Dunnett's Test

Background

A one-way ANOVA tests the null hypothesis (H0) that all the k treatment means are equal; that is,

against the alternative hypothesis (HI) that at least one of the means is different from the others. The difficulty is that if Ho is rejected, it is not known which mean differs from the others. It is possible to run t tests on all possible pairs of means (e.g., A vs. B, A vs. C, B vs. C). However, if there are five groups, this would result in 10 t tests (in general, there are k × (k −1) / 2 pairs). Moreover, the tests are not independent, because anyone mean enters into a number of comparisons, and there is a common estimate of the experimental error. As a result, the probability of a Type I error (that is, concluding that there is a significant difference when in fact there is not one) increases beyond 5% to an unknown extent. The various post hoc tests are attempts to control this family-wise error rate and constrain it to 5%.

The majority of the post hoc tests compare each group mean against every other group mean. One, called Scheffe's test, goes further, and allows the user to compare combinations of groups (e.g., A+B vs. C, A+C vs. B, A+B vs. C+D). Dunnett's test is limited to the situation in which one group is a control or reference condition (R), and each of the other (experimental group) means is compared to it.

Dunnett's Test

Rather than one null hypothesis, Dunnett's test has as many null hypotheses as there are experimental groups. If there are three such groups (A, B, and C), then the null and alternative hypotheses are:

Each hypothesis is tested against a critical value, called q′, using the formula:

where the subscript i refers to each of the experimental groups, and MSerror is the mean square for the error term, taken from the ANOVA table. The value of q′ is then compared against a critical value in a special table, the researcher knowing the number of groups (including the reference group) and the degrees of freedom of MSerror The form of the equation is very similar to that of the NewmanKeuls test. However, it accounts for the fact that there are a smaller number of comparisons because the various experimental groups are not compared with each other. Consequently, it is more powerful than other post hoc tests.

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