The defining relation for a design is determined by the words in the generating rules. A word is represented by a product of factors, but it is interpreted as the elementwise product of the entries in the design matrix for those columns. A defining relation consists of words whose product is a column of ones, called the identity.
Generating Rules for the Standard 25-3 Design shows the default-generating rules for a 25-3 design (five factors and eight runs).
Generating Rules for the Standard 25-3 Design
 • If the +/- box is selected, the sign associated with the generating rule is positive and the corresponding word equals the identity.
 • If the +/- box is not selected, the sign associated with the generating rule is negative and the corresponding word equals minus the identity.
The principal fraction of a full factorial design is the fractional factorial design obtained by setting all the defining relations equal to the identity. By default, the factorial design that JMP provides is the principal fraction. Notice that the +/- box is selected by default for all generating rules, so that each word in the defining relation equals the identity.
 • To change the generating rules, select the appropriate boxes.
 •
Generating Rules for the Standard 25-3 Design shows two columns of check boxes:
 • The first column represents the word Temperature = Feed Rate*Catalyst*Stir Rate.
 • The second column represents the word Concentration = Catalyst*Stir Rate.
Define I to represent a column consisting of the values +1. Because all factor levels are -1 or +1, the word in the first column is equivalent to Temperature*Feed Rate*Catalyst*Stir Rate = I. The word in the second column is equivalent to Concentration*Catalyst*Stir Rate = I. Together, these give the defining relations for the 25-3 design:
I = Temperature*Feed Rate*Catalyst*Stir Rate = Concentration*Catalyst*Stir Rate
Temperature = Feed Rate*Catalyst*Stir Rate = Feed Rate*Concentration
The second equality follows from replacing Catalyst*Stir Rate by Concentration using the second generating rule.
Now, post-multiply the first and third expressions by Concentration to obtain the following expression:
Temperature*Concentration = Feed Rate*Concentration*Concentration
Because the column for Concentration in the design matrix contains values of -1 and +1, the term Concentration*Concentration represents a column of +1 values. The expression becomes the first alias relation shown in the Aliasing of Effects outline:
Temperature*Concentration = Feed Rate*I = Feed Rate

Help created on 9/19/2017