Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects

Content: Mixed Linear Models: Syntax, Theory, and Methods An Opinionated Survey of Methods for Mixed Linear Models Mixed linear models in the standard formulation Conventional analysis of the mixed linear model Bayesian analysis of the mixed linear model Conventional and Bayesian approaches compared A few words about computing Two More Tools: Alternative Formulation, Measures of Complexity Alternative formulation: The ''constraint-case'' formulation Measuring the complexity of a mixed linear model fit Richly Parameterized Models as Mixed Linear Models Penalized Splines as Mixed Linear Models Penalized splines: Basis, knots, and penalty More on basis, knots, and penalty Mixed linear model representation Additive Models and Models with Interactions Additive models as mixed linear models Models with interactions Spatial Models as Mixed Linear Models Geostatistical models Models for areal data Two-dimensional penalized splines Time-Series Models as Mixed Linear Models Example: Linear growth model Dynamic linear models in some generality Example of a multi-component DLM Two Other Syntaxes for Richly Parameterized Models Schematic comparison of the syntaxes Gaussian Markov random fields Likelihood inference for models with unobservables From Linear Models to Richly Parameterized Models: Mean Structure Adapting Diagnostics from Linear Models Preliminaries Added variable plots Transforming variables Case influence Residuals Puzzles from Analyzing Real Datasets Four puzzles Overview of the next three chapters A Random Effect Competing with a Fixed Effect Slovenia data: Spatial confounding Kids and crowns: Informative cluster size Differential Shrinkage The simplified model and an overview of the results Details of derivations Conclusion: What might cause differential shrinkage? Competition between Random Effects Collinearity between random effects in three simpler models Testing hypotheses on the optical-imaging data and DLM models Discussion Random Effects Old and New Old-style random effects New-style random effects Practical consequences Conclusion Beyond Linear Models: Variance Structure Mysterious, Inconvenient, or Wrong Results from Real Datasets Periodontal data and the ICAR model Periodontal data and the ICAR with two classes of neighbor pairs Two very different smooths of the same data Misleading zero variance estimates Multiple maxima in posteriors and restricted likelihoods Overview of the remaining chapters Re-Expressing the Restricted Likelihood: Two-Variance Models The re-expression Examples A tentative collection of tools Exploring the Restricted Likelihood for Two-Variance Models Which vj tell us about which variance? Two mysteries explained Extending the Re-Expressed Restricted Likelihood Restricted likelihoods that can and can't be re-expressed Expedients for restricted likelihoods that can't be re-expressed Zero Variance Estimates Some observations about zero variance estimates Some thoughts about tools Multiple Maxima in the Restricted Likelihood and Posterior Restricted likelihoods with multiple local maxima Posteriors with multiple modes
Abstract: ''This book covers a wide range of statistical models, including hierarchical, hierarchical generalized linear, linear mixed, dynamic linear, smoothing, spatial, and longitudinal. It presents a framework for expressing these richly parameterized models together as well as tools for exploring and interpreting the results of fitting the models to data. It extends the standard theory of linear models and illustrates the advantages and disadvantages of various theories. The book also examines surprising or undesirable results arising in the use of the models to analyze real data sets from collaborative research''