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Preface | |

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Preface to the First Edition | |

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Introduction | |

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Models | |

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Linear models (LM) and linear mixed models (LMM) | |

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Generalized models (GLMs and GLMMs) | |

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Factors, Levels, Cells, Effects and Data | |

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Fixed Effects Models | |

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Example 1: Placebo and a drug | |

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Example 2: Comprehension of humor | |

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Example 3: Four dose levels of a drug | |

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Random Effects Models | |

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Example 4: Clinics | |

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Notation | |

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Example 5: Ball bearings and calipers | |

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Linear Mixed Models (LMMs) | |

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Example 6: Medications and clinics | |

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Example 7: Drying methods and fabrics | |

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Example 8: Potomac River Fever | |

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Regression models | |

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Longitudinal data | |

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Example 9: Osteoarthritis Initiative | |

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Model equations | |

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Fixed or Random? | |

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Example 10: Clinical trials | |

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Making a decision | |

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Inference | |

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Estimation | |

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Testing | |

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Prediction | |

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Computer Software | |

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Exercises | |

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One-Way Classifications | |

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Normality and Fixed Effects | |

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Model | |

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Estimation by ML | |

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Generalized likelihood ratio test | |

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Confidence intervals | |

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Hypothesis tests | |

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Normality, Random Effects and MLE | |

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Model | |

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Balanced data | |

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Unbalanced data | |

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Bias | |

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Sampling variances | |

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Normality, Random Effects and Reml | |

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Balanced data | |

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Unbalanced data | |

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More on Random Effects and Normality | |

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Tests and confidence intervals | |

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Predicting random effects | |

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Binary Data: Fixed Effects | |

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Model equation | |

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Likelihood | |

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ML equations and their solutions | |

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Likelihood ratio test | |

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The usual chi-square test | |

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Large-sample tests and confidence intervals | |

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Exact tests and confidence intervals | |

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Example: Snake strike data | |

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Binary Data: Random Effects | |

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Model equation | |

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Beta-binomial model | |

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Logit-normal model | |

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Probit-normal model | |

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Computing | |

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Exercises | |

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Single-Predictor Regression | |

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Introduction | |

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Normality: Simple Linear Regression | |

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Model | |

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Likelihood | |

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Maximum likelihood estimators | |

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Distributions of MLEs | |

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Tests and confidence intervals | |

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Illustration | |

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Normality: A Nonlinear Model | |

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Model | |

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Likelihood | |

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Maximum likelihood estimators | |

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Distributions of MLEs | |

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Transforming Versus Linking | |

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Transforming | |

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Linking | |

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Comparisons | |

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Random Intercepts: Balanced Data | |

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The model | |

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Estimating [mu] and [beta] | |

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Estimating variances | |

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Tests of hypotheses - using LRT | |

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Illustration | |

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Predicting the random intercepts | |

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Random Intercepts: Unbalanced Data | |

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The model | |

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Estimating [mu] and [beta] when variances are known | |

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Bernoulli - Logistic Regression | |

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Logistic regression model | |

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Likelihood | |

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ML equations | |

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Large-sample tests and confidence intervals | |

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Bernoulli - Logistic with Random Intercepts | |

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Model | |

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Likelihood | |

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Large-sample tests and confidence intervals | |

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Prediction | |

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Conditional Inference | |

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Exercises | |

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Linear Models (LMs) | |

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A General Model | |

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A Linear Model for Fixed Effects | |

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Mle Under Normality | |

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Sufficient Statistics | |

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Many Apparent Estimators | |

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General result | |

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Mean and variance | |

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Invariance properties | |

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Distributions | |

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Estimable Functions | |

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Introduction | |

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Definition | |

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Properties | |

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Estimation | |

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A Numerical Example | |

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Estimating Residual Variance | |

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Estimation | |

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Distribution of estimators | |

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The One- and Two-Way Classifications | |

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The one-way classification | |

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The two-way classification | |

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Testing Linear Hypotheses | |

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Likelihood ratio test | |

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Wald test | |

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t-Tests and Confidence Intervals | |

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Unique Estimation Using Restrictions | |

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Exercises | |

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Generalized Linear Models (GLMs) | |

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Introduction | |

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Structure of the Model | |

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Distribution of y | |

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Link function | |

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Predictors | |

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Linear models | |

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Transforming Versus Linking | |

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Estimation by Maximum Likelihood | |

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Likelihood | |

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Some useful identities | |

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Likelihood equations | |

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Large-sample variances | |

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Solving the ML equations | |

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Example: Potato flour dilutions | |

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Tests of Hypotheses | |

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Likelihood ratio tests | |

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Wald tests | |

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Illustration of tests | |

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Confidence intervals | |

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Illustration of confidence intervals | |

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Maximum Quasi-Likelihood | |

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Introduction | |

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Definition | |

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Exercises | |

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Linear Mixed Models (LMMs) | |

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A General Model | |

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Introduction | |

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Basic properties | |

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Attributing Structure to Var(y) | |

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Example | |

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Taking covariances between factors as zero | |

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The traditional variance components model | |

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An LMM for longitudinal data | |

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Estimating Fixed Effects for V Known | |

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Estimating Fixed Effects for V Unknown | |

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Estimation | |

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Sampling variance | |

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Bias in the variance | |

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Approximate F-statistics | |

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Predicting Random Effects for V Known | |

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Predicting Random Effects for V Unknown | |

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Estimation | |

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Sampling variance | |

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Bias in the variance | |

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Anova Estimation of Variance Components | |

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Balanced data | |

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Unbalanced data | |

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Maximum Likelihood (ML) Estimation | |

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Estimators | |

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Information matrix | |

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Asymptotic sampling variances | |

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Restricted Maximum Likelihood (REML) | |

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Estimation | |

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Sampling variances | |

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Notes and Extensions | |

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ML or REML? | |

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Other methods for estimating variances | |

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Appendix for Chapter 6 | |

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Differentiating a log likelihood | |

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Differentiating a generalized inverse | |

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Differentiation for the variance components model | |

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Exercises | |

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Generalized Linear Mixed Models | |

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Introduction | |

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Structure of the Model | |

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Conditional distribution of y | |

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Consequences of Having Random Effects | |

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Marginal versus conditional distribution | |

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Mean of y | |

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Variances | |

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Covariances and correlations | |

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Estimation by Maximum Likelihood | |

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Likelihood | |

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Likelihood equations | |

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Other Methods of Estimation | |

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Penalized quasi-likelihood | |

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Conditional likelihood | |

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Simpler models | |

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Tests of Hypotheses | |

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Likelihood ratio tests | |

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Asymptotic variances | |

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Wald tests | |

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Score tests | |

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Illustration: Chestnut Leaf Blight | |

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A random effects probit model | |

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Exercises | |

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Models for Longitudinal Data | |

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Introduction | |

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A Model for Balanced Data | |

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Prescription | |

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Estimating the mean | |

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Estimating V[subscript 0] | |

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A Mixed Model Approach | |

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Fixed and random effects | |

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Variances | |

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Random Intercept and Slope Models | |

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Variances | |

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Within-subject correlations | |

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Predicting Random Effects | |

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Uncorrelated subjects | |

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Uncorrelated between, and within, subjects | |

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Uncorrelated between, and autocorrelated within | |

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Random intercepts and slopes | |

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Estimating Parameters | |

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The general case | |

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Uncorrelated subjects | |

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Uncorrelated between, and autocorrelated within, subjects | |

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Unbalanced Data | |

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Example and model | |

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Uncorrelated subjects | |

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Models for Non-Normal Responses | |

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Covariances and correlations | |

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Estimation | |

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Prediction of random effects | |

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Binary responses, random intercepts and slopes | |

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A Summary of Results | |

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Balanced data | |

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Unbalanced data | |

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Appendix | |

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For Section 8.4a | |

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For Section 8.4b | |

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Exercises | |

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Marginal Models | |

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Introduction | |

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Examples of Marginal Regression Models | |

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Generalized Estimating Equations | |

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Models with marginal and conditional interpretations | |

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Contrasting Marginal and Conditional Models | |

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Exercises | |

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Multivariate Models | |

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Introduction | |

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Multivariate Normal Outcomes | |

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Non-Normally Distributed Outcomes | |

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A multivariate binary model | |

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A binary/normal example | |

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A Poisson/Normal Example | |

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Correlated Random Effects | |

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Likelihood-Based Analysis | |

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Example: Osteoarthritis Initiative | |

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Notes and Extensions | |

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Missing data | |

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Efficiency | |

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Exercises | |

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Nonlinear Models | |

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Introduction | |

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Example: Corn Photosynthesis | |

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Pharmacokinetic Models | |

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Computations for Nonlinear Mixed Models | |

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Exercises | |

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Departures from Assumptions | |

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Introduction | |

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Incorrect Model for Response | |

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Omitted covariates | |

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Misspecified link functions | |

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Misclassified binary outcomes | |

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Informative cluster sizes | |

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Incorrect Random Effects Distribution | |

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Incorrect distributional family | |

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Correlation of covariates and random effects | |

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Covariate-dependent random effects variance | |

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Diagnosing Misspecification | |

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Conditional likelihood methods | |

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Between/within cluster covariate decompositions | |

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Specification tests | |

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Nonparametric maximum likelihood | |

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A Summary of Results | |

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Exercises | |

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Prediction | |

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Introduction | |

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Best Prediction (BP) | |

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The best predictor | |

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Mean and variance properties | |

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A correlation property | |

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Maximizing a mean | |

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Normality | |

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Best Linear Prediction (BLP) | |

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BLP(u) | |

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Example | |

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Derivation | |

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Ranking | |

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Linear Mixed Model Prediction (BLUP) | |

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BLUE(X[beta]) | |

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BLUP(t'X[beta] + s'u) | |

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Two variances | |

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Other derivations | |

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Required Assumptions | |

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Estimated Best Prediction | |

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Henderson's Mixed Model Equations | |

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Origin | |

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Solutions | |

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Use in ML estimation of variance components | |

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Appendix | |

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Verification of (13.5) | |

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Verification of (13.7) and (13.8) | |

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Exercises | |

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Computing | |

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Introduction | |

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Computing ML Estimates for LMMs | |

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The EM algorithm | |

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Using E[uy] | |

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Newton-Raphson method | |

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Computing ML Estimates for GLMMs | |

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Numerical quadrature | |

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EM algorithm | |

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Markov chain Monte Carlo algorithms | |

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Stochastic approximation algorithms | |

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Simulated maximum likelihood | |

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Penalized Quasi-Likelihood and Laplace | |

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Iterative Bootstrap Bias Correction | |

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Exercises | |

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Some Matrix Results | |

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Vectors and Matrices of Ones | |

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Kronecker (or Direct) Products | |

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A Matrix Notation in Terms of Elements | |

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Generalized Inverses | |

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Definition | |

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Generalized inverses of X'X | |

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Two results involving X(X'V[superscript -1]X)[superscript -]X'V[superscript -1] | |

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Solving linear equations | |

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Rank results | |

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Vectors orthogonal to columns of X | |

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A theorem for K' with K'X being null | |

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Differential Calculus | |

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Definition | |

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Scalars | |

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Vectors | |

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Inner products | |

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Quadratic forms | |

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Inverse matrices | |

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Determinants | |

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Some Statistical Results | |

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Moments | |

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Conditional moments | |

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Mean of a quadratic form | |

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Moment generating function | |

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Normal Distributions | |

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Univariate | |

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Multivariate | |

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Quadratic forms in normal variables | |

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Exponential Families | |

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Maximum Likelihood | |

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The likelihood function | |

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Maximum likelihood estimation | |

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Asymptotic variance-covariance matrix | |

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Asymptotic distribution of MLEs | |

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Likelihood Ratio Tests | |

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MLE Under Normality | |

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Estimation of [beta] | |

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Estimation of variance components | |

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Asymptotic variance-covariance matrix | |

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Restricted maximum likelihood (REML) | |

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References | |

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Index | |