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Chapter 11: Finding Your Way through Two-Way ANOVA 205

temperatures 1, 2, and 3, respectively. Each boxplot shows the results of the
whiteness levels for the five shirts washed under that combination of detergent
and water temp.

Looking at these plots you can see that each detergent reacts differently with
different water temperatures. For example, Brand A does best in cold water
(water temp level 1) and worst in warm water (water temp level 2), while
Brand C is just the opposite, having the highest scores in warm water and the
lowest in cold water. Each detergent does best/worst under a different com-
bination of water temperatures. You can really see why the interaction term
in this model is significant!

Now which combination of detergent and water temperature does the best?
If you look at the plots, Brand B in cold water looks really good, and so does
Brand C in warm water, closely followed perhaps by Brand D in hot water.
This is where Tukey’s multiple comparisons come in.
Running multiple comparisons on all 12 combinations of detergent and water
temperature, you confirm that the three top combinations identified are all
significantly higher than all the others (because their sample means were
higher and their differences from all the other means had p-values less than
0.05). But the top three can’t be distinguished from each other (because the
p-values for the differences between them all exceed 0.05). Tukey also tells
you that the three worst combinations are Brand A in warm water, Brand B
in hot water, and Brand D in cold water. And they’re all at the bottom of the
barrel together (their means are significantly lower than all the rest but can’t
be distinguished from each other). So no single combination can claim all the
bragging rights or shoulder all the blame.

You can imagine the many other comparisons that you could make from here
to put the other combinations in some sort of order, but I think the best and
worst are the most interesting for this case. It’s like the fashion police com-
menting on what the stars wear on awards night. (Whatever they do wear,
let’s hope their statistician told them which brand of detergent to use and
what water temperature to wash it in!)

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