Informative Hypotheses: Theory and Practice for Behavioral and Social Scientists

Overview

When scientists formulate their theories, expectations, and hypotheses, they often use statements like: &'grave;I expect mean A to be bigger than means B and C"; &'grave;I expect that the relation between Y and both X1 and X2 is positive"; and &'grave;I expect the relation between Y and X1 to be stronger than the relation between Y and X2". Stated otherwise, they formulate their expectations in terms of inequality constraints among the parameters in which they are ...

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Overview

When scientists formulate their theories, expectations, and hypotheses, they often use statements like: &'grave;I expect mean A to be bigger than means B and C"; &'grave;I expect that the relation between Y and both X1 and X2 is positive"; and &'grave;I expect the relation between Y and X1 to be stronger than the relation between Y and X2". Stated otherwise, they formulate their expectations in terms of inequality constraints among the parameters in which they are interested, that is, they formulate Informative Hypotheses.

There is currently a sound theoretical foundation for the evaluation of informative hypotheses using Bayes factors, p-values and the generalized order restricted information criterion. Furthermore, software that is often free is available to enable researchers to evaluate the informative hypotheses using their own data. The road is open to challenge the dominance of the null hypothesis for contemporary research in behavioral, social, and other sciences.

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Meet the Author

Since 2003, Herbert Hoijtink has been Professor of Applied Bayesian Statistics at Utrecht University. He works in the Faculty of Social Sciences where he does research, teaches and provides statistical advice to behavioral (US spelling)and social scientists. In 2005, he received the prestigious VICI grant from the Netherlands Organisation for Scientific Research. This grant enabled him to establish a research group with the purpose to develop the statistical theory and corresponding software such that behavioral and social scientists will be able to evaluate informative hypotheses. The achievements of this group are presented in this book. Further information about the author can be found at http://tinyurl.com/hoijtink.

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Table of Contents

INTRODUCTION
An Introduction to Informative Hypotheses
Introduction
Analysis of Variance
Analysis of Covariance .
Multiple Regression
Epistemology and Overview of the Book
Appendix A: Effect Size Determination for Multiple Regression

The Multivariate Normal Linear Model
Introduction
The Multivariate Normal Linear Model
Multivariate One Sided Testing
Multivariate Treatment Evaluation
Multivariate Regression
Repeated Measures Analysis
Other Options
Appendix A: Example Data for Multivariate Regression

BAYESIAN EVALUATION OF INFORMATIVE HYPOTHESES
An Introduction to Bayesian Evaluation of Informative Hypotheses
Introduction .
Density of the Data, Prior and Posterior .
Bayesian Evaluation of Informative Hypotheses
Specifying the Parameters of Prior Distributions
Discussion .
Appendix A: Density of the Data, Prior and Posterior Distribution
Appendix B: Derivation of the Bayes Factor and Prior Sensitivity .
Appendix C: Using BIEMS
for a two group ANOVA

The J Group ANOVA Model
Introduction
Simple Constraints
One Informative Hypothesis
Constraints on Combinations of Means .
Ordered Means with Effect Sizes
About Equality Constraints
Discussion

Sample Size Determination: AN(C)OVA and Multiple Regression
Introduction
Sample Size Determination
ANOVA: Comparison of an Informative with the Null Hypothesis
ANOVA: Comparison of an Informative Hypothesis with its Complement
ANCOVA
Signed Regression Coecients: Informative versus Null Hypothesis
Signed Regression Coecients: Informative Hypothesis versus Complement
Signed Regression Coecients: Including Effect Sizes
Comparing Regression Coecients
Discussion .
Appendix A: Bayes Factors for Parameters on the Boundary of H1 and H1c

Appendix B: Command Files for GenMVLData

Sample Size Determination: The Multivariate Normal Linear Model
Introduction
Sample Size Determination: Error Probabilities
Multivariate One Sided Testing
Multivariate Treatment Evaluation
Multivariate Regression .
Repeated Measures Analysis
Discussion .
Appendix A: GenMVLData and BIEMS
: Multivariate One Sided Testing
Appendix B: GenMVLData and BIEMS
: Multivariate Treatment Evaluation
Appendix C: GenMVLData and BIEMS
: Multivariate Regression
Appendix D: GenMVLData and BIEMS
: Repeated Measures Analysis

OTHER MODELS, OTHER APPROACHES AND SOFTWARE
Beyond the Multivariate Normal Linear Model
Introduction
Contingency Tables
Multilevel Models
Latent Class Analysis
A General Frame Work
Appendices: Sampling Using Winbugs

Other Approaches
Introduction
Resume: Bayesian Evaluation of Informative Hypotheses
Null Hypothesis Signi cance Testing
The Order Restricted Information Criterion
Discussion
Appendix A: Data and Command File for
Confirmatory ANOVA

Software
Introduction
Software Packages
New Developments

STATISTICAL FOUNDATIONS
Foundations of Bayesian Evaluation of Informative Hypotheses
Introduction
The Bayes Factor
The Prior Distribution
The Posterior Distribution
Estimation of the Bayes Factor
Discussion
Appendix A: Density of the Data of Various Statistical Models
Appendix B: Unconstrained Prior Distributions Used in Book and Software
Appendix C: Probability Distributions Used in Appendices A and B
References
Index

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