Interpreting statistical data as evidence, Statistical Evidence: A Likelihood Paradigm focuses on the law of likelihood, fundamental to solving many of the problems associated with interpreting data in this way. Statistics has long neglected this principle, resulting in a seriously defective methodology. This book redresses the balance, explaining why science has clung to a defective methodology despite its well-known defects. After examining the strengths and weaknesses of the work of Neyman and Pearson and the Fisher paradigm, the author proposes an alternative paradigm which provides, in the law of likelihood, the explicit concept of evidence missing from the other paradigms. At the same time, this new paradigm retains the elements of objective measurement and control of the frequency of misleading results, features which made the old paradigms so important to science. The likelihood paradigm leads to statistical methods that have a compelling rationale and an elegant simplicity, no longer forcing the reader to choose between frequentist and Bayesian statistics.

The First Principle

Introduction

The Law of Likelihood

Three Questions

Towards Verification

Relativity of Evidence

Strength of Evidence

Counterexamples

Testing Simple Hypotheses

Composite Hypotheses

Another Counterexample

Irrelevance of the Sample Space

The Likelihood Principle

Evidence and Uncertainty

Summary

Exercises

Neyman-Pearson Theory

Introduction

Neyman-Pearson Statistical Theory

Evidential Interpretation of Results of Neyman-Pearson Decision Procedures

Neyman-Pearson Hypothesis Testing in Planning Experiments: Choosing the Sample Size

Summary

Exercises

Fisherian Theory

Introduction

A Method for Measuring Statistical Evidence: The Test of Significance

The Rationale for Significance Tests

Troubles with p-Values

Rejection Trials

A Sample of Interpretations

The Illogic of Rejection Trials

Confidence Sets from Rejection Trials

Alternative Hypothesis in Science

Summary

Paradigms for Statistics

Introduction

Three Paradigms

An Alternative Paradigm

Probabilities of Weak and Misleading Evidence: Normal Distribution Mean

Understanding the Likelihood Paradigm

Evidence about a Probability: Planning a Clinical Trial and Interpreting the Results

Summary

Exercises

Resolving the Old Paradoxes

Introduction

Why is Power of Only 0.80 OK?

Peeking at Data Repeated Tests

Testing More than One Hypothesis

What's Wrong with One-SIded Tests?

Must the Significance Level be Predetermined?

And is the Strength of Evidence Limited by the Researcher's Expectations?

Summary

Looking at Likelihoods

Introduction

Evidence about Hazard Rates in Two Factories

Evidence about an Odds Ration

A Standardized Mortality Rate

Evidence about a Finite Population Total

Determinants of Plans to Attend College

Evidence about the Probabilities in a 2x2x2x2 Table

Evidence from a Community Intervention Study of Hypertension

Effects of Sugars on Growth of Pea Sections: Analysis of Variance

Summary

Exercises

Nuisance Parameters

Introduction

Orthogonal Parameters

Marginal Likelihoods

Conditional Likelihoods

Estimated Likelihoods

Profile Likelihoods

Synthetic Conditional Likelihoods

Summary

Exercises

Bayesian Statistical Inference

Introduction

Bayesian Statistical Models

Subjectivity in Bayesian Models

The Trouble with Bayesian Statistics

Are Likelihood Methods Bayesian?

Objective Bayesian Inference

Bayesian Integrated Likelihoods

Summary

Appendix: The Paradox of the Ravens

"...provides the explicit concept of evidence missing from the other approaches."

-Aslib Book Guide

"…the book is well written and readable."

--Hoben Thomas, Journal of Mathematical Psychology

"This (hardback) book provides a very readable discussion of a possible alternative to both the Neyman-Pearson and the Fisherian approaches to the problem of interpreting data as evidence…present this area of work in a accessible manner with a clear readable style. The main ideas are made easy to understand and well illustrated with some interesting examples, including in an appendix the paradox of the ravens. Diagrams and tables are well used in this respect and the number of formulae is kept low, which aids readability…provides a well-presented discussion of an interesting new way of looking at data which would be accessible to most with some understanding of statistics. For this reason I would recommend it to a library."

--Thomas Chadwick, University of Newcastle, Biometrics