Complete defining relationship

S.3.3 Complete Defining Relationship for Fractional Factorial Experiments... 

Example 4.5 Complete Defining Relationship and Confounding Pattern for a Half-Fractional Factorial Example... 

Continuing with the same fractional factorial experiment as in Example 4.4, determine the complete defining relationship and the complete confounding pattern for the experiment. The defining relationship is given as... 

Since n is odd, all the possible confounded variables have been found. Thus, the resolution of this method is V, since no second-order terms are confounded with each other. As well, note that the length of the smallest word in the complete defining relationship is 5, which is equal to the resolution. Such a relationship between the resolution and the smallest word in the complete defining relationship always holds. 

In order to obtain the complete defining relation, the two generators can be multiplied together to give the complete defining relationship as... 

Multiplying the complete defining relationship by each of the variables singly gives... 

Determine the complete defining relationship, the complete confounding pattern, and the model that can be fit given the confounding pattern. 

The complete defining relationship for this experiment can be found as follows. First, the given generator has to be converted to give... 

Consider the factorial design shown in Table 4.2. What are the independent factors, the dependent factors, the generators, the complete defining relationship, the resolution, and the aliases for A and for AB What type of factorial design is it ... 

Complete Defining Relationship The complete defining relationship is obtained as follows ... 

Resolution Since the smallest word in the complete defining relationship is 4 letters (factors) long, the resolution is IV, that is, the second-order interactions are confounded with each other. 

Complete Description This is a 2 y factorial experiment with the complete defining relationship 1 = ABCD. Note that a complete description requires that all the necessary parameters (/, p, resolution, and complete defining relationship) be provided. 

Using this data set, determine the generators, the complete defining relationship, and the type of experiment. 

After some experimentation, it can be determined that the basic factors are A, B, C, and D, while the dependent factor is E. The generator for this experiment can be written as E = ABCD. The complete defining relationship is then 1 = ABCDE. Based on the above analysis, it can be concluded that this is a A-fractional factorial experiment with a resolution of V, that is, 2y . It should be noted here that not all of the parameters can be estimated since they will be confounded with others. Without going into the details here, all of the zero-, first-, and second-order interactions are estimable. They will be confounded with various higher-order interactions. 

Consider a 2" factorial experiment with the complete defining relationship I = ABCD. Determine an appropriate blocking pattern for this experiment given that AB is known to be zero, and determine the new complete defining relationship. 

If the complete defining relationship is I = ABCD = ADE = ABF, then the resolution of this design is IV. 

---