Confounding in Fractional Factorial Experiments

One of the most important concepts in fractional factorial design is confounding or aliasing. Confounding occurs when two or more interactions share the same column space, that is, the column entries for the interactions are the same. Only a full factorial experiment does not have confounding. By reducing the number of experiments performed, not all of the parameters can be estimated. In a p- ractional experiment. 

In order to determine the maimer in which the different variables are confounded, it is first necessary to consider two mathematical concepts identity vector and modular arithmetic. 

For an orthogonal basis, let 7 be the defined as a vector of 1 s. The vector I forms the basis for the constant term, / o, in the factorial experiment. Irrespective of the factorial design, the vector 7 can be treated as representing the identity vector for the system under pointwise multiplication denoted by Q  

Modular arithmetic denoted as x mod y, where x is the divisor and y is the dividend (or base), seeks to determine the remainder when x is divided by y, for example, 7 mod 2 will be equal to 1, since the remainder when 7 is divided by 2 is 1 (7 = 3x2+ ). When seeking to determine the confounding pattern in fractional factorial experiments and higher-order terms are encountered, then reduction of these terms is performed using Z-base modular arithmetic, where /, as before, is the number of levels in the design. 

Consider a 3-level design with the term Determine the reduced form. 

---