ANOVA and ANCOVA: A GLM Approach

360 Pages·2011·16.33 MB·English

by  Andrew Rutherford

ANOVA and ANCOVA

ANOVA and ANCOVA
A GLM Approach
Second Edition
ANDREW RUTHERFORD
Keele University
School of Psychology
Staffordshire, United Kingdom
)WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee
to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone],
fax [phone], or on the web at www. copyright com. Requests to the Publisher for permission
should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken,
NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best
efforts in preparing this book, they make no representations or warranties with respect to the accuracy
or completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be
suitable for your situation. You should consult with a professional where appropriate. Neither the
publisher nor author shall be liable for any loss of profit or any other commercial damages, including
but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact
our Customer Care Department within the United States at [phone], outside the
United States at [phone] or fax [phone].
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print
may not be available in electronic formats. For more information about Wiley products, visit our web
site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Rutherford, Andrew, 1958-
ANOVA and ANCOVA : a GLM approach / Andrew Rutherford. - 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-38555-5 (cloth)
1. Analysis of variance. 2. Analysis of covariance. 3. Linear models (Statistics) I.
Title.
QA279.R879 2011
519.5'38-dc22
2010018486
Printed in the Singapore
10 9 8 7 6 5 4 3 21
Contents
Acknowledgments xiii
1 An Introduction to General Linear Models:
Regression, Analysis of Variance, and Analysis of Covariance 1
1.1 Regression, Analysis of Variance, and Analysis of Covariance 1
1.2 A Pocket History of Regression, ANOVA, and ANCOVA 2
1.3 An Outline of General Linear Models (GLMs) 3
1.3.1 Regression 4
1.3.2 Analysis of Variance 5
1.3.3 Analysis of Covariance 5
1.4 The "General" in GLM 6
1.5 The "Linear" in GLM 8
1.6 Least Squares Estimates 11
1.7 Fixed, Random, and Mixed Effects Analyses 12
1.8 The Benefits of a GLM Approach to ANOVA and ANCOVA 13
1.9 The GLM Presentation 14
1.10 Statistical Packages for Computers 15
2 Traditional and GLM Approaches to Independent Measures
Single Factor ANOVA Designs 17
2.1 Independent Measures Designs 17
2.2 Balanced Data Designs 19
2.3 Factors and Independent Variables 20
2.4 An Outline of Traditional ANOVA for Single Factor Designs 21
2.5 Variance 23
2.6 Traditional ANOVA Calculations for Single Factor Designs 25
2.7 Confidence Intervals 30
2.8 GLM Approaches to Single Factor ANOVA 31
2.8.1 Experimental Design GLMs 31
2.8.2 Estimating Effects by Comparing Full and Reduced
Experimental Design GLMs 37
2.8.3 Regression GLMs 41
2.8.4 Schemes for Coding Experimental Conditions 41
2.8.4.1 Dummy Coding 41
2.8.4.2 Why Only (p - 1) Variables Are Used to
Represent All Experimental Conditions? 44
2.8.4.3 Effect Coding 47
2.8.5 Coding Scheme Solutions to the Overparameterization
Problem 50
2.8.6 Cell Mean GLMs 50
2.8.7 Experimental Design Regression and Cell Mean GLMs 51
Comparing Experimental Condition Means, Multiple
Hypothesis Testing, Type 1 Error, and a Basic Data
Analysis Strategy 53
3.1 Introduction 53
3.2 Comparisons Between Experimental Condition Means 55
3.3 Linear Contrasts 56
3.4 Comparison Sum of Squares 57
3.5 Orthogonal Contrasts 58
3.6 Testing Multiple Hypotheses 62
3.6.1 Type 1 and Type 2 Errors 63
3.6.2 Type 1 Error Rate Inflation with Multiple Hypothesis
Testing 65
3.6.3 Type 1 Error Rate Control and Analysis Power 66
3.6.4 Different Conceptions of Type 1 Error Rate 68
3.6.4.1 Testwise Type 1 Error Rate 68
3.6.4.2 Family wise Type 1 Error Rate 69
3.6.4.3 Experimentwise Type 1 Error Rate 70
3.6.4.4 False Discovery Rate 70
3.6.5 Identifying the "Family" in Family wise Type 1 Error
Rate Control 71
3.6.6 Logical and Empirical Relations 72
3.6.6.1 Logical Relations 72
3.6.6.2 Empirical Relations 74
3.7 Planned and Unplanned Comparisons 76
CONTENTS vii
3.7.1 Direct Assessment of Planned Comparisons 77
3.7.2 Contradictory Results with ANOVA Omnibus F-tests
and Direct Planned Comparisons 78
3.8 A Basic Data Analysis Strategy 79
3.8.1 ANOVA First? 79
3.8.2 Strong and Weak Type 1 Error Control 80
3.8.3 Stepwise Tests 81
3.8.4 Test Power 82
3.9 The Three Basic Stages of Data Analysis 83
3.9.1 Stage 1 83
3.9.2 Stage 2 83
3.9.2.1 Rom's Test 83
3.9.2.2 Shaffer's R Test 84
3.9.2.3 Applying Shaffer's R Test After a Significant
F-test 86
3.9.3 Stage 3 89
3.10 The Role of the Omnibus F-Test 91
Measures of Effect Size and Strength of Association,
Power, and Sample Size 93
4.1 Introduction 93
4.2 Effect Size as a Standardized Mean Difference 94
4.3 Effect Size as Strength of Association (SOA) 96
4.3.1 SOA for Specific Comparisons 98
4.4 Small, Medium, and Large Effect Sizes 99
4.5 Effect Size in Related Measures Designs 99
4.6 Overview of Standardized Mean Difference
and SOA Measures of Effect Size 100
4.7 Power 101
4.7.1 Influences on Power 101
4.7.2 Uses of Power Analysis 103
4.7.3 Determining the Sample Size Needed
to Detect the Omnibus Effect 104
4.7.4 Determining the Sample Size Needed to Detect
Specific Effects 107
4.7.5 Determining the Power Level of a Planned
or Completed Study 109
4.7.6 The Fallacy of Observed Power 110
viii
CONTENTS
GLM Approaches to Independent Measures Factorial Designs 111
5.1 Factorial Designs 111
5.2 Factor Main Effects and Factor Interactions 112
5.2.1 Estimating Effects by Comparing Full and
Reduced Experimental Design GLMs 117
5.3 Regression GLMs for Factorial ANOVA 121
5.4 Estimating Effects with Incremental Analysis 123
5.4.1 Incremental Regression Analysis 124
5.4.1.1 Step 1 124
5.4.1.2 Step 2 124
5.4.1.3 Step 3 125
5.5 Effect Size Estimation 126
5.5.1 SOA for Omnibus Main and Interaction Effects 126
5.5.1.1 Complete ω2 for Main and Interaction Effects 126
5.5.1.2 Partial ω for Main and Interaction Effects 127
5.5.2 Partial ω for Specific Comparisons 127
5.6 Further Analyses 128
5.6.1 Main Effects: Encoding Instructions and Study Time 128
5.6.2 Interaction Effect: Encoding Instructions x Study Time 131
5.6.2.1 Simple Effects: Comparing the Three Levels
of Factor B at al, and at a2 132
5.6.2.2 Simple Effects: Comparing the Two Levels
of Factor A at bl, at b2, and at b3 135
5.7 Power 136
5.7.1 Determining the Sample Size Needed to Detect
Omnibus Main Effects and Interactions 136
5.7.2 Determining the Sample Size Needed
to Detect Specific Effects 138
GLM Approaches to Related Measures Designs 139
6.1 Introduction 139
6.1.1 Randomized Block Designs 140
6.1.2 Matched Sample Designs 141
6.1.3 Repeated Measures Designs 141
6.2 Order Effect Controls in Repeated Measures Designs 144
6.2.1 Randomization 144
6.2.2 Counterbalancing 144
6.2.2.1 Crossover Designs 144
6.2.2.2 Latin Square Designs 145