1 Analysis of Variance

This first section is about comparing several groups at once. The two-sample \(t\) test answers the question for two groups; with \(k\) groups there are \(\binom {k}{2}\) pairs, and testing them all separately makes a false positive somewhere almost certain. The analysis of variance replaces those many tests with one, by splitting the total variation in the data into a part attributable to the treatments and a part attributable to everything else, and comparing them.

The section builds the two designs the rest of the course refers back to: the one-way layout, in which units are assigned to treatments and nothing else is recorded, and the randomised block design, in which a known source of nuisance variation is used rather than ignored. The distributional facts are stated here and used; they are proved in Section 2.4, once the necessary matrix theory is available.

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