A contrast is a linear combination of the parameters associated with a term where the coefficients sum to zero. Contrasts facilitate the testing of linear combinations of parameters. For example, consider a completely randomized design with four treatment groups: Placebo, Low Dose, Medium Dose, and High Dose. It is possible to test the hypothesis that the average of the dosed groups is the same as the average of the placebo group. One could write this hypothesis as:
or, equivalently:
Both of these are contrasts since the sum of the coefficients is zero. Note that both contrasts make the same statement. This is true generally: contrasts are invariant under changes in scaling.
Contrasts are tested using the Wald (1943) test statistic:
where:
is the estimator of b.
is the estimator of V.
L must be a matrix such that each row is in the row space of X.
This requirement is enforced in the wizard. The Wald statistic is compared to an F distribution with rank(L) numerator degrees of freedom and denominator degrees of freedom estimated by the program or user-specified.