Checking orthogonality

The set of contrasts we considered in the More Information page was:

₁ versus ₂    and    [₁ + ₂] / 2   versus ₃

For an orthogonal set, the following conditions must be met:

1. The contrast coefficients for each contrast must sum to 0.
2. The cross products of coefficients for each pair of contrasts must sum to 0.

For the set above, the sum of each set of coefficients [1 −1 + 0] and [1/2 + 1/2 −1] is equal to zero.
Similarly the sum of their cross products [(1 × 1/2) + (−1 × 1/2) + (0 × -1)] is equal to zero.

The following set of comparisons is not orthogonal:

₁ versus ₂    and    ₂ versus ₃

Rewriting this set of contrasts with their coefficients we get:

1₁ − 1₂ + 0₃

0₁ + 1₂ − 1₃

The sum of each set of coefficients is still equal to zero, but the sum of their cross products is not equal to zero, namely [(1 × 0) + (−1 × +1) + (0 × −1)] = −1.

Note that this simple method for checking orthogonality is only valid if each group has the same sample size.