Table Of ContentStatistical Power Analysis
A Simple and General Model
for Traditional and Modern Hypothesis Tests
Second Edition
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Statistical Power Analysis
A Simple and General Model
for Traditional and Modern Hypothesis Tests
Second Edition
Kevin R. Murphy
Pennsylvania State University
Brett Myors
Griffith University
LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS
Mahwah, New Jersey London
Copyright © 2004 by Lawrence Erlbaum Associates, Inc.
All rights reserved. No part of this book may be repro-
duced in any form, by photostat, microform, retrieval sys-
tem, or any other means, without prior written permission
of the publisher.
Lawrence Erlbaum Associates, Inc., Publishers
10 Industrial Avenue
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Cover design by Kathryn Houghtaling Lacey
Library of Congress Cataloging-in-Publication Data
Statistical power analysis : a simple and general model for
traditional and modern hypothesis tests, second edition,
by Kevin R. Murphy and Brett Myors.
Includes bibliographical references and index.
ISBN 0-8058-4525-9 (cloth : alk. paper)
ISBN 0-8058-4526-7 (pbk. : alk. paper)
Copyright information for this volume can be obtained by
contacting the Library of Congress
2004
—dc21 2004000000
CIP
Books published by Lawrence Erlbaum Associates are
printed on acid-free paper, and their bindings are chosen for
strength and durability.
Printed in the United States of America
10 9 8 7 6 5 4 3 21
Disclaimer:
This eBook does not include the ancillary media that was
packaged with the original printed version of the book.
Contents
Preface vii
1 The Power of Statistical Jests 1
The Structure of Statistical Tests 2
The Mechanics of Power Analysis 7
Statistical Power of Research in the Social and Behavioral Sciences 15
Using Power Analysis 17
Hypothesis Tests Versus Confidence Intervals 20
Conclusions 21
2 A Simple and General Model for Power Analysis 22
The General Linear Model, the F Statistic, and Effect Size 24
The F Distribution and Power 25
Translating Common Statistics and Effect Size Measures Into F 30
Alternatives to the Traditional Null Hypothesis 33
Minimum-Effect Tests as Alternatives to Traditional
Null Hypothesis Tests 36
Analytic and Tabular Methods of Power Analysis 41
Using the One-Stop F Table 43
The One-Stop PV Table 48
The One-Stop F Calculator 49
Effect Size Conventions for Defining Minimum-Effect Hypotheses 51
Conclusions 53
3 Using Power Analyses 55
Estimating the Effect Size 56
Four Applications of Statistical Power Analysis 59
V
ri_
CONTENTS
Conclusions 68
4 Multi-Factor ANOVA and Repeated-Measures Studies 69
The Factorial Analysis of Variance 70
Repeated Measures Designs 76
The Multivariate Analysis of Variance 82
Conclusions 83
5 Illustrative Examples 84
Simple Statistical Tests 84
Statistical Tests in Complex Experiments 90
Conclusions 96
6 The Implications of Power Analyses 98
Tests of the Traditional Null Hypothesis 99
Tests of Minimum-Effect Hypotheses 100
Power Analysis: Benefits, Costs, and Implications
for Hypothesis Testing 104
Conclusions 110
References 113
Appendix A - Working With the Noncentral F Distribution 117
Appendix B - One-Stop FTable 119
Appendix C - One-Stop PV Table 131
Appendix D - df Needed for Power = .80 (a = .05) 149
in Tests of Traditional Null Hypothesis
Appendix E - df Needed for Power =.80 (a = .05) 153
in Tests of the Hypothesis That Treatments Account
for 1% or Less of the Variance in Outcomes
Author Index 157
Subject Index 159
Preface
One of the most common statistical procedures in the behavioral and
social sciences is to test the hypothesis that treatments or interven-
tions have no effect, or that the correlation between two variables is
equal to zero, and so on (i.e., tests of the null hypothesis). Researchers
have long been concerned with the possibility that they will reject the
null hypothesis when it is in fact correct (i.e., make a Type I error), and
an extensive body of research and data-analytic methods exists to help
understand and control these errors. Substantially less attention has
been devoted to the possibility that researchers will fail to reject the
null hypothesis, when in fact treatments, interventions, and so forth,
have some real effect (i.e., make a Type II error). Statistical tests that
fail to detect the real effects of treatments or interventions might sub-
stantially impede the progress of scientific research.
The statistical power of a test is the probability that it will lead you to
reject the null hypothesis when that hypothesis is in fact wrong. Be-
cause most statistical tests are done in contexts where treatments have
at least some effect (although it might be minuscule), power often trans-
lates into the probability that the test will lead to a correct conclusion
about the null hypothesis. Viewed in this light, it is obvious why re-
searchers have become interested in the topic of statistical power, and
in methods of assessing and increasing the power of their tests.
This book presents a simple and general model for statistical
power analysis based on the widely used F statistic. A wide variety of
statistics used in the social and behavioral sciences can be thought of
as special applications of the general linear model (e.g., t tests, analy-
sis of variance and covariance, correlation, multiple regression), and
the F statistic can be used in testing hypotheses about virtually any of
these specialized applications. The model for power analysis laid out
here is quite simple, and it illustrates how these analyses work and
how they can be applied to problems of study design, to evaluating
others' research, and even to problems such as choosing the appro-
priate criterion for defining statistically significant outcomes.
vii
viii PREFACE
In response to criticisms of traditional null hypothesis testing, sev-
eral researchers have developed methods for testing what is referred
to as a minimum-effect hypothesis (i.e., the hypothesis that the effect
of treatments, interventions, etc. exceeds some specific minimal
level). This is the first book to discuss in detail the application of
power analysis to both traditional null hypothesis tests and to mini-
mum-effect tests. It shows how the same basic model applies to both
types of testing, and illustrates applications of power analysis to both
traditional null hypothesis tests (i.e., tests of the hypothesis that treat-
ments have no effect) and to minimum-effect tests (i.e., tests of the hy-
pothesis that the effects of treatments exceeds some minimal level). A
single table is used to conduct both significance tests and power anal-
yses for traditional and for minimum-effect tests (The One-Stop F Ta-
ble, presented in Appendix B), and some relatively simple procedures
are presented that may be used to ask a series of important and so-
phisticated questions about the research.
This book is intended for a wide audience, and so presentations
are kept simple and nontechnical wherever possible. For example,
Appendix A presents some fairly daunting statistical formulas, but it
also shows how a researcher with little expertise or interest in statis-
tical analysis could quickly obtain the values needed to carry out
power analyses for any range of hypotheses. Similarly, the first three
chapters of this book present a few formulas, but the reader who
skips them entirely will still be able to follow the ideas being pre-
sented in this book.
Finally, most of the examples presented herein are drawn from the
social and behavioral sciences, as are many of the generalizations
about statistical methods that are most likely to be used. In part, this
reflects our biases (we are both psychologists), but it also reflects the
fact that issues related to power analysis have been widely discussed
in this literature over the last several years. Researchers in other ar-
eas may find that some of the specific advice offered here does not ap-
ply as well to them, but the general principles articulated in this book
should be useful to researchers in a wide range of disciplines.
This second edition includes a number of features that were not
part of our first edition. First, a chapter (chap. 4) dealing with power
analysis in multifactor analysis of variance (ANOVA), including re-
peated measures designs, has been added. Multifactor ANOVA is very
common in the behavioral and social sciences, and whereas the con-
ceptual issues in power analysis are quite similar in factorial ANOVA
as in other methods of analysis, there are several features of ANOVA
that require special attention, and this topic deserves treatment in a
separate chapter. Second, a "One-Stop PV Table" has been included,
which presents the same information as in the One-Stop F Table,
