Richly parameterized linear models: additive, time series, and spatial models using random effects
Hodges, James S.
- Boca Raton CRC Press 2014
- xxxviii, 431 p.
- Chapman & Hall/CRC Texts in Statistical Science .
Table of Contents:
1. 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
2. Two More Tools: Alternative Formulation, Measures of Complexity Alternative formulation: The "constraint-case" formulation Measuring the complexity of a mixed linear model fit
3. 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
4. Additive Models and Models with Interactions Additive models as mixed linear models Models with interactions
5. Spatial Models as Mixed Linear Models Geostatistical models Models for areal data Two-dimensional penalized splines
6. Time-Series Models as Mixed Linear Models Example: Linear growth model Dynamic linear models in some generality Example of a multi-component DLM
7. Two Other Syntaxes for Richly Parameterized Models Schematic comparison of the syntaxes Gaussian Markov random fields Likelihood inference for models with unobservables
8. From Linear Models to Richly Parameterized Models: Mean Structure Adapting Diagnostics from Linear Models Preliminaries Added variable plots Transforming variables Case influence Residuals
9. Puzzles from Analyzing Real Datasets Four puzzles Overview of the next three chapters
10. A Random Effect Competing with a Fixed Effect Slovenia data: Spatial confounding Kids and crowns: Informative cluster size
11. Differential Shrinkage The simplified model and an overview of the results Details of derivations Conclusion: What might cause differential shrinkage?
12. Competition between Random Effects Collinearity between random effects in three simpler models Testing hypotheses on the optical-imaging data and DLM models Discussion
13. Random Effects Old and New Old-style random effects New-style random effects Practical consequences Conclusion
14. 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
15. Re-Expressing the Restricted Likelihood: Two-Variance Models The re-expression Examples A tentative collection of tools
16. Exploring the Restricted Likelihood for Two-Variance Models Which vj tell us about which variance? Two mysteries explained
17. 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
18. Zero Variance Estimates Some observations about zero variance estimates Some thoughts about tools
19. Multiple Maxima in the Restricted Likelihood and Posterior Restricted likelihoods with multiple local maxima Posteriors with multiple modes
A First Step toward a Unified Theory of Richly Parameterized Linear Models
Using mixed linear models to analyze data often leads to results that are mysterious, inconvenient, or wrong. Further compounding the problem, statisticians lack a cohesive resource to acquire a systematic, theory-based understanding of models with random effects.
Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects takes a first step in developing a full theory of richly parameterized models, which would allow statisticians to better understand their analysis results. The author examines what is known and unknown about mixed linear models and identifies research opportunities.
The first two parts of the book cover an existing syntax for unifying models with random effects. The text explains how richly parameterized models can be expressed as mixed linear models and analyzed using conventional and Bayesian methods.
In the last two parts, the author discusses oddities that can arise when analyzing data using these models. He presents ways to detect problems and, when possible, shows how to mitigate or avoid them. The book adapts ideas from linear model theory and then goes beyond that theory by examining the information in the data about the mixed linear model’s covariance matrices.
Each chapter ends with two sets of exercises. Conventional problems encourage readers to practice with the algebraic methods and open questions motivate readers to research further. Supporting materials, including datasets for most of the examples analyzed, are available on the author’s website.