Main effect of factor A: ME(A) = z̄(A+) − z̄(A−), where z̄(A+) is the average
of the zi values observed at A+ and z̄(A−) is similarly defined. Interaction effect
between factors A and B:
INT(A, B) = 1
2
[ME(B|A+) − ME(B|A−)]
= 1
2
[ME(A|B+) − ME(A|B−)]
= 1
2
{z̄(A + |B+) + z̄(A − |B−)}
− 1
2
{z̄(A + |B−) + z̄(A − |B+)},
where ME(B|A+) = z̄(B + |A+) − z̄(B − |A+) is the conditional main effect of
factor B at the + level of factor A, and other ME terms can be similarly defined.
A significant interaction between A and B suggests that the effect of one factor
depends on the level of the other factor. For k factors define their interaction
in terms of the (k − 1)-factor interactions:
INT(A1, A2, ..., Ak) = 1
2
INT(A1, A2, ..., Ak−1|Ak+)
− 1
2
INT(A1, A2, ..., Ak−1|Ak−),
which can be written as z̄+ − z̄−, where z̄+ is the average of the z values at
the factorial combinations whose product of its levels is +; z̄− is similarly
defined. These factorial effects can be estimated via regression analysis by
using a model like (4.15).