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Chapter 11: Finding Your Way through Two-Way ANOVA 199

in a two-way ANOVA is an F-test based on the ideas of one-way ANOVA (see
Chapter 9 for more on this).

To conduct the F-tests for these terms, you basically want to see whether
more of the total variability in the y’s can be explained by the term you’re
testing compared to what’s left in the error term. A large value of F means
that the term you’re testing is significant.

First, you test whether the interaction term AB is significant. To do this, you
use the test statistic , which has an F-distribution with (i – 1) * (j – 1)
degrees of freedom from MS (mean sum of squares for the interaction term
AB
of A and B) and i * j * (m – 1) degrees of freedom from MSE (mean sum of
squares for error), respectively. (Recall that i and j are the number of levels
of A and B, and m is the sample size at each combination of A and B.)

If the interaction term isn’t significant, you take the AB term out of the model,
and you can explore the effects of Factors A and B separately regarding the
response variable y.
The test for Factor A uses the test statistic , which has an
F-distribution with i – 1 degrees of freedom from MS (mean sum of squares
A
for Factor A) and i * j * (m – 1) degrees of freedom from MSE (mean sum of
squares for error), respectively.
Testing for Factor B uses the test statistic , which has an
F-distribution with j – 1 and i * j * (m – 1) degrees of freedom. (See Chapter 9
for all the details on F-tests, MSE, and degrees of freedom.)
The results you can get from testing the terms of the ANOVA model are the
same as those represented in Figure 11-1. They’re all provided in Minitab
output outlined in the next section, including their sum of squares, degrees of
freedom, mean sum of squares, and p-values for their appropriate F-tests.

Running the Two-Way ANOVA Table

The ANOVA table for two-way ANOVA includes the same elements as the
ANOVA table for one-way ANOVA (see Chapter 9). But where in the one-way
ANOVA you have one line for Factor A’s contributions, now you add lines for
the effects of Factor B and the interaction term AB. Minitab calculates the
ANOVA table for you as part of the output from running a two-way ANOVA.

In this section, you figure out how to interpret the results of a two-way
ANOVA, assess the model’s fit, and use a multiple comparisons procedure, all
using the drug data study.

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