1Non Linear mixed effects ordinary differential equations models. M. Prague - SISTM - NLME-ODE September 27,

Size: px
Start display at page:

Download "1Non Linear mixed effects ordinary differential equations models. M. Prague - SISTM - NLME-ODE September 27,"

Transcription

1 GDR MaMoVi 2017 Parameter estimation in Models with Random effects based on Ordinary Differential Equations: A bayesian maximum a posteriori approach. Mélanie PRAGUE, Daniel COMMENGES & Rodolphe THIÉBAUT - SISTM September 27, 2017

2 1Non Linear mixed effects ordinary differential equations models M. Prague - SISTM - NLME-ODE September 27,

3 Available Data come from clinical trials and observational studies Longitudinal data Y ijk : patient i, time j and biomarker k M. Prague - SISTM - NLME-ODE September 27,

4 Mathematical Model for mechanistic models Compartiments Biologiques Compartiment Q T T V Signification CD4 Quiescents CD4 Activés CD4 Activés Infectés Virions M. Prague - SISTM - NLME-ODE September 27,

5 Mathematical Model for mechanistic models Dynamique des cellules T (CD4 infectés) dt dt = γvt µ T T Paramètre Signification µ T Taux de décès des cellules T γ Infectivité : Taux d infection des cellules T par les virions M. Prague - SISTM - NLME-ODE September 27,

6 Mathematical Model for mechanistic models Target cells model dq dt = λ µ Q Q αq + ρt dt dt = αq ρt µ T T γvt dt dt = γvt µ T T dv dt = πt µ V V M. Prague - SISTM - NLME-ODE September 27,

7 Statistical Model for mechanistic models Target cells model Mixte effects models on parameters ( ) ξ i = α i, λ i,..., γ i 0, µ i V ξ l i = φ l + zl i (t)β l + }{{} ωl i (t)ul i }{{} Effets fixes Effets aléatoires u i N (0, I q) M. Prague - SISTM - NLME-ODE September 27,

8 Observational Model for mechanistic models Among, X(t ij, ξ i ) = (Q(t ij, ξ i ), T (t ij, ξ i ), T (t ij, ξ i ), V (t ij, ξ i )) We only observe (with measurement errors): Viral load : CD4 count : Y ij1 = log 10 (V ) + ɛ ij1 Y ij2 = (Q + T + T ) ɛ ij2 ɛ ijm N (0, σ 2 m) Donc, g 1(.) = log 10 (.) g 2(.) = (.) 0.25 M. Prague - SISTM - NLME-ODE September 27,

9 Parameters of interest We want to estimate more than 15 parameters: θ = {λ, µ Q, α, ρ, µ T, γπ, µ T, µ V } {{ } Effet fixes, β 1,..., β r }{{} Covariates effects, σ 1,..., σ s }{{}, Σ 1,..., Σ k }{{} Random effects Measurement errors } There are sometimes problems of identifiability 1 This approach is unbiased more efficient than marginal structural models in presence of dynamic treatment regimens 2 1 [1] Guedj et al. (2010), Bull. Math. Biol. 2 [2] Prague et al. (2016), Biometrics. M. Prague - SISTM - NLME-ODE September 27,

10 2Bayesian penalised likelihood estimation M. Prague - SISTM - Estimation September 27,

11 Existing methods Method Ref. Software Non parametric Functional analysis [Ramsay et al. 2012] - Non Bayesian parametric FOCE [Pinheiro et Bates 1995] R Bayesian SAEM [Kuhn et al MONOLIX Lavielle et al. 2007] Bayesian MCMC [Lunn et al 2000 WinBUGS Huang et al. 2011] Bayesian penalized likelihood [Guedj et al 2007; NIMROD 3 Prague et al. 2013] 3 [3] Prague et al. (2013), Comp. Meth. and Prog. in Biomed. M. Prague - SISTM - Estimation September 27,

12 Penalization for the log-likelihood It is possible to have an approximate idea of the value of biological parameters and treatment effects, for example from previous in vitro experiment or analysis of studies. Normal approximation of the posterior of previous analysis can be used as new prior for analysis as in a sequential bayesian meta-analysis 4 : J(θ) = { 9 φj E 0 (φ } 2 j) var0 (φ j) j=1 n TRT + j=1 { βj E 0 (β j) } 2 var0 (β j) 4 [4] Prague et al. (2016) Journal de la statistique francaise M. Prague - SISTM - Estimation September 27,

13 Penalized likelihood computation (1) Individual likelihood (censorship δ ij = I Yij1 <ζ) L Fi u i = { [ 1 exp 1 ( Yij1 g 1(X(t ij, ξ ) i 2 ]} 1 δij )) j,1 σ 1 (2π) 2 σ 1 { ( )} ζ g1(x(t ij, ξ i δij ) Φ σ 1 [ 1 exp 1 ( Yij2 g 2(X(t ij, ξ ) i 2 ] )) σ 2 (2π) 2 σ 2 j,2 Φ Repartition function of a Normal law. ODE Solver (dlsode Fortran) - [Radhakrishnan et Hindmarsh (1993)] M. Prague - SISTM - Estimation September 27,

14 Penalized likelihood computation (2) - Observed individual likelihood L Oi = R q L Fi u i (u)φ(u)du, with φ N (0, I q) Numerical integration: Adaptive Gaussian Quadrature Numerical integration:[genz et Keister (1996)] - Penalized log-likelihood L P O = i n log (L Oi ) J(θ) Parallel computing: Each computation L Oi are independent. M. Prague - SISTM - Estimation September 27,

15 Robust-Variance Scoring (RVS) We use a Newton-Raphson-like algorithm to maximize the penalized likelihood. Score computation (Gradients approximation) ( ) n L P Oi U O(θ k ) = θ θ k ODE solver (dlsode Fortran) i=1 Sensitivity Equation of ODE systems Adaptive Gaussian Quadrature Parallel computing M. Prague - SISTM - Estimation September 27,

16 Robust-Variance Scoring (RVS) Computation of G (Approximation of the Hessian H) H(θ k ) G(θ k ) = i n ( UOi (θ k )U O i (θ k ) ) ν n U(θ k)u (θ k ) + 2 J(θ) θ 2 Switch to a Marquardt-Levenberg algorithm [Marquardt, JSIAM, 1963] when the RVS algorithm does not provide maximization for multiple iterations. M. Prague - SISTM - Estimation September 27,

17 Convergence criteria Stabilization of parameters estimates : Stabilization of log-likelihood : θ (k+1) θ k < η 1 L P O(θ (k+1) ) L P O(θ k ) < η 2 Relative Distance to Maximum (main) : RDM(θ k ) = U(θ k)g 1 (θ k )U (θ k ) m < η 3 M. Prague - SISTM - Estimation September 27,

18 4Some Illustration M. Prague - SISTM - Illustration September 27,

19 Example in pharmacokinetics Pharmacokinetics model 1 compartment: Pharmacokinetics model 2 compartments: Label GI CP GT Name Gastro-intestinal tract Plasma compartment Tissue compartment M. Prague - SISTM - Illustration September 27,

20 Simulations - FOCE is not stable and less accurate (Laplace integration) - MCMC is more computationally demanding than NIMROD - NIMROD gives more efficient results than MCMC - NIMROD sometimes achieve estimation where MCMC fails Failure Time Empirical Overall Overall (%) (s) SE Abs. Bias RMSE FOCE MCMC NIMROD M. Prague - SISTM - Illustration September 27,

21 Properties of RDM Log likelihood RDM Number of iterations M. Prague - SISTM - Illustration September 27,

22 Real Data: The PUZZLE study [Raguin et al. 2004] Explain the 600 mg Amprenavir (APV) concentrations in blood (A CP ) in 39 HIV infected patients Longitudinal data {0, 1/2, 1 1/2, 2, 3, 4, 6, 8, 10} hours M. Prague - SISTM - Illustration September 27,

23 5Conclusion M. Prague - SISTM - Conclusion September 27,

24 Existing and perspectives Increase the dimension of the mechanistic models: Limited number of inter-individual variability (random effects). - Investigate alternative algorithms: Explore Kalman filters. M. Prague - SISTM - Conclusion September 27,

25 References 1. Guedj, J., Thiébaut, R., and Commenges, D. (2010). Practical identifiability of HIV dynamics models. Bulletin of mathematical biology, 69(8), Prague, M., Commenges, D., Gran, J. M., Ledergerber, B., Young, J., Furrer, H., and Thiébaut, R. (2016). Dynamic models for estimating the effect of HAART on CD4 in observational studies: Application to the Aquitaine Cohort and the Swiss HIV Cohort Study. Biometrics, 73(1), Prague M., Commenges D., Guedj J., Drylewicz J., Thiébaut R. (2013) NIMROD: A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations. Computer methods and Programs in Biomedecine 111(2) Prague M. (2016) Dynamical modeling for Optimization of treatment in HIV infected patients. Invited paper in Statistical French Society journal. 157(2), M. Prague - SISTM - Conclusion September 27,

26 MERCI SISTM Inria, Bordeaux, Sud-ouest, France

Stochastic approximation EM algorithm in nonlinear mixed effects model for viral load decrease during anti-hiv treatment

Stochastic approximation EM algorithm in nonlinear mixed effects model for viral load decrease during anti-hiv treatment Stochastic approximation EM algorithm in nonlinear mixed effects model for viral load decrease during anti-hiv treatment Adeline Samson 1, Marc Lavielle and France Mentré 1 1 INSERM E0357, Department of

More information

Estimation and Model Selection in Mixed Effects Models Part I. Adeline Samson 1

Estimation and Model Selection in Mixed Effects Models Part I. Adeline Samson 1 Estimation and Model Selection in Mixed Effects Models Part I Adeline Samson 1 1 University Paris Descartes Summer school 2009 - Lipari, Italy These slides are based on Marc Lavielle s slides Outline 1

More information

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects Panhard Xavière

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects Panhard Xavière Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects Panhard Xavière * Modèles et mé thodes de l'évaluation thérapeutique des maladies chroniques INSERM : U738, Universit

More information

Mixed-Effects Biological Models: Estimation and Inference

Mixed-Effects Biological Models: Estimation and Inference Mixed-Effects Biological Models: Estimation and Inference Hulin Wu, Ph.D. Dr. D.R. Seth Family Professor & Associate Chair Department of Biostatistics School of Public Health University of Texas Health

More information

Bayesian 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 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 information

Score test for random changepoint in a mixed model

Score test for random changepoint in a mixed model Score test for random changepoint in a mixed model Corentin Segalas and Hélène Jacqmin-Gadda INSERM U1219, Biostatistics team, Bordeaux GDR Statistiques et Santé October 6, 2017 Biostatistics 1 / 27 Introduction

More information

TGDR: An Introduction

TGDR: An Introduction TGDR: An Introduction Julian Wolfson Student Seminar March 28, 2007 1 Variable Selection 2 Penalization, Solution Paths and TGDR 3 Applying TGDR 4 Extensions 5 Final Thoughts Some motivating examples We

More information

Longitudinal + Reliability = Joint Modeling

Longitudinal + Reliability = Joint Modeling Longitudinal + Reliability = Joint Modeling Carles Serrat Institute of Statistics and Mathematics Applied to Building CYTED-HAROSA International Workshop November 21-22, 2013 Barcelona Mainly from Rizopoulos,

More information

Density Estimation. Seungjin Choi

Density Estimation. Seungjin Choi Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/

More information

Nonlinear Mixed Effects Models

Nonlinear Mixed Effects Models Nonlinear Mixed Effects Modeling Department of Mathematics Center for Research in Scientific Computation Center for Quantitative Sciences in Biomedicine North Carolina State University July 31, 216 Introduction

More information

Normalising constants and maximum likelihood inference

Normalising constants and maximum likelihood inference Normalising constants and maximum likelihood inference Jakob G. Rasmussen Department of Mathematics Aalborg University Denmark March 9, 2011 1/14 Today Normalising constants Approximation of normalising

More information

Heterogeneous shedding of influenza by human subjects and. its implications for epidemiology and control

Heterogeneous shedding of influenza by human subjects and. its implications for epidemiology and control 1 2 3 4 5 6 7 8 9 10 11 Heterogeneous shedding of influenza by human subjects and its implications for epidemiology and control Laetitia Canini 1*, Mark EJ Woolhouse 1, Taronna R. Maines 2, Fabrice Carrat

More information

A Hierarchical Bayesian Approach for Parameter Estimation in HIV Models

A Hierarchical Bayesian Approach for Parameter Estimation in HIV Models A Hierarchical Bayesian Approach for Parameter Estimation in HIV Models H. T. Banks, Sarah Grove, Shuhua Hu, and Yanyuan Ma Center for Research in Scientific Computation North Carolina State University

More information

A comparison of estimation methods in nonlinear mixed effects models using a blind analysis

A comparison of estimation methods in nonlinear mixed effects models using a blind analysis A comparison of estimation methods in nonlinear mixed effects models using a blind analysis Pascal Girard, PhD EA 3738 CTO, INSERM, University Claude Bernard Lyon I, Lyon France Mentré, PhD, MD Dept Biostatistics,

More information

Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications

Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications Hulin Wu, Ph.D. Dr. D.R. Seth Family Professor & Associate Chair Department of Biostatistics, School of

More information

Parametric Models. Dr. Shuang LIANG. School of Software Engineering TongJi University Fall, 2012

Parametric Models. Dr. Shuang LIANG. School of Software Engineering TongJi University Fall, 2012 Parametric Models Dr. Shuang LIANG School of Software Engineering TongJi University Fall, 2012 Today s Topics Maximum Likelihood Estimation Bayesian Density Estimation Today s Topics Maximum Likelihood

More information

Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to HIV dynamics model

Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to HIV dynamics model Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to HIV dynamics model Adeline Samson 1, Marc Lavielle 2, France Mentré 1 1 INSERM U738, Paris, France;

More information

Design of HIV Dynamic Experiments: A Case Study

Design of HIV Dynamic Experiments: A Case Study Design of HIV Dynamic Experiments: A Case Study Cong Han Department of Biostatistics University of Washington Kathryn Chaloner Department of Biostatistics University of Iowa Nonlinear Mixed-Effects Models

More information

Modelling geoadditive survival data

Modelling geoadditive survival data Modelling geoadditive survival data Thomas Kneib & Ludwig Fahrmeir Department of Statistics, Ludwig-Maximilians-University Munich 1. Leukemia survival data 2. Structured hazard regression 3. Mixed model

More information

Integrated approaches for analysis of cluster randomised trials

Integrated approaches for analysis of cluster randomised trials Integrated approaches for analysis of cluster randomised trials Invited Session 4.1 - Recent developments in CRTs Joint work with L. Turner, F. Li, J. Gallis and D. Murray Mélanie PRAGUE - SCT 2017 - Liverpool

More information

Linear Mixed Models. One-way layout REML. Likelihood. Another perspective. Relationship to classical ideas. Drawbacks.

Linear 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 information

Sparse Linear Models (10/7/13)

Sparse 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 information

I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S (I S B A)

I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S (I S B A) I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2011/01 1101 BAYESIAN

More information

Description of UseCase models in MDL

Description of UseCase models in MDL Description of UseCase models in MDL A set of UseCases (UCs) has been prepared to illustrate how different modelling features can be implemented in the Model Description Language or MDL. These UseCases

More information

Likelihood-Based Methods

Likelihood-Based Methods Likelihood-Based Methods Handbook of Spatial Statistics, Chapter 4 Susheela Singh September 22, 2016 OVERVIEW INTRODUCTION MAXIMUM LIKELIHOOD ESTIMATION (ML) RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION (REML)

More information

Bayesian Regression Linear and Logistic Regression

Bayesian Regression Linear and Logistic Regression When we want more than point estimates Bayesian Regression Linear and Logistic Regression Nicole Beckage Ordinary Least Squares Regression and Lasso Regression return only point estimates But what if we

More information

Modeling Real Estate Data using Quantile Regression

Modeling Real Estate Data using Quantile Regression Modeling Real Estate Data using Semiparametric Quantile Regression Department of Statistics University of Innsbruck September 9th, 2011 Overview 1 Application: 2 3 4 Hedonic regression data for house prices

More information

Multilevel Statistical Models: 3 rd edition, 2003 Contents

Multilevel 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 information

COMP90051 Statistical Machine Learning

COMP90051 Statistical Machine Learning COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Trevor Cohn 2. Statistical Schools Adapted from slides by Ben Rubinstein Statistical Schools of Thought Remainder of lecture is to provide

More information

MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems

MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Integrating Mathematical and Statistical Models Recap of mathematical models Models and data Statistical models and sources of

More information

Construction of an Informative Hierarchical Prior Distribution: Application to Electricity Load Forecasting

Construction of an Informative Hierarchical Prior Distribution: Application to Electricity Load Forecasting Construction of an Informative Hierarchical Prior Distribution: Application to Electricity Load Forecasting Anne Philippe Laboratoire de Mathématiques Jean Leray Université de Nantes Workshop EDF-INRIA,

More information

Generative Models and Stochastic Algorithms for Population Average Estimation and Image Analysis

Generative Models and Stochastic Algorithms for Population Average Estimation and Image Analysis Generative Models and Stochastic Algorithms for Population Average Estimation and Image Analysis Stéphanie Allassonnière CIS, JHU July, 15th 28 Context : Computational Anatomy Context and motivations :

More information

Fitting Multidimensional Latent Variable Models using an Efficient Laplace Approximation

Fitting Multidimensional Latent Variable Models using an Efficient Laplace Approximation Fitting Multidimensional Latent Variable Models using an Efficient Laplace Approximation Dimitris Rizopoulos Department of Biostatistics, Erasmus University Medical Center, the Netherlands d.rizopoulos@erasmusmc.nl

More information

ML estimation: Random-intercepts logistic model. and z

ML estimation: Random-intercepts logistic model. and z ML estimation: Random-intercepts logistic model log p ij 1 p = x ijβ + υ i with υ i N(0, συ) 2 ij Standardizing the random effect, θ i = υ i /σ υ, yields log p ij 1 p = x ij β + σ υθ i with θ i N(0, 1)

More information

Stochastic Analogues to Deterministic Optimizers

Stochastic Analogues to Deterministic Optimizers Stochastic Analogues to Deterministic Optimizers ISMP 2018 Bordeaux, France Vivak Patel Presented by: Mihai Anitescu July 6, 2018 1 Apology I apologize for not being here to give this talk myself. I injured

More information

Regularization in Cox Frailty Models

Regularization in Cox Frailty Models Regularization in Cox Frailty Models Andreas Groll 1, Trevor Hastie 2, Gerhard Tutz 3 1 Ludwig-Maximilians-Universität Munich, Department of Mathematics, Theresienstraße 39, 80333 Munich, Germany 2 University

More information

Econometrics I, Estimation

Econometrics I, Estimation Econometrics I, Estimation Department of Economics Stanford University September, 2008 Part I Parameter, Estimator, Estimate A parametric is a feature of the population. An estimator is a function of the

More information

ICML Scalable Bayesian Inference on Point processes. with Gaussian Processes. Yves-Laurent Kom Samo & Stephen Roberts

ICML Scalable Bayesian Inference on Point processes. with Gaussian Processes. Yves-Laurent Kom Samo & Stephen Roberts ICML 2015 Scalable Nonparametric Bayesian Inference on Point Processes with Gaussian Processes Machine Learning Research Group and Oxford-Man Institute University of Oxford July 8, 2015 Point Processes

More information

Using Estimating Equations for Spatially Correlated A

Using 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 information

Restricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model

Restricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model Restricted Maximum Likelihood in Linear Regression and Linear Mixed-Effects Model Xiuming Zhang zhangxiuming@u.nus.edu A*STAR-NUS Clinical Imaging Research Center October, 015 Summary This report derives

More information

Extension of the SAEM algorithm for nonlinear mixed. models with two levels of random effects

Extension of the SAEM algorithm for nonlinear mixed. models with two levels of random effects Author manuscript, published in "Biostatistics 29;1(1):121-35" DOI : 1.193/biostatistics/kxn2 Extension of the SAEM algorithm for nonlinear mixed models with two levels of random effects Xavière Panhard

More information

The Bayesian approach to inverse problems

The Bayesian approach to inverse problems The Bayesian approach to inverse problems Youssef Marzouk Department of Aeronautics and Astronautics Center for Computational Engineering Massachusetts Institute of Technology ymarz@mit.edu, http://uqgroup.mit.edu

More information

F. Combes (1,2,3) S. Retout (2), N. Frey (2) and F. Mentré (1) PODE 2012

F. Combes (1,2,3) S. Retout (2), N. Frey (2) and F. Mentré (1) PODE 2012 Prediction of shrinkage of individual parameters using the Bayesian information matrix in nonlinear mixed-effect models with application in pharmacokinetics F. Combes (1,2,3) S. Retout (2), N. Frey (2)

More information

Marginal density. If the unknown is of the form x = (x 1, x 2 ) in which the target of investigation is x 1, a marginal posterior density

Marginal density. If the unknown is of the form x = (x 1, x 2 ) in which the target of investigation is x 1, a marginal posterior density Marginal density If the unknown is of the form x = x 1, x 2 ) in which the target of investigation is x 1, a marginal posterior density πx 1 y) = πx 1, x 2 y)dx 2 = πx 2 )πx 1 y, x 2 )dx 2 needs to be

More information

SAEMIX, an R version of the SAEM algorithm for parameter estimation in nonlinear mixed effect models

SAEMIX, an R version of the SAEM algorithm for parameter estimation in nonlinear mixed effect models SAEMIX, an R version of the SAEM algorithm for parameter estimation in nonlinear mixed effect models Emmanuelle Comets 1,2 & Audrey Lavenu 2 & Marc Lavielle 3 1 CIC 0203, U. Rennes-I, CHU Pontchaillou,

More information

CSC 2541: Bayesian Methods for Machine Learning

CSC 2541: Bayesian Methods for Machine Learning CSC 2541: Bayesian Methods for Machine Learning Radford M. Neal, University of Toronto, 2011 Lecture 10 Alternatives to Monte Carlo Computation Since about 1990, Markov chain Monte Carlo has been the dominant

More information

Between-Subject and Within-Subject Model Mixtures for Classifying HIV Treatment Response

Between-Subject and Within-Subject Model Mixtures for Classifying HIV Treatment Response Progress in Applied Mathematics Vol. 4, No. 2, 2012, pp. [148 166] DOI: 10.3968/j.pam.1925252820120402.S0801 ISSN 1925-251X [Print] ISSN 1925-2528 [Online] www.cscanada.net www.cscanada.org Between-Subject

More information

Pharmacometrics : Nonlinear mixed effect models in Statistics. Department of Statistics Ewha Womans University Eun-Kyung Lee

Pharmacometrics : Nonlinear mixed effect models in Statistics. Department of Statistics Ewha Womans University Eun-Kyung Lee Pharmacometrics : Nonlinear mixed effect models in Statistics Department of Statistics Ewha Womans University Eun-Kyung Lee 1 Clinical Trials Preclinical trial: animal study Phase I trial: First in human,

More information

Dynamic System Identification using HDMR-Bayesian Technique

Dynamic System Identification using HDMR-Bayesian Technique Dynamic System Identification using HDMR-Bayesian Technique *Shereena O A 1) and Dr. B N Rao 2) 1), 2) Department of Civil Engineering, IIT Madras, Chennai 600036, Tamil Nadu, India 1) ce14d020@smail.iitm.ac.in

More information

Mark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation.

Mark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation. CS 189 Spring 2015 Introduction to Machine Learning Midterm You have 80 minutes for the exam. The exam is closed book, closed notes except your one-page crib sheet. No calculators or electronic items.

More information

CTDL-Positive Stable Frailty Model

CTDL-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 information

dans les modèles à vraisemblance non explicite par des algorithmes gradient-proximaux perturbés

dans les modèles à vraisemblance non explicite par des algorithmes gradient-proximaux perturbés Inférence pénalisée dans les modèles à vraisemblance non explicite par des algorithmes gradient-proximaux perturbés Gersende Fort Institut de Mathématiques de Toulouse, CNRS and Univ. Paul Sabatier Toulouse,

More information

EnKF-based particle filters

EnKF-based particle filters EnKF-based particle filters Jana de Wiljes, Sebastian Reich, Wilhelm Stannat, Walter Acevedo June 20, 2017 Filtering Problem Signal dx t = f (X t )dt + 2CdW t Observations dy t = h(x t )dt + R 1/2 dv t.

More information

Introduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf

Introduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf 1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf 2013-14 We know that X ~ B(n,p), but we do not know p. We get a random sample

More information

Parametric Inference Maximum Likelihood Inference Exponential Families Expectation Maximization (EM) Bayesian Inference Statistical Decison Theory

Parametric Inference Maximum Likelihood Inference Exponential Families Expectation Maximization (EM) Bayesian Inference Statistical Decison Theory Statistical Inference Parametric Inference Maximum Likelihood Inference Exponential Families Expectation Maximization (EM) Bayesian Inference Statistical Decison Theory IP, José Bioucas Dias, IST, 2007

More information

Bayesian linear regression

Bayesian linear regression Bayesian linear regression Linear regression is the basis of most statistical modeling. The model is Y i = X T i β + ε i, where Y i is the continuous response X i = (X i1,..., X ip ) T is the corresponding

More information

Analysing repeated measurements whilst accounting for derivative tracking, varying within-subject variance and autocorrelation: the xtiou command

Analysing repeated measurements whilst accounting for derivative tracking, varying within-subject variance and autocorrelation: the xtiou command Analysing repeated measurements whilst accounting for derivative tracking, varying within-subject variance and autocorrelation: the xtiou command R.A. Hughes* 1, M.G. Kenward 2, J.A.C. Sterne 1, K. Tilling

More information

Sequential Monte Carlo Methods for Bayesian Computation

Sequential Monte Carlo Methods for Bayesian Computation Sequential Monte Carlo Methods for Bayesian Computation A. Doucet Kyoto Sept. 2012 A. Doucet (MLSS Sept. 2012) Sept. 2012 1 / 136 Motivating Example 1: Generic Bayesian Model Let X be a vector parameter

More information

Sequential Monte Carlo Methods for Bayesian Model Selection in Positron Emission Tomography

Sequential Monte Carlo Methods for Bayesian Model Selection in Positron Emission Tomography Methods for Bayesian Model Selection in Positron Emission Tomography Yan Zhou John A.D. Aston and Adam M. Johansen 6th January 2014 Y. Zhou J. A. D. Aston and A. M. Johansen Outline Positron emission tomography

More information

Robust design in model-based analysis of longitudinal clinical data

Robust design in model-based analysis of longitudinal clinical data Robust design in model-based analysis of longitudinal clinical data Giulia Lestini, Sebastian Ueckert, France Mentré IAME UMR 1137, INSERM, University Paris Diderot, France PODE, June 0 016 Context Optimal

More information

Semiparametric Mixed Effects Models with Flexible Random Effects Distribution

Semiparametric Mixed Effects Models with Flexible Random Effects Distribution Semiparametric Mixed Effects Models with Flexible Random Effects Distribution Marie Davidian North Carolina State University davidian@stat.ncsu.edu www.stat.ncsu.edu/ davidian Joint work with A. Tsiatis,

More information

Lecture 3 September 1

Lecture 3 September 1 STAT 383C: Statistical Modeling I Fall 2016 Lecture 3 September 1 Lecturer: Purnamrita Sarkar Scribe: Giorgio Paulon, Carlos Zanini Disclaimer: These scribe notes have been slightly proofread and may have

More information

Approximate Bayesian Computation

Approximate Bayesian Computation Approximate Bayesian Computation Michael Gutmann https://sites.google.com/site/michaelgutmann University of Helsinki and Aalto University 1st December 2015 Content Two parts: 1. The basics of approximate

More information

Numerical Solutions of ODEs by Gaussian (Kalman) Filtering

Numerical Solutions of ODEs by Gaussian (Kalman) Filtering Numerical Solutions of ODEs by Gaussian (Kalman) Filtering Hans Kersting joint work with Michael Schober, Philipp Hennig, Tim Sullivan and Han C. Lie SIAM CSE, Atlanta March 1, 2017 Emmy Noether Group

More information

Stat 710: Mathematical Statistics Lecture 12

Stat 710: Mathematical Statistics Lecture 12 Stat 710: Mathematical Statistics Lecture 12 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 12 Feb 18, 2009 1 / 11 Lecture 12:

More information

OPTIMAL ESTIMATION of DYNAMIC SYSTEMS

OPTIMAL ESTIMATION of DYNAMIC SYSTEMS CHAPMAN & HALL/CRC APPLIED MATHEMATICS -. AND NONLINEAR SCIENCE SERIES OPTIMAL ESTIMATION of DYNAMIC SYSTEMS John L Crassidis and John L. Junkins CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London

More information

Comparison of multiple imputation methods for systematically and sporadically missing multilevel data

Comparison 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 information

ABC methods for phase-type distributions with applications in insurance risk problems

ABC methods for phase-type distributions with applications in insurance risk problems ABC methods for phase-type with applications problems Concepcion Ausin, Department of Statistics, Universidad Carlos III de Madrid Joint work with: Pedro Galeano, Universidad Carlos III de Madrid Simon

More information

Accurate Maximum Likelihood Estimation for Parametric Population Analysis. Bob Leary UCSD/SDSC and LAPK, USC School of Medicine

Accurate Maximum Likelihood Estimation for Parametric Population Analysis. Bob Leary UCSD/SDSC and LAPK, USC School of Medicine Accurate Maximum Likelihood Estimation for Parametric Population Analysis Bob Leary UCSD/SDSC and LAPK, USC School of Medicine Why use parametric maximum likelihood estimators? Consistency: θˆ θ as N ML

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models Mixed effects models - Part IV Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

Recap. Vector observation: Y f (y; θ), Y Y R m, θ R d. sample of independent vectors y 1,..., y n. pairwise log-likelihood n m. weights are often 1

Recap. Vector observation: Y f (y; θ), Y Y R m, θ R d. sample of independent vectors y 1,..., y n. pairwise log-likelihood n m. weights are often 1 Recap Vector observation: Y f (y; θ), Y Y R m, θ R d sample of independent vectors y 1,..., y n pairwise log-likelihood n m i=1 r=1 s>r w rs log f 2 (y ir, y is ; θ) weights are often 1 more generally,

More information

Computational Methods Short Course on Image Quality

Computational Methods Short Course on Image Quality Computational Methods Short Course on Image Quality Matthew A. Kupinski What we Will Cover Sources of randomness Computation of linear-observer performance Computation of ideal-observer performance Miscellaneous

More information

Causality through the stochastic system approach

Causality through the stochastic system approach Causality through the stochastic system approach Daniel Commenges INSERM, Centre de Recherche Epidémiologie et Biostatistique, Equipe Biostatistique, Bordeaux http://sites.google.com/site/danielcommenges/

More information

Model Selection in Bayesian Survival Analysis for a Multi-country Cluster Randomized Trial

Model Selection in Bayesian Survival Analysis for a Multi-country Cluster Randomized Trial Model Selection in Bayesian Survival Analysis for a Multi-country Cluster Randomized Trial Jin Kyung Park International Vaccine Institute Min Woo Chae Seoul National University R. Leon Ochiai International

More information

ON THE CONSEQUENCES OF MISSPECIFING ASSUMPTIONS CONCERNING RESIDUALS DISTRIBUTION IN A REPEATED MEASURES AND NONLINEAR MIXED MODELLING CONTEXT

ON THE CONSEQUENCES OF MISSPECIFING ASSUMPTIONS CONCERNING RESIDUALS DISTRIBUTION IN A REPEATED MEASURES AND NONLINEAR MIXED MODELLING CONTEXT ON THE CONSEQUENCES OF MISSPECIFING ASSUMPTIONS CONCERNING RESIDUALS DISTRIBUTION IN A REPEATED MEASURES AND NONLINEAR MIXED MODELLING CONTEXT Rachid el Halimi and Jordi Ocaña Departament d Estadística

More information

Tutorial on Approximate Bayesian Computation

Tutorial on Approximate Bayesian Computation Tutorial on Approximate Bayesian Computation Michael Gutmann https://sites.google.com/site/michaelgutmann University of Helsinki Aalto University Helsinki Institute for Information Technology 16 May 2016

More information

Case Study in the Use of Bayesian Hierarchical Modeling and Simulation for Design and Analysis of a Clinical Trial

Case Study in the Use of Bayesian Hierarchical Modeling and Simulation for Design and Analysis of a Clinical Trial Case Study in the Use of Bayesian Hierarchical Modeling and Simulation for Design and Analysis of a Clinical Trial William R. Gillespie Pharsight Corporation Cary, North Carolina, USA PAGE 2003 Verona,

More information

Review Article Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues

Review Article Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues Journal of Probability and Statistics Volume 2012, Article ID 640153, 17 pages doi:10.1155/2012/640153 Review Article Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and

More information

Introduction to Machine Learning

Introduction to Machine Learning Introduction to Machine Learning Linear Regression Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574 1

More information

Connections between score matching, contrastive divergence, and pseudolikelihood for continuous-valued variables. Revised submission to IEEE TNN

Connections between score matching, contrastive divergence, and pseudolikelihood for continuous-valued variables. Revised submission to IEEE TNN Connections between score matching, contrastive divergence, and pseudolikelihood for continuous-valued variables Revised submission to IEEE TNN Aapo Hyvärinen Dept of Computer Science and HIIT University

More information

Discussion of Maximization by Parts in Likelihood Inference

Discussion of Maximization by Parts in Likelihood Inference Discussion of Maximization by Parts in Likelihood Inference David Ruppert School of Operations Research & Industrial Engineering, 225 Rhodes Hall, Cornell University, Ithaca, NY 4853 email: dr24@cornell.edu

More information

σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =

σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) = Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,

More information

Graphical Models for Collaborative Filtering

Graphical Models for Collaborative Filtering Graphical Models for Collaborative Filtering Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Sequence modeling HMM, Kalman Filter, etc.: Similarity: the same graphical model topology,

More information

A State Space Model Approach for HIV Infection Dynamics

A State Space Model Approach for HIV Infection Dynamics A State Space Model Approach for HIV Infection Dynamics Jiabin Wang a, Hua Liang b, and Rong Chen a Mathematical models have been proposed and developed to model HIV dynamics over the past few decades,

More information

Data assimilation with and without a model

Data assimilation with and without a model Data assimilation with and without a model Tyrus Berry George Mason University NJIT Feb. 28, 2017 Postdoc supported by NSF This work is in collaboration with: Tim Sauer, GMU Franz Hamilton, Postdoc, NCSU

More information

Bayesian Additive Regression Tree (BART) with application to controlled trail data analysis

Bayesian Additive Regression Tree (BART) with application to controlled trail data analysis Bayesian Additive Regression Tree (BART) with application to controlled trail data analysis Weilan Yang wyang@stat.wisc.edu May. 2015 1 / 20 Background CATE i = E(Y i (Z 1 ) Y i (Z 0 ) X i ) 2 / 20 Background

More information

General Regression Model

General Regression Model Scott S. Emerson, M.D., Ph.D. Department of Biostatistics, University of Washington, Seattle, WA 98195, USA January 5, 2015 Abstract Regression analysis can be viewed as an extension of two sample statistical

More information

ADAPTIVE EXPERIMENTAL DESIGNS. Maciej Patan and Barbara Bogacka. University of Zielona Góra, Poland and Queen Mary, University of London

ADAPTIVE EXPERIMENTAL DESIGNS. Maciej Patan and Barbara Bogacka. University of Zielona Góra, Poland and Queen Mary, University of London ADAPTIVE EXPERIMENTAL DESIGNS FOR SIMULTANEOUS PK AND DOSE-SELECTION STUDIES IN PHASE I CLINICAL TRIALS Maciej Patan and Barbara Bogacka University of Zielona Góra, Poland and Queen Mary, University of

More information

Lecture 8: Bayesian Estimation of Parameters in State Space Models

Lecture 8: Bayesian Estimation of Parameters in State Space Models in State Space Models March 30, 2016 Contents 1 Bayesian estimation of parameters in state space models 2 Computational methods for parameter estimation 3 Practical parameter estimation in state space

More information

Basic math for biology

Basic math for biology Basic math for biology Lei Li Florida State University, Feb 6, 2002 The EM algorithm: setup Parametric models: {P θ }. Data: full data (Y, X); partial data Y. Missing data: X. Likelihood and maximum likelihood

More information

Learning Parameters of Undirected Models. Sargur Srihari

Learning Parameters of Undirected Models. Sargur Srihari Learning Parameters of Undirected Models Sargur srihari@cedar.buffalo.edu 1 Topics Log-linear Parameterization Likelihood Function Maximum Likelihood Parameter Estimation Simple and Conjugate Gradient

More information

Contrasting Marginal and Mixed Effects Models Recall: two approaches to handling dependence in Generalized Linear Models:

Contrasting Marginal and Mixed Effects Models Recall: two approaches to handling dependence in Generalized Linear Models: Contrasting Marginal and Mixed Effects Models Recall: two approaches to handling dependence in Generalized Linear Models: Marginal models: based on the consequences of dependence on estimating model parameters.

More information

Statistical Inference and Methods

Statistical 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 information

Multivariate Survival Analysis

Multivariate 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 information

DESIGN EVALUATION AND OPTIMISATION IN CROSSOVER PHARMACOKINETIC STUDIES ANALYSED BY NONLINEAR MIXED EFFECTS MODELS

DESIGN EVALUATION AND OPTIMISATION IN CROSSOVER PHARMACOKINETIC STUDIES ANALYSED BY NONLINEAR MIXED EFFECTS MODELS DESIGN EVALUATION AND OPTIMISATION IN CROSSOVER PHARMACOKINETIC STUDIES ANALYSED BY NONLINEAR MIXED EFFECTS MODELS Thu Thuy Nguyen, Caroline Bazzoli, France Mentré UMR 738 INSERM - University Paris Diderot,

More information

Introduction 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 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 information

Bayesian Logistic Regression

Bayesian Logistic Regression Bayesian Logistic Regression Sargur N. University at Buffalo, State University of New York USA Topics in Linear Models for Classification Overview 1. Discriminant Functions 2. Probabilistic Generative

More information

Meta-Analysis for Diagnostic Test Data: a Bayesian Approach

Meta-Analysis for Diagnostic Test Data: a Bayesian Approach Meta-Analysis for Diagnostic Test Data: a Bayesian Approach Pablo E. Verde Coordination Centre for Clinical Trials Heinrich Heine Universität Düsseldorf Preliminaries: motivations for systematic reviews

More information

Bayesian non-parametric model to longitudinally predict churn

Bayesian 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 information