Two-level fractional factorial designs 2k–q
Modelling by Experimental Designs

Fractional factorial designs are factorial designs that allow all factors to be studied, but with a reduced number of trials compared to full factorial designs. A fractional factorial design has 2 times fewer or 4 times fewer or 2q times fewer trials than the corresponding full factorial. At the end of a fractional factorial design, we have a system of n equations with p unknowns, with p greater than n. We can't solve such a system. We don't know how to solve such a system. Since we can't increase the number of equations, we have to decrease the number of unknowns. This is achieved using an artifice: we group the unknowns together and solve the system for these groups of unknowns. These groups of unknowns are called "contrasts" and the unknowns are said to be "aliased into the contrasts".