2 factor anova

from Dallals page Little Handbook of Statistical Practice

dairy farmer wished to determine which type of feed will produce the greatest yield of milk. From the research literature she is able to determine the mean milk output for each of the breeds she owns for each type of feed she is considering. As a practical matter, she can use only one type of feed for her herd.
Since she can use only one type of feed, she wants the one that will produce the greatest yield from her herd. She wants the feed type that produces the greatest yield when averaged over all breeds, even if it means using a feed that is not optimal for a particular breed. (In fact, it is easy to construct examples where the feed-type that is best on average is not the best for any breed!) The dairy farmer is interested in what the main effects have to say even in the presence of the interaction. She wants to compare
where the means are obtained by averaging over breed.
For the sake of rigor, it is worth remarking that this assumes the herd is composed of equal numbers of each breed. Otherwise, the feed-types would be compared through weighted averages with weights determined by the composition of the herd. For example, suppose feed A is splendid for Jerseys but mundane for Holsteins while feed B is splendid for Holsteins but mundane for Jerseys. Finally, let feed C be pretty good for both. In a mixed herd, feed C would be the feed of choice. If the composition of the herd were to become predominantly Jerseys, A might be the feed of choice with the gains in the Jerseys more than offsetting the losses in the Holsteins. A similar argument applies to feed B and a herd that is predominantly Holsteins.