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

394 CHAPTER 12 | Introduction to Analysis of Variance
For example, with k = 3, we would compare μ 1
vs. μ 2
, then μ 2
vs. μ 3
, and then μ 1
vs. μ 3
.
In each case, we are looking for a significant mean difference. The process of conducting
pairwise comparisons involves performing a series of separate hypothesis tests, and each of
these tests includes the risk of a Type I error. As you do more and more separate tests, the
risk of a Type I error accumulates and is called the experimentwise alpha level (see p. 370).
We have seen, for example, that a research study with three treatment conditions produces
three separate mean differences, each of which could be evaluated using a post hoc
test. If each test uses α = .05, then there is a 5% risk of a Type I error for the first posttest,
another 5% risk for the second test, and one more 5% risk for the third test. Although
the probability of error is not simply the sum across the three tests, it should be clear that
increasing the number of separate tests definitely increases the total, experimentwise probability
of a Type I error.
Whenever you are conducting posttests, you must be concerned about the experimentwise
alpha level. Statisticians have worked with this problem and have developed several
methods for trying to control Type I errors in the context of post hoc tests. We will consider
two alternatives.
■ Tukey’s Honestly Significant Difference (HSD) Test
The first post hoc test we consider is Tukey’s HSD test. We selected Tukey’s HSD test
because it is a commonly used test in psychological research. Tukey’s test allows you to
compute a single value that determines the minimum difference between treatment means
that is necessary for significance. This value, called the honestly significant difference, or
HSD, is then used to compare any two treatment conditions. If the mean difference exceeds
Tukey’s HSD, you conclude that there is a significant difference between the treatments.
Otherwise, you cannot conclude that the treatments are significantly different. The formula
for Tukey’s HSD is
HSD 5 qÎ MS within
n
(12.15)
The q value used in
Tukey’s HSD test is
called a Studentized
range statistic.
EXAMPLE 12.5
where the value of q is found in Table B.5 (Appendix B, p. 656), MS within treatments
is the
within-treatments variance from the ANOVA, and n is the number of scores in each treatment.
Tukey’s test requires that the sample size, n, be the same for all treatments. To locate
the appropriate value of q, you must know the number of treatments in the overall experiment
(k), the degrees of freedom for MS within treatments
(the error term in the F-ratio), and you
must select an alpha level (generally the same α used for the ANOVA).
To demonstrate the procedure for conducting post hoc tests with Tukey’s HSD, we use the
data from Example 12.2, which are summarized in Table 12.6. Note that the table displays
summary statistics for each sample and the results from the overall ANOVA. With k = 3
treatments, n = 6, and α = .05, you should find that the value of q for the test is q = 3.67
(see Table B.5). Therefore, Tukey’s HSD is
HSD 5 qÎ MS within
n
5 3.67Î 5.87
6 5 3.63
Thus, the mean difference between any two samples must be at least 3.63 to be significant.
Using this value, we can make the following conclusions:
1. Treatment A is significantly different from treatment B (M A
– M B
= 4.00).
2. Treatment A is also significantly different from treatment C (M A
– M C
= 5.00).
3. Treatment B is not significantly different from treatment C (M B
– M C
= 1.00). ■