In an augmented design, if the original experiment includes block factors, when conducting an augmented design for the original experiment, it is found that the number of experimental runs recommended for the augmented design (after adding model terms) is not simply the sum of the number of runs in

[#1]

In an augmented design, if the original experiment includes block factors, when conducting an augmented design for the original experiment, it is found that the number of experimental runs recommended for the augmented design (after adding model terms) is not simply the sum of the number of runs in the original experiment and the number of newly added model terms. How should the number of recommended experimental runs be calculated when performing an augmented design that includes block factors?

[#2]

Hi @CompositeCamel5,

Indeed, if you augment a design already including a blocking factor, the recommended number of runs for the augmented design will depend on the number of newly added terms as well as the number of runs per block that you can do.

Here is a simple example, with a 12-runs screening design for 3 continuous factors, 1 three-levels categorical factor, and 1 blocking factor (with 4 runs per block).
I have 8 terms to estimate: 1 intercept, 3 main effects for each of the three continuous factors, 2 effects for the three-levels categorical factors (the third level effect can be deduced from the two estimated : L1+L2+L3=0), and 2 block effects : for 12 runs, I need 3 blocks of 4 runs, so I need to estimate 2 block effects, the third one being deduced from the two others, similar situation as for the categorical factor.

If I want to augment this design :

• By keeping all the factors (including blocking factor) and stay with the same assumed model, the recommended number of runs is 22, but I can reduce this number to 15 or 16 (to have full 4-runs blocks), as I only need to estimate one new added term (a block effect level) compared to the 12-runs initial design. The 22 recommended runs provide more power and precision to the estimation of effects, but you can reduce this suggestion.
• By keeping all the factors (including block factor) and augmenting my model with 2-factors interaction terms, JMP still recommends 22 runs, and I now have to estimate 11 new terms : 9 interaction terms and 2 new block effect levels (as I have now 6 blocks in my design with this 22 runs recommandation). Regarding the 12 initial runs from my original design, that represents an addition of "only" 10 runs, since I had some degree of freedom left in my original design. If you had started with a minimal run screening design with 7 runs instead of 12, JMP would have recommended 23 runs in this scenario of model augmentation, as you need to have more runs to estimate precisely main effects and the addition of interaction terms in the model augments the required number of runs, so you also need to add some new block effect to estimate.

I hope this explanation and example will help you better figure out how the augment design platform works,

[#3]

I am very sorry, but I still haven’t fully understood. Let me take an example: suppose the original experiment involves 5 continuous factors and 1 blocking factor, with 2 experimental runs per block, totaling 16 experimental runs. Now, when conducting an augmented design for it—if we add one interaction term (between continuous factors) to the original model, the recommended number of experimental runs becomes 20; if we add two interaction terms (between continuous factors), the recommended number of experimental runs increases to 24. Could you please explain why this happens? And how is the recommended number of runs calculated?

[#4]

As illustrated in the example (or figure) provided：

[#5]

Hi @CompositeCamel5,

Ok, let's see with your example.

You have in the original design a 16-runs screening design, where you need to estimate 12 terms: 1 intercept, 5 main effects, and 7 block effects (since you have 8 blocks of 2 runs in this original design).

• If you add one interaction term, you'll need to estimate 1 interaction term as well as one additional block effect in the case of adding 2 runs. But by adding only these 2 runs (total 18 runs), you'll have no degree of freedom left to estimate the Std error of these effects (p-values, etc...), since you only add 2 runs for 2 new terms. Hence JMP recommands adding another block with 2 runs, so that you have 4 additional runs for 3 new terms (1 interaction term and 2 block effects). In total, you'll have 20 runs, with 15 terms to estimate.
• If you add two interaction terms, you'll need to estimate 2 new interaction terms, so you need at least 4 new runs because each of these interaction terms will need a new block of 2 runs. But you'll be in the same situation as before, with 4 new runs for 4 new effects to estimate (2 interaction and 2 new block effects to estimate), so no degree of freedom left to estimate. So JMP recommends adding 6 runs (3 blocks of 2 runs), that enables to estimate 5 new effects : 2 interaction terms and 3 new block effects.

Note that you may find this explanation with an error message when trying to force the number of runs below the recommandation proposed by JMP (here for example 1):

Hope this explanation may solve your question,

[#6]

Here is the **formal English translation** of your full paragraph — preserving **all original meaning and detail**, without any shortening or simplification: --- **Formal English Translation:** Thank you very much for your patient explanation. However, I still have the following questions that are quite confusing to me: 1. In the design with *five continuous factors and one blocking factor (two runs per block, for a total of 16 runs)*, when performing an augmented design by adding one interaction term, your explanation states that adding only two additional runs (for a total of 18 runs) would leave no degrees of freedom to estimate the standard errors (and thus the p-values) of these effects. Why would there be no degrees of freedom available for estimating these effects and their p-values? Wouldn’t there still be four degrees of freedom remaining for estimation, since (18 - 14 = 4)?

2. In the design with *five continuous factors and one blocking factor (four runs per block, for a total of 12 runs)*, when augmenting the design by adding one model term, the recommended number of runs after augmentation remains 12. When adding two or three model terms, the recommended number of runs also remains 12. Only when adding four model terms does the recommended number of runs increase to 16, and when adding nine model terms, the recommended number of runs becomes 23. Is the principle behind this change in recommended run numbers the same as in the previous example?

3. In the design with *five continuous factors and one blocking factor (four runs per block, for a total of 12 runs)*, when nine interaction terms (without any power terms) are added, the recommended number of runs becomes 23; however, when the nine interaction terms include two power terms, the recommended number of runs decreases to 22. What is the reason for this difference?

[#7]

Hi @CompositeCamel5,

1. No, you can't group all individual runs together and compare the total number of degree of freedoms brought by the augmented design to the total number of terms. The original design didn't include the interaction term in the model, so the 16 runs are already used for the main effects models only. If you add new terms (directly in the model panel or indirectly through adding new blocks), you have to add specific new runs for estimating these effects. Hence my answer, with 18 runs you will have no degree of freedom left, because you will have 2 new runs to estimate 2 new terms (a block effect and the interaction effect). 
2. In this second situation, you start with the original design with 12 runs to estimate 8 terms: 1 intercept, 5 main effects, and 2 block effects (because in a 12-runs design involving a blocking factor with 4 runs per block, you'll need 3 blocks, so 2 block effects to estimate). So you have 4 degree of freedom left.
   As you have created your design with more runs than needed and the design is a factorial one, the remaining runs are other combinations of factor levels that could be used for either more precise estimation of main effects, or for detecting significant interactions between 2 factors (you can have a look at the Color Map on Correlations in Evaluate Designs platform, or using the script "Evaluate design" once your design is created) : 
   
   

   As there is no perfect confounding between main effect and interaction terms, even if you add one, two or three new effects, you have the possibility to detect and estimate them without adding runs, as these effects are not fully confounded/aliased to other terms and you have enough degree of freedom to estimate them (you have 4 DFs left).
   
   I don't have the same results as you have : For four terms, JMP recommends by default 15 runs (creating a last incomplete block with 3 runs), for five terms it recommends 16, for six terms it recommends 19, for seven terms it recommends 20, for eight terms it is 23, and for nine terms it is 32.

3. I don't have the same results. In this situation of adding 7 interaction terms and 2 power effects, JMP recommends 24 runs, which is in accordance with all the explanations and calculations I have detailed in this answer and the previous ones. You'll have 20 terms to estimate : 1 intercept, 5 main effects, 7 interaction effects, 2 power effects, and 5 block effects (because you have 6 blocks).

Hope this answer will solve your question(s),