduction to Regression and Analysis of Variance Generalized Linear Models I Jonathan Taylor - p. 1/16 Today s class Poisson regression. Residuals for diagnostics. Exponential families.
More informationDepartamento de Economía Universidad de Chile
Departamento de Economía Universidad de Chile GRADUATE COURSE SPATIAL ECONOMETRICS November 14, 16, 17, 20 and 21, 2017 Prof. Henk Folmer University of Groningen Objectives The main objective of the course
More informationSTA 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 informationAn Extended BIC for Model Selection
An Extended BIC for Model Selection at the JSM meeting 2007 - Salt Lake City Surajit Ray Boston University (Dept of Mathematics and Statistics) Joint work with James Berger, Duke University; Susie Bayarri,
More informationBayesian shrinkage approach in variable selection for mixed
Bayesian shrinkage approach in variable selection for mixed effects s GGI Statistics Conference, Florence, 2015 Bayesian Variable Selection June 22-26, 2015 Outline 1 Introduction 2 3 4 Outline Introduction
More informationGibbs 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 informationComparison 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 informationSTAT 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 informationMeasurement Error and Linear Regression of Astronomical Data. Brandon Kelly Penn State Summer School in Astrostatistics, June 2007
Measurement Error and Linear Regression of Astronomical Data Brandon Kelly Penn State Summer School in Astrostatistics, June 2007 Classical Regression Model Collect n data points, denote i th pair as (η
More informationFixed and Random Effects Selection in Linear and Logistic Models
Biometrics 63, 690 698 September 2007 DOI: 10.1111/j.1541-0420.2007.00771.x Fixed and Random Effects Selection in Linear and Logistic Models Satkartar K. Kinney Institute of Statistics and Decision Sciences,
More informationBayesian 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 informationModel 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 informationExtending causal inferences from a randomized trial to a target population
Extending causal inferences from a randomized trial to a target population Issa Dahabreh Center for Evidence Synthesis in Health, Brown University issa dahabreh@brown.edu January 16, 2019 Issa Dahabreh
More informationPlausible Values for Latent Variables Using Mplus
Plausible Values for Latent Variables Using Mplus Tihomir Asparouhov and Bengt Muthén August 21, 2010 1 1 Introduction Plausible values are imputed values for latent variables. All latent variables can
More informationAnalysing 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 informationGeneralized 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 informationBayesian Mixture Modeling
University of California, Merced July 21, 2014 Mplus Users Meeting, Utrecht Organization of the Talk Organization s modeling estimation framework Motivating examples duce the basic LCA model Illustrated
More informationHierarchical Generalized Linear Models. ERSH 8990 REMS Seminar on HLM Last Lecture!
Hierarchical Generalized Linear Models ERSH 8990 REMS Seminar on HLM Last Lecture! Hierarchical Generalized Linear Models Introduction to generalized models Models for binary outcomes Interpreting parameter
More informationReview of Panel Data Model Types Next Steps. Panel GLMs. Department of Political Science and Government Aarhus University.
Panel GLMs Department of Political Science and Government Aarhus University May 12, 2015 1 Review of Panel Data 2 Model Types 3 Review and Looking Forward 1 Review of Panel Data 2 Model Types 3 Review
More informationIntroduction 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 informationSupplementary 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 informationHierarchical 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 informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and
More informationLecture 5: LDA and Logistic Regression
Lecture 5: and Logistic Regression Hao Helen Zhang Hao Helen Zhang Lecture 5: and Logistic Regression 1 / 39 Outline Linear Classification Methods Two Popular Linear Models for Classification Linear Discriminant
More informationNonparametric Bayes Uncertainty Quantification
Nonparametric Bayes Uncertainty Quantification David Dunson Department of Statistical Science, Duke University Funded from NIH R01-ES017240, R01-ES017436 & ONR Review of Bayes Intro to Nonparametric Bayes
More informationRonald Christensen. University of New Mexico. Albuquerque, New Mexico. Wesley Johnson. University of California, Irvine. Irvine, California
Texts in Statistical Science Bayesian Ideas and Data Analysis An Introduction for Scientists and Statisticians Ronald Christensen University of New Mexico Albuquerque, New Mexico Wesley Johnson University
More informationThe Polya-Gamma Gibbs Sampler for Bayesian. Logistic Regression is Uniformly Ergodic
he Polya-Gamma Gibbs Sampler for Bayesian Logistic Regression is Uniformly Ergodic Hee Min Choi and James P. Hobert Department of Statistics University of Florida August 013 Abstract One of the most widely
More informationLinear 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 informationThe Wishart distribution Scaled Wishart. Wishart Priors. Patrick Breheny. March 28. Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/11
Wishart Priors Patrick Breheny March 28 Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/11 Introduction When more than two coefficients vary, it becomes difficult to directly model each element
More informationMultilevel Statistical Models: 3 rd edition, 2003 Contents
Multilevel Statistical Models: 3 rd edition, 2003 Contents Preface Acknowledgements Notation Two and three level models. A general classification notation and diagram Glossary Chapter 1 An introduction
More informationFixed and random effects selection in linear and logistic models
Fixed and random effects selection in linear and logistic models Satkartar K. Kinney Institute of Statistics and Decision Sciences, Duke University, Box 9051, Durham, North Carolina 7705, U.S.A. email:
More information1 Data Arrays and Decompositions
1 Data Arrays and Decompositions 1.1 Variance Matrices and Eigenstructure Consider a p p positive definite and symmetric matrix V - a model parameter or a sample variance matrix. The eigenstructure is
More informationStatistical Analysis of List Experiments
Statistical Analysis of List Experiments Graeme Blair Kosuke Imai Princeton University December 17, 2010 Blair and Imai (Princeton) List Experiments Political Methodology Seminar 1 / 32 Motivation Surveys
More informationA Study into Mechanisms of Attitudinal Scale Conversion: A Randomized Stochastic Ordering Approach
A Study into Mechanisms of Attitudinal Scale Conversion: A Randomized Stochastic Ordering Approach Zvi Gilula (Hebrew University) Robert McCulloch (Arizona State) Ya acov Ritov (University of Michigan)
More informationBinary Choice Models Probit & Logit. = 0 with Pr = 0 = 1. decision-making purchase of durable consumer products unemployment
BINARY CHOICE MODELS Y ( Y ) ( Y ) 1 with Pr = 1 = P = 0 with Pr = 0 = 1 P Examples: decision-making purchase of durable consumer products unemployment Estimation with OLS? Yi = Xiβ + εi Problems: nonsense
More informationMonte Carlo Techniques for Regressing Random Variables. Dan Pemstein. Using V-Dem the Right Way
: Monte Carlo Techniques for Regressing Random Variables Workshop Goals (Re)familiarize you with Monte Carlo methods for estimating functions of random variables, integrating/marginalizing (Re)familiarize
More informationThe Mixture Approach for Simulating New Families of Bivariate Distributions with Specified Correlations
The Mixture Approach for Simulating New Families of Bivariate Distributions with Specified Correlations John R. Michael, Significance, Inc. and William R. Schucany, Southern Methodist University The mixture
More informationCourse Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model
Course Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 1: August 22, 2012
More informationStatistics 203: Introduction to Regression and Analysis of Variance Course review
Statistics 203: Introduction to Regression and Analysis of Variance Course review Jonathan Taylor - p. 1/?? Today Review / overview of what we learned. - p. 2/?? General themes in regression models Specifying
More informationDirichlet Process Mixtures of Generalized Linear Models
Dirichlet Process Mixtures of Generalized Linear Models Lauren A. Hannah Department of Operations Research and Financial Engineering Princeton University Princeton, NJ 08544, USA David M. Blei Department
More informationBayesian Areal Wombling for Geographic Boundary Analysis
Bayesian Areal Wombling for Geographic Boundary Analysis Haolan Lu, Haijun Ma, and Bradley P. Carlin haolanl@biostat.umn.edu, haijunma@biostat.umn.edu, and brad@biostat.umn.edu Division of Biostatistics
More informationMeasurement 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,..., θ(2),..., θ(n)
Likelihoods for Multivariate Binary Data Log-Linear Model We have 2 n 1 distinct probabilities, but we wish to consider formulations that allow more parsimonious descriptions as a function of covariates.
More information