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Part III: Comparing Many Means with ANOVA
degrees of freedom for total is 3 2 4 – 1 = 23; and degrees of freedom for
* *
error is 3 2 (4 – 1) = 18.
* *
Here are the answers to match the graphs from Figure 11-1 with the output
from Figure 11-2:
In the ANOVA table for Figure 11-2a, you see that the interaction term
isn’t significant (p-value = 0.526), so the main effects can be studied. The
p-values for dosage and times taken are 0.000 and 0.001, indicating both
factors A and B respectively are significant; this matches the plot in
Figure 11-1a.
In Figure 11-2b, you see that the p-value for interaction is significant
(p-value = 0.000) so you can’t examine the main effects of factors A and B
(in other words, don’t look at their p-values). This represents the situa-
tion in Figure 11-1e.
Figure 11-2c shows nothing is significant (p-value for interaction term is
0.513; p-values for main effects of A (dosage) and B (times taken) are
0.926 and 0.416, respectively). These results coincide with Figure 11-1d.
Figure 11-2d matches Figure 11-1b, with no interaction effect (p-value =
0.899), dosage (factor A) is significant (p-value = 0.000), and times per
day (factor B) isn’t (p-value = 0.207).
Figure 11-2e matches Figure 11-c. Dosage * times per day is not signifi-
cant (p-value = 0.855); times per day is significant with p-value 0.000 but
not dosage level (p-value = 0.855).
Assessing the fit
2
To assess the fit of the two-way ANOVA models, you can use the R adjusted
(see Chapter 5). The higher this number is, the better (the maximum is 100
percent or 1.00). Notice that all the ANOVA tables in Figure 11-2 show a fairly
2
high R adjusted except for Figure 11-2c. In this table, none of the terms was
significant.
Multiple comparisons
In the case where you find that an interaction effect is statistically significant,
you can conduct multiple comparisons to see which combinations of factors
A and B create different results in the response. The same ideas hold here as
those for Chapter 10 on multiple comparisons, except the tests are performed
on all i * j interactions.
To perform multiple comparisons for a two-way ANOVA by using Minitab, enter
your responses (data) in Column 1, your levels of Factor A in Column 2, and
your levels of Factor B in Column 3. Choose Stat>ANOVA>General Linear Model.
In the Responses box, enter your Column 1 variable. In Model, enter 1 <space> 2
<space> 1*2 (for the main effects and the interaction effect, respectively; here
<space> means leave a space where indicated). Click on Comparisons. In Terms,
enter columns 2 and 3. Check the Method you want to use for your multiple
comparisons (see Chapter 10). Click OK.