Return to Example 10.2, with the data in Table 10.2.
(a) Test the hypothesis that the tests have the same distribution for the scores.
(b) Use pairwise comparisons to compare the tests, forcing the family-wise significance level at 5%.
Example 10.2
An educational specialist is aware of three different vocabulary tests that are available for prekindergarten-aged children. The specialist suspects that these tests tend to give different results. There are at least two ways the specialist could conduct an experiment to check this suspicion. For example, the specialist could take a large group of children and randomly divide them into three groups, with each group given a different test. These independent samples would be compared using the one-way ANOVA, as in Chapter 6. Because the differences among individual children are very large, the withingroup variances will tend to be large. Unless the samples are very large, this will tend to swamp any small differences in the tests.
Instead, the specialist decides to select a smaller group of children and give each child all three of the tests. To avoid “learning effects” or “test fatigue,” the tests are given in random order two weeks apart. This is an RB design, and the data are given in Table 10.2. Notice that if there were only two tests, we would compare them using a paired t test.
The layout of the data in Table 10.2 suggests a two-factor ANOVA with only one observation per cell. In fact, that is how the mechanics of the calculations will be carried out. Unlike Chapter 9, though, the factor Child is a random effect while Test is a fixed effect (see Chapter 6). Hence, the interpretation of the results will be somewhat different.