Taking All of the Quadratic and Binary Effects into Account in Screening Design

[#1]

Hi, 

I conduct an optimization study using JMP Pro 17. My problem has 13 design variables and I try to perform a Screening Design.

I perform Screening Design from the ‘DOE --> Classical --> Two Level Screening --> Screening Design’ menu. As the Screening type I choose ‘choose from a list of fractional factorial designs’ (Picture1_DOE_ScreeningDesign).

The options I chose next are as follows:

‘Number of Runs: 32,

Block Size: -,

Deisgn Type: Fractional Factorial,

Resolution-what is estimable: 4- Some 2-factor interactions,

Run Order: Keep the Same,

Number of Center Points: 1’. 

When I click on ‘Make Table’, the window ‘Fractional Factorial’ pops up (left part of Picture2_FractionalFactorial_FitModel).

After conducting the 33 experiments listed, I fill the Y column and I fit a model to the problem from the ‘Analyze --> Fit Model’ menu to conduct the Screening Design. On the ‘Fit Model’ window, JMP lists the default parameters on the ‘Construct Model Effects’, which includes main effects and some of the binary interactions (right part of Picture2_FractionalFactorial_FitModel). Additional parameters can also be added to this list by using Add and Cross buttons. When I try to add all of the binary and quadratic terms (Picture3_AllEffects_ParameterList), and hit the Run button, I see that some terms cause singularity, given in the Singularity Details list (Figure: Picture4_AllEffects_SingularityList or Picture5_X1X12_X2X5).

However, no singularity is obtained when only of the terms in each line of Singularity List is added to the ‘Construct Model Effects’ list, and each term on a single line results with the same Logworth value. What I guess is that, JMP builds the experiment set in ‘Fractional Factorial’ such that the effects of the terms given in one line of the singularity list are already taken into account. For example, 33 experiments are listed such that the effect (Logworth value) of ‘X1*X1 and X2*X2’ or ‘X1*X12 and X5*X2’ will always be the same. (Figure: Picture5_X1X1 and Figure: Picture6_X2X2 or Figure: Picture7_X1X12 and Figure8: Picture_X2X5)

If that is the case, I need a physical explanation on why I can not take the individual effects of all binary and quadratic effects into account. Is it possible to conduct a 2 level (by taking only the minimum and maximum values for each design variable) Screening Design study and taking all of the binary and quadratic interactions into account using JMP? Maybe choosing different options from the menu would help me.

Thank you for the answers in advance.

Best regards,

Ceyda Kavak

PS: I recently came across a similar discussion (Solved: No quadratic effects in Response surface experiment - JMP User Community), hence I would like to tag @statman . Thank you for your time and consideration.

[#2 · Accepted solution]

Hi @ceydakavak : The simple answer to your question " I need a physical explanation on why I can not take the individual effects of all binary and quadratic effects into account." is: You only have 33 runs.  With 33 runs you can't hope to estimate 104 parameters (13 main effects, 13 quadratic effects, and.78 pairwise interactions). You only have 33-1 = 32 total degrees of freedom so that is the maximum number of effects you can estimate: all of the main effects and some of the two-way interactions; effects on a common row in the singularity details are indistinguishable from each other, so you can only estimate only one per row. From a purely statistical point of view, there is no way to know which effect on a given row is correct (that perhaps comes via your understanding of the science/process). Screening designs are typically main effects only to see which main effects are active. You then take that subset of main effects into a further study to perhaps optimize etc. via considering interactions, quadratic effects, etc.

i.e., you are asking far too much from 13 factors with 33 runs.     

[#3]

Hi @MRB3855 , thank you for your detailed reply. 

I tried another case where I have 3 design variables, and I perform 9 experiments (8+1 center point), where degree of freedom is 8. I tried to add the quadratic interactions of X1 and X3 into account, which makes the number of the parameters to be estimated, 8. When I tried to fit a model, I still get singularity in the solution, as you can see in the attachment (3variables_fitmodel.JPG). Should not I expect to get a solution for 8 parameters to be estimated, when the degree of freedom is also 8? I also attached the screenshot where the resolution I chose is visible (3variables_designlist.JPG).

[#4]

Hi @ceydakavak : Yes: X1, X2, X3, X1*X2, X1*X3, X2*X3, X1*X2*X3, and one of either X1*X1, X2*X2, X3*X3.

But…that can be problematic on a lot of levels. And it is like fitting a straight line through two points. I refer you back to @statman ‘s post.

[#5 · Accepted solution]

"If that is the case, I need a physical explanation on why I can not take the individual effects of all binary and quadratic effects into account. Is it possible to conduct a 2 level (by taking only the minimum and maximum values for each design variable) Screening Design study and taking all of the binary and quadratic interactions into account using JMP? Maybe choosing different options from the menu would help me."

As explained by MRB, You do not have sufficient DF's to include all of the desired effects in the model (hence singularity).  When JMP encounters this, it will continue on , however it will drop terms from last entered from the model until all of the DF's are assigned.  This is why you still get a fit model output.

Can you separate all 2nd order linear and non-linear effects (what you describe as binary and quadratic)? Yes, but it will take sufficiently more treatments to do this. When using fractional factorials, there are Principles we use:

1. Scarcity:  There are only a small number of significantly active effects for a useful predictive model

2. Hierarchy:  1st>2nd>>3rd>>>4th order effects.  Build models in hierarchical order

3. Heredity:  Mostly for analysis in trying to help advise separating aliased effects of the same order.  In order for  2nd+ order effect to be active, at least one parent (main effect) must be active.

The fourth principle, and perhaps the most important, is the effects need to make engineering or scientific sense.  I suggest predicting these possible effects before running the experiment.  This reduces the effect of bias in interpretation.

 My suggestion, since you are screening, is to prioritize which effects you really predict to be active.  Don't try to separate everything in the first study.  This is inefficient\ (what is the likelihood all 13 factors, all 2nd order interactions and all quadratic effects are active and significant?  What is the likelihood you are testing the factors at optimum levels?). Instead, design the screening experiment with the idea you will be iterating.  The first experiment is designed to help design a better experiment (e.g., smaller list of active factors and better level setting).  If you are suspicious of non-linear effects in the design space, it may be possible to add center point runs to test this hypothesis efficiently.

[#6]

Hi @statman , thank you for your detailed reply.

Removing all the binary interactions from the default parameter list (fitmodel_default.JPG) and adding two quadratic terms (fitmodel_modified.JPG) reduces the total number of parameters to be estimated. But I still get singularity (fractionalfactorial_modified.JPG). In addition to the number of parameters to be estimated, choosing only one interaction from each line of the singularity list seems crucial. And taking individual terms in each line into account separately would result with the same effect. So it seems like which interaction I chose from each line ("predicting these possible effects before running the experiment") has no importance on the model fit for the input and output relation of the problem.

[#7]

The you used JMP to design the experiment, it created an Aliasing of Effects table:

You can only have one term from the "string" of aliases in the model. Example string:

X1*X2

= X5*X12 = X6*X7 = X8*X9 = X10*X11

If you put more than one of those terms in the model, you will get singularity as you are asking JMP to analyze the same DF more than once.  Again, if you used JMP to design the experiment, you should see a list of items in the top left corner with green arrows.  Click on the DOE Dialog arrow and you will see the aliasing structure.

"So it seems like which interaction I chose from each line ("predicting these possible effects before running the experiment") has no importance on the model fit for the input and output relation of the problem."

You did NOT predict the rank order of model effects prior to running the experiment.  Had you done this, you could have designed an experiment to ensure those predicted effects were not aliased.

[#8]

Hi @ceydakavak  : in addition to what @statman  said, with your design you can only estimate one quadratic effect: X1*X1, or X2*X2, or X3*X3, etc. If you try to include more than one of those you will have a singularity, regardless of how many degrees of freedom are taken up.  The ability to estimate one of them comes from the center point. There is no statistical way to know which one is correct though. That would perhaps come from your understanding of the science etc.

[#9]

Thank you @MRB3855 and @statman for your time!

[#10]

Thank you @statman and @MRB3855 for your time!