Frailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Mela. P.


 Francine Blake
 11 months ago
 Views:
Transcription
1 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
2 Outlines of the talk Introduction Spatial Frailty Modelling Lattice models Bayesian implementation Bayesian model choice Application to Minnessota Mortality Model fitting Mapping summaries Model checking Neonatal versus postneonatal mortality Comparison of Spatial Frailty and Logistic Regression Models Concluding Remarks
3 Introduction Timetoevent data, Most of the time, grouped into clusters; clinical sites, geographic regions. Hierarchical modelling approach using stratumspecific frailties is often appropriate for this type of data. Let t ij be the time to event or censoring for subject j in stratum i, j = 1,..., n i, i = 1,..., I. Let x ij be a vector of individualspecific covariates. In the frailty setting h(t ij ; X ij ) = h 0 (t ij ) exp(x β) (1) h(t ij ; X ij ) = h 0 (t ij ) exp(x β + W i ) (2)
4 Introduction cont.. W i iid N(0, σ 2 ) Normal distribution is used for the frailty model to facilitate correlation structure between them. Introducing parametric model on the base line (h 0 )( Weibull model) makes it hierarchical frailty model. The parametric frailty model becomes; h(t ij ; X ij ) = ρt ρ 1 ij exp(x β + W i ) (3) By introducing the prior distributions on ρ, βandσ, the bayesian implementation can be handled.
5 Introduction cont.. Such spatial arrangement of the strata are modeled in two ways. Geostatistical approaches, where we use the exact geographic locations (e.g. latitude and longitude) of the strata. Lattice approaches, where we use only the positions of the strata relative to each other.
6 2. Spatial Frailty Modelling 2.1 Geostatistical Models These Models assume that the random process of interest Y (s) is indexed continuously by s throughout a space D. Predict the unobserved value Y (t) at some target location t, given observations Y {Y (s i )} at known source locations s i, i = 1,..., I. Y µ, θ N I (µ, H(θ)), θ = (σ 2, φ) (4) Where N I indicates I dimensional normal distribution with stationary mean level µ and H(θ) ii is the covariance between Y (s i ) and Y (s i ).
7 Assuming isotropic setting H(θ) ii = σ 2 exp{ φd ii }, σ 2 > 0, φ > 0. (5) apply the geostatistical model (4) and (5) to the random frailties W i with the spatial structure. Adding prior distributions for ρ, β, andθ completes a Bayesian specification using (3) and (6). W Θ N I (0, H(Θ)) (6)
8 Lattice Models In this model assumption W is defined only on discretely indexed regions such that the regions form a partition of the space D, usually incorporate information about the adjacency of regions. W λ CAR(λ) (7) The most common form of this prior (Bernardinelli and Montomoli, 1992) has joint distribution proportional to λ I /2 exp{ λ (W i W 2 i )} iadji λ I /2 exp{ λ 2 I m i W i (W i W i )} i=1 where i adj i denotes that regions i and i are adjacent, Wi is
9 where i adj i denotes that regions i and i are adjacent, W i is the average of the W i i that are adjacent to Wi, and m i is the number of these adjacencies. This CAR prior is a member of the class of pairwise difference priors (Besag et al., 1995), which are identified only up to an additive constant. To permit the data to identify an intercept term 0 in the hazard function (2), we also add the constraint I i=1 W i=0. The prior specification is then, W i W i i N( W i, And put the gamma hyperprior distribution for λ. 1 λm i ). (8)
10 Bayesian Implementation Letting γ ij be a death indicator, the joint posterior distribution of interest is given by P(β, W, ρ, θ t, x, γ) L(β, W, ρ; t, x, γ)p(w Θ)P(β)P(ρ)P(θ) (9) I n i L(β, W, ρ, θ t, x, γ) {ρt ρ 1 ij exp(β T X ij + W i } γ ij (10) i=1 j=1 exp{ t ρ 1 ij exp(β T X ij + W i )} The model specification in the Bayesian setup is completed by assigning prior distributions for β, ρ, and Θ.
11 2.4 Bayesian Model Choice Bayes factors are notoriously difficult to compute, and the Bayes factor is only defined when the marginal density of y under each model is proper. Deviance information criterion have been used for model selection. The deviance Statistics is: D(θ) = 2 log f (y θ) + 2 log h(y) (11) Where h(y) is some standardize function. The DIC is defined as: DIC = D + P D Where D and P D are E θ y (D), E θ y (D) D(E θ y (θ)), respectively.
12 3. Application to Minnesota Infant Mortality 3.1 Model fitting The data were obtained from the linked birthdeath records data registry kept by the Minnesota Department of Health. The data comprise live births occurring during the years 1992 to 1996 followed through the first year of life. The covariate information such as birth weight, sex, race, mothers age, and the mothers total number of previous births have been incorporated in this study. The contiguous county neighbor structure as well as the latitude and longitude of the centroids have been taken. This information is important to implement both the geostatistical and the lattice models.
13 3.1 Model fitting In addition, they investigate the nonspatial frailty model (2), as well as a simple nonhierarchical (no frailty) model, which simply sets W i = 0 for all i. Metropolis random walk steps with Gaussian proposals were used for sampling from the full conditionals for β, while Hastings independence steps with gamma proposals were used for updating ρ. For the geostatistical modeling of the W i, they used the isotropic exponential correlation function. The intercounty distances (d ii ) are computed using the coordinates of the centroids of the counties.
14 Model fitting (Cont.) For the exponential correlation function, the quantity 3 φ may be thought of as a measure of the effective isotropic range, i.e. the distance beyond which the correlation between the observations drops to less than Here they adopt a vague IG(2, 0.01) prior for σ 2, ensuring a mean of 100 but infinite variance. For φ we take a vague G(0.01, 100) prior, having mean 1 but variance 100. For the lattice model, they use the CAR distribution for the W i,putting a prior for the smoothness parameter λ.
15 3.1 Model fitting By using DIC and effective model size pd(from the table), for the nofrailty model we can see that a pd is 8.72, very close to the actual number of parameters(nine). The DIC values suggest that each of these models is substantially better than the nofrailty model, despite their increased size.
16 Model Fitting Figure: Ha Figure: Sa
17 In all three models, all of the predictors are significant at the 0.05 level. Boys have a higher hazard of death during the first year of life. Evidence of the modest amount of spatial similarity in our dataset is provided by the posterior median for φ in the geostatistical model (Table 4); its value of implies a median effective spatial range of 3/0.043 = 70 km. Indeed, this provides some reason why our geostatistical and CAR results should be so similar, since in most cases, borrowing strength from counties having centroids within 70 km will be nearly the same as borrowing strength from adjacent counties. A benefit of fitting the spatial CAR structure is seen in the reduction of the length of the 95% credible intervals for the covariates in the spatial models compared to the i.i.d model.
18 3.2 Mapping summaries Figures 3 and 4 map(from the Paper!!!) the posterior medians of the W i under the nonspatial (i.i.d. frailties) and CAR models, respectively, in the case where no covariates x are included in the model. The fitted i.i.d model indicates excess mortality in the north, which is accentuated and extended to a generally increasing pattern from south to north by the CAR model. This trend, combined with the clear emergence of the Minneapolis (county 27) and St Paul (county 62) urban area, strongly suggests the need for fitting covariates in our model, most of which vary spatially.
19 3.3 Model checking Figure: 1, Boxplots of posterior median frailties, i.i.d. and CAR models with and without covariates
20 From Figure 1, Posterior median frailties for the four cases (IID no covariates, CAR no covariates, IID with covariates, CAR with covariates): The tightness of the full CAR boxplot suggests this model is best at reducing the need for the frailty terms.
21 Neonatal versus postneonatal mortality Neonatal (death within the first 28 days) and Post Neonatal (Death between 29 and 365). So these two data sets are fitted separately using CAR frailty model. Figure: Ha Figure: Sa
22 Sex, birthweight and total births are significant for both groups. while mothers age and native American race are significant only for the postneonatal group, and black and unknown race are significant only for the neonatal group. Thus the two groups differ in ways that are both intuitive and substantively intriguing.
23 Comparison of Spatial Frailty and Logistic Regression Models Since the dataset does not have any censored, competing risks, or any reason other than the end of the study, there is no ambiguity in defining a binary survival outcome to use logistic regression model. That is, we replace the event time data t ij with an indicator of whether the subject did (Y ij=0 ) or did not (Y ij=1 ) survive the first year. Letting p ij = (Y ij=1 )), then model is logit(p ij ) = β ˆ ij X + W i (12) with the usual flat prior for ˆβ and an i.i.d., CAR, or geostatistical prior for the W i. When the probability of death is very small, as it is in the case of infant mortality, the log odds and log relative risk become even more similar.
24 Comparison of Spatial Frailty cont. When the probability of death is very small, as it is in the case of infant mortality, the log odds and log relative risk become even more similar. Figure: Posterior medians of the frailties Wi (horizontal axis) versus posterior medians of the logistic random effects W i (vertical axis). Plotting character is county number
25 Concluding Remark Several hierarchical approaches to frailty modeling for spatially correlated survival data have been discussed. Previous work by Carlin and Hodges (1999) suggests a generalization of our basic model (3) to h(t ij ; X ij ) = ρ i t ρ i 1 ij exp(x β + W i ) (13) That is, they allow two sets of random effects: the existing frailty parameters W i, and a new set of shape parameters ρ i. This then allows both the overall level and the shape of the hazard function over time to vary from county to county.
26 Sudipto B. et al., Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota. Biostatistics, pp , Carlin P. et al., Hierarchical Proportional Hazards Regression Models for Highly Stratified Data. Biometricss, , 1999.
Multivariate spatial modeling
Multivariate spatial modeling Pointreferenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21 Multivariate spatial modeling Pointreferenced
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 informationspbayes: An R Package for Univariate and Multivariate Hierarchical Pointreferenced Spatial Models
spbayes: An R Package for Univariate and Multivariate Hierarchical Pointreferenced Spatial Models Andrew O. Finley 1, Sudipto Banerjee 2, and Bradley P. Carlin 2 1 Michigan State University, Departments
More informationSpatioTemporal Threshold Models for Relating UV Exposures and Skin Cancer in the Central United States
SpatioTemporal Threshold Models for Relating UV Exposures and Skin Cancer in the Central United States Laura A. Hatfield and Bradley P. Carlin Division of Biostatistics School of Public Health University
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 informationModels for spatial data (cont d) Types of spatial data. Types of spatial data (cont d) Hierarchical models for spatial data
Hierarchical models for spatial data Based on the book by Banerjee, Carlin and Gelfand Hierarchical Modeling and Analysis for Spatial Data, 2004. We focus on Chapters 1, 2 and 5. Georeferenced data arise
More informationFrailty Modeling for clustered survival data: a simulation study
Frailty Modeling for clustered survival data: a simulation study IAA Oslo 2015 Souad ROMDHANE LaREMFiQ  IHEC University of Sousse (Tunisia) souad_romdhane@yahoo.fr Lotfi BELKACEM LaREMFiQ  IHEC University
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics, School of Public
More informationHierarchical Modelling for Univariate and Multivariate Spatial Data
Hierarchical Modelling for Univariate and Multivariate Spatial Data p. 1/4 Hierarchical Modelling for Univariate and Multivariate Spatial Data Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationAnalysis of Cure Rate Survival Data Under Proportional Odds Model
Analysis of Cure Rate Survival Data Under Proportional Odds Model Yu Gu 1,, Debajyoti Sinha 1, and Sudipto Banerjee 2, 1 Department of Statistics, Florida State University, Tallahassee, Florida 32310 5608,
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
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 informationHypothesis Testing. Econ 690. Purdue University. Justin L. Tobias (Purdue) Testing 1 / 33
Hypothesis Testing Econ 690 Purdue University Justin L. Tobias (Purdue) Testing 1 / 33 Outline 1 Basic Testing Framework 2 Testing with HPD intervals 3 Example 4 Savage Dickey Density Ratio 5 Bartlett
More informationCTDLPositive Stable Frailty Model
CTDLPositive 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 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 informationPhysician Performance Assessment / Spatial Inference of Pollutant Concentrations
Physician Performance Assessment / Spatial Inference of Pollutant Concentrations Dawn Woodard Operations Research & Information Engineering Cornell University Johns Hopkins Dept. of Biostatistics, April
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 informationBIOS 312: Precision of Statistical Inference
and Power/Sample Size and Standard Errors BIOS 312: of Statistical Inference Chris Slaughter Department of Biostatistics, Vanderbilt University School of Medicine January 3, 2013 Outline Overview and Power/Sample
More informationAnalysis of Marked Point Patterns with Spatial and Nonspatial Covariate Information
Analysis of Marked Point Patterns with Spatial and Nonspatial Covariate Information p. 1/27 Analysis of Marked Point Patterns with Spatial and Nonspatial Covariate Information Shengde Liang, Bradley
More informationComparing Noninformative Priors for Estimation and Prediction in Spatial Models
Environmentrics 00, 1 12 DOI: 10.1002/env.XXXX Comparing Noninformative Priors for Estimation and Prediction in Spatial Models Regina Wu a and Cari G. Kaufman a Summary: Fitting a Bayesian model to spatial
More informationSurvival Regression Models
Survival Regression Models David M. Rocke May 18, 2017 David M. Rocke Survival Regression Models May 18, 2017 1 / 32 Background on the Proportional Hazards Model The exponential distribution has constant
More informationWhy experimenters should not randomize, and what they should do instead
Why experimenters should not randomize, and what they should do instead Maximilian Kasy Department of Economics, Harvard University Maximilian Kasy (Harvard) Experimental design 1 / 42 project STAR Introduction
More informationHierarchical Modeling and Analysis for Spatial Data
Hierarchical Modeling and Analysis for Spatial Data Bradley P. Carlin, Sudipto Banerjee, and Alan E. Gelfand brad@biostat.umn.edu, sudiptob@biostat.umn.edu, and alan@stat.duke.edu University of Minnesota
More informationUsing Estimating Equations for Spatially Correlated A
Using Estimating Equations for Spatially Correlated Areal Data December 8, 2009 Introduction GEEs Spatial Estimating Equations Implementation Simulation Conclusion Typical Problem Assess the relationship
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 informationSeparate and Joint Modeling of Longitudinal and Event Time Data Using Standard Computer Packages
Separate and Joint Modeling of Longitudinal and Event Time Data Using Standard Computer Packages Xu GUO and Bradley P. CARLIN Many clinical trials and other medical and reliability studies generate both
More informationStatistics 352: Spatial statistics. Jonathan Taylor. Department of Statistics. Models for discrete data. Stanford University.
352: 352: Models for discrete data April 28, 2009 1 / 33 Models for discrete data 352: Outline Dependent discrete data. Image data (binary). Ising models. Simulation: Gibbs sampling. Denoising. 2 / 33
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 informationLecture 5: Spatial probit models. James P. LeSage University of Toledo Department of Economics Toledo, OH
Lecture 5: Spatial probit models James P. LeSage University of Toledo Department of Economics Toledo, OH 43606 jlesage@spatialeconometrics.com March 2004 1 A Bayesian spatial probit model with individual
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics,
More informationTechnical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models
Technical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models Christopher Paciorek, Department of Statistics, University
More informationEffects of Residual Smoothing on the Posterior of the Fixed Effects in DiseaseMapping Models
Biometrics 62, 1197 1206 December 2006 DOI: 10.1111/j.15410420.2006.00617.x Effects of Residual Smoothing on the Posterior of the Fixed Effects in DiseaseMapping Models Brian J. Reich, 1, James S. Hodges,
More informationLattice Data. Tonglin Zhang. Spatial Statistics for Point and Lattice Data (Part III)
Title: Spatial Statistics for Point Processes and Lattice Data (Part III) Lattice Data Tonglin Zhang Outline Description Research Problems Global Clustering and Local Clusters Permutation Test Spatial
More informationBayesian Learning. HT2015: SC4 Statistical Data Mining and Machine Learning. Maximum Likelihood Principle. The Bayesian Learning Framework
HT5: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford http://www.stats.ox.ac.uk/~sejdinov/sdmml.html Maximum Likelihood Principle A generative model for
More informationMetropolisHastings Algorithm
Strength of the Gibbs sampler MetropolisHastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to
More informationLongitudinal + Reliability = Joint Modeling
Longitudinal + Reliability = Joint Modeling Carles Serrat Institute of Statistics and Mathematics Applied to Building CYTEDHAROSA International Workshop November 2122, 2013 Barcelona Mainly from Rizopoulos,
More informationBayesian inference. Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark. April 10, 2017
Bayesian inference Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark April 10, 2017 1 / 22 Outline for today A genetic example Bayes theorem Examples Priors Posterior summaries
More informationMarkov Chain Monte Carlo in Practice
Markov Chain Monte Carlo in Practice Edited by W.R. Gilks Medical Research Council Biostatistics Unit Cambridge UK S. Richardson French National Institute for Health and Medical Research Vilejuif France
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 informationSemiparametric Varying Coefficient Models for Matched CaseCrossover Studies
Semiparametric Varying Coefficient Models for Matched CaseCrossover Studies Ana Maria OrtegaVilla Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial
More informationBayesian Inference. Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Ausín and Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative
More informationLecture 12: Application of Maximum Likelihood Estimation:Truncation, Censoring, and Corner Solutions
Econ 513, USC, Department of Economics Lecture 12: Application of Maximum Likelihood Estimation:Truncation, Censoring, and Corner Solutions I Introduction Here we look at a set of complications with the
More informationThe Multilevel Logit Model for Binary Dependent Variables Marco R. Steenbergen
The Multilevel Logit Model for Binary Dependent Variables Marco R. Steenbergen January 2324, 2012 Page 1 Part I The Single Level Logit Model: A Review Motivating Example Imagine we are interested in voting
More informationHierarchical Multiresolution Approaches for Dense PointLevel Breast Cancer Treatment Data
Hierarchical Multiresolution Approaches for Dense PointLevel Breast Cancer Treatment Data Shengde Liang, Sudipto Banerjee, Sally Bushhouse, Andrew Finley, and Bradley P. Carlin 1 Correspondence author:
More informationPracticum : Spatial Regression
: Alexandra M. Schmidt Instituto de Matemática UFRJ  www.dme.ufrj.br/ alex 2014 Búzios, RJ, www.dme.ufrj.br Exploratory (Spatial) Data Analysis 1. Nonspatial summaries Numerical summaries: Mean, median,
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and NonParametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationKernel density estimation in R
Kernel density estimation in R Kernel density estimation can be done in R using the density() function in R. The default is a Guassian kernel, but others are possible also. It uses it s own algorithm to
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 informationProbabilistic machine learning group, Aalto University Bayesian theory and methods, approximative integration, model
Aki Vehtari, Aalto University, Finland Probabilistic machine learning group, Aalto University http://research.cs.aalto.fi/pml/ Bayesian theory and methods, approximative integration, model assessment and
More informationBayesian Inference on Joint Mixture Models for SurvivalLongitudinal Data with Multiple Features. Yangxin Huang
Bayesian Inference on Joint Mixture Models for SurvivalLongitudinal Data with Multiple Features Yangxin Huang Department of Epidemiology and Biostatistics, COPH, USF, Tampa, FL yhuang@health.usf.edu January
More informationThe STS Surgeon Composite Technical Appendix
The STS Surgeon Composite Technical Appendix Overview Surgeonspecific riskadjusted operative operative mortality and major complication rates were estimated using a bivariate randomeffects logistic
More informationvariability of the model, represented by σ 2 and not accounted for by Xβ
Posterior Predictive Distribution Suppose we have observed a new set of explanatory variables X and we want to predict the outcomes ỹ using the regression model. Components of uncertainty in p(ỹ y) variability
More informationThe Relationship Between the Power Prior and Hierarchical Models
Bayesian Analysis 006, Number 3, pp. 55 574 The Relationship Between the Power Prior and Hierarchical Models MingHui Chen, and Joseph G. Ibrahim Abstract. The power prior has emerged as a useful informative
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 informationBAYESIAN INFLUENCE DIAGNOSTIC METHODS FOR PARAMETRIC REGRESSION MODELS
BAYESIAN INFLUENCE DIAGNOSTIC METHODS FOR PARAMETRIC REGRESSION MODELS Hyunsoon Cho A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of
More informationRandom Effects Models for Network Data
Random Effects Models for Network Data Peter D. Hoff 1 Working Paper no. 28 Center for Statistics and the Social Sciences University of Washington Seattle, WA 981954320 January 14, 2003 1 Department of
More informationTied survival times; estimation of survival probabilities
Tied survival times; estimation of survival probabilities Patrick Breheny November 5 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/22 Introduction Tied survival times Introduction Breslow approximation
More informationDynamic Prediction of Disease Progression Using Longitudinal Biomarker Data
Dynamic Prediction of Disease Progression Using Longitudinal Biomarker Data Xuelin Huang Department of Biostatistics M. D. Anderson Cancer Center The University of Texas Joint Work with Jing Ning, Sangbum
More informationDirichlet process Bayesian clustering with the R package PReMiuM
Dirichlet process Bayesian clustering with the R package PReMiuM Dr Silvia Liverani Brunel University London July 2015 Silvia Liverani (Brunel University London) Profile Regression 1 / 18 Outline Motivation
More informationOptimal rules for timing intercourse to achieve pregnancy
Optimal rules for timing intercourse to achieve pregnancy Bruno Scarpa and David Dunson Dipartimento di Statistica ed Economia Applicate Università di Pavia Biostatistics Branch, National Institute of
More informationGaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008
Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:
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 informationAn Introduction to Spatial Statistics. Chunfeng Huang Department of Statistics, Indiana University
An Introduction to Spatial Statistics Chunfeng Huang Department of Statistics, Indiana University Microwave Sounding Unit (MSU) Anomalies (Monthly): 19792006. Iron Ore (Cressie, 1986) Raw percent data
More informationQuantile POD for HitMiss Data
Quantile POD for HitMiss Data YewMeng Koh a and William Q. Meeker a a Center for Nondestructive Evaluation, Department of Statistics, Iowa State niversity, Ames, Iowa 50010 Abstract. Probability of detection
More informationLISA Short Course Series Generalized Linear Models (GLMs) & Categorical Data Analysis (CDA) in R. Liang (Sally) Shan Nov. 4, 2014
LISA Short Course Series Generalized Linear Models (GLMs) & Categorical Data Analysis (CDA) in R Liang (Sally) Shan Nov. 4, 2014 L Laboratory for Interdisciplinary Statistical Analysis LISA helps VT researchers
More informationA Bayesian Probit Model with Spatial Dependencies
A Bayesian Probit Model with Spatial Dependencies Tony E. Smith Department of Systems Engineering University of Pennsylvania Philadephia, PA 19104 email: tesmith@ssc.upenn.edu James P. LeSage Department
More informationBayesian Learning (II)
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning (II) Niels Landwehr Overview Probabilities, expected values, variance Basic concepts of Bayesian learning MAP
More informationBeyond MCMC in fitting complex Bayesian models: The INLA method
Beyond MCMC in fitting complex Bayesian models: The INLA method Valeska Andreozzi Centre of Statistics and Applications of Lisbon University (valeska.andreozzi at fc.ul.pt) European Congress of Epidemiology
More informationModels for Count and Binary Data. Poisson and Logistic GWR Models. 24/07/2008 GWR Workshop 1
Models for Count and Binary Data Poisson and Logistic GWR Models 24/07/2008 GWR Workshop 1 Outline I: Modelling counts Poisson regression II: Modelling binary events Logistic Regression III: Poisson Regression
More informationRidge regression. Patrick Breheny. February 8. Penalized regression Ridge regression Bayesian interpretation
Patrick Breheny February 8 Patrick Breheny HighDimensional Data Analysis (BIOS 7600) 1/27 Introduction Basic idea Standardization Largescale testing is, of course, a big area and we could keep talking
More informationMachine Learning 2017
Machine Learning 2017 Volker Roth Department of Mathematics & Computer Science University of Basel 21st March 2017 Volker Roth (University of Basel) Machine Learning 2017 21st March 2017 1 / 41 Section
More informationAreal Unit Data Regular or Irregular Grids or Lattices Large Pointreferenced Datasets
Areal Unit Data Regular or Irregular Grids or Lattices Large Pointreferenced Datasets Is there spatial pattern? Chapter 3: Basics of Areal Data Models p. 1/18 Areal Unit Data Regular or Irregular Grids
More informationChapter 2. Data Analysis
Chapter 2 Data Analysis 2.1. Density Estimation and Survival Analysis The most straightforward application of BNP priors for statistical inference is in density estimation problems. Consider the generic
More informationAdvanced Methods for Agricultural and Agroenvironmental. Emily Berg, Zhengyuan Zhu, Sarah Nusser, and Wayne Fuller
Advanced Methods for Agricultural and Agroenvironmental Monitoring Emily Berg, Zhengyuan Zhu, Sarah Nusser, and Wayne Fuller Outline 1. Introduction to the National Resources Inventory 2. Hierarchical
More informationA Generalized Global Rank Test for Multiple, Possibly Censored, Outcomes
A Generalized Global Rank Test for Multiple, Possibly Censored, Outcomes Ritesh Ramchandani Harvard School of Public Health August 5, 2014 Ritesh Ramchandani (HSPH) Global Rank Test for Multiple Outcomes
More informationFrailty Models and Copulas: Similarities and Differences
Frailty Models and Copulas: Similarities and Differences KLARA GOETHALS, PAUL JANSSEN & LUC DUCHATEAU Department of Physiology and Biometrics, Ghent University, Belgium; Center for Statistics, Hasselt
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS Parametric Distributions Basic building blocks: Need to determine given Representation: or? Recall Curve Fitting Binary Variables
More informationST440/540: Applied Bayesian Statistics. (9) Model selection and goodnessoffit checks
(9) Model selection and goodnessoffit checks Objectives In this module we will study methods for model comparisons and checking for model adequacy For model comparisons there are a finite number of candidate
More informationMoger, TA; Haugen, M; Yip, BHK; Gjessing, HK; Borgan, Ø. Citation Lifetime Data Analysis, 2010, v. 17, n. 3, p
Title A hierarchical frailty model applied to twogeneration melanoma data Author(s) Moger, TA; Haugen, M; Yip, BHK; Gjessing, HK; Borgan, Ø Citation Lifetime Data Analysis, 2010, v. 17, n. 3, p. 445460
More informationComparing Noninformative Priors for Estimation and. Prediction in Spatial Models
Comparing Noninformative Priors for Estimation and Prediction in Spatial Models Vigre Semester Report by: Regina Wu Advisor: Cari Kaufman January 31, 2010 1 Introduction Gaussian random fields with specified
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 informationMotivation Scale Mixutres of Normals Finite Gaussian Mixtures SkewNormal 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 informationA new strategy for metaanalysis of continuous covariates in observational studies with IPD. Willi Sauerbrei & Patrick Royston
A new strategy for metaanalysis of continuous covariates in observational studies with IPD Willi Sauerbrei & Patrick Royston Overview Motivation Continuous variables functional form Fractional polynomials
More informationChapter 4 Regression Models
23.August 2010 Chapter 4 Regression Models The target variable T denotes failure time We let x = (x (1),..., x (m) ) represent a vector of available covariates. Also called regression variables, regressors,
More informationWorstCase Bounds for Gaussian Process Models
WorstCase Bounds for Gaussian Process Models Sham M. Kakade University of Pennsylvania Matthias W. Seeger UC Berkeley Abstract Dean P. Foster University of Pennsylvania We present a competitive analysis
More informationSpatial Stochastic Volatility for Lattice Data
Spatial Stochastic Volatility for Lattice Data Jun Yan Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 2242, U.S.A. March 22, 06 Abstract Spatial heteroskedasticity may
More informationNORGES TEKNISKNATURVITENSKAPELIGE UNIVERSITET
NORGES TEKNISKNATURVITENSKAPELIGE UNIVERSITET Approximate Bayesian Inference for nonhomogeneous Poisson processes with application to survival analysis by Rupali Akerkar, Sara Martino and Håvard Rue PREPRINT
More informationChapter 4  Fundamentals of spatial processes Lecture notes
TK4150  Intro 1 Chapter 4  Fundamentals of spatial processes Lecture notes Odd Kolbjørnsen and Geir Storvik January 30, 2017 STK4150  Intro 2 Spatial processes Typically correlation between nearby sites
More informationLatent 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 informationOverall Objective Priors
Overall Objective Priors Jim Berger, Jose Bernardo and Dongchu Sun Duke University, University of Valencia and University of Missouri Recent advances in statistical inference: theory and case studies University
More informationExample using R: Heart Valves Study
Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion R and WinBUGS Examples p. 1/27 Example using R: Heart Valves
More informationMultinomial Logistic Regression Models
Stat 544, Lecture 19 1 Multinomial Logistic Regression Models Polytomous responses. Logistic regression can be extended to handle responses that are polytomous, i.e. taking r>2 categories. (Note: The word
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 informationECE531 Homework Assignment Number 6 Solution
ECE53 Homework Assignment Number 6 Solution Due by 8:5pm on Wednesday 3Mar Make sure your reasoning and work are clear to receive full credit for each problem.. 6 points. Suppose you have a scalar random
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 informationSpatial Analysis of Incidence Rates: A Bayesian Approach
Spatial Analysis of Incidence Rates: A Bayesian Approach Silvio A. da Silva, Luiz L.M. Melo and Ricardo Ehlers July 2004 Abstract Spatial models have been used in many fields of science where the data
More informationBayesian modelling. HansPeter Helfrich. University of Bonn. TheodorBrinkmannGraduate School
Bayesian modelling HansPeter Helfrich University of Bonn TheodorBrinkmannGraduate School H.P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 1 / 22 Overview 1 Bayesian modelling
More informationAdvanced Machine Learning
Advanced Machine Learning Nonparametric Bayesian Models Learning/Reasoning in Open Possible Worlds Eric Xing Lecture 7, August 4, 2009 Reading: Eric Xing Eric Xing @ CMU, 20062009 Clustering Eric Xing
More informationBayesian Classification Methods
Bayesian Classification Methods Suchit Mehrotra North Carolina State University smehrot@ncsu.edu October 24, 2014 Suchit Mehrotra (NCSU) Bayesian Classification October 24, 2014 1 / 33 How do you define
More information