Summary

The abstract concept of "information" can be quantified and this has lead to many important advances in the analysis of data in the empirical sciences. This text focuses on a science philosophy based on "multiple working hypotheses" and statistical models to represent them. The fundamental science question relates to the empirical evidence for hypotheses in this set'a formal strength of evidence. Kullback-Leibler information is the information lost when a model is used to approximate full reality. Hirotugu Akaike found a link between K-L information (a cornerstone of information theory) and the maximized log-likelihood (a cornerstone of mathematical statistics). This combination has become the basis for a new paradigm in model based inference. The text advocates formal inference from all the hypotheses/models in the a priori set'multimodel inference. This compelling approach allows a simple ranking of the science hypothesis and their models. Simple methods are introduced for computing the likelihood of model i, given the data; the probability of model i, given the data; and evidence ratios. These quantities represent a formal strength of evidence and are easy to compute and understand, given the estimated model parameters and associated quantities (e.g., residual sum of squares, maximized log-likelihood, and covariance matrices). Additional forms of multimodel inference include model averaging, unconditional variances, and ways to rank the relative importance of predictor variables. This textbook is written for people new to the information-theoretic approaches to statistical inference, whether graduate students, post-docs, or professional in various universities, agencies or institutes. Readers are expected to have a background in general statistical principles, regression analysis, and some exposure to likelihood methods. This is not an elementary text as it assumes reasonable competence in modeling and parameter estimation.

Author Biography

David R. Anderson retired recently from serving as a senior scientist with the U.S. Geological Survey and professor in the Department of Fish, Wildlife, and Conservation Biology at Colorado State University. He has an emeritus professorship at CSU and is president of the Applied Information Company in Fort Collins.

Preface	p. vii
About the Author	p. xvii
Glossary	p. xix
Introduction: Science Hypotheses and Science Philosophy	p. 1
Some Science Background	p. 1
Multiple Working Hypotheses	p. 3
Bovine TB Transmission in Ferrets	p. 4
Approaches to Scientific Investigations	p. 6
Experimental Studies	p. 7
Descriptive Studies	p. 8
Confirmatory Studies	p. 8
Science Hypothesis Set Evolves	p. 10
Null Hypothesis Testing	p. 11
Evidence and Inferences	p. 12
Hardening of Portland Cement	p. 13
What Does Science Try to Provide?	p. 14
Remarks	p. 15
Exercises	p. 17
Data and Models	p. 19
Data	p. 19
Hardening of Portland Cement Data	p. 22
Bovine TB Transmission in Ferrets	p. 23
What Constitutes a "Data Set"?	p. 24
Models	p. 25
True Models (An Oxymoron)	p. 27
The Concept of Model Parameters	p. 28
Parameter Estimation	p. 29
Principle of Parsimony	p. 30
Tapering Effect Sizes	p. 33
Case Studies	p. 33
Models of Hardening of Portland Cement Data	p. 33
Models of Bovine TB Transmission in Ferrets	p. 35
Additional Examples of Modeling	p. 36
Modeling Beak Lengths	p. 37
Modeling Dose Response in Flour Beetles	p. 41
Modeling Enzyme Kinetics	p. 44
Data Dredging	p. 45
The Effect of a Flood on European Dippers: Modeling Contrasts	p. 46
Traditional Null Hypothesis Testing	p. 46
Information-Theoretic Approach	p. 47
Remarks	p. 48
Exercises	p. 49
Information Theory and Entropy	p. 51
Kullback-Leibler Information	p. 52
Linking Information Theory to Statistical Theory	p. 54
Akaike's Information Criterion	p. 55
The Bias Correction Term	p. 57
Why Multiply by -2?	p. 57
Parsimony is Achieved as a by-Product	p. 58
Simple vs. Complex Models	p. 59
AIC Scale	p. 60
A Second-Order Bias Correction: AICc	p. 60
Regression Analysis	p. 61
Additional Important Points	p. 62
Differences Among AICc Values	p. 62
Nested vs. Nonnested Models	p. 63
Data and Response Variable Must Remain Fixed	p. 63
AICc is not a "Test"	p. 64
Data Dredging Using AICc	p. 64
Keep all the Model Terms	p. 64
Missing Data	p. 65
The "Pretending Variable"	p. 65
Cement Hardening Data	p. 66
Interpreting AICc Values	p. 66
What if all the Models are Bad?	p. 67
Prediction from the Best Model	p. 68
Ranking the Models of Bovine Tuberculosis in Ferrets	p. 69
Other Important Issues	p. 70
Takeuchi's Information Criterion	p. 70
Problems When Evaluating Too Many Candidate Models	p. 71
The Parameter Count K and Parameters that Cannot be Uniquely Estimated	p. 71
Cross Validation and AICc	p. 72
Science Advances as the Hypothesis Set Evolves	p. 72
Summary	p. 73
Remarks	p. 74
Exercises	p. 80
Quantifying the Evidence About Science Hypotheses	p. 83
[Delta subscript i] Values and Ranking	p. 84
Model Likelihoods	p. 86
Model Probabilities	p. 87
Evidence Ratios	p. 89
Hardening of Portland Cement	p. 91
Bovine Tuberculosis in Ferrets	p. 93
Return to Flather's Models and R[superscript 2]	p. 94
The Effect of a Flood on European Dippers	p. 95
More about Evidence and Inference	p. 98
Summary	p. 100
Remarks	p. 101
Exercises	p. 103
Multimodel Inference	p. 105
Model Averaging	p. 106
Model Averaging for Prediction	p. 107
Model Averaging Parameter Estimates Across Models	p. 108
Unconditional Variances	p. 110
Examples Using the Cement Hardening Data	p. 112
Averaging Detection Probability Parameters in Occupancy Models	p. 115
Relative Importance of Predictor Variables	p. 118
Rationale for Ranking the Relative Importance of Predictor Variables	p. 119
An Example Using the Cement Hardening Data	p. 119
Confidence Sets on Models	p. 121
Summary	p. 122
Remarks	p. 122
Exercises	p. 124
Advanced Topics	p. 125
Overdispersed Count Data	p. 126
Lack of Independence	p. 126
Parameter Heterogeneity	p. 126
Estimation of a Variance Inflation Factor	p. 127
Coping with Overdispersion in Count Data	p. 127
Overdispersion in Data on Elephant Seals	p. 128
Model Selection Bias	p. 129
Understanding the Issue	p. 129
A Solution to the Problem of Model Selection Bias	p. 130
Multivariate AICc	p. 133
Model Redundancy	p. 133
Model Selection in Random Effects Models	p. 134
Use in Conflict Resolution	p. 135
Analogy with the Flip of a Coin	p. 136
Conflict Resolution Protocol	p. 137
A Hypothetical Example: Hen Clam Experiments	p. 138
Remarks	p. 140
Summary	p. 141
The Science Question	p. 142
Collection of Relevant Data	p. 143
Mathematical Models	p. 143
Data Analysis	p. 144
Information and Entropy	p. 144
Quantitative Measures of Evidence	p. 144
Inferences	p. 145
Post Hoc Issues	p. 146
Final Comment	p. 146
Appendices	p. 147
Likelihood Theory	p. 147
Expected Values	p. 155
Null Hypothesis Testing	p. 157
Bayesian Approaches	p. 158
The Bayesian Information Criterion	p. 160
Common Misuses and Misinterpretations	p. 162
References	p. 167
Index	p. 181
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