Sequential Monte Carlo methods in Bayesian joint models for longitudinal and time-to-event data

The statistical analysis of the information generated by medical follow-up is a very important challenge in the field of personalised medicine. As the evolutionary course of a patient's disease progresses, its medical follow-up generates more and more information that should be processed immediately in order to review and update its prognosis and treatment. Our objective in this thesis focuses on this update process through sequential inference methods for joint models of longitudinal and time-to-event data from a Bayesian perspective. More specifically, we propose the use of sequential Monte Carlo methods for static parameter joint models in order to update the posterior distribution of the parameters, hyperparameters, and random effects with the intention of reducing computation time in each update of the inferential process. Our proposal is very general and can be easily applied to most popular joint models approaches. We illustrate our research with two different studies: (i) a joint model for longitudinal data with informative dropout simulated through an own novel mechanism, and (ii) a joint model with competing risk events for a real problem about patients receiving mechanical ventilation in intensive care units.

[1]  K. Jain.  Textbook of Personalized Medicine , 2009, Springer New York.

[2]  Raymond J. Carroll,et al.  Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process , 1988 .

[3]  P. Diggle,et al.  Analysis of Longitudinal Data. , 1997 .

[4]  Eric Moulines,et al.  Comparison of resampling schemes for particle filtering , 2005, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005..

[5]  Anthony N. Pettitt,et al.  A Sequential Monte Carlo Algorithm to Incorporate Model Uncertainty in Bayesian Sequential Design , 2014 .

[6]  Menggang Yu,et al.  Individual Prediction in Prostate Cancer Studies Using a Joint Longitudinal Survival–Cure Model , 2008 .

[7]  Xihong Lin,et al.  A penalized likelihood approach to joint modeling of longitudinal measurements and time-to-event data , 2008 .

[8]  J. Ball,et al.  Statistics review 12: Survival analysis , 2004, Critical care.

[9]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[10]  S. E. Ahmed,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 2008, Technometrics.

[11]  T. Loredo.  From Laplace to Supernova SN 1987A: Bayesian Inference in Astrophysics , 1990 .

[12]  M Ganjali,et al.  Bayesian Joint Modeling of Longitudinal Measurements and Time-to-Event Data Using Robust Distributions , 2014, Journal of biopharmaceutical statistics.

[13]  Paul Fearnhead,et al.  An Adaptive Sequential Monte Carlo Sampler , 2010, 1005.1193.

[14]  Harald Niederreiter.  Some current issues in quasi-Monte Carlo methods , 2003, J. Complex..

[15]  James O. Berger,et al.  The interplay of Bayesian and frequentist analysis , 2004 .

[16]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[17]  Simon J. Godsill,et al.  An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo , 2007, Proceedings of the IEEE.

[18]  Anastasios A. Tsiatis,et al.  Joint Modeling of Longitudinal and Time-to-Event Data : An Overview , 2004 .

[19]  Rui Martins,et al.  Bayesian joint modeling of longitudinal and spatial survival AIDS data , 2016, Statistics in medicine.

[20]  C. Armero,et al.  Bayesian joint modeling of bivariate longitudinal and competing risks data: An application to study patient‐ventilator asynchronies in critical care patients , 2017, Biometrical journal. Biometrische Zeitschrift.

[21]  K. Bailey,et al.  Analysing changes in the presence of informative right censoring caused by death and withdrawal. , 1988, Statistics in medicine.

[22]  Melania Pintilie,et al.  Competing Risks: A Practical Perspective , 2006 .

[23]  Peter J Diggle,et al.  Joint modelling of repeated measurement and time-to-event data: an introductory tutorial. , 2015, International journal of epidemiology.

[24]  G. Molenberghs,et al.  Longitudinal data analysis , 2008 .

[25]  George Casella,et al.  A Short History of Markov Chain Monte Carlo: Subjective Recollections from Incomplete Data , 2008, 0808.2902.

[26]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[27]  Xin Huang,et al.  A joint model of longitudinal and competing risks survival data with heterogeneous random effects and outlying longitudinal measurements. , 2010, Statistics and its interface.

[28]  M. Wulfsohn,et al.  Modeling the Relationship of Survival to Longitudinal Data Measured with Error. Applications to Survival and CD4 Counts in Patients with AIDS , 1995 .

[29]  Joseph G. Ibrahim,et al.  Missing data methods in longitudinal studies: a review , 2009 .

[30]  Anabel Forte,et al.  Bayesian joint modeling for assessing the progression of chronic kidney disease in children , 2018, Statistical methods in medical research.

[31]  Hedibert F. Lopes,et al.  Particle filters and Bayesian inference in financial econometrics , 2011 .

[32]  Tomasz Burzykowski,et al.  Linear Mixed-Effects Models Using R: A Step-by-Step Approach , 2013 .

[33]  David Madigan,et al.  A Sequential Monte Carlo Method for Bayesian Analysis of Massive Datasets , 2003, Data Mining and Knowledge Discovery.

[34]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[35]  Joseph G. Ibrahim,et al.  Bayesian Survival Analysis , 2004 .

[36]  Maurizio Ferconi,et al.  The Theory That Would Not Die , 2013 .

[37]  A. Rotnitzky,et al.  Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis by DANIELS, M. J. and HOGAN, J. W , 2009 .

[38]  Piotr Indyk,et al.  Learning Approximate Sequential Patterns for Classification , 2009, J. Mach. Learn. Res..

[39]  S Bandyopadhyay,et al.  A review of multivariate longitudinal data analysis , 2011, Statistical methods in medical research.

[40]  Haesook T. Kim.  Cumulative Incidence in Competing Risks Data and Competing Risks Regression Analysis , 2007, Clinical Cancer Research.

[41]  Andrew Gelman,et al.  The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo , 2011, J. Mach. Learn. Res..

[42]  Gang Li,et al.  A Bayesian approach to joint analysis of longitudinal measurements and competing risks failure time data , 2007, Statistics in medicine.

[43]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[44]  Benjamin A Clegg,et al.  Sequence learning , 1998, Trends in Cognitive Sciences.

[45]  Dynamic prediction of outcome for patients with severe aortic stenosis: application of joint models for longitudinal and time-to-event data , 2015, BMC Cardiovascular Disorders.

[46]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[47]  K M Leung,et al.  Censoring issues in survival analysis. , 1997, Annual review of public health.

[48]  Ali Satty,et al.  Selection and pattern mixture models for modelling longitudinal data with dropout: An application study , 2013 .

[49]  Noureddine Zerhouni,et al.  Particle filter-based prognostics: Review, discussion and perspectives , 2016 .

[50]  Andrew Thomas,et al.  WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility , 2000, Stat. Comput..

[51]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[52]  Francisco Escolano,et al.  Structural, Syntactic, and Statistical Pattern Recognition , 2016, Lecture Notes in Computer Science.

[53]  Geert Molenberghs,et al.  The analysis of multivariate longitudinal data: A review , 2014, Statistical methods in medical research.

[54]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[55]  D. Kleinbaum,et al.  Survival Analysis: A Self-Learning Text. , 1996 .

[56]  B. Muthén,et al.  Applications Of Continuous-Time Survival In Latent Variable Models For The Analysis Of Oncology Randomized Clinical Trial Data Using Mplus , 2009 .

[57]  Bradley P Carlin,et al.  Separate and Joint Modeling of Longitudinal and Event Time Data Using Standard Computer Packages , 2004 .

[58]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[59]  Bradley P. Carlin,et al.  Combining Dynamic Predictions From Joint Models for Longitudinal and Time-to-Event Data Using Bayesian Model Averaging , 2013, 1303.2797.

[60]  Li Su,et al.  Dynamic predictions using flexible joint models of longitudinal and time‐to‐event data , 2017, Statistics in medicine.

[61]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[62]  Erica E. M. Moodie,et al.  Statistical Methods for Dynamic Treatment Regimes: Reinforcement Learning, Causal Inference, and Personalized Medicine , 2013 .

[63]  Christian P. Robert,et al.  Bayesian Computational Tools , 2013, 1304.2048.

[64]  Arnaud Doucet,et al.  An overview of sequential Monte Carlo methods for parameter estimation in general state-space models , 2009 .

[65]  Fredrik Gustafsson,et al.  On Resampling Algorithms for Particle Filters , 2006, 2006 IEEE Nonlinear Statistical Signal Processing Workshop.

[66]  P. Moral,et al.  On adaptive resampling strategies for sequential Monte Carlo methods , 2012, 1203.0464.

[67]  Mark D Schluchter,et al.  Shared parameter models for the joint analysis of longitudinal data and event times , 2006, Statistics in medicine.

[68]  Nicolas Chopin,et al.  Sequential Monte Carlo on large binary sampling spaces , 2011, Statistics and Computing.

[69]  Jennifer S K Chan,et al.  Bayesian informative dropout model for longitudinal binary data with random effects using conditional and joint modeling approaches , 2016, Biometrical journal. Biometrische Zeitschrift.

[70]  Raghavan Murugan,et al.  Movement towards personalised medicine in the ICU. , 2015, The Lancet. Respiratory medicine.

[71]  Nikolaos Limnios,et al.  Statistical models and methods for reliability and survival analysis , 2013 .

[72]  Elisa Lee,et al.  Statistical Methods for Survival Data Analysis: Lee/Survival Data Analysis , 2003 .

[73]  N M Laird,et al.  Increasing efficiency from censored survival data by using random effects to model longitudinal covariates , 1998, Statistical methods in medical research.

[74]  N M Laird,et al.  Mixture models for the joint distribution of repeated measures and event times. , 1997, Statistics in medicine.

[75]  N. Chopin,et al.  Sequential Quasi-Monte Carlo , 2014, 1402.4039.

[76]  R Henderson,et al.  Joint modelling of longitudinal measurements and event time data. , 2000, Biostatistics.

[77]  Cécile Proust-Lima,et al.  Development and validation of a dynamic prognostic tool for prostate cancer recurrence using repeated measures of posttreatment PSA: a joint modeling approach. , 2009, Biostatistics.

[78]  Drew D. Creal.  A Survey of Sequential Monte Carlo Methods for Economics and Finance , 2012 .

[79]  Wei Liu,et al.  Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues , 2012 .

[80]  Christian P. Robert,et al.  The Bayesian choice : from decision-theoretic foundations to computational implementation , 2007 .

[81]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[82]  D. Rubin,et al.  Statistical Analysis with Missing Data , 1988 .

[83]  N M Laird,et al.  Model-based approaches to analysing incomplete longitudinal and failure time data. , 1997, Statistics in medicine.

[84]  David Barber,et al.  Bayesian reasoning and machine learning , 2012 .

[85]  Eleni-Rosalina Andrinopoulou,et al.  Combined dynamic predictions using joint models of two longitudinal outcomes and competing risk data , 2017, Statistical methods in medical research.

[86]  Keith R. Abrams,et al.  Joint Modeling of Longitudinal and Survival Data , 2013 .

[87]  R. Douc,et al.  Long-term stability of sequential Monte Carlo methods under verifiable conditions , 2012, 1203.6898.

[88]  Matthew J. Beal.  Variational algorithms for approximate Bayesian inference , 2003 .

[89]  J. Stoyanov.  The Oxford Handbook of Nonlinear Filtering , 2012 .

[90]  Nicolas Chopin,et al.  SMC2: an efficient algorithm for sequential analysis of state space models , 2011, 1101.1528.

[91]  H Putter,et al.  Tutorial in biostatistics: competing risks and multi‐state models , 2007, Statistics in medicine.

[92]  Joseph G Ibrahim,et al.  Joint modeling of survival and longitudinal non‐survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group , 2015, Statistics in medicine.

[93]  Yangxin Huang,et al.  Bayesian inference on joint models of HIV dynamics for time‐to‐event and longitudinal data with skewness and covariate measurement errors , 2011, Statistics in medicine.

[94]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[95]  E. Antman,et al.  Population and personalized medicine in the modern era. , 2014, JAMA.

[96]  Rizopoulos Dimitris,et al.  Joint Modeling of Longitudinal and Time-to-Event Data , 2014 .

[97]  D. Spiegelhalter,et al.  Modelling Complexity: Applications of Gibbs Sampling in Medicine , 1993 .

[98]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[99]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[100]  E. Ziegel.  Lifetime Data: Models in Reliability and Survival Analysis , 2010 .

[101]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[102]  Geert Molenberghs,et al.  Missing Data in Clinical Studies , 2007 .

[103]  A Gelman,et al.  A case study on the choice, interpretation and checking of multilevel models for longitudinal binary outcomes. , 2001, Biostatistics.

[104]  G. Molenberghs,et al.  Linear Mixed Models for Longitudinal Data , 2001 .

[105]  P. Moral,et al.  Sequential Monte Carlo samplers , 2002, cond-mat/0212648.

[106]  R. Michael Alvarez,et al.  Event History Modeling: A Guide for Social Scientists , 2004 .

[107]  Laurence L. George,et al.  The Statistical Analysis of Failure Time Data , 2003, Technometrics.

[108]  Virginie Rondeau,et al.  Dynamic prediction of risk of death using history of cancer recurrences in joint frailty models , 2013, Statistics in medicine.

[109]  Robert Tibshirani,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd Edition , 2001, Springer Series in Statistics.

[110]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[111]  P. Fearnhead,et al.  An improved particle filter for non-linear problems , 1999 .

[112]  D.,et al.  Regression Models and Life-Tables , 2022 .

[113]  W. Gilks,et al.  Following a moving target—Monte Carlo inference for dynamic Bayesian models , 2001 .

[114]  J. Klein,et al.  Survival Analysis: Techniques for Censored and Truncated Data , 1997 .

[115]  Hongtu Zhu,et al.  Bayesian Influence Measures for Joint Models for Longitudinal and Survival Data , 2012, Biometrics.

[116]  Dimitris Rizopoulos,et al.  Dynamic Predictions and Prospective Accuracy in Joint Models for Longitudinal and Time‐to‐Event Data , 2011, Biometrics.

[117]  Menggang Yu,et al.  JOINT LONGITUDINAL-SURVIVAL-CURE MODELS AND THEIR APPLICATION TO PROSTATE CANCER , 2004 .

[118]  Giovanni Petris,et al.  Dynamic Linear Models with R , 2009 .

[119]  Mei-Jie Zhang,et al.  Modeling cumulative incidence function for competing risks data , 2008, Expert review of clinical pharmacology.

[120]  Trevor Hastie,et al.  An Introduction to Statistical Learning , 2013, Springer Texts in Statistics.

[121]  Ji-Qian Fang,et al.  Advanced Medical Statistics , 2015 .

[122]  James M. McGree,et al.  A pseudo-marginal sequential Monte Carlo algorithm for random effects models in Bayesian sequential design , 2016, Stat. Comput..

[123]  M. Kenward,et al.  Handbook of Missing Data Methodology , 2019 .

[124]  D. Madigan,et al.  A one-pass sequential Monte Carlo method for Bayesian analysis of massive datasets , 2006 .

[125]  Cécile Proust-Lima,et al.  Joint modelling of multivariate longitudinal outcomes and a time-to-event: A nonlinear latent class approach , 2009, Comput. Stat. Data Anal..

[126]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo in Practice: A Roundtable Discussion , 1998 .

[127]  M. P. Wand,et al.  Generalised linear mixed model analysis via sequential Monte Carlo sampling , 2008, 0810.1163.

[128]  N. G. Best,et al.  Dynamic conditional independence models and Markov chain Monte Carlo methods , 1997 .

[129]  D. Zeng,et al.  Recent development on statistical methods for personalized medicine discovery , 2013, Frontiers of Medicine.

[130]  Harry Vereecken,et al.  Multivariate and Multiscale Data Assimilation in Terrestrial Systems: A Review , 2012, Sensors.

[131]  Joseph G. Ibrahim,et al.  BAYESIAN METHODS FOR JOINT MODELING OF LONGITUDINAL AND SURVIVAL DATA WITH APPLICATIONS TO CANCER VACCINE TRIALS , 2004 .

[132]  Darrell Whitley,et al.  A genetic algorithm tutorial , 1994, Statistics and Computing.

[133]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[134]  Martyn Plummer,et al.  JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling , 2003 .

[135]  D. Rubin.  INFERENCE AND MISSING DATA , 1975 .

[136]  Geert Verbeke,et al.  Conditional Linear Mixed Models , 2001 .

[137]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[138]  Thomas Augustin,et al.  A Review on Joint Models in Biometrical Research , 2009 .

[139]  Robert E. Weiss,et al.  Modeling Longitudinal Data , 2005 .

[140]  N. Chopin.  Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference , 2004, math/0508594.

[141]  Mark J. Schiefsky.  Hippocrates on ancient medicine , 2005 .

[142]  Ameet Talwalkar,et al.  Foundations of Machine Learning , 2012, Adaptive computation and machine learning.

[143]  Geert Molenberghs,et al.  Multiple‐Imputation‐Based Residuals and Diagnostic Plots for Joint Models of Longitudinal and Survival Outcomes , 2010, Biometrics.

[144]  Thomas L. Griffiths,et al.  Win-Stay, Lose-Sample: A simple sequential algorithm for approximating Bayesian inference , 2014, Cognitive Psychology.

[145]  B. Carlin,et al.  Bayesian Model Choice Via Markov Chain Monte Carlo Methods , 1995 .

[146]  Yangxin Huang,et al.  Jointly modeling time-to-event and longitudinal data: a Bayesian approach , 2014, Stat. Methods Appl..

[147]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[148]  Ping Wang,et al.  Joint modeling of longitudinal outcomes and survival using latent growth modeling approach in a mesothelioma trial , 2012, Health Services and Outcomes Research Methodology.

[149]  James M. McGree,et al.  Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data , 2013, Comput. Stat. Data Anal..

[150]  T. Loredo.  Promise of Bayesian Inference for Astrophysics , 1992 .

[151]  Inês Sousa,et al.  A review on joint modelling of longitudinal measurements and time-to-event , 2011 .

[152]  Benoit Liquet,et al.  Estimation of extended mixed models using latent classes and latent processes: the R package lcmm , 2015, 1503.00890.

[153]  Wei Shen,et al.  JMFit: A SAS Macro for Joint Models of Longitudinal and Survival Data. , 2016, Journal of statistical software.

[154]  Seung Il Kim,et al.  Personalized Medicine in Breast Cancer: A Systematic Review , 2012, Journal of breast cancer.

[155]  D. Thomas,et al.  Simultaneously modelling censored survival data and repeatedly measured covariates: a Gibbs sampling approach. , 1996, Statistics in medicine.

[156]  Hui Zhang,et al.  Sequential Monte Carlo methods for parameter estimation in nonlinear state-space models , 2012, Comput. Geosci..

[157]  W. D. Ray.  4. Modelling Survival Data in Medical Research , 1995 .

[158]  J. Klein,et al.  Statistical Models Based On Counting Process , 1994 .

[159]  J Jack Lee,et al.  Bayesian clinical trials in action , 2012, Statistics in medicine.

[160]  C Armero,et al.  Bayesian joint ordinal and survival modeling for breast cancer risk assessment , 2016, Statistics in medicine.

[161]  Andrew Gelman,et al.  Data Analysis Using Regression and Multilevel/Hierarchical Models , 2006 .

[162]  Jean-Michel Marin,et al.  Adaptive importance sampling in general mixture classes , 2007, Stat. Comput..

[163]  M. Wulfsohn,et al.  A joint model for survival and longitudinal data measured with error. , 1997, Biometrics.

[164]  Eric Moulines,et al.  Inference in hidden Markov models , 2010, Springer series in statistics.

[165]  Dimitris Rizopoulos,et al.  JM: An R package for the joint modelling of longitudinal and time-to-event data , 2010 .

[166]  Dimitris Rizopoulos,et al.  Joint Models for Longitudinal and Time-to-Event Data: With Applications in R , 2012 .

[167]  Joseph G Ibrahim,et al.  Basic concepts and methods for joint models of longitudinal and survival data. , 2010, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[168]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[169]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[170]  M. Niranjan,et al.  Sequential support vector machines , 1999, Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No.98TH8468).

[171]  J. Ibrahim,et al.  A Bayesian semiparametric joint hierarchical model for longitudinal and survival data. , 2003, Biometrics.

[172]  M Ganjali,et al.  A Copula Approach to Joint Modeling of Longitudinal Measurements and Survival Times Using Monte Carlo Expectation-Maximization with Application to AIDS Studies , 2015, Journal of biopharmaceutical statistics.