In a customized design, if block factors are included in the factors and RSM (Response Surface Methodology) is incorporated into the model, how is the minimum number of experiments calculated?

[#1]

Hello, regarding the case where the number of experimental replicates is set to 2 and the minimum number of runs is 12, may I ask why the number of runs is not increased to avoid the occurrence of singularity? Is it acceptable for the design to remain singular?

[#2]

Hi @CompositeCamel5,

A singular design is indeed not recommended, as you won't be able to differentiate and estimate terms in your model.
However, it's not an impossible task if you follow the three principles behind the analysis of DoE : Effect Hierarchy, Effect Heredity and Effect Sparsity. You can start building your model with domain expertise and following these principles, starting with the identification of active main effects before including higher order terms to improve model performances and better respect regression model assumptions.

The pattern in the number of runs recommended by JMP is here a bit surprising, but I think this is a tradeoff (and manageable risk) of not increasing the design size from 12 to 18 for "just" one degree of freedom (information) missing. There are enough estimation methods and ways to deal with this situation, and effect sparsity principle should help you analyze such "supersaturated" situations, as not every effects will be important/significant.

Hope this answer will help you,

[#3]

Thank you for your patient explanation.
I’m still not fully clear about how the minimum and default number of runs are calculated in Custom Design after adding replicate runs.
I’ve summarized the logic in a table, but I still haven’t figured out exactly how the minimum number of runs changes as the number of replicates increases.
Could you please explain each case in my table?
Thank you very much!

[#4]

Hi @CompositeCamel5,

I think I have been patient enough on your two topics, this one and In an augmented design, if the original experiment includes block factors, when conducting an augmen... , so I won't deep-dive into each of your 5 use cases with all the different settings you have. 

For all the situations you have listed, you have my responses to help you : without replication, the minimum number of runs corresponds to the number of terms to be estimated ; with replication and/or when adding centre points, you need to have the minimum number of runs depending on your model + the number of replicates or centre points you have specified as a minimum.

Best,