information:classification
Contrasts, main effects and interactions

The expected response to a given treatment combination is called a cell mean, usually denoted using the Greek letter μ. (The term cell is borrowed from its use in tables of data.) This notation is illustrated here for the 2 × 3 experiment.

A contrast in cell means is a linear combination of cell means in which the coefficients sum to 0. Contrasts are of interest in themselves, and are the building blocks by which main effects and interactions are defined.

In the 2 × 3 experiment illustrated here, the expression

μ 11 − μ 12 {\displaystyle \mu _{11}-\mu _{12}}

is a contrast that compares the mean responses of the treatment combinations 11 and 12. (The coefficients here are 1 and –1.) The contrast

μ 11 + μ 12 + μ 13 − μ 21 − μ 22 − μ 23 {\displaystyle \mu _{11}+\mu _{12}+\mu _{13}-\mu _{21}-\mu _{22}-\mu _{23}}

is said to belong to the main effect of factor A as it contrasts the responses to the "1" level of factor A {\displaystyle A} with those for the "2" level. The main effect of A is said to be absent if this expression equals 0.

Interaction in a factorial experiment is the lack of additivity between factors, and is also expressed by contrasts. In the 2 × 3 experiment, the contrasts

μ 11 − μ 12 − μ 21 + μ 22 {\displaystyle \mu _{11}-\mu _{12}-\mu _{21}+\mu _{22}} and μ 11 − μ 13 − μ 21 + μ 23 {\displaystyle \mu _{11}-\mu _{13}-\mu _{21}+\mu _{23}}

belong to the A × B interaction; interaction is absent (additivity is present) if these expressions equal 0. Additivity may be viewed as a kind of parallelism between factors, as illustrated in the Analysis section below.

Since it is the coefficients of these contrasts that carry the essential information, they are often displayed as column vectors. For the example above, such a table might look like this:

The columns of such a table are called contrast vectors: their components add up to 0. Each effect is determined by both the pattern of components in its columns and the number of columns.

The patterns of components of these columns reflect the general definitions given by Bose:

Similar definitions hold for interactions of more than two factors. In the 2 × 3 example, for instance, the pattern of the A column follows the pattern of the levels of factor A, indicated by the first component of each cell.

The number of columns needed to specify each effect is the degrees of freedom for the effect, and is an essential quantity in the analysis of variance. The formula is as follows:

The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively.