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Chapter 9: Going One-Way with Analysis of Variance
With the critical value approach, any F-statistic that lies beyond the critical
value results in rejecting Ho, no matter how far or close to the line it is. If
your F-statistic is beyond the value found in Table A-5, then you reject Ho
and say at least two of the treatments (or populations) have different means.
What’s next?
After you’ve rejected Ho in the F-test and concluded that not all the popula-
tions means are the same, your next question may be: Which ones are differ-
ent? You can answer that question by using a statistical technique called
multiple comparisons. Statisticians use many different multiple comparison
procedures to further explore the means themselves after the F-test has been
rejected. I discuss and apply some of the more common multiple comparison
techniques in Chapter 10.
Checking the Fit of the ANOVA Model 175
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 the data, see
2
2
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 9-3.
2
The value of R measures the percentage of the variability in the response vari-
able (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
2
population being compared). A high value of R (say above 80 percent) means
2
this model fits well. The value of R adjusted, the preferred measure, takes R 2
and adjusts it for the number of variables in the model. In the case of one-way
2
ANOVA, you have only one variable, the factor due to treatment so R and R 2
2
2
adjusted won’t be very far apart. For more on R and R adjusted, see Chapter 5.
2
For the watermelon seed spitting data, the value of R adjusted (as found in
the last row of Figure 9-3) is only 53.97 percent. That means age group (while
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 age group alone explains only a
little over half of what’s going on in the seed spitting distances, you may find
other variables you can examine in addition to age group, making an even
better model.