9.6 Determining the Confounding Pattern
Before proceeding with the fractional factorial designs, we need a method to determine how the factors and interactions are confounded. Without this knowledge it is not possible to interpret the results. One way to do it is to use what is called the defining relation of the design.
Firstly, we may note that multiplying one of the columns in the matrix with itself yields a column where every element equals one. (In vector multiplication each element of one vector is multiplied with the corresponding element in the other.) A column of ones is called an identity vector and is denoted I. It gets its name from the fact that multiplying a vector with I will return the vector itself. When D is confounded with ABC we get ABCD = DD = I, which is the defining relationship of our design:
This relationship can be used to identify the confounding pattern. Multiplying a column in the matrix by ABCD returns the column that it is confounded with. For example, A times ABCD equals IBCD, which is equivalent to BCD. The defining relation also gives that A times ABCD equals AI, or A. This means that A is confounded with BCD.
We could do the same thing with an interaction column; for example, AB = ABI. According to the defining relation this is equivalent to ABABCD = AABBCD = IICD = CD, meaning that AB is confounded with CD.
To turn these calculations into ...