UNTIL RECENTLY, MOST APPLIED STATISTICAL WORKas it is actually carried out in analyzing
experimental results was primarily concerned with minimizing (or at least controlling) the
probability of a Type I error (a). When designing experiments, people tend to ignore the
very important fact that there is a probability (b) of another kind of error, Type II errors.
Whereas Type I errors deal with the problem of finding a difference that is not there, Type
II errors concern the equally serious problem of not finding a difference that is there. When
we consider the substantial cost in time and money that goes into a typical experiment, we
could argue that it is remarkably short-sighted of experimenters not to recognize that they
may, from the start, have only a small chance of finding the effect they are looking for, even
if such an effect does exist in the population.
There are very good historical reasons why investigators have tended to ignore Type II
errors. Cohen places the initial blame on the emphasis Fisher gave to the idea that the null
hypothesis was either true or false, with little attention to. Although the Neyman-Pearson
approach does emphasize the importance of , Fisher’s views have been very influential.
In addition, until recently, many textbooks avoided the problem altogether, and those books
that did discuss power did so in ways that were not easily understood by the average reader.
Cohen, however, discussed the problem clearly and lucidly in several publications.^1 Cohen
(1988) presents a thorough and rigorous treatment of the material. In Welkowitz, Ewen,
and Cohen (2000) the material is treated in a slightly simpler way through the use of an ap-
proximation technique. That approach is the one adopted in this chapter. Two extremely
good papers that are very accessible and that provide useful methods are by Cohen (1992a,
1992b). You should have no difficulty with either of these sources, or, for that matter, with
any of the many excellent papers Cohen published on a wide variety of topics not neces-
sarily directly related to this particular one.
Speaking in terms of Type II errors is a rather negative way of approaching the problem,
since it keeps reminding us that we might make a mistake. The more positive approach
would be to speak in terms of power,which is defined as the probability of correctly reject-
ing a false when a particular alternative hypothesis is true. Thus, power 51 2b. A more
powerful experiment is one that has a better chance of rejecting a false than does a less
powerful experiment.
In this chapter we will take the approach of Welkowitz, Ewen, and Cohen (2000) and
work with an approach that gives a good approximation of the true power of a test. This
approximation is an excellent one, especially in light of the fact that we do not really care
whether the power is .85 or .83, but rather whether it is near .80 or nearer to .30. Cohen
(1988) takes a more detailed approach; rather than working with an approximation, he
works with more exact probabilities. That approach requires much more extensive tables
but produces answers very similar to the ones that we will obtain here. However, it does
not make a great deal of sense to work through extensive tables when the alternative is to
use simple software programs that have been developed to automate power calculations.
The method that I will use makes clear the concepts involved in power calculations, and if
you wish more precise answers you can download, very good, free, software. An excel-
lent program named G*Power by Faul and Erdfelder is available on the Internet at
http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/ and there are both Macintosh
and DOS programs at that site. In what follows I will show power calculations by hand, but
then will show the results of using G*Power and the advantages that the program offers.H 0
H 0
H 1
H 1
226 Chapter 8 Power
(^1) A somewhat different approach is taken by Murphy and Myors (1998), who base all of their power calculations
on the Fdistribution. The Fdistribution appears throughout this book, and virtually all of the statistics covered in
this book can be transformed to a F. The Murphy and Myors approach is worth examining, and will give results
very close to the results we find in this chapter.
power