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### Summary

### Summary

Examining the major approaches to hypothesis testing and model selection, this book blends statistical theory with recommendations for practice, illustrated with real-world social science examples. It systematically compares classical (frequentist) and Bayesian approaches, showing how they are applied, exploring ways to reconcile the differences between them, and evaluating key controversies and criticisms. The book also addresses the role of hypothesis testing in the evaluation of theories, the relationship between hypothesis tests and confidence intervals, and the role of prior knowledge in Bayesian estimation and Bayesian hypothesis testing. Two easily calculated alternatives to standard hypothesis tests are discussed in depth: the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The companion website ([ital]www.guilford.com/weakliem-materials[/ital]) supplies data and syntax files for the book's examples.

### Author Notes

David L. Weakliem is Professor of Sociology at the University of Connecticut.

### Table of Contents

1 Hypothesis Testing and Model Selection | p. 1 |

1.1 Introduction | p. 1 |

1.2 Standard Procedure of Hypothesis Testing | p. 4 |

1.2.1 One Parameter | p. 4 |

1.2.2 Multiple Parameters | p. 5 |

1.2.3 Lihelihood Ratio and Wald Tests | p. 6 |

1.2.4 Types of Hypotheses | p. 7 |

1.2.5 Non-Nested Hypotheses | p. 8 |

1.3 Model Selection | p. 9 |

1.4 Purpose and Plan of the Book | p. 10 |

2 Hypothesis Testing: Criticisms and Alternatives | p. 12 |

2.1 Hypothesis Testing and Its Discontents | p. 12 |

2.2 Uses of Hypothesis Tests | p. 13 |

2.2.1 Conclusions about Parameters of interest | p. 13 |

2.2.2 Choice of Control Variables | p. 14 |

2.2.3 Primary Structure | p. 15 |

2.2.4 Secondary Structure | p. 15 |

2.2.5 Goodness of Fit | p. 16 |

2.3 Criticisms of Conventional Hypothesis Testing | p. 17 |

2.3.1 Sampling | p. 17 |

2.3.2 Credibility of Point Null Hypotheses | p. 18 |

2.3.3 Ranking of Models | p. 19 |

2.3.4 'Flexible versus Inflexible Interpretation of Significance | p. 20 |

2.3.5 Arbitrary Nature of Significance Levels | p. 21 |

2.3.6 Effects of Sample Size | p. 22 |

2.3.7 Lack of Symmetry | p. 24 |

2.3.8 Likelihood Principle | p. 25 |

2.4 Implications of the Criticisms | p. 26 |

2.5 Alternatives to Conventional Tests | p. 27 |

2.6 Examples | p. 29 |

2.6.1 Economic Growth | p. 30 |

2.6.2 Development and Fertility | p. 32 |

2.6.1 Comparative Social Mobility | p. 34 |

2.6.4 Race and Voting Choices | p. 37 |

2.7 Summary and Conclusions | p. 41 |

Recommended Reading | p. 42 |

3 The Classical Approach | p. 43 |

3.1 Random Sampling and Classical Tests | p. 43 |

3.2 Two Approaches to Hypothesis Tests | p. 46 |

3.2.1 Significance Tests | p. 46 |

3.2.2 The Neyman-Pearson Approach | p. 50 |

3.3 Confidence Intervals | p. 51 |

3.4 Choosing a Significance Level | p. 53 |

3.4.1 One-Tailed Tests | p. 57 |

3.4.2 Tests with Multiple Degrees of Freedom | p. 60 |

3.5 Comparison to Conventional Practice | p. 61 |

3.5.1 Hypothesis Tests in Small Samples | p. 62 |

3.6 Implications of Choosing an ¿ Level | p. 63 |

3.7 Other Kinds of Errors | p. 65 |

3.8 Example of Choosing an a Level | p. 67 |

3.9 Evaluation of Criticisms | p. 68 |

3.9.1 Sampling | p. 68 |

3.9.2 Credibility of Point Null Hypotheses | p. 69 |

3.9.3 Ranking of Models | p. 69 |

3.9.4 Flexible versus Rigid Interpretation of Significance Levels | p. 70 |

3.9.5 Arbitrary Nature of Significance Levels | p. 70 |

3.9.6 Effects of Sample Size | p. 70 |

3.9.7 Each of Symmetry | p. 71 |

3.9.8 Likelihood Principle | p. 71 |

3.10 Summary and Conclusions | p. 72 |

Recommended Reading | p. 74 |

4 Bayesian Hypothesis Tests | p. 75 |

4.1 Bayes's Theorem | p. 75 |

4.2 Bayesian Estimation | p. 77 |

4.2.1 Accuracy of Bayesian and Classical Estimates | p. 80 |

4.3 Bayes Factors | p. 80 |

4.3.1 Bayes Factors for General Alternative Hypotheses | p. 82 |

4.3.2 Prior Distributions and Bayes Factors | p. 83 |

4.3.3 Influence of Sample Size on Bayes Factors | p. 85 |

4.4 Bayesian Confidence Intervals and Bayes Factors | p. 88 |

4.5 Approaches to Bayesian Hypothesis Testing | p. 91 |

4.6 The Unit Information Prior | p. 92 |

4.6.1 An Example of the Unit Information Prior | p. 92 |

4.6.2 Evaluation | p. 94 |

4.7 Limits on Bayes Factors | p. 97 |

4.7.1 Most Favorable Bayes Factors and Evidence for the Null Hypothesis | p. 101 |

4.8 Bayes Factors for Multiple Parameters | p. 102 |

4.9 Summary and Conclusions | p. 105 |

Recommended Reading | p. 106 |

5 The Akaike Information Criterion | p. 108 |

5.1 Information | p. 108 |

5.2 Prediction and Model Selection | p. 109 |

5.3 The AIC | p. 110 |

5.3.1 The AIC in Small Samples | p. 111 |

5.3.2 The AIC and Related Criteria for Regression Models | p. 112 |

5.4 Consistency and Efficiency | p. 113 |

5.5 Cross-Validattan and the AIC | p. 114 |

5.6 A Classical Perspective on the AIC | p. 115 |

5.7 A Bayesian Perspective on the AIC | p. 116 |

5.8 A General Class of Model Selection Criteria | p. 119 |

5.9 Summary and Conclusions | p. 120 |

Recommended Reading | p. 122 |

6 Three-Way Decisions | p. 123 |

6.1 Substantive and Statistical Hypotheses | p. 123 |

6.2 Bayes Factors for Directional Hypotheses | p. 125 |

6.2.1 Sample Size | p. 128 |

6.3 Bayes Factors for Three-Way Decisions | p. 129 |

6.3.1 Posterior Probabilities in Three-Way Choices | p. 131 |

6.3.2 Sample Size and Bayes Factors for Three-Way Choices | p. 132 |

6.3.3 Prior Distributions for Directional Hypotheses | p. 133 |

6.4 Summary and Conclusions | p. 136 |

Recommended Reading | p. 137 |

7 Model Selection | p. 139 |

7.1 Introduction | p. 139 |

7.2 Bayesian Model Selection | p. 140 |

7.2.1 Model Averaging | p. 143 |

7.2.2 Bayesian Model Selection and Bayesian Estimation | p. 144 |

7.3 The Value of Model Selection | p. 145 |

7.4 The Risks of Model Selection | p. 146 |

7.4.1 Simple to Complex or Complex to Simple? | p. 148 |

7.5 Examples of Model Selection | p. 149 |

7.5.1 Social Mobility in 16 Nations | p. 149 |

7.5.2 Social Mobility in Britain and the United States | p. 152 |

7.5.3 Interactions between State and Race in Voting Choices | p. 155 |

7.6 Summary and Conclusions | p. 156 |

Recommended Reading | p. 157 |

8 Hypothesis Tests | p. 158 |

8.1 Hypothesis Tests and the Strength of Evidence | p. 158 |

8.2 When Should Hypotheses Be Tested? | p. 162 |

8.3 The Role of Hypothesis Tests | p. 165 |

8.3.1 Specification and Goodness-of-Fit Tests | p. 167 |

8.3.2 Testing and Replication | p. 167 |

8.4 Overfitting | p. 168 |

8.5 Hypothesis Tests and the Development of Theory | p. 170 |

8.5.1 Making a Theory Elaborate | p. 173 |

8.5.2 Evaluation of Multiple Implications | p. 174 |

8.6 Summary and Conclusions | p. 176 |

Recommended Reading | p. 178 |

References | p. 179 |

Author Index | p. 191 |

Subject Index | p. 195 |

About the Author | p. 202 |