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

Sabda

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`?

Victor_G

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-Effects-Screening/m-p/843161 
  • 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,