Peritz Procedure

The Peritz procedure can be applied after a significant overall F test in a one-way ANOVA. A series of additional F tests can be used to determine the significance of the difference between each of the possible pairs of means. Among k means, there will be k(k − 1)/2 pairs of means. If the usual assumptions of an ANOVA are satisfied, then the probability of one or more Type I errors is limited to the nominal level, α, of the overall F test. This will be true when each mean is based on the same sample size, N, and when sample sizes differ no matter how large the difference[Page 761] in sample sizes may be. For a wide variety of conditions, the Peritz procedure is more powerful than any other procedure for pairwise testing of means.

The application of the Peritz procedure can be accomplished by testing all partitions of the k means into subsets. For a particular pair of means to be found significantly different, it must be the case that a significant F test is found for every partition of the k means in which the two means in question are included in the same subset. However, progressively smaller subsets must be tested at more stringent levels. For example, if five means are partitioned into two subsets of the first three means and the last two means, a separate F test would be applied to the two subsets. The first subset would be tested at level (3/5)α and the second subset at level (2/5)α. That is, the first subset of three means includes three of the five means. It is tested at 3/5 times α. The second subset of two means includes two of the five means. It is tested at 2/5 times α.

If either of these tests is significant at the designated level, then the partition is rejected. Nonsignificance of the partition requires that a number of pairs cannot be significantly different. For example, there are three pairs among the means in the first subset (1, 2), (1, 3), and (2, 3). There is also the single pair in the second subset, (4, 5). All four pairs must fail to differ significantly if both F tests for this partition fail to be significant. Of course, all other partitions must be tested as well.

If the five means are partitioned into three subsets of the first two means, the third and fourth means, and the single fifth mean, then the first two means would be tested at level (2/4)α and the second subset at (2/4)α. The denominator of each fraction is taken as the number of means included in subsets of at least two means.

Slightly greater power can be obtained for the Peritz procedure by using slightly less stringent levels in each subset of the partition. For example, consider the case of five means partitioned into two subsets of the first three means and the last two means. Instead of testing the first subset of three means at level (3/5)α, it is possible to test that subset at level 1 – (1 – α)3/5. Similarly, instead of testing the second subset at level (2/5)α, it would be tested at level 1 – (1 – α)2/5. Again, the partition would be rejected if and only if at least one of the F tests is significant at the designated level.

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