### Finding the Alias Structure of a Fractional Factorial

The entire set of aliases in a fractional factorial design is called the alias structure of the design. There is a fairly easy method for writing down the alias structure of a fractional design. This method depends on some simple observations about multiplying columns of +1's and -1's:

1. The letter I denotes the column consisting entirely of +1's.
2. Note that any column multiplied by itself yields column I. For example, A·A = A2 = I, B·B = B2 = I, and so forth.
3. Multiplying column I by any other column does not change the column. For example, A·I = I·A = A.

Using these facts, we can obtain the alias structure of any fractional factorial as follows:

1. First, write the p assignments of additional factors in equation form. These p equations are called the design generators.
2. Multiply each generator from step 1 by its left side to put each generator into the form I = w, where w is a "word" composed of several letters representing particular experimental factors (e.g., D = ABC becomes I = ABCD). It is also possible tocreate words with "-" signs, such as D = -ABC. If this is done, the resulting design will use a different fraction of the runs from the full 2k design.
3. Letting I = w1, I = w2, ..., I = wp denote the p design generators from step 2, form all possible products of the words wi (one at a time, two at a time, three at a time, etc.). Use the fact that squares of factors can be eliminated (e.g., A2 = I and multiplying by I does not change anything). There will be a total of 2p words formed. This collection is called the defining relation of the design.
4. Multiply each word in the defining relation by all 2k - 1 effects based on k factors. Use the fact that squares of factors cancel out to simplify the products. The result is called the alias structure of the design.

Example:

Question: Determine the alias structure of the 25-2 design with the generators D = ABC and I = ACE.

Solution:

1. The design generators are D = ABC and E = AC.
2. The generators in standard form are I = ABCD and I = ACE
3. Only one product of the above words is possible, namely, (ABCD)(ACE) = A2BC2DE = BDE.
4. The defining relation is I = ACE = BDE = ABCD
5. Multiplying each of the 25-1 effects by the defining relation gives
• I = ACE = BDE = ABCD
• A = CE = ABDE = BCD
• B = DE = ACD = ABCE
• C = AE = ABD = BCDE
• D = BE = ABC = ACDE
• E = AC = BD = ABCDE
• AB = CD = ADE = BCE
• AD = BC = ABE = CDE
Note that each main effect is now aliased with a two-factor interaction in this design.