Joint Modeling of Longitudinal and Time-to-Event Data MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F Bunea, V Isham, N Keiding, T Louis, R L Smith, and H Tong 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Stochastic Population Models in Ecology and Epidemiology M.S Barlett (1960) Queues D.R Cox and W.L Smith (1961) Monte Carlo Methods J.M Hammersley and D.C Handscomb (1964) The Statistical Analysis of Series of Events D.R Cox and P.A.W Lewis (1966) Population Genetics W.J Ewens (1969) Probability, Statistics and Time M.S Barlett (1975) Statistical Inference S.D Silvey (1975) The Analysis of Contingency Tables B.S Everitt (1977) Multivariate Analysis in Behavioural Research A.E Maxwell (1977) Stochastic Abundance Models S Engen (1978) Some Basic Theory for Statistical Inference E.J.G Pitman (1979) Point Processes D.R Cox and V Isham (1980) Identification of Outliers D.M Hawkins (1980) Optimal Design S.D Silvey (1980) Finite Mixture Distributions B.S Everitt and D.J Hand (1981) Classification A.D Gordon (1981) Distribution-Free Statistical Methods, 2nd edition J.S Maritz (1995) Residuals and Influence in Regression R.D Cook and S Weisberg (1982) Applications of Queueing Theory, 2nd edition G.F Newell (1982) Risk Theory, 3rd edition R.E Beard, T Pentikäinen and E Pesonen (1984) Analysis of Survival Data D.R Cox and D Oakes (1984) An Introduction to Latent Variable Models B.S Everitt (1984) Bandit Problems D.A Berry and B Fristedt (1985) Stochastic Modelling and Control M.H.A Davis and R Vinter (1985) The Statistical Analysis of Composition Data J Aitchison (1986) Density Estimation for Statistics and Data Analysis B.W Silverman (1986) Regression Analysis with Applications G.B Wetherill (1986) Sequential Methods in Statistics, 3rd edition G.B Wetherill and K.D Glazebrook (1986) Tensor Methods in Statistics P McCullagh (1987) Transformation and Weighting in Regression R.J Carroll and D Ruppert (1988) Asymptotic Techniques for Use in Statistics O.E Bandorff-Nielsen and D.R Cox (1989) Analysis of Binary Data, 2nd edition D.R Cox and E.J Snell (1989) Analysis of Infectious Disease Data N.G Becker (1989) Design and Analysis of Cross-Over Trials B Jones and M.G Kenward (1989) Empirical Bayes Methods, 2nd edition J.S Maritz and T Lwin (1989) Symmetric Multivariate and Related Distributions K.T Fang, S Kotz and K.W Ng (1990) Generalized Linear Models, 2nd edition P McCullagh and J.A Nelder (1989) Cyclic and Computer Generated Designs, 2nd edition J.A John and E.R Williams (1995) Analog Estimation Methods in Econometrics C.F Manski (1988) Subset Selection in Regression A.J Miller (1990) Analysis of Repeated Measures M.J Crowder and D.J Hand (1990) Statistical Reasoning with Imprecise Probabilities P Walley (1991) Generalized Additive Models T.J Hastie and R.J Tibshirani (1990) Inspection Errors for Attributes in Quality Control N.L Johnson, S Kotz and X Wu (1991) The Analysis of Contingency Tables, 2nd edition B.S Everitt (1992) The Analysis of Quantal Response Data B.J.T Morgan (1992) Longitudinal Data with Serial Correlation—A State-Space Approach R.H Jones (1993) 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 Differential Geometry and Statistics M.K Murray and J.W Rice (1993) Markov Models and Optimization M.H.A Davis (1993) Networks and Chaos—Statistical and Probabilistic Aspects O.E Barndorff-Nielsen, J.L Jensen and W.S Kendall (1993) Number-Theoretic Methods in Statistics K.-T Fang and Y Wang (1994) Inference and Asymptotics O.E Barndorff-Nielsen and D.R Cox (1994) Practical Risk Theory for Actuaries C.D Daykin, T Pentikäinen and M Pesonen (1994) Biplots J.C Gower and D.J Hand (1996) Predictive Inference—An Introduction S Geisser (1993) Model-Free Curve Estimation M.E Tarter and M.D Lock (1993) An Introduction to the Bootstrap B Efron and R.J Tibshirani (1993) Nonparametric Regression and Generalized Linear Models P.J Green and B.W Silverman (1994) Multidimensional Scaling T.F Cox and M.A.A Cox (1994) Kernel Smoothing M.P Wand and M.C Jones (1995) Statistics for Long Memory Processes J Beran (1995) Nonlinear Models for Repeated Measurement Data M Davidian and D.M Giltinan (1995) Measurement Error in Nonlinear Models R.J Carroll, D Rupert and L.A Stefanski (1995) Analyzing and Modeling Rank Data J.J Marden (1995) Time Series Models—In Econometrics, Finance and Other Fields D.R Cox, D.V Hinkley and O.E Barndorff-Nielsen (1996) Local Polynomial Modeling and its Applications J Fan and I Gijbels (1996) Multivariate Dependencies—Models, Analysis and Interpretation D.R Cox and N Wermuth (1996) Statistical Inference—Based on the Likelihood A Azzalini (1996) Bayes and Empirical Bayes Methods for Data Analysis B.P Carlin and T.A Louis (1996) Hidden Markov and Other Models for Discrete-Valued Time Series I.L MacDonald and W Zucchini (1997) Statistical Evidence—A Likelihood Paradigm R Royall (1997) Analysis of Incomplete Multivariate Data J.L Schafer (1997) Multivariate Models and Dependence Concepts H Joe (1997) Theory of Sample Surveys M.E Thompson (1997) Retrial Queues G Falin and J.G.C Templeton (1997) Theory of Dispersion Models B Jørgensen (1997) Mixed Poisson Processes J Grandell (1997) Variance Components Estimation—Mixed Models, Methodologies and Applications P.S.R.S Rao (1997) Bayesian Methods for Finite Population Sampling G Meeden and M Ghosh (1997) Stochastic Geometry—Likelihood and computation O.E Barndorff-Nielsen, W.S Kendall and M.N.M van Lieshout (1998) Computer-Assisted Analysis of Mixtures and Applications—Meta-Analysis, Disease Mapping and Others D Böhning (1999) Classification, 2nd edition A.D Gordon (1999) Semimartingales and their Statistical Inference B.L.S Prakasa Rao (1999) Statistical Aspects of BSE and vCJD—Models for Epidemics C.A Donnelly and N.M Ferguson (1999) Set-Indexed Martingales G Ivanoff and E Merzbach (2000) The Theory of the Design of Experiments D.R Cox and N Reid (2000) Complex Stochastic Systems O.E Barndorff-Nielsen, D.R Cox and C Klüppelberg (2001) Multidimensional Scaling, 2nd edition T.F Cox and M.A.A Cox (2001) Algebraic Statistics—Computational Commutative Algebra in Statistics G Pistone, E Riccomagno and H.P Wynn (2001) Analysis of Time Series Structure—SSA and Related Techniques N Golyandina, V Nekrutkin and A.A Zhigljavsky (2001) Subjective Probability Models for Lifetimes Fabio Spizzichino (2001) Empirical Likelihood Art B Owen (2001) Statistics in the 21st Century Adrian E Raftery, Martin A Tanner, and Martin T Wells (2001) Accelerated Life Models: Modeling and Statistical Analysis Vilijandas Bagdonavicius and Mikhail Nikulin (2001) 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 Subset Selection in Regression, Second Edition Alan Miller (2002) Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M Ryan (2002) Components of Variance D.R Cox and P.J Solomon (2002) Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G Kenward (2003) Extreme Values in Finance, Telecommunications, and the Environment Bärbel Finkenstädt and Holger Rootzén (2003) Statistical Inference and Simulation for Spatial Point Processes Jesper Møller and Rasmus Plenge Waagepetersen (2004) Hierarchical Modeling and Analysis for Spatial Data Sudipto Banerjee, Bradley P Carlin, and Alan E Gelfand (2004) Diagnostic Checks in Time Series Wai Keung Li (2004) Stereology for Statisticians Adrian Baddeley and Eva B Vedel Jensen (2004) Gaussian Markov Random Fields: Theory and Applications H˚avard Rue and Leonhard Held (2005) Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition Raymond J Carroll, David Ruppert, Leonard A Stefanski, and Ciprian M Crainiceanu (2006) Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood Youngjo Lee, John A Nelder, and Yudi Pawitan (2006) Statistical Methods for Spatio-Temporal Systems Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007) Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007) Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis Michael J Daniels and Joseph W Hogan (2008) Hidden Markov Models for Time Series: An Introduction Using R Walter Zucchini and Iain L MacDonald (2009) ROC Curves for Continuous Data Wojtek J Krzanowski and David J Hand (2009) Antedependence Models for Longitudinal Data Dale L Zimmerman and Vicente A Núđez-Antón (2009) Mixed Effects Models for Complex Data Lang Wu (2010) Intoduction to Time Series Modeling Genshiro Kitagawa (2010) Expansions and Asymptotics for Statistics Christopher G Small (2010) Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010) Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010) Simultaneous Inference in Regression Wei Liu (2010) Robust Nonparametric Statistical Methods, Second Edition Thomas P Hettmansperger and Joseph W McKean (2011) Statistical Inference: The Minimum Distance Approach Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park (2011) Smoothing Splines: Methods and Applications Yuedong Wang (2011) Extreme Value Methods with Applications to Finance Serguei Y Novak (2012) Dynamic Prediction in Clinical Survival Analysis Hans C van Houwelingen and Hein Putter (2012) Statistical Methods for Stochastic Differential Equations Mathieu Kessler, Alexander Lindner, and Michael Sørensen (2012) Maximum Likelihood Estimation for Sample Surveys R L Chambers, D G Steel, Suojin Wang, and A H Welsh (2012) Mean Field Simulation for Monte Carlo Integration Pierre Del Moral (2013) Analysis of Variance for Functional Data Jin-Ting Zhang (2013) Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition Peter J Diggle (2013) Constrained Principal Component Analysis and Related Techniques Yoshio Takane (2014) Randomised Response-Adaptive Designs in Clinical Trials Anthony C Atkinson and Atanu Biswas (2014) Theory of Factorial Design: Single- and Multi-Stratum Experiments Ching-Shui Cheng (2014) Quasi-Least Squares Regression Justine Shults and Joseph M Hilbe (2014) Data Analysis and Approximate Models: Model Choice, Location-Scale, Analysis of Variance, Nonparametric Regression and Image Analysis Laurie Davies (2014) Dependence Modeling with Copulas Harry Joe (2014) Hierarchical Modeling and Analysis for Spatial Data, Second Edition Sudipto Banerjee, Bradley P Carlin, and Alan E Gelfand (2014) 136 Sequential Analysis: Hypothesis Testing and Changepoint Detection Alexander Tartakovsky, Igor Nikiforov, and Michèle Basseville (2015) 137 Robust Cluster Analysis and Variable Selection Gunter Ritter (2015) 138 Design and Analysis of Cross-Over Trials, Third Edition Byron Jones and Michael G Kenward (2015) 139 Introduction to High-Dimensional Statistics Christophe Giraud (2015) 140 Pareto Distributions: Second Edition Barry C Arnold (2015) 141 Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data Paul Gustafson (2015) 142 Models for Dependent Time Series Granville Tunnicliffe Wilson, Marco Reale, John Haywood (2015) 143 Statistical Learning with Sparsity: The Lasso and Generalizations Trevor Hastie, Robert Tibshirani, and Martin Wainwright (2015) 144 Measuring Statistical Evidence Using Relative Belief Michael Evans (2015) 145 Stochastic Analysis for Gaussian Random Processes and Fields: With Applications Vidyadhar S Mandrekar and Leszek Gawarecki (2015) 146 Semialgebraic Statistics and Latent Tree Models Piotr Zwiernik (2015) 147 Inferential Models: Reasoning with Uncertainty Ryan Martin and Chuanhai Liu (2016) 148 Perfect Simulation Mark L Huber (2016) 149 State-Space Methods for Time Series Analysis: Theory, Applications and Software Jose Casals, Alfredo Garcia-Hiernaux, Miguel Jerez, Sonia Sotoca, and A Alexandre Trindade (2016) 150 Hidden Markov Models for Time Series: An Introduction Using R, Second Edition Walter Zucchini, Iain L MacDonald, and Roland Langrock (2016) 151 Joint Modeling of Longitudinal and Time-to-Event Data Robert M Elashoff, Gang Li, and Ning Li (2016) Monographs on Statistics and Applied Probability 151 Joint Modeling of Longitudinal and Time-to-Event Data Robert M Elashoff, Gang Li, and Ning Li UCLA Departments of Biostatistics and Biomathematics Los Angeles, California, USA CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20160607 International Standard Book 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Cook-type distance, 201 Corrected score approach, 119 Covariate-dependent dropout, 79 Cox proportional hazards model, 105–107, 119, 154, 167, 169 Cox regression, 69, 125 Cox’s proportional hazards model, Deviance Information Criterion (DIC), 155, 156, 160, 170 Dirichlet process, 168 Dirichlet process (DP) prior, 205 Disease-free survival (DFS), 184, 187 Dummy variable, 26 Balanced, 17 Baseline survival function, 120 Bayes Information Criterion (BIC), 174, 175 Bayesian approach, 73, 89, 108, 115, 137, 157, 187, 201 Bernoulli distribution, 109, 205 Best linear unbiased predictor (BLUP), 22 Bootstrap, 73, 83, 122, 123 Canonical link, 28 Cause-specific competing risks model, 155 Cause-specific hazards model, 138, 156 Cholesky decomposition, 23, 151, 152 Class-specific proportional hazard model, 173 Class-specific recurrent event model, 183 Competing risk, 5, 137, 148, 152, 156 Complete-data likelihood, 143 Complete-data log-likelihood, 144 Conditional independence, 128 Conditional linear model, 82 Eigenvalue, 193 Expectation-Maximization (EM) algorithm, 71, 74, 80, 89, 101, 108, 112, 113, 121, 127, 140, 143, 144, 148, 174, 179, 183 Exponential family, 20, 26–28 Fixed effect, 20–22, 28, 39, 206 Gamma distribution, 127 Gaussian distribution, 144 Gaussian process, 70, 71, 75, 95, 107, 178, 198, 199 Gaussian quadrature, 28, 29, 72, 90, 101 Gaussian–Hermite quadrature, 102 Generalized autoregressive parameters (GARP), 152, 153, 155 239 240 Generalized estimating equation, 29–31, 33, 34, 37 Generalized linear mixed effects model (GLMM), 166, 168 Gibbs sampler, 72, 90, 117, 154, 187, 205, 206, 211 Hannan–Quinn criterion (HQ), 114 Hessian matrix, 174 Heterogeneous, 153, 155, 157 Highest probability density (HPD), 187 Homogeneous, 153, 155, 157 Ignorable, 1, 17, 19, 191 Index of sensitivity to nonignorability (ISNI), 192, 196 Informative, 4, 17, 125, 199 Informative dropout, 93, 202 Integrated Ornstein–Uhlenbeck (IOU) process, 107 Intermittent, 99, 100 Interval-censored, 96 Joint distribution, 18 Joint model, 70, 72, 74, 112 Kaplan-Meier, 169, 184, 187 Laplace method, 102 Last observation carried forward, 36 Latent pattern mixture model, 125, 126, 172, 181 Latent process, 71 Latent random effect, 68 Latent random effects model, 130, 177 Likelihood approach, 115 Likelihood ratio test, 22, 29 Linear mixed effects model (LMM), 2, 20, 22, 23, 28, 31, 32, 37, 38, 68, 99, 102, 141, 168, 172 Linear predictor, 28 Linear regression, 21 INDEX Link function, 28, 37, 197 Marginal distribution, 22, 67 Marginal models, 20 Markov chain Monte Carlo (MCMC), 37, 38, 72, 73, 117–119, 151, 153, 154, 156 Maximum likelihood, 20–22, 28, 80, 93, 148, 173, 179 Maximum likelihood estimator, 35 Mean-zero stochastic process, 107 Metropolis–Hastings (MH) sampling, 72, 117, 151, 154 Missing at random (MAR), 18, 25, 31, 33, 34, 37, 80, 82, 88, 191–195, 210 Missing Completely at Random (MCAR), 18, 29, 33–35, 37, 39, 88, 89, 191, 194 Missing not at random (MNAR), 19, 25, 69, 88, 141, 191–194, 196, 202, 210 Mixed effects models, 20 Mixture model, 67, 77 Monotone, 33, 36, 100 Monte Carlo EM (MCEM) algorithm, 72, 89, 90, 179 Multimodality, 23 Multiple imputation, 35, 201 Multivariate normal distribution, 28, 70, 104, 139, 147, 168, 209 Multivariate normal mixed effects model, 184 Multivariate survival data, 183 Mutually independent, 21 Naive imputation method, 106 Newton–Raphson algorithm, 90, 102, 174 Newton-Raphson algorithm, 140 Non-ignorable, 1, 18, 19, 68, 100, 146, 167, 191 Non-informative, 17, 68 Nonparametric baseline hazard, 75 Normal distribution, 21, 80, 205 INDEX Normally distributed longitudinal data, 20 Ordinary least squares, 82 Outcome-dependent dropout, 79 Outcome-dependent dropout model, 87 Overall survival (OS), 184, 187 Partial maximum likelihood estimator, 33 Partial proportional odds model, 146 Pattern-mixture model, 78–80, 86, 87, 191 Penalized quasi-likelihood, 29, 174 Poisson process, 185 Predictive mean matching method, 36 Propensity score method, 36 Proportional hazards frailty model, 100, 104 Proportional transition intensity model, 213 Protective estimates, 80 Pseudo-likelihood method, 34 Random effect, 20–22, 25, 26, 28, 29, 69, 74, 112, 113, 147, 202, 206 Random effect-dependent dropout, 79, 82 Random intercept, 21 Random-effect dependent dropout model, 97 Random-effects mixture model, 79 Response, 18, 20 Restricted maximum likelihood, 20–22, 24 Retrospective, 98 Second order extension of estimating equation, 31 Selection model, 67, 68, 73, 86, 87, 191 Seminonparametric(SNP), 113 Semivariogram, 23 241 Sensitivity analysis, xviii, 80, 156, 192 Shared parameter model, 68–70, 97, 166 Single imputation, 35, 36 Spike-and-slab prior, 202, 204 Subject-dependent covariate, 155, 157 t-distribution, 23, 144, 160 Taylor series, 29, 72, 99, 101, 102, 197, 201 Time-dependent covariate, 1, 69, 105, 119, 172 Time-to-event data, 1, 67, 166 Unstructured homogeneous two-random-slope model, 159 Variance-covariance matrix, 21, 37, 70, 73, 88, 107, 111, 113, 126, 153, 172 Vertex Exchange Method (VEM), 74 Wald’s test, 22, 24 Weighted generalized estimating equation, 33 Weighted least squares, 82 Weighted loess curve, 201 Working correlation matrix, 30, 31 Zero-inflated mixture, 210 Zero-inflated mixture prior, 203, 209 ... 3.7.4 3.7.5 3.7.6 3.8 xiii Further Topics Overview of Joint Models for Longitudinal and Time- toEvent Data 4.1 66 67 Joint Models of Longitudinal Data and an Event Time 67 4.1.1 Selection Models 68... MacDonald, and Roland Langrock (2016) 151 Joint Modeling of Longitudinal and Time- to- Event Data Robert M Elashoff, Gang Li, and Ning Li (2016) Monographs on Statistics and Applied Probability 151 Joint. . .Joint Modeling of Longitudinal and Time- to- Event Data MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F Bunea, V Isham, N Keiding, T Louis, R L Smith, and H Tong 10 11