Longitudinal Data Analysis (eBook)
X, 141 Seiten
Springer Singapore (Verlag)
978-981-10-0077-5 (ISBN)
This book provides a new analytical approach for dynamic data repeatedly measured from multiple subjects over time. Random effects account for differences across subjects. Auto-regression in response itself is often used in time series analysis. In longitudinal data analysis, a static mixed effects model is changed into a dynamic one by the introduction of the auto-regression term. Response levels in this model gradually move toward an asymptote or equilibrium which depends on covariates and random effects. The book provides relationships of the autoregressive linear mixed effects models with linear mixed effects models, marginal models, transition models, nonlinear mixed effects models, growth curves, differential equations, and state space representation. State space representation with a modified Kalman filter provides log likelihoods for maximum likelihood estimation, and this representation is suitable for unequally spaced longitudinal data. The extension to multivariate longitudinal data analysis is also provided. Topics in medical fields, such as response-dependent dose modifications, response-dependent dropouts, and randomized controlled trials are discussed. The text is written in plain terms understandable for researchers in other disciplines such as econometrics, sociology, and ecology for the progress of interdisciplinary research.
Ikuko Funatogawa, The Institute of Statistical MathematicsTakashi Funatogawa, Chugai Pharmaceutical Co. Ltd.
Preface 6
Contents 8
1 Longitudinal Data and Linear Mixed Effects Models 12
1.1 Longitudinal Data 12
1.2 Linear Mixed Effects Models 14
1.3 Examples of Linear Mixed Effects Models 15
1.3.1 Means at Each Time Point with Random Intercept 16
1.3.2 Group Comparison Based on Means at Each Time Point with Random Intercept 18
1.3.3 Means at Each Time Point with Unstructured Variance Covariance 20
1.3.4 Linear Time Trend Models with Random Intercept and Random Slope 21
1.3.5 Group Comparison Based on Linear Time Trend Models with Random Intercept and Random Slope 23
1.4 Mean Structures and Variance Covariance Structures 24
1.4.1 Mean Structures 24
1.4.2 Variance Covariance Structures 25
1.5 Inference 29
1.5.1 Maximum Likelihood Method 29
1.5.2 Variances of Estimates of Fixed Effects 32
1.5.3 Prediction 32
1.5.4 Goodness of Fit for Models 34
1.5.5 Estimation and Test Using Contrast 34
1.6 Vector Representation 35
References 36
2 Autoregressive Linear Mixed Effects Models 38
2.1 Autoregressive Models of Response Itself 38
2.1.1 Introduction 38
2.1.2 Response Changes in Autoregressive Models 40
2.1.3 Interpretation of Parameters 43
2.2 Examples of Autoregressive Linear Mixed Effects Models 45
2.2.1 Example Without Covariates 46
2.2.2 Example with Time-Independent Covariates 47
2.2.3 Example with a Time-Dependent Covariate 48
2.3 Autoregressive Linear Mixed Effects Models 49
2.3.1 Autoregressive Form 49
2.3.2 Representation of Response Changes with Asymptotes 53
2.3.3 Marginal Form 55
2.4 Variance Covariance Structures 56
2.4.1 AR(1) Error and Measurement Error 56
2.4.2 Variance Covariance Matrix Induced by Random Effects 59
2.4.3 Variance Covariance Matrix Induced by Random Effects and Random Errors 61
2.4.4 Variance Covariance Matrix for Asymptotes 62
2.5 Estimation in Autoregressive Linear Mixed Effects Models 63
2.5.1 Likelihood of Marginal Form 63
2.5.2 Likelihood of Autoregressive Form 64
2.5.3 Indirect Methods Using Linear Mixed Effects Models 65
2.6 Models with Autoregressive Error Terms 67
References 69
3 Case Studies of Autoregressive Linear Mixed Effects Models: Missing Data and Time-Dependent Covariates 70
3.1 Example with Time-Independent Covariate: PANSS Data 70
3.2 Missing Data 72
3.2.1 Missing Mechanism 72
3.2.2 Model Comparison: PANSS Data 74
3.3 Example with Time-Dependent Covariate: AFCR Data 79
3.4 Response-Dependent Modification of Time-Dependent Covariate 83
References 85
4 Multivariate Autoregressive Linear Mixed Effects Models 87
4.1 Multivariate Longitudinal Data and Vector Autoregressive Models 87
4.1.1 Multivariate Longitudinal Data 87
4.1.2 Vector Autoregressive Models 88
4.2 Multivariate Autoregressive Linear Mixed Effects Models 90
4.2.1 Example of Bivariate Autoregressive Linear Mixed Effects Models 90
4.2.2 Autoregressive Form and Marginal Form 92
4.2.3 Representation of Response Changes with Equilibria 95
4.2.4 Variance Covariance Structures 96
4.2.5 Estimation 98
4.3 Example with Time-Dependent Covariate: PTH and Ca Data 100
4.4 Multivariate Linear Mixed Effects Models 104
4.5 Appendix 106
4.5.1 Direct Product 106
4.5.2 Parameter Transformation 106
References 107
5 Nonlinear Mixed Effects Models, Growth Curves, and Autoregressive Linear Mixed Effects Models 109
5.1 Autoregressive Models and Monomolecular Curves 109
5.2 Autoregressive Linear Mixed Effects Models and Monomolecular Curves with Random Effects 114
5.3 Nonlinear Mixed Effects Models 115
5.3.1 Nonlinear Mixed Effects Models 115
5.3.2 Estimation 117
5.4 Nonlinear Curves 118
5.4.1 Exponential Functions 119
5.4.2 Gompertz Curves 119
5.4.3 Logistic Curves 120
5.4.4 Emax Models and Logistic Curves 122
5.4.5 Other Nonlinear Curves 123
5.5 Generalization of Growth Curves 124
References 127
6 State Space Representations of Autoregressive Linear Mixed Effects Models 128
6.1 Time Series Data 128
6.1.1 State Space Representations of Time Series Data 129
6.1.2 Steps for Kalman Filter for Time Series Data 130
6.2 Longitudinal Data 132
6.2.1 State Space Representations of Longitudinal Data 132
6.2.2 Calculations of Likelihoods 133
6.3 Autoregressive Linear Mixed Effects Models 134
6.3.1 State Space Representations of Autoregressive Linear Mixed Effects Models 134
6.3.2 Steps for Modified Kalman Filter for Autoregressive Linear Mixed Effects Models 137
6.3.3 Steps for Calculating Standard Errors and Predicted Values of Random Effects 140
6.3.4 Another Representation 141
6.4 Multivariate Autoregressive Linear Mixed Effects Models 141
6.5 Linear Mixed Effects Models 143
6.5.1 State Space Representations of Linear Mixed Effects Models 143
6.5.2 Steps for Modified Kalman Filter 145
References 147
Index 148
| Erscheint lt. Verlag | 4.2.2019 |
|---|---|
| Reihe/Serie | JSS Research Series in Statistics | SpringerBriefs in Statistics |
| Zusatzinfo | X, 141 p. 27 illus. |
| Verlagsort | Singapore |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Informatik |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Mathematik / Informatik ► Mathematik ► Computerprogramme / Computeralgebra | |
| Mathematik / Informatik ► Mathematik ► Statistik | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Schlagworte | Autoregressive • dynamic • Longitudinal • mixed effects • State Space |
| ISBN-10 | 981-10-0077-8 / 9811000778 |
| ISBN-13 | 978-981-10-0077-5 / 9789811000775 |
| Haben Sie eine Frage zum Produkt? |
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