Orthogonal Contrasts

Each null hypothesis is a linear combination of the
treatment means. A set of linear combinations of this
type is called a set of orthogonal contrasts.

A set of linear combinations must satisfy two
mathematical properties in order to be orthogonal
contrasts:
1) The sum of the coefficients in each linear contrast
must sum to zero, and
2) The sum of the products of the corresponding
coefficients in any two contrasts must equal zero.

A set of contrasts is orthogonal if every pair of contrasts
is orthogonal. An experiment with A treatments can
have several sets of mutually orthogonal contrasts, but
each set is limited to A - 1 possibilities.

Two contrasts are orthogonal if the sum of the products of corresponding coefficients (i.e. coefficients for the same means) adds to zero.

Formally, the definition of contrast is expressed below, using the notation m_i for the i-th treatment mean:

Simple contrasts includes the case of the difference between two factor means, such as m₁ - m₂ . If one wishes to compare treatments 1 and 2 with treatment 3, one way of expressing this is by: m₁ + m₂ - 2m_3. Note that

m₁ - m₂ has coefficients +1, -1
m₁ + m₂ - 2m₃ has coefficients +1, +1, -2.
These coefficients sum to zero.

As an example of orthogonal contrasts note the three contrasts defined by the table below, where the rows denote coefficients for the column treatment means. The following is true:

1. The sum of the coefficients for each contrast is zero.

2. The sum of the products of coefficients of each pair of contrasts is also 0 (orthogonality property).

3. The first two contrasts are simply pair wise comparisons, the third one involves all the treatments.

As might be expected, contrasts are estimated by taking the same linear combination of treatment mean estimates. In other words: