Plan de mélange_Choix interaction

[#1]

Bonjour, 

Dans le cas d'un plan de mélange à 8 MP, je souhaiterais savoir comment faire pour choisir les meilleures interactions possible avec un faible nombre d'essais? En effet, je n'arrive pas à trouver les interactions intéressantes afin de prédire de manière correcte mon modèle. Je peux avoir un R², RMSE et résidus de student qui me semblent bons pour ma réponse mais le VIF est toujours très élevé et je n'arrive pas à le réduire de manière suffisante. 

Je vous met en PJ la matrice d'essais si besoin. 

La réponse est un temps de fabrication :

          - 0 min = fabrication instantanée

          - 1500min = fabrication impossible => choix arbitraire comme valeur 

Merci pour votre aide

[#2]

Hi @SquaresJackal10,

There are a lot of questions in your post. Here are some answers from me:

• Regarding modeling from Mixture designs: The topic of modeling with mixture designs is a broad topic already well discussed in this forum. You may be interested in the following discussions :
  Backward regression for Mixture DOE analysis with regular (non pro) JMP? 
  How to use the effect summary effectively for a mixture DOE? 
  Analysis of a Mixture DOE with stepwise regression Analysis Strategies for Constrained Mixture and Mixture Process Experiments Using JMP(R) Pro 14 ( 20... 
  
  As the emphasis of this type of design is more on predictivity and optimization than screening/statistical significance, I would create, refine, compare and select models based on predictive accuracy (RMSE) or Information criterion (like AICc / BIC), but not based on p-values (model or individual effects). 
  You have chosen a model-based mixture design (not a Space-Filling approach), so you already have assumed a possible complete model you would like to investigate. Because of this assumed model, I would recommend in the analysis to start from the full model with the possible terms you have assumed in the design creation, and start removing terms in the model (except main effects), based on the predictive performance of the model (RMSE for example), NOT on individual p-values/logworth of each term (because of multicollinearity/correlation among mixture factors, no intercept in this type of model, p-values/logworth are not a valid metric for model selection).
  To help you with this task, you can use the platform Generalized Regression (JMP Pro) with different estimation methods, like Pruned Forward Regression and Best Subset for example. These estimation methods help finding a suitable model based on an information criterion, balancing model complexity with model accuracy.
  

• About the VIFs: You'll always have high VIFs in a Mixture designs as the mixture factors are not independent, but linked through the mixture constraint: in your example, the sum of mixture factors MP1 + MP2 + MP3 + MP4 + MP5 + MP6 + MP7 + MP 8 = 1.
  Due to this mixture constraints (+ any other constraints you may add), the factors are not independent : changing the level of a mixture factor has an impact on others. It creates a multicollinearity situation that increases VIFs for your mixture factors.
  

• Regarding your "arbitrary value" (detection limit): It seems you have a situation of detection limit. From the JMP Help: "The Detection Limits column property defines bounds beyond which the response cannot be measured. You can use these limits to specify a censored response in either the Distribution or Generalized Regression platform."
  Not considering this censoring and using only biased values in the modeling can have consequences: the mean can be biased from the true mean, and not considering this censoring my lead you to a different model. Having the detection limit property column for your response helps, as it provides information to the model that response values above a certain threshold (here I have used 1000) are not really "certain", so the model can do error without being strongly penalized for these errors. This is why the Generalized Regression platform would be strongly recommended here, to support this type of censoring.
  As an example, I have run the analysis with GenReg on your data with and without detection limits (upper limit set at 1000) property column. The outcomes are different:
  • On the estimates and terms present in the model (models built in the same way in both cases, GenReg with Pruned Forward Regression estimation based on AICc criterion):
    
    In the model built with censoring, several interactions are added (MP2xMP6, MP3xMP5 and MP4xMP5) compared to the model without censoring
    
    
  • On the optimum found: when using the Profiler in both situations (with and without censoring), the optimum found is different (different values for MP2, MP5 and MP6, linked to the interactions missing from the model without censoring):
    

You can learn more about detection limits and censoring here: Limits of Detection (LoD) - New in JMP Pro 16 - User Community.

If you don't have JMP Pro, you could do a two-step modeling approach: 

1. Create a binary response (production possible / production impossible) and use a logistic model to predict the classes.
2. For the class "production possible" only, use a regression model.

This way you can still get results with a reliable approach.

Note that further refinement could be done in the modeling done here, as a time response may need a different distribution set in the GenReg platform (possibly a zero-inflated distribution type like ZI-Gamma to handle the large number of zero values). Please find attached the datatable with the scripts used in my post (fit GenReg with Normal distribution in both cases, comparison of models, + two steps approach and GenReg with ZI-Gamma).

Final remark: when augmenting an original mixture design, you don't need to add a constraint that the sum of the mixture factors should be between 0,999 and 1,001 for example. The fact that the factors have the Mixture column property already enforce this constraint in the design creation, no matter if you're using the Custom design or augmenting an original mixture design.

Hope this answer will help you,

PS: Make sure your whole datatable is anonymized when sharing tables with the Community; in this case, some of the scripts lead to opening the original datatable, with the original factor names. I have removed these sensitive scripts in the document shared.