Model Selection and Inference: A Practical Information Theoretic Approach / Edition 1

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Overview

This book is unique in that it covers the philosophy of model-based data analysis and a strategy for the analysis of empirical data. Model selection, under the information theoretic approach presented here, attempts to identify the (likely) best model, orders the models from best to worst, and measures the plausibility ("calibration") that each model is really the best as an inference. Model selection methods are extended to allow inference from more than a single "best" model. Several methods are given that allow the uncertainty as to which model is "best" to be incorporated into estimates of precision. An array of examples are given to illustrate various technical issues. This is an applied book written primarily for biologists and statisticians using models for making inferences from empirical data. Research biologists working either in the field or in the laboratory will find simple methods that are likely to be useful in their investigations. Applied statisticians will find the information theoretic methods presented here quite useful and a superior alternative, especially for observational studies. People interested in the empirical sciences will find this material useful as it offers an alternative to hypothesis testing and Bayesian approaches.
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Product Details

  • ISBN-13: 9780387985046
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 1/28/1998
  • Edition description: Older Edition
  • Edition number: 1
  • Pages: 372
  • Product dimensions: 6.30 (w) x 9.46 (h) x 0.88 (d)

Table of Contents

Preface
1. Introduction
Objectives of Book
Background Material
Inference, Given a Model
The Critical Issue: "What Model To Use?"
Science Inputs - Formulation of Set of Candidate Models
Models Versus Full Reality
A "Best Approximating Model"
Overview of Models in the Biological Sciences
Types of Models Commonly Used in the Biological Sciences
Likelihood and Least Squares Theory
Data Dredging
Some Trends
Inference and the Principle of Parsimony
Over-fitting to Achieve a Good Model Fit
The Principle of Parsimony
Model Selection Methods
Model Selection Uncertainty
Summary
2. Information Theory and Log-Likelihood Models - A Basis for Model Selection and Inference
The Distance of Discrepancy Between Two Models
f, Truth, Full Reality and "True Models"
g, Approximating Models
Kullback-Leibler Distance (or Information)
Truth, f, Drops Out as a Constant
Akaike's Information Criterion
Akaike's Predictive Expected Log-Likelihood
Important Refinements to AIC
A Second Order AIC
Modification to AIC for Overdispersed Count Data
A Useful Analogy
Some History
The G-statistic and K-L Information
Further Insights
Entropy
A Summary
Further Comments
A Heuristic Interpretation
Interpreting Differences Among AIC Values
Non-nested Models
Model Selection Uncertainty
AIC When Different Data Sets are to be Compared
Order Not Important in Computing AIC Values
Hypothesis Testing Is Still Important
Comparisons with Other Criteria
Information Criteria That Are Estimates of K-L Information
Criteria That Are Consistent for K
Contrasts
Return to Flather's Models
Summary
3. Practical Use of the Information Theoretic Approach
Computation and Interpretation of AIC Values
Example 1 - Cement Hardening Data
Set of Candidate Models
Some Results and Comparisons
A Summary
Example 2 - Time Distribution of an Insecticide Added to a Simulated Ecosystem
Set of Candidate Models
Some Results
Example 3 - Nestling Starlings
Experimental Scenario
Monte Carlo Data
Set of Candidate Models
Data Analysis Results
Further Insights Into the First 14 Nested Models
Hypothesis Testing and Information Theoretic Approaches Have Different Selection Frequencies
Further Insights Following Final Model Selection
Why Not Always Use the Global Model for Inference?
Example 4 - Sage Grouse Survival
Introduction
Set of Candidate Models
Model Selection
Hypothesis Tests for Year-Dependent Survival Probabilities
Hypothesis Testing Versus AIC in Model Selection
A Class of Intermediate Models
Example 5 - Resource Utilization of Anolis Lizards
Set of Candidate Models
Comments on Analysis Method
Some Tentative Results
Summary
4. Model Selection Uncertainty with Examples
Introduction
Methods for Assessing Model Selection Uncertainty
AIC Differences and a Confidence Set on the K-L Best Model
Likelihood for a Model and Akaike Weights
More Options for a Confidence Set on the K-L Best Model
&Dgr;_i, Model Selection Probabilities, and the Bootstrap
Concepts of Parameter Estimation and Model Selection Uncertainty
Including Model Selection Uncertainty in Estimator Sampling Variance
Unconditional Confidence Intervals
Uncertainty of Variable Selection
Model Redundancy
Recommendations
Examples
Cement Data
Anolis Lizards in Jamaica
Simulated Starling Experiment
Sage Grouse in North Park
Sakamoto et al.'s (1986) Simulated Data
Pine Wood Data
Summary
5. Monte Carlo Insights and Extended Examples
Introduction
Survival Models
A Chain Binomial Survival Model
An Extended Survival Model
Further Survival Models
General Linear Model
[mention step up and step down and stepwise testing, all subsets, etc., etc.]
[K-L, vs. MSE vs. AIC and AICc]
Estimation of Density from Line Transect Sampling
Density Estimation Background
Line Transect Sampling of Kangaroos at Wallaby Creek
Analysis of Wallaby Creek Data
Bootstrap Analysis
Summary
6. Statistical Theory
Useful Preliminaries
A General Derivation of AIC
General K-L Based Model Selection: TIC
Analytical Computations of TIC
Bootstrap Estimation of TIC
AICc, a Second Order Improvement
Derivation of AICc
Lack of Uniqueness of AICc
Derivation of AIC for the Exponential Family of Distributions
Evaluation of tr(J(o)[I((o)]-1) and Its Estimator
Comparison of AIC vs. TIC in a Very Simple Setting
Evaluation Under Logistic Regression
Evaluation Under Multinomially Distributed Count Data
Evaluation Under Poisson Distributed Data
Evaluation for Fixed-effects Normality-Based Linear Models
Additional Results and Considerations
Selection Simulation for Nested Models
Simulation of the Distribution of ( p
Does AIC Over-fit?
Can Selection be Improved Based on all the ( i?
Linear Regression, AIC and Mean Square Error
AIC and Random Coeeficient Models
AICc and Models for Multivariate Data
There is No True TICc
Kullback-Leibler Information Relationship to the Fisher Information Matrix
Entropy and Jaynes MaxEnt Principle
Akaike Weights, (i, versus Selection Probabilities, (i
Model Goodness of Fit After Selection
Kullback-Leibler Information is Always >= 0
Summary
7. Summary
The Scientific Question and the Collection of Data
Actual Thinking and A Priori Modeling
The Basis of Objective Model Selection
The Principle of Parsimony
Information Criteria as Estimates of Relative Kullback-Leibler Information
Ranking and Calibrating Alternative Models
Model Selection Uncertainty
More on Inferences
Final Thoughts
8. References
9. Index
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