Orthogonal Comparisons (3 of 5)

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If there are "a" groups in an experiment, then it is possible to make up a-1 mutually orthogonal comparisons among the means. There are many ways to make up a set of a-1 orthogonal comparisons. There is no way to make up a set of more than a-1 comparisons. Two sets of 3 mutually orthogonal comparisons among four means are shown below.

             Set 1          Set 2
Mean 1    1   1   1       3   0   0
Mean 2    1  -1  -1      -1   2   0
Mean 3   -1   1  -1      -1  -1   1
Mean 4   -1  -1   1      -1  -1  -1
 

The value of Σa_ib_i for any pair of comparisons in either set is zero. For instance, for Comparisons 2 and 3 of Set 1,

Σ a_ib_i = (1)(1) + (-1)(-1) + (1)(-1) + (-1)(1) = 1 + 1 + (-1) + (-1) = 0.