Chapter 10 Analysis of Variance 423degrees of freedom for cola brand, DFI are the interaction degrees of freedom,
and DFE are the degrees of freedom for the error term, then
Total degrees of freedom 5 DFT 1 DFC 1 DFI 1 DFE
The next column of the two-way ANOVA table displays the mean square
of each of the factors (equal to the sum of squares divided by the degrees of
freedom). These are
Type 1,880.00
Cola 61,250.17
Interaction 1,634.46
Error 1,839.31
These values are the variances in foam volume within the various factors.
The largest variance is displayed in the cola factor; this indicates that this is
where the greatest difference in foam volume lies. The mean square value for
the error term 1839.31 is an estimate of s^2 , the variance in foam volume after
accounting for the factors of cola brand, type, and the interaction between
the two. In other words, after accounting for these effects in your model,
the typical deviation—or standard deviation—in foam volume is about
! 18405 42.9.
As with one-way ANOVA, the next column of the table displays the ratio of
each mean square to the mean square of the error term. These ratios follow a
F(m, n) distribution, where m is the degrees of freedom of the factor (type, cola
or interaction) and n is the degrees of freedom of the error term. By comparing
these values to the F distribution, Excel calculates the p values (cells F25:F27)
for each of the three effects in the model. Examine fi rst the interaction p value,
which is .455 (cell F27)—much greater than .05 and not even close to indicat-
ing signifi cance at the 5% level. This confi rms what we suspected from view-
ing the interaction plot. Now let’s look at the type and cola factors.
The column or cola effect is highly signifi cant, with a p value of 5.84 3
10211 (cell F26). This is less than .05, so there is a significant difference
among colas at the 5% level (because the p value is less than .001, there
is signifi cance at the 0.1% level, too). However, the p value is .318 for the
sample or type effect (cell F25), so there is no signifi cant difference between
diet and regular.
These quantitative conclusions from the analysis of variance are in agree-
ment with the qualitative conclusions drawn from the boxplot: There is a
signifi cant difference in foam volume between cola brands, but not between
cola types. Nor does there appear to be an interaction between cola brand
and type in how they infl uence foam volume.
Finally, how much of the total variation in foam volume has been explained
by the two-way ANOVA model? Recall that the coeffi cient of determination
(R^2 value) is equal to the fraction of the total sum of squares that is explained
by the sums of squares of the various factors. In this case that value is
(^1) 1880.00 1 183,750.50 1 4903.38 2
264,106.46