Analysis model for categorical and quantitative factors in full factorial design without center points

[#1]

Hi,

I am working in a experiment with categorical and continuous factors  each at 4 levels without center points in full factorial design with 3 replicates (4*4*3). For the response variables, when plotted it is non-linear  and tried to apply second order polynomial model for data analysis.  Do you think this is right way? Does only using  full factorial classical model (linear regression ) with scaled continuous factor help to data analysis when my response is non-linear?  IS it not correct to use polynomial regression model because my model is not RSM or my  design is without center points?

So, what  is the better way to do the data analysis here`?

[#2]

Hi @Sabda,

Welcome in the Community !

There is a lot to answer in your question(s):

• From what I understand, you have specified 4 levels for your continuous and categorical factors. That means your model could include polynomial terms for the continuous effects up to cubic terms (order 3). You indeed need k+1 levels to estimate kth-order polynomial terms. So you could assume a model with main effects, interaction effects, quadratic and cubic polynomial effects (and even partial cubic effects) for the continuous factors, in the limit of the degree of freedoms available to estimate these terms. Even if you don't have middle levels for your continuous factors, you can estimate quadratic effects as you have tested more than 2 levels (min/max) for these factors. So even if you didn't specify a RSM model in the "traditional" way (by specifying main effects, interactions and quadratic effects in the Model panel or directly by clicking on "RSM"), the model you're able to fit with this model can clearly help estimate a response surface.
   
• Centre points are useful to check for curvature, reduce the prediction error in the center of the factor region and test for lack of fit due to nonlinear effects, but they are not helpful to identify the responsible quadratic effect. So adding centre points wouldn't have helped you fit quadratic effects, only give you the opportunity to check for curvature (but you can already do this since you have 4 levels for your continuous factors) and test Lack of Fit (but you can also already use this test since you have replicates).
  
  
• Here is how I would proceed:
  • Visualize your results with Graph Builder, to catch some patterns of interest (for example non-linear ones). Note that in Graph Builder, you can use Line of Fit to see which fit and degree seems to be adequate for the relationship you're observing. This can help later in determining the adequate model terms to include.
  • Try to fit a simple model (full factorial) and evaluate how the fit is. Check the residuals, the lack of fit test to better understand which next steps seem appropriate : is the lack of fit test significant and the pattern of the residuals not random (and with your earlier observations, are some higher order terms needed) ?
  • Based on your observations and first model fitting, look for improvement by adding model terms that may improve model performances. You can also give a try to The Fit Two Level Screening Platform, which can help you determine which main effects, interactions and possibly higher order terms to include in the model. I know the name of this platform is Two Level screening, but I used this platform with success to identify higher order terms : https://community.jmp.com/t5/Groupe-francophone-des/D%C3%A9couverte-des-plans-OML-Orthogonal-Main-Ef... 
  • Finally, iterate to come to a final model and evaluate it completely, by looking at R² and R² adjusted (is the gap low between these values), RMSE (and compare it to your experimental uncertainty), AICc and BIC information criterion (available in Summary of Fit, and compare the values to your first model to compare how much complexity has been added in this final model and how much it has improved the model ; the lower the better), and check the residuals, to make sure a transformation may not be needed (like a Box-Cox Y Transformation).
    Do take time to visualize the results of the model, compare actual vs. predicted values, check prediction profiler... 
  • Confirm the relevance of your model by doing other experiments that will serve as validation runs.

If you need a more detailed guidance, maybe you could share an anonimized version of your dataset ?

Hope this answer will help you,

[#3]

Dear Victor, 

Thank you for the answer and I had previously worked on simple full factorial model . Also plotting the graph, I got curvature with varying levels  for continuous factor. So, I also tried with second order polynomial order. Doing so, the R2 did not change that much (2%).  I will try again  to validate what I did before and get back to you with detailed results  to help me choose the right approach. 

Thank you,

Best,

Sabina 

[#4]

Hi @Sabda,

It's difficult if ot impossible to help you further without looking at the design structure and results.
To follow up on @statman advice of using continuous factors, you can have a look at these two Discovery Summit presentations :

• Coding with Continuous and Mixture Variables to Explore More of the Input Space (2022-US-45MP-1103): https://community.jmp.com/t5/Abstracts/Coding-with-Continuous-and-Mixture-Variables-to-Explore-More-... : This talk shows how switching from categorical to continuous factors the definition of the design space enable to have a broader inference space where it is possible to find solutions.
• Increase Efficiency and Model Applicability Domain When Testing Options That Are at First Glance Multilevel Categorical Factors : https://community.jmp.com/t5/Abstracts/Increase-Efficiency-and-Model-Applicability-Domain-When-Testi... : This talk shows how using many descriptors for chemical structures and reducing the dimensionality through PCA can help select continuous factors that allow to create a DoE built on a large design space.

If you are working with chemicals and are interested in the last talk, you can also check this conversation Efficient DOE of one multi-level (3+) categorical variable and many continuous variables where I explain the process to go from categorical chemical factors to continuous ones using chemical descriptors.

Hope these links may help you,

[#5]

It is virtually impossible to provide sound advice with the limited information given.  Victor has done a great job providing general advice.  I'm not sure why you would want a complex polynomial (>quadratic)?  It is typically non-sensical to develop a parametric model with categorical factors.  There is no continuum.  So you might first decide if the categorical factor(s) is significant and are there any interactions between the categorical and the continuous variable(s).  Then choose the level for categorical and develop a model for the continuous variable(s).