Hodges Richly Parameterized Linear Models

Additive, Time Series, and Spatial Models Using Random Effects
Erscheinungsjahr 2013
ISBN: 978-1-4398-6684-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 469 Seiten

Reihe: Chapman & Hall/CRC Texts in Statistical Science

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.

Zielgruppe

Researchers and graduate students in statistics, biostatistics, medicine, epidemiology, engineering, and social sciences.

Autoren/Hrsg.

Hodges, James S.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

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