How are the default number of runs calculated in Custom DOE? Specifically I-optimality criteria.

[#1]

Looking at the guidelines of the custom DOE from JMP, i have found this description of the default runs (attached image) but I dont know exactly how is the default number of runs calculated? especially if there are different factor levels or different type of factors (discrete or categorical) Any help is appreciated.

P.S: I have read previous posts that mention that JMP adds atleast 4 runs to the minimum but that doesn't always seem to be the case.

[#2]

Hi @statistic_nerd,

There are indeed several responses on this topic in the following discussions :

Custom Design: Default Number of Runs 

how JMP design the number of runs for Default under DOE (custom design)

Plan optimal // facteurs catégoriels 

After several experimentations and observations, here is what I found as guidelines for understanding the default number of runs proposed by JMP :

Depending on the minimum number of runs based on your assumed model, JMP will try to add minimum 4 runs to this minimum number of runs and ensure a balanced design. If this number of recommended runs is not balanced regarding the levels of some factors, JMP will add a number of runs to satisfy a good balance across all factors depending on their levels number. For example, if you have one categorical factor at 5 levels and another one at 3 levels, the minimum number of runs is 7 for a main effects model. Adding 4 runs won't help balance the design across all factors levels (11 is not a multiple of 5 and 3), so it will recommend a multiple of these 2 numbers, and the lowest number will be recommended, here 15.

For continuous factors only and main effects model, the default number of runs change by size 4 : from 2 to 3 factors, the default number of runs is 8, from 4 to 7 factors it's 12, from 8 to 11 it's 16, etc...

The number of runs is not affected by the choice of optimality criterion, as the calculated runs number is done prior computing the matrix. The recommended number of runs is influenced by the factors types, number of levels and the assumed model (as it impacts the minimum number of runs). The optimality criterion can however modify the treatments combination of factors levels in the added recommended runs (the experimental matrix) : I-optimality will favor new runs that can help minimize the average variance prediction over the design space for example. See Optimality Criteria for more info.

Hope this answer will help you,

[#3]

Hi @Victor_G ,

Thank you for your prompt and detailed response. I have two follow up questions:

1. What if there is an interaction between the continous and categorical factor, then how is it treated in terms of levels and the number of runs? Are they included in the calculation for the default number of runs or is that number based on the main effects? for example categorical 3 levels, discrete 4 levels, and an interaction effect between the two.

2. Can you further elaborate about the discrete levels changing with multiple of 4? For example if there is a main effect with 3 discrete levels, and another with 8 discrete levels, how would their default number of runs be different than main effect with 4 levels and main effect with 8 levels?

[#4]

Hi @statistic_nerd,

I mentioned model with main effects in my previous response, as the assumed model you specify has an impact on the number of minimum runs needed, and thus, on the recommended default number of runs.

Let's take an example with a simple design involving a 3-levels categorical factor and a 5-levels categorical factor. 

• When the assumed model is comprised only of main effects, the minimum number of runs is 7, and the recommended number of runs is 15 (lowest multiple of 3 and 5, higher than 7+4), to make sure the number of runs is balanced across all levels configuration.
• If you assume a model with main effects and the 2-factors interaction, the minimum number of runs is 15, and the recommended number of runs is 30 (lowest multiple of 3 and 5, higher than 15+4).

To go back to your initial scenarii :

1. With one categorical 3-levels factor and one discrete 4-levels factor, with a model comprising of main effects and 2-factors interaction, the minimum number of runs is 6, and recommended number of runs is 12, as it's the lowest multiple of 3 and 4 that is higher than 6+4.
   For k-levels categorical factor, you need to estimate k-1 main effects estimates.
   For k-levels discrete numeric factor, you need to estimate at least 1 main effect, and if possible you can augment the number of effect to be estimated with higher order effect up to k-1 polynomial order effect, so from 1 to k-1 model terms (by default JMP proposes the estimability of these polynomial effects as "If Possible", so it's not counting them in the minimum number of runs required, unless you switch the estimability to "Necessary").
2. I mentioned the size 4 for designs and model using continuous factors only. If you create a design with one 3-levels discrete numeric factor and one 8-levels discrete numeric factor, the minimum number of runs for main effects model will be 3 (intercept + one main effect for each factor), and recommended number of runs will be 24, lowest multiple of 3 and 8. Adding an interaction term will increase the minimum number of runs to 4 and not change the recommended number of runs 24, already the lowest multiple of 3 and 8 higher than 4 (or 3).
   If you specify the higher order effect estimability in the model as "Necessary", you will need as many runs as terms to estimate, so minimum will be 8 (or 9 with interaction between the two factors), as JMP limits the order to 5 for polynomial effect, but minimum is 10 (or 11) if you go up to polynomial 7th order term for the 8-levels discrete numeric factor, and since 24 is the lowest multiple of 3 and 8 (and higher than 10+4  / 11+4), the recommended number of runs will still be 24.

Hope this small example will help you,