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Joint modeling of longitudinal and time to event data

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  • Cover

  • Half Title

  • Title Page

  • Copyright Page

  • Dedication

  • Table of Contents

  • Preface

  • 1: Introduction and Examples

    • 1.1 Introduction

      • 1.1.1 Scleroderma Lung Study

      • 1.1.2 Stroke Study: the NINDS rt-PA trial

      • 1.1.3 ENABLE II Study

      • 1.1.4 Milk Protein Trial

      • 1.1.5 ACTG study

      • 1.1.6 Medfly Fecundity Data

      • 1.1.7 Bladder Cancer Study

      • 1.1.8 Renal Graft Failure Study

      • 1.1.9 PAQUID Study

      • 1.1.10 Rat Data

      • 1.1.11 AIDS Clinical Trial

  • 2: Methods for Longitudinal Measurements with Ignorable Missing Data

    • 2.1 Introduction

    • 2.2 Missing Data Mechanisms

    • 2.3 Linear and Generalized Linear Mixed Effects Models

      • 2.3.1 Linear Mixed Effects Models

        • General Form of Linear Mixed Effects Models

        • Estimation and Inference

      • 2.3.2 Generalized Linear Mixed Effects Models

        • Model Assumptions

        • Estimation and Inference

    • 2.4 Generalized Estimating Equations

      • 2.4.1 General Theory

      • 2.4.2 Weighted Generalized Estimating Equations

    • 2.5 Further topics

      • 2.5.1 Multivariate Longitudinal Data Analysis

      • 2.5.2 Pseudo-Likelihood Methods for Longitudinal Data

      • 2.5.3 Missing Data Imputation

  • 3: Methods for Time-to-Event Data

    • 3.1 Right Censoring

    • 3.2 Survival Function and Hazard Function

    • 3.3 Estimation of a Survival Function

      • 3.3.1 The Kaplan–Meier Estimate

      • 3.3.2 Asymptotic Inference

        • Confidence Intervals for S(t)

        • Transformation-Based Confidence Intervals for S(t)

        • Nonparametric Likelihood Ratio Condence Intervals for S(t)

    • 3.4 Cox's Semiparametric Multiplicative Hazards Model

      • 3.4.1 Model Formulation

      • 3.4.2 Partial Likelihood

      • 3.4.3 Estimation of β and Λ0(t) = ʃt0 λ0(s)ds

      • 3.4.4 Prediction of a Conditional Survival Function

        • Time-Independent Covariates

        • Time-Dependent Covariates

      • 3.4.5 Remark on Cox's Model with Intermittently Measured Time-Dependent Covariates and Measurement Error

    • 3.5 Accelerated Failure Time Models with Time-Independent Covariates

      • 3.5.1 Parametric AFT Models

      • 3.5.2 Semiparametric AFT Model

        • Synthetic Data Method

        • The Buckley–James Method

        • Linear Rank Method

    • 3.6 Accelerated Failure Time Model with Time-Dependent Covariates

      • 3.6.1 Model Formulation

      • 3.6.2 Rank-Based Estimation

    • 3.7 Methods for Competing Risks Data

      • 3.7.1 Basic Quantities for Competing Risks Data

      • 3.7.2 Latent Variable Representation of Competing Risks Data

      • 3.7.3 Estimation of the Cumulative Cause-Specific Hazard and Cumulative Incidence

      • 3.7.4 Regression Models for a Cause-Specific Hazard

        • Multiplicative Cause-Specific Hazards Model

        • Accelerated Failure Time Model

      • 3.7.5 Regression Models for Cumulative Incidence

        • Multiplicative Subdistribution Hazards Model

      • 3.7.6 Joint Inference of Cause-Specific Hazard and Cumulative Incidence

    • 3.8 Further Topics

  • 4: Overview of Joint Models for Longitudinal and Time-to-Event Data

    • 4.1 Joint Models of Longitudinal Data and an Event Time

      • 4.1.1 Selection Models

        • Shared Parameter Models

        • Missingness in Y

        • Extensions of Shared Parameter Models

        • Likelihood and Parameter Estimation

        • Standard Error Estimation

        • Bayesian Approaches

        • Nonparametric Distributions for Random Effects in Joint Models

      • 4.1.2 Mixture Models

        • Pattern-Mixture Models

        • Missing-Data Mechanisms in Pattern-Mixture Models

        • Random-Effects Mixture Models

        • Terminal Decline Models

      • 4.1.3 Remarks on Selection and Mixture Models

    • 4.2 Joint Models with Discrete Event Times and Monotone Missingness

      • 4.2.1 Outcome-Dependent Dropout Models

        • Model Formulation

        • Parameter Estimation and Inference

      • 4.2.2 Random-Effects Dependent Dropout Models

    • 4.3 Longitudinal Data with Both Monotone and Intermittent Missing Values

      • 4.3.1 Model Formulation for Monotone and Intermittent Missing Data

      • 4.3.2 Likelihood and Estimation

    • 4.4 Event Time Models with Intermittently Measured Time-Dependent Covariates

      • 4.4.1 Cox Models with Intermittently Measured Time-Dependent Covariates

        • Conditional Score Approach

        • Likelihood-Based Methods

        • Bayesian Methods

        • Corrected Score Approach

      • 4.4.2 Accelerated Failure Time Models with Intermittently Measured Time-Dependent Covariates

    • 4.5 Longitudinal Data with Informative Observation Times

      • 4.5.1 Latent Pattern Mixture Models

        • Model Specification

        • Estimation and EM Algorithm

      • 4.5.2 Latent Random Effects Models

        • Model and Inference

    • 4.6 Dynamic Prediction in Joint Models

  • 5: Joint Models for Longitudinal Data and Continuous Event Times from Competing Risks

    • 5.1 Joint Analysis of Longitudinal Data and Competing Risks

      • 5.1.1 The Model Formulation

      • 5.1.2 Estimation and Inference Procedure

    • 5.2 A Robust Model with t-Distributed Random Errors

    • 5.3 Ordinal Longitudinal Outcomes with Missing Data Due to Multiple Failure Types

      • 5.3.1 Model Formulation

      • 5.3.2 Estimation and Inference

    • 5.4 Bayesian Joint Models with Heterogeneous Random Effects

      • 5.4.1 Model Specification

      • 5.4.2 Estimation and Inference

      • 5.4.3 A Robust Joint Model with Heterogeneous Random Effects and t-Distributed Random Errors

    • 5.5 Accelerated Failure Time Models for Competing Risks

  • 6: Joint Models for Multivariate Longitudinal and Survival Data

    • 6.1 Joint Models for Multivariate Longitudinal Outcomes and an Event Time

      • 6.1.1 Random-Effects Models

      • 6.1.2 Latent Pattern Mixture Models

        • Model Formulation

        • Parameter Estimation

    • 6.2 Joint Models for Recurrent Events and Longitudinal Data

      • 6.2.1 Random-Effects Models

      • 6.2.2 Latent Pattern Mixture Models

    • 6.3 Joint Models for Multivariate Survival and Longitudinal Data

  • 7: Further Topics

    • 7.1 Joint Models and Missing Data: Assumptions, Sensitivity Analysis, and Diagnostics

      • 7.1.1 Sensitivity Analysis

        • A Local Influence Approach

        • An Index of Sensitivity to Nonignorability

      • 7.1.2 Joint Model Diagnostics

    • 7.2 Variable Selection in Joint Models

      • 7.2.1 Spike-and-Slab Priors

        • Posterior Distribution of Indicator Variables

        • Posterior Probability of Selecting an Effect

      • 7.2.2 Zero-Inflated Mixture Priors

    • 7.3 Joint Multistate Models

    • 7.4 Joint Models for Cure Rate Survival Data

    • 7.5 Sample Size and Power Estimation for Joint Models

  • Appendices

  • A Software to Implement Joint Models

  • Bibliography

  • Index

Nội dung

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 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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

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