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Charts for Experiments with Linear Models, Second Edition

Robert E. Odeh, Martin Fox

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780367402921
• eBook ISBN: 9781000147926
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

A guide to testing statistical hypotheses for readers familiar with the Neyman-Pearson theory of hypothesis testing including the notion of power, the general linear hypothesis (multiple regression) problem, and the special case of analysis of variance. The second edition (date of first not mentione

PART ONE

Chapter 1

INTRODUCTION

We assume that users of these charts are familiar with the Neyman-Pearson theory of hypothesis testing including the notion of power. We also assume familiarity with the general linear hypothesis (multiple regression) problem as well as the special case of analysis of variance.

The remainder of this chapter consists of discussions of the power of F-tests and the ranges of parameter values covered by the charts.

1.1 Power of the F-test

When the F-test is used in general linear hypothesis (e.g, analysis of variance, regression), the power of the test or, equivalently, the operating characteristic (1-power), is a function of a quantity known as the noncentrality parameter. As a first step in defining the noncentrality parameter, we give three equivalent computations of a quantity denoted by S*. The numerator and denominator degrees of freedom are denoted by f₁ and f₂, respectively.

1. Replace each observation in the numerator sum of squares in the F-statistic by its expected value.
2. The numerator E(MS) is σ² + S*/f₁
3. Express the problem in canonical form. The observations are Y_i (i = 1,..., N), which are independent random variables, normally distributed with means μ_i (i = 1,..., N) and common variance σ² where μ₁ = ... = μ_f2 = 0.

The hypothesis is that μ_{f2 + 1} = … = μ_{f2 + f1} = 0. Then,

$$S^{*}=\underset{i=f_2+1}{\overset{f_2+f_1}{\mathrm{\Sigma }}}{\mu }_i^2(1.1)$$

The noncentrality parameter is

$$\phi =\sqrt{\frac{S*}{{\sigma }^2(f_1+1)}}(1.2)$$

Fix the level...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Preface to the Second Edition
• Preface to the First Edition
• Table of Contents
• Original Half Title
• Part One
• Part Two
• Part Three
• Index