Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

370 CHAPTER 12 | Introduction to Analysis of Variance
For this reason, researchers often make a distinction between the testwise alpha level
and the experimentwise alpha level. The testwise alpha level is simply the alpha level you
select for each individual hypothesis test. The experimentwise alpha level is the total probability
of a Type I error accumulated from all of the separate tests in the experiment. As the
number of separate tests increases, so does the experimentwise alpha level.
DEFINITIONS
The testwise alpha level is the risk of a Type I error, or alpha level, for an individual
hypothesis test.
When an experiment involves several different hypothesis tests, the experimentwise
alpha level is the total probability of a Type I error that is accumulated from all of
the individual tests in the experiment. Typically, the experimentwise alpha level is
substantially greater than the value of alpha used for any one of the individual tests.
For example, an experiment involving three treatments would require three separate
t tests to compare all of the mean differences:
Test 1 compares treatment I vs. treatment II.
Test 2 compares treatment I vs. treatment III.
Test 3 compares treatment II vs. treatment III.
If all tests use α = .05, then there is a 5% risk of a Type I error for the first test, a 5% risk for
the second test, and another 5% risk for the third test. The three separate tests accumulate to
produce a relatively large experimentwise alpha level. The advantage of ANOVA is that it performs
all three comparisons simultaneously in one hypothesis test. Thus, no matter how many
different means are being compared, ANOVA uses one test with one alpha level to evaluate the
mean differences and thereby avoids the problem of an inflated experimentwise alpha level.
■ The Test Statistic for ANOVA
The test statistic for ANOVA is very similar to the t statistics used in earlier chapters. For
the t statistic, we first computed the standard error, which measures the difference between
two sample means that is reasonable to expect if there is no treatment effect (that is, if H 0
is true). Then we computed the t statistic with the following structure:
obtained difference between two sample means
t 5
standard error sthe difference expected with no treatment effectd
For ANOVA, however, we want to compare differences among two or more sample
means. With more than two samples, the concept of “difference between sample means”
becomes difficult to define or measure. For example, if there are only two samples and they
have means of M = 20 and M = 30, then there is a 10-point difference between the sample
means. Suppose, however, that we add a third sample with a mean of M = 35. Now how
much difference is there between the sample means? It should be clear that we have a problem.
The solution to this problem is to use variance to define and measure the size of the
differences among the sample means. Consider the following two sets of sample means:
Set 1 Set 2
M 1
= 20 M 1
= 28
M 2
= 30 M 2
= 30
M 3
= 35 M 3
= 31