Multi-level Models: Idea
|
|
- Teresa Warner
- 5 years ago
- Views:
Transcription
1 Review of
2 Review Introduction to multi-level models The two-stage normal-normal model Two-stage linear models with random effects Three-stage linear models Two-stage logistic regression with random effects Three stage logistic regression
3 Level: Multi-level Models: Idea Predictor Variables Person s Income Family Income Percent poverty in neighborhood Response Alcohol Abuse 4. State support of the poor
4 Key Points Multi-level Models: Have covariates from many levels and their interactions Acknowledge correlation among observations from within a level (cluster) Random effect MLMs condition on unobserved random effects to account for the correlation Assumptions about the random effects determine the nature of the within cluster correlations Information can be borrowed across clusters (levels) to improve individual estimates
5 Fixed and Random Effects Standard regression models: ε ij ~ N(0,σ 2 ) Y ij = µ + ε ij E(Y ij )=µ (overall average) Y ij = µ + b * j + ε ij A random effects model: E(Y ij )=θ j (observed school avgs) Fixed Effects Y ij b j = µ + b j + ε ij, where: b j ~ N(0,τ 2 ) Random Effects:
6 Testing in Schools: Shrinkage Plot score D ire ct S a m p le E sts B a ye s S hrunk E sts b * j µ b j scho o l
7 Relative Risks for Six Largest Cities y j σ j σ j 2 City RR Estimate (% per 10 micrograms/ml Statistical Standard Error Statistical Variance Los Angeles New York Chicago Dallas/Ft Worth Houston San Diego Approximate values read from graph in Daniels, et al AJE
8 Two-stage normal normal model RR estimate in city j True RR in city j y j = θ j + ε j ε j ~ N(0,σ j 2 ) θ j ~ N(θ,τ 2 ) Within city statistical Uncertainty (known) Heterogeneity across cities in the true RR
9 Two Extremes Natural variance >> Statistical variance Weights wj approximately constant Use ordinary mean of estimates regardless of their relative precision Statistical variance >> Natural variance Weight each estimator inversely proportional to its statistical variance
10 Empirical Bayes Estimation ˆ θ = λ y + (1 λ )y j j j j λ j = τ 2 τ 2 + σ j 2
11 C ity -s p e c ific M L E s (L e ft) a n d E m p ir ic a l B a y e s E s tim a te s (R ig h t) Percent Change * * * * * * c ity
12 Key Ideas Better to use data for all cities to estimate the relative risk for a particular city Reduce variance by adding some bias Smooth compromise between city specific estimates and overall mean Empirical-Bayes estimates depend on measure of natural variation Assess sensitivity to estimate of NV
13 Inner-London School data: How effective are the different schools? (gcse.dat,chap 3) Outcome: score exam at age 16 (gcse) Data are clustered within schools Covariate: reading test score at age 11 prior enrolling in the school (lrt) Goal: to examine the relationship between the score exam at age 16 and the score at age 11 and to investigate how this association varies across schools
14 Linear regression model with random intercept and random slope Y ij = b 0 j + b 1 j x ij + ε ij b ~ N(β,τ 2 ) 0 j 0 1 b ~ N(β,τ 2 ) 1 j 1 2 centered cov(b 0 j,b 1 j ) = τ 12
15 Empirical Bayes Prediction (xtmixed reff*,reffects) In stata we can calculate: ( b 0 j, b 1 j ) EB: borrow strength across schools ( ˆ b 0 j, ˆ b 1 j ) MLE: DO NOT borrow strength across Schools
16 Fig 3.10: EB predictions of school-specific lines
17 Three levels models In three levels models the clusters themselves are nested in superclusters, forming a hierarchical structure. For example, we might have repeated measurement occasions (units) for patients (clusters) who are clustered in hospitals (superclusters).
18
19 Table 1.1: Peak respiratory flow rate measured on two occasions using both the Wright and the Mini Wright meter ( Bland and Altma, Lancet 1986) Level 1: occasion (i) Level 2: method (j) Level 3: individual (k)
20 Model 3: three-level variance component models y = β + ζ (2) + ζ (3) + ε ijk 1 jk k ijk ε ijk ~ N(0,σ 2 ) ζ (2) ~ N(0,τ 2 ) jk 2 ζ k (3) ~ N(0,τ 3 2 ) account for between-method within-subject heterogeneity Variance of the measurements across the two methods for the same subject Variance of the measurements across subjects
21 ML models for binary data
22 Marginal and Individual Probabilities Marginal (ordinary) logistic regression models the overall (populationaveraged) probabilities Random effects logistic regression models the individual (subject-specific) probabilities
23 Marginal and Individual probabilities A:Marginal Logistic regression logit{ P(y ij =1 x ij )}= β 1 + β 2 x ij marginal prob B:Random Intercept Logistic regression logit{ P(y ij =1 x ij,ς j )}= (β * 1 + ς j ) + β * 2 x ij individual prob ς j ~ N(0,τ 2 )
24 Average of individual level probabilities IS NOT equal to marginal probability P * ( y =1 x )= ij ij = P( y =1 x,ς )φ(ς ;0, ˆ ij ij j j τ 2 )dς = j exp(β 1 * + ς j + β 2 * x ij ) 1+ exp(β 1 * + ς j + β 2 * x ij ) φ(ς j;0, ˆ τ 2 )dς j exp(β 1 + β 2 x ij ) Normal density 1+ exp(β 1 + β 2 x ij )
25 Figure 4.11: Subject-specific versus population averaged logistic regression Pop average slope is attenuated with respect to the subject-specific slopes
26 What is profiling? Outline Definitions Statistical challenges Centrality of multi-level analysis Fitting Multilevel Models with Winbugs: A toy example on institutional ranking Profiling medical care providers: a case-study Hierarchical logistic regression model Performance measures Comparison with standard approaches
27 Borrowing strength Reliability of hospital-specific estimates: because of difference in hospital sample sizes, the precision of the hospital-specific estimates may vary greatly. Large differences between observed and expected mortality rates at hospitals with small sample sizes may be due primarily to sampling variability Implement shrinkage estimation methods: hospitals performances with small sample size will be shrunk toward the mean more heavily
28 Toy example on using WinBUGS for hospital performance ranking
29 Hierarchical logistic regression model I: patient level, within-provider model Patient-level logistic regression model with random intercept and random slope II: between-providers model Hospital-specific random effects are regressed on hospital-specific characteristics
30 Interpretation of the Random Effects A model with a random intercept indicate inter hospital differences in baseline mortality rates A model with random slope indicate that the effect of clinical burden (patient severity) on mortality differs across hospitals
31 Posterior distributions of the ranks who is the worst?
32 In summary Multilevel models are a natural approach to analyze data collected at different level of aggregation Provide an easy framework to model sources of variability (within county, across counties, within regions etc..) Allow to incorporate covariates at the different levels to explain heterogeneity within clusters and estimate cross-level interactions Allow flexibility in specifying the distribution of the random effects, which for example, can take into account spatially correlated latent variables (only in Winbugs)
Lecture 1 Introduction to Multi-level Models
Lecture 1 Introduction to Multi-level Models Course Website: http://www.biostat.jhsph.edu/~ejohnson/multilevel.htm All lecture materials extracted and further developed from the Multilevel Model course
More informationmultilevel modeling: concepts, applications and interpretations
multilevel modeling: concepts, applications and interpretations lynne c. messer 27 october 2010 warning social and reproductive / perinatal epidemiologist concepts why context matters multilevel models
More informationEstimation and Centering
Estimation and Centering PSYED 3486 Feifei Ye University of Pittsburgh Main Topics Estimating the level-1 coefficients for a particular unit Reading: R&B, Chapter 3 (p85-94) Centering-Location of X Reading
More informationPrevious lecture. P-value based combination. Fixed vs random effects models. Meta vs. pooled- analysis. New random effects testing.
Previous lecture P-value based combination. Fixed vs random effects models. Meta vs. pooled- analysis. New random effects testing. Interaction Outline: Definition of interaction Additive versus multiplicative
More informationPropensity Score Weighting with Multilevel Data
Propensity Score Weighting with Multilevel Data Fan Li Department of Statistical Science Duke University October 25, 2012 Joint work with Alan Zaslavsky and Mary Beth Landrum Introduction In comparative
More informationMultilevel modelling of fish abundance in streams
Multilevel modelling of fish abundance in streams Marco A. Rodríguez Université du Québec à Trois-Rivières Hierarchies and nestedness are common in nature Biological data often have clustered or nested
More informationFrailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Mela. P.
Frailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Melanie M. Wall, Bradley P. Carlin November 24, 2014 Outlines of the talk
More informationMarginal versus conditional effects: does it make a difference? Mireille Schnitzer, PhD Université de Montréal
Marginal versus conditional effects: does it make a difference? Mireille Schnitzer, PhD Université de Montréal Overview In observational and experimental studies, the goal may be to estimate the effect
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 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 informationSample Size and Power Considerations for Longitudinal Studies
Sample Size and Power Considerations for Longitudinal Studies Outline Quantities required to determine the sample size in longitudinal studies Review of type I error, type II error, and power For continuous
More informationGENERALIZED LINEAR MIXED MODELS AND MEASUREMENT ERROR. Raymond J. Carroll: Texas A&M University
GENERALIZED LINEAR MIXED MODELS AND MEASUREMENT ERROR Raymond J. Carroll: Texas A&M University Naisyin Wang: Xihong Lin: Roberto Gutierrez: Texas A&M University University of Michigan Southern Methodist
More informationAppendix A. Numeric example of Dimick Staiger Estimator and comparison between Dimick-Staiger Estimator and Hierarchical Poisson Estimator
Appendix A. Numeric example of Dimick Staiger Estimator and comparison between Dimick-Staiger Estimator and Hierarchical Poisson Estimator As described in the manuscript, the Dimick-Staiger (DS) estimator
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 informationInstructions: Closed book, notes, and no electronic devices. Points (out of 200) in parentheses
ISQS 5349 Final Spring 2011 Instructions: Closed book, notes, and no electronic devices. Points (out of 200) in parentheses 1. (10) What is the definition of a regression model that we have used throughout
More informationSemiparametric Generalized Linear Models
Semiparametric Generalized Linear Models North American Stata Users Group Meeting Chicago, Illinois Paul Rathouz Department of Health Studies University of Chicago prathouz@uchicago.edu Liping Gao MS Student
More informationLecture 10: Introduction to Logistic Regression
Lecture 10: Introduction to Logistic Regression Ani Manichaikul amanicha@jhsph.edu 2 May 2007 Logistic Regression Regression for a response variable that follows a binomial distribution Recall the binomial
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 informationA framework for developing, implementing and evaluating clinical prediction models in an individual participant data meta-analysis
A framework for developing, implementing and evaluating clinical prediction models in an individual participant data meta-analysis Thomas Debray Moons KGM, Ahmed I, Koffijberg H, Riley RD Supported by
More informationReview. Timothy Hanson. Department of Statistics, University of South Carolina. Stat 770: Categorical Data Analysis
Review Timothy Hanson Department of Statistics, University of South Carolina Stat 770: Categorical Data Analysis 1 / 22 Chapter 1: background Nominal, ordinal, interval data. Distributions: Poisson, binomial,
More informationAn Introduction to Multilevel Models. PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012
An Introduction to Multilevel Models PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012 Today s Class Concepts in Longitudinal Modeling Between-Person vs. +Within-Person
More informationJoint Modeling of Longitudinal Item Response Data and Survival
Joint Modeling of Longitudinal Item Response Data and Survival Jean-Paul Fox University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede,
More informationSpatio-Temporal Modelling of Credit Default Data
1/20 Spatio-Temporal Modelling of Credit Default Data Sathyanarayan Anand Advisor: Prof. Robert Stine The Wharton School, University of Pennsylvania April 29, 2011 2/20 Outline 1 Background 2 Conditional
More informationBayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling
Bayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling Jon Wakefield Departments of Statistics and Biostatistics University of Washington 1 / 37 Lecture Content Motivation
More informationA (Brief) Introduction to Crossed Random Effects Models for Repeated Measures Data
A (Brief) Introduction to Crossed Random Effects Models for Repeated Measures Data Today s Class: Review of concepts in multivariate data Introduction to random intercepts Crossed random effects models
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 informationLab 3: Two levels Poisson models (taken from Multilevel and Longitudinal Modeling Using Stata, p )
Lab 3: Two levels Poisson models (taken from Multilevel and Longitudinal Modeling Using Stata, p. 376-390) BIO656 2009 Goal: To see if a major health-care reform which took place in 1997 in Germany was
More informationCommunity Health Needs Assessment through Spatial Regression Modeling
Community Health Needs Assessment through Spatial Regression Modeling Glen D. Johnson, PhD CUNY School of Public Health glen.johnson@lehman.cuny.edu Objectives: Assess community needs with respect to particular
More informationSTAT5044: 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 informationPart 7: Hierarchical Modeling
Part 7: Hierarchical Modeling!1 Nested data It is common for data to be nested: i.e., observations on subjects are organized by a hierarchy Such data are often called hierarchical or multilevel For example,
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 information36-463/663: Multilevel & Hierarchical Models
36-463/663: Multilevel & Hierarchical Models (P)review: in-class midterm Brian Junker 132E Baker Hall brian@stat.cmu.edu 1 In-class midterm Closed book, closed notes, closed electronics (otherwise I have
More informationMultilevel Modeling Day 2 Intermediate and Advanced Issues: Multilevel Models as Mixed Models. Jian Wang September 18, 2012
Multilevel Modeling Day 2 Intermediate and Advanced Issues: Multilevel Models as Mixed Models Jian Wang September 18, 2012 What are mixed models The simplest multilevel models are in fact mixed models:
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Class (or 3): Summary of steps in building unconditional models for time What happens to missing predictors Effects of time-invariant predictors
More informationFor more information about how to cite these materials visit
Author(s): Kerby Shedden, Ph.D., 2010 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution Share Alike 3.0 License: http://creativecommons.org/licenses/by-sa/3.0/
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 informationIntroduction to Within-Person Analysis and RM ANOVA
Introduction to Within-Person Analysis and RM ANOVA Today s Class: From between-person to within-person ANOVAs for longitudinal data Variance model comparisons using 2 LL CLP 944: Lecture 3 1 The Two Sides
More informationLongitudinal Modeling with Logistic Regression
Newsom 1 Longitudinal Modeling with Logistic Regression Longitudinal designs involve repeated measurements of the same individuals over time There are two general classes of analyses that correspond to
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 informationBinomial Model. Lecture 10: Introduction to Logistic Regression. Logistic Regression. Binomial Distribution. n independent trials
Lecture : Introduction to Logistic Regression Ani Manichaikul amanicha@jhsph.edu 2 May 27 Binomial Model n independent trials (e.g., coin tosses) p = probability of success on each trial (e.g., p =! =
More informationMixed Models for Longitudinal Ordinal and Nominal Outcomes
Mixed Models for Longitudinal Ordinal and Nominal Outcomes Don Hedeker Department of Public Health Sciences Biological Sciences Division University of Chicago hedeker@uchicago.edu Hedeker, D. (2008). Multilevel
More informationHabilitationsvortrag: Machine learning, shrinkage estimation, and economic theory
Habilitationsvortrag: Machine learning, shrinkage estimation, and economic theory Maximilian Kasy May 25, 218 1 / 27 Introduction Recent years saw a boom of machine learning methods. Impressive advances
More informationShared Random Parameter Models for Informative Missing Data
Shared Random Parameter Models for Informative Missing Data Dean Follmann NIAID NIAID/NIH p. A General Set-up Longitudinal data (Y ij,x ij,r ij ) Y ij = outcome for person i on visit j R ij = 1 if observed
More informationRate Maps and Smoothing
Rate Maps and Smoothing Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu Outline Mapping Rates Risk
More informationNinth ARTNeT Capacity Building Workshop for Trade Research "Trade Flows and Trade Policy Analysis"
Ninth ARTNeT Capacity Building Workshop for Trade Research "Trade Flows and Trade Policy Analysis" June 2013 Bangkok, Thailand Cosimo Beverelli and Rainer Lanz (World Trade Organization) 1 Selected econometric
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 informationModelling heterogeneous variance-covariance components in two-level multilevel models with application to school effects educational research
Modelling heterogeneous variance-covariance components in two-level multilevel models with application to school effects educational research Research Methods Festival Oxford 9 th July 014 George Leckie
More informationBayesian decision procedures for dose escalation - a re-analysis
Bayesian decision procedures for dose escalation - a re-analysis Maria R Thomas Queen Mary,University of London February 9 th, 2010 Overview Phase I Dose Escalation Trial Terminology Regression Models
More informationHigh-Throughput Sequencing Course
High-Throughput Sequencing Course DESeq Model for RNA-Seq Biostatistics and Bioinformatics Summer 2017 Outline Review: Standard linear regression model (e.g., to model gene expression as function of an
More informationBiostatistics Workshop Longitudinal Data Analysis. Session 4 GARRETT FITZMAURICE
Biostatistics Workshop 2008 Longitudinal Data Analysis Session 4 GARRETT FITZMAURICE Harvard University 1 LINEAR MIXED EFFECTS MODELS Motivating Example: Influence of Menarche on Changes in Body Fat Prospective
More informationWeakly informative priors
Department of Statistics and Department of Political Science Columbia University 23 Apr 2014 Collaborators (in order of appearance): Gary King, Frederic Bois, Aleks Jakulin, Vince Dorie, Sophia Rabe-Hesketh,
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 informationIntroducing Generalized Linear Models: Logistic Regression
Ron Heck, Summer 2012 Seminars 1 Multilevel Regression Models and Their Applications Seminar Introducing Generalized Linear Models: Logistic Regression The generalized linear model (GLM) represents and
More informationMultivariate Survival Analysis
Multivariate Survival Analysis Previously we have assumed that either (X i, δ i ) or (X i, δ i, Z i ), i = 1,..., n, are i.i.d.. This may not always be the case. Multivariate survival data can arise in
More informationMachine Learning Linear Classification. Prof. Matteo Matteucci
Machine Learning Linear Classification Prof. Matteo Matteucci Recall from the first lecture 2 X R p Regression Y R Continuous Output X R p Y {Ω 0, Ω 1,, Ω K } Classification Discrete Output X R p Y (X)
More informationLecture 2: Poisson and logistic regression
Dankmar Böhning Southampton Statistical Sciences Research Institute University of Southampton, UK S 3 RI, 11-12 December 2014 introduction to Poisson regression application to the BELCAP study introduction
More informationRon Heck, Fall Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October 20, 2011)
Ron Heck, Fall 2011 1 EDEP 768E: Seminar in Multilevel Modeling rev. January 3, 2012 (see footnote) Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October
More informationMeasures of Association and Variance Estimation
Measures of Association and Variance Estimation Dipankar Bandyopadhyay, Ph.D. Department of Biostatistics, Virginia Commonwealth University D. Bandyopadhyay (VCU) BIOS 625: Categorical Data & GLM 1 / 35
More informationChapter 11. Correlation and Regression
Chapter 11. Correlation and Regression The word correlation is used in everyday life to denote some form of association. We might say that we have noticed a correlation between foggy days and attacks of
More informationRidge regression. Patrick Breheny. February 8. Penalized regression Ridge regression Bayesian interpretation
Patrick Breheny February 8 Patrick Breheny High-Dimensional Data Analysis (BIOS 7600) 1/27 Introduction Basic idea Standardization Large-scale testing is, of course, a big area and we could keep talking
More informationNELS 88. Latent Response Variable Formulation Versus Probability Curve Formulation
NELS 88 Table 2.3 Adjusted odds ratios of eighth-grade students in 988 performing below basic levels of reading and mathematics in 988 and dropping out of school, 988 to 990, by basic demographics Variable
More informationWell-developed and understood properties
1 INTRODUCTION TO LINEAR MODELS 1 THE CLASSICAL LINEAR MODEL Most commonly used statistical models Flexible models Well-developed and understood properties Ease of interpretation Building block for more
More informationCluster investigations using Disease mapping methods International workshop on Risk Factors for Childhood Leukemia Berlin May
Cluster investigations using Disease mapping methods International workshop on Risk Factors for Childhood Leukemia Berlin May 5-7 2008 Peter Schlattmann Institut für Biometrie und Klinische Epidemiologie
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 informationBAYESIAN MODEL FOR SPATIAL DEPENDANCE AND PREDICTION OF TUBERCULOSIS
BAYESIAN MODEL FOR SPATIAL DEPENDANCE AND PREDICTION OF TUBERCULOSIS Srinivasan R and Venkatesan P Dept. of Statistics, National Institute for Research Tuberculosis, (Indian Council of Medical Research),
More informationReview of CLDP 944: Multilevel Models for Longitudinal Data
Review of CLDP 944: Multilevel Models for Longitudinal Data Topics: Review of general MLM concepts and terminology Model comparisons and significance testing Fixed and random effects of time Significance
More informationWeakly informative priors
Department of Statistics and Department of Political Science Columbia University 21 Oct 2011 Collaborators (in order of appearance): Gary King, Frederic Bois, Aleks Jakulin, Vince Dorie, Sophia Rabe-Hesketh,
More informationIntroduction to lnmle: An R Package for Marginally Specified Logistic-Normal Models for Longitudinal Binary Data
Introduction to lnmle: An R Package for Marginally Specified Logistic-Normal Models for Longitudinal Binary Data Bryan A. Comstock and Patrick J. Heagerty Department of Biostatistics University of Washington
More informationDescribing Change over Time: Adding Linear Trends
Describing Change over Time: Adding Linear Trends Longitudinal Data Analysis Workshop Section 7 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section
More informationA Brief and Friendly Introduction to Mixed-Effects Models in Linguistics
A Brief and Friendly Introduction to Mixed-Effects Models in Linguistics Cluster-specific parameters ( random effects ) Σb Parameters governing inter-cluster variability b1 b2 bm x11 x1n1 x21 x2n2 xm1
More informationSparse Linear Models (10/7/13)
STA56: Probabilistic machine learning Sparse Linear Models (0/7/) Lecturer: Barbara Engelhardt Scribes: Jiaji Huang, Xin Jiang, Albert Oh Sparsity Sparsity has been a hot topic in statistics and machine
More informationStatistics 572 Semester Review
Statistics 572 Semester Review Final Exam Information: The final exam is Friday, May 16, 10:05-12:05, in Social Science 6104. The format will be 8 True/False and explains questions (3 pts. each/ 24 pts.
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 informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Topics: What happens to missing predictors Effects of time-invariant predictors Fixed vs. systematically varying vs. random effects Model building
More informationCIMAT Taller de Modelos de Capture y Recaptura Known Fate Survival Analysis
CIMAT Taller de Modelos de Capture y Recaptura 2010 Known Fate urvival Analysis B D BALANCE MODEL implest population model N = λ t+ 1 N t Deeper understanding of dynamics can be gained by identifying variation
More informationPropensity Score Analysis with Hierarchical Data
Propensity Score Analysis with Hierarchical Data Fan Li Alan Zaslavsky Mary Beth Landrum Department of Health Care Policy Harvard Medical School May 19, 2008 Introduction Population-based observational
More informationBayesian methods for missing data: part 1. Key Concepts. Nicky Best and Alexina Mason. Imperial College London
Bayesian methods for missing data: part 1 Key Concepts Nicky Best and Alexina Mason Imperial College London BAYES 2013, May 21-23, Erasmus University Rotterdam Missing Data: Part 1 BAYES2013 1 / 68 Outline
More informationLecture 4: Generalized Linear Mixed Models
Dankmar Böhning Southampton Statistical Sciences Research Institute University of Southampton, UK S 3 RI, 11-12 December 2014 An example with one random effect An example with two nested random effects
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More informationRandom Intercept Models
Random Intercept Models Edps/Psych/Soc 589 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Spring 2019 Outline A very simple case of a random intercept
More informationON THE USE OF HIERARCHICAL MODELS
ON THE USE OF HIERARCHICAL MODELS IN METHOD COMPARISON STUDIES Alessandra R. Brazzale LADSEB-CNR alessandra.brazzale@ladseb.pd.cnr.it TIES 2002 Genova, June 18-22, 2002 1 Credits Alberto Salvan (LADSEB-CNR,
More informationList of Supplemental Figures
Online Supplement for: Weather-Related Mortality: How Heat, Cold, and Heat Waves Affect Mortality in the United States, GB Anderson and ML Bell, Epidemiology List of Supplemental Figures efigure 1. Distribution
More informationWeb-based Supplementary Material for A Two-Part Joint. Model for the Analysis of Survival and Longitudinal Binary. Data with excess Zeros
Web-based Supplementary Material for A Two-Part Joint Model for the Analysis of Survival and Longitudinal Binary Data with excess Zeros Dimitris Rizopoulos, 1 Geert Verbeke, 1 Emmanuel Lesaffre 1 and Yves
More information1-bit Matrix Completion. PAC-Bayes and Variational Approximation
: PAC-Bayes and Variational Approximation (with P. Alquier) PhD Supervisor: N. Chopin Bayes In Paris, 5 January 2017 (Happy New Year!) Various Topics covered Matrix Completion PAC-Bayesian Estimation Variational
More informationBIAS OF MAXIMUM-LIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY
BIAS OF MAXIMUM-LIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY Ingo Langner 1, Ralf Bender 2, Rebecca Lenz-Tönjes 1, Helmut Küchenhoff 2, Maria Blettner 2 1
More informationSampling bias in logistic models
Sampling bias in logistic models Department of Statistics University of Chicago University of Wisconsin Oct 24, 2007 www.stat.uchicago.edu/~pmcc/reports/bias.pdf Outline Conventional regression models
More informationRecent Developments in Multilevel Modeling
Recent Developments in Multilevel Modeling Roberto G. Gutierrez Director of Statistics StataCorp LP 2007 North American Stata Users Group Meeting, Boston R. Gutierrez (StataCorp) Multilevel Modeling August
More informationChapter 20: Logistic regression for binary response variables
Chapter 20: Logistic regression for binary response variables In 1846, the Donner and Reed families left Illinois for California by covered wagon (87 people, 20 wagons). They attempted a new and untried
More informationExpression Data Exploration: Association, Patterns, Factors & Regression Modelling
Expression Data Exploration: Association, Patterns, Factors & Regression Modelling Exploring gene expression data Scale factors, median chip correlation on gene subsets for crude data quality investigation
More informationDIAGNOSTICS FOR STRATIFIED CLINICAL TRIALS IN PROPORTIONAL ODDS MODELS
DIAGNOSTICS FOR STRATIFIED CLINICAL TRIALS IN PROPORTIONAL ODDS MODELS Ivy Liu and Dong Q. Wang School of Mathematics, Statistics and Computer Science Victoria University of Wellington New Zealand Corresponding
More informationLinear Regression With Special Variables
Linear Regression With Special Variables Junhui Qian December 21, 2014 Outline Standardized Scores Quadratic Terms Interaction Terms Binary Explanatory Variables Binary Choice Models Standardized Scores:
More informationIncorporating published univariable associations in diagnostic and prognostic modeling
Incorporating published univariable associations in diagnostic and prognostic modeling Thomas Debray Julius Center for Health Sciences and Primary Care University Medical Center Utrecht The Netherlands
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 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 informationContinuous Time Survival in Latent Variable Models
Continuous Time Survival in Latent Variable Models Tihomir Asparouhov 1, Katherine Masyn 2, Bengt Muthen 3 Muthen & Muthen 1 University of California, Davis 2 University of California, Los Angeles 3 Abstract
More informationA Joint Model with Marginal Interpretation for Longitudinal Continuous and Time-to-event Outcomes
A Joint Model with Marginal Interpretation for Longitudinal Continuous and Time-to-event Outcomes Achmad Efendi 1, Geert Molenberghs 2,1, Edmund Njagi 1, Paul Dendale 3 1 I-BioStat, Katholieke Universiteit
More informationSTA 216, GLM, Lecture 16. October 29, 2007
STA 216, GLM, Lecture 16 October 29, 2007 Efficient Posterior Computation in Factor Models Underlying Normal Models Generalized Latent Trait Models Formulation Genetic Epidemiology Illustration Structural
More informationMultilevel Modeling: A Second Course
Multilevel Modeling: A Second Course Kristopher Preacher, Ph.D. Upcoming Seminar: February 2-3, 2017, Ft. Myers, Florida What this workshop will accomplish I will review the basics of multilevel modeling
More informationNeutral Bayesian reference models for incidence rates of (rare) clinical events
Neutral Bayesian reference models for incidence rates of (rare) clinical events Jouni Kerman Statistical Methodology, Novartis Pharma AG, Basel BAYES2012, May 10, Aachen Outline Motivation why reference
More informationCalculating Effect-Sizes. David B. Wilson, PhD George Mason University
Calculating Effect-Sizes David B. Wilson, PhD George Mason University The Heart and Soul of Meta-analysis: The Effect Size Meta-analysis shifts focus from statistical significance to the direction and
More information