What is a defining relation in a two-level factorial design?

The defining relation is the total collection of terms that are held constant to define the fraction in a fractional factorial design. The defining relation is used to calculate the alias structure that describes the confounding in fractional factorial designs.

Here is design information for a fractional factorial design with five factors (A, B, C, D, and E):

Defining Relation: I = ABD = ACE = BCDE

Alias Structure I + ABD + ACE + BCDE A + BD + CE + ABCDE B + AD + CDE + ABCE C + AE + BDE + ABCD D + AB + BCE + ACDE E + AC + BCD + ABDE BC + DE + ABE + ACD BE + CD + ABC + ADE

Minitab uses the defining relation to calculate each line in the alias table. Any letter multiplied by itself is the identity, I (that is, A * A = I). I multiplied by any letter is the same letter (e.g., I * A = A). For example, to obtain the aliases for factor A, multiply all terms in the defining relation by A.

The identity column I is always a column of 1’s (in coded units). Therefore, since I = ABD in our example, the product of the columns A, B, D is a column of 1’s. The same is true for ACE and BCDE.