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Part III: Analyzing Variance with ANOVA
Checking the Fit of the ANOVA Model

As with any other model, you must determine how well the ANOVA model
fits before you can use its results with confidence. In the case of ANOVA,
the model basically boils down to a treatment variable (also known as the
population you’re in) plus an error term. To assess how well that model fits
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the data, see the values of R and R adjusted on the last line of the ANOVA
output below the ANOVA table. For the seed-spitting data, you see those
values at the bottom of Figure 9-4.
✓ The value of R measures the percentage of the variability in the
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response variable (y) explained by the explanatory variable (x). In the
case of ANOVA, the x variable is the factor due to treatment (where the
treatment can represent a population being compared). A high value of
R (say, above 80 percent) means this model fits well.
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✓ The value of R adjusted, the preferred measure, takes R and adjusts
it for the number of variables in the model. In the case of one-way
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ANOVA, you have only one variable, the factor due to treatment, so R
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and R adjusted won’t be very far apart. For more on R and R adjusted,
see Chapter 6.
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For the watermelon seed-spitting data, the value of R adjusted (as found
in the last row of Figure 9-4) is only 53.97 percent. That means age group
(shown to be statistically significant by the F-test; see the section “Making
conclusions from ANOVA”) explains just over half of the variability in the
watermelon seed-spitting distances. Because of that connection, you may
find other variables you can examine in addition to age group, making an
even better model for predicting how far those seeds will go.
As you see in Figure 9-1, the results of the t-test done to compare the spitting
distances of males and females in the section “Comparing Two Means with
a t-Test” show that males and females were significantly different on mean
seed-spitting distances (p-value = 0.039 < 0.05). So I would venture a guess
that if you include gender as well as age group, thereby creating what statisti-
cians call a two-factor ANOVA (or two-way ANOVA), the resulting model would
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fit the data even better, resulting in higher values of R and R adjusted.
(Chapter 11 walks you through the two-way ANOVA.)

7/23/09 9:31:31 PM
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15_466469-ch09.indd 170 7/23/09 9:31:31 PM