In the screening plan, a generator is used to determine which effects are confounding each other.
In JMP, you can check which generators are being used by looking at the checkboxes displayed under "Change generation rules" on the plan creation screen.
Notice
In JMP, when creating a screening plan, the generation rule that minimizes deviation under the specified conditions is automatically selected.
Using "Change generation rules" allows you to intentionally alter the confounding structure, but this may result in the system no longer being minimally divergent or the resolution changing.
The concept of a generator
A generator is a set of rules used in a classic 2^(kp) factorial design to generate the remaining factors from the underlying factors.
In other words, it can be thought of as a rule that determines which effects are confounded by each other.
A defining relation can be derived from the generator, and this defining relation can then be used to find alias relationships.
In JMP, the generator is interpreted by reading the following two elements vertically:
- The effect of checking the generation rule.
- Factors assigned to it
In the case of a 6-factor resolution IV (Resolution IV)
Factors: A, B, C, D, E, F
Plan: Resolution 4 (Partial factorial design (16 experiment trials = 2^(6-2)), no blocks)
In the initial "Change generation rules" setting, the factors selected by JMP to minimize deviations are checked.
The four "factors" A through D are the basic columns, and a 2^4 full factorial design is created using these four columns (in a classical 2^(kp) partial factorial design, the number of experiments is a power of 2).
Then, we define the relationship with the remaining two columns (E and F) using "Change Generation Rules".
At this time, the following generators are configured in JMP:
Multiplying both sides of these two generators by E and F respectively reveals the two defining relationships of this design (the product of identical factors is I (identity)).
"Definitional relationships" based on the product of generators
By combining existing definitional relationships, we can obtain new definitional relationships.
Since the product of identical factors is I (identity),
Using this, BCDE × ACDF = ABEF.
Based on the above, the final definitions of this plan are as follows:
- BCDE = I
- ACDF = I
- ABEF = I
Since all the definitions in this plan consist of four letters, this plan is Resolution 4.
Generally, the resolution of a two-level partial factorial plan is equal to the length of the shortest word included in the definitional relationship.
*A word is expressed as the product of factors such as BCDE, ACDF, or ABEF.
Changing the generator alters the confounding structure, even at the same resolution.
In this case, the plan that minimizes the number of shortest words included in the definitional relationships
is called the Minimum Aberration Design. In minimum deviation designs, confounding from lower-order effects is minimized within designs of the same resolution.
Methods for confirming main effects and confounding
By multiplying any effect by both sides of the definition relationship, we can find the effects that confound that effect.
For example, in the case of main effect A:
- A × I = A × BCDE
- A = ABCDE
- A × I = A × ACDF
- A = CDF
- A × I = A × ABEF
- A = BEF
From these results, we can see that the main effect A is confounded by A BCDE, CDF, and BEF.
*Due to the resolution of 4, the main effect is not confounded by the two-factor interaction. When we actually find the alias structure of A, we get A = CDF = BEF = ABCDE , which shows that the main effect A is confounded by the three-factor and five-factor interactions.
Confounding of two-factor interactions
In Resolution 4, there is a possibility of confounding between two-factor interactions.
Similar to the case of main effects, confounding relationships can be determined by multiplying the two-factor interaction by the definitional relationship.
Example: Which effects confound AB?
Multiplying both sides of each definition relation by AB, we get the following:
- AB × I = AB × BCDE
- AB = ACDE (4 times)
- AB × I = AB × ACDF
- AB = BCDF (4 times)
- AB × I = AB × ABEF
- AB = EF (2 factors)
Therefore, it can be seen that AB is confounded with ACDE, BCDF, and EF .
Final confounding relationships
Performing the same operation on all two-factor interactions yields the following confounding relationships:
- AB = EF
- AC = DF
- AD = CF
- AE = BF
- AF = BE = CD
- BC = DE
- BD = CE
This information is displayed under "Confounding Relationships of Effects."
supplement
- In JMP screening plans, it is not possible to directly specify individual confounding relationships.
- A confounding relationship is a complete confounding relationship that is uniquely determined by the generation rule (generator).
- Therefore, it is important to check the "confounding relationships between effects" after creating the plan and to understand which effects are confounding each other.
Reference (JMP Online Help)
