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Chapter 11: Finding Your Way through Two-Way ANOVA 203

combinations are: Brand A in cold water, Brand A in warm water, Brand A
in hot water, Brand B in cold water, Brand B in warm water, Brand B in hot
water, and so on.

The resulting two-way ANOVA model looks like this: y = b + w + bw + e,
i j ij
where b represents the brand of detergent, w represents the water tempera-
ture, y represents the whiteness of the clothes after washing, and bw repre-
ij
sents the interaction of brand i of detergent (i = A, B, C, D) and temperature j
of the water (j = 1, 2, 3). (Note that e represents the amount of variation in the
y values (whiteness) that isn’t explained by either brand or temperature.)
Suppose you decide to run the experiment five times on each of the 12 com-
binations, which means 60 observations. (That’s 60 t-shirts to wash — hey,
it’s a dirty job but someone’s got to do it!) The results of the two-way ANOVA
are shown in Figure 11-3.

ANOVA Table: Clothing Example
Source DF SS MS F P
Brand 3 22.983 7.6611 20.89 0.000
Water 2 1.433 0.7167 1.95 0.153
Figure 11-3: Interaction 6 308.167 51.3611 140.08 0.000
ANOVA Error 48 17.600 0.3667
table for Total 59 350.183
the clothing
example. S = 0.6055 R-Sq = 94.97% R-Sq(adj) = 93.82%

Note that the degrees of freedom (DF) for Brand, Water, Interaction, Error,
and Total were arrived at from the following:

✓ DF for brand: 4 – 1 = 3
✓ DF for water temperature: 3 – 1 = 2
✓ DF for interaction term: (4 – 1) * (3 – 1) = 6
✓ DF for error: 60 – (4 * 3) = 48
✓ DF for total: n – 1 = 60 – 1 = 59

Looking at the ANOVA table in Figure 11-3, you can see that the model fits the
2
data very well, with R adjusted equal to 93.82 percent. The interaction term
(brand of detergent interacting with water temperature) is significant, with a
p-value of 0.000. This means you can’t look separately at the effect of brand
of detergent or water temperature separately. One brand of detergent isn’t
always best, and one water temperature is not always best; it’s the combina-
tion of the two that has different effects.

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