# Richly parameterized linear models: additive, time series, and spatial models using random effects

##### By: Hodges, James S.

Series: Chapman & Hall/CRC Texts in Statistical Science. Publisher: Boca Raton CRC Press 2014Description: xxxviii, 431 p.ISBN: 9781439866832.Subject(s): Regression analysis - Textbooks | Linear models (Statistics) - TextbooksDDC classification: 519.536 Summary: 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. (https://www.crcpress.com/Richly-Parameterized-Linear-Models-Additive-Time-Series-and-Spatial-Models/Hodges/p/book/9781439866832)Item type | Current location | Item location | Collection | Call number | Status | Date due | Barcode |
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Books | Vikram Sarabhai Library | Slot 1427 (0 Floor, East Wing) | Non-fiction | 519.536 H6R4 (Browse shelf) | Available | 192274 |

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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.

(https://www.crcpress.com/Richly-Parameterized-Linear-Models-Additive-Time-Series-and-Spatial-Models/Hodges/p/book/9781439866832)

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