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]

Hi @CompositeCamel5,

Welcome in the Community !

When you enter a block factor in your Custom design, this block factor is considered as a fixed effect, so your model will have a new main effect to estimate. Since there are no interactions between the fixed block effects and other factors, or any higher order effect for the block effect, the number of runs will change based on the number of runs for each block in your design.

• For example, if you have a RSM model for 3 continuous factors, you have to estimate 1 intercept, 3 main effects, 3 interaction terms and 3 quadratic effects, so the minimum of runs is 10.
• If you add a blocking factor with 6 runs per block, the minimum number of blocks you have is 2 and the minimum number of runs is 11 : you need to estimate the same effects as before, and add one effect for each level of your blocking factor (except one). So for k blocks, you need to estimate k-1 main effects corresponding to the different blocks (the last block main effect can be estimated from the others). In this case with 2 blocks, this only adds one main effect to estimate, so we only need 11 runs (instead of 10) to estimate the block effects. JMP accepts a small imbalance between blocks (in this case, having a block with 6 runs and another one with 5 runs).
• If you add a blocking factor with 3 runs per block, you now need to estimate 10 terms + 1 term per block (-1). As you already have a minimum of 10 runs without block, that means you need at least 4 blocks to distribute you runs (so minimum 12 runs, 4 blocks of 3 runs). But by doing so, you now increase the number of terms to estimate to 13 (since you have 4 blocks, so 3 main effects terms to estimate), which is superior to the number of minimum runs mentioned before. So that means you need to increase the number of blocks to 5, with 14 terms to estimate, and JMP mentions the minimum number of runs to 15, so that each block is complete and balanced. 

Hope this answer will help you,

[#2]

I really appreciate your answer and would like to ask one more question: On this basis, if the number of center points is incorporated, how is the minimum number of experiments calculated? That is to say, in a customized design, if block factors are included among the factors, Response Surface Methodology (RSM) is integrated into the model, and the number of center points is added, how should the minimum number of experiments be calculated?

[#3 · Accepted solution]

Hi @CompositeCamel5,

If you add centre points, it can change the minimum number of runs in the case where you don't have any spare runs to estimate the terms in your model. Centre points can estimate curvature effect but not allocate this curvature effect precisely to one specific quadratic effect (it can be allocated to any of the quadratic effect).

In the last example with the design with 5 blocks of 3 runs, you can add up to 2 centre points without changing the number of runs : as you need to estimate 14 terms, you can add 1 centre point (to get to 15 total minimum runs), and you can add one centre point that will estimate one of the quadratic effect in replacement of one design point in the matrix (the other quadratic effects will be estimated with specific points). But since you can't allocate specifically centre points to specific quadratic effects, if you increase the number of centre points to 3, then the number of runs increase to 16 : As you need to estimate 14 terms, one of these run can be a centre point for one of the quadratic effect and you need 2 more runs for the 2 extra centre points specified. A small imbalance is accepted here.

Hope this further helps you in your understanding, 

[#4]

Thank you very much for your explanation — it was really helpful to me! To deepen my understanding, could you please provide two further examples? For instance, when a blocking factor is included and RSM terms are added to the model, how would the minimum number of runs be calculated when the number of runs per block is 5, and when it is 6, with different numbers of center points added?

[#5 · Accepted solution]

Ok @CompositeCamel5, I'll try my best.

If we take the first example, 3 continuous factor and a RSM model design, and make the number of runs/block and centre points vary :

• Without any blocking or centre points, the minimum number of runs is 10, because you have 10 terms to estimate in the RSM model : 1 intercept, 3 main effects, 3 interaction effects and 3 quadratic effects.
• For 5 runs/block, the minimum number of runs is 15 for 1 to 4 centre points specified. If you specified 5 centre points (or more), the minimum number of runs change to 16 (or more).
  It's normal, following the explanation from before: with 15 runs by default and 5 runs/block, JMP recommends 3 blocks, so you have to estimate 12 terms (10 from the RSM model and 2 for the block effects). So you can add freely 3 centre points (because of the 3 remaining runs not allowed to estimation of these 12 terms), and you can increase the number of centre points to 4 without any augmentation of the minimum number of runs because one of the runs used for the estimation of one specific quadratic effect can be replaced by the last centre points. But you can't estimate (and differentiate) more than one quadratic effect with the use of centre points, so if you increase the number of centre points to 5, you now need 16 runs, because the added centre point won't be helpful to estimate another quadratic effect, so you need an extra run for this estimation.
• For 6 runs/block, the minimum number of runs is 11 for 0 to 1 centre point specified, 12 runs for 2 centre points, 13 for 3 centre points, etc... You have to estimate 10 terms as before, and with 6 runs per block, you only need 2 blocks, so one extra main effect term for block to estimate. So you have 11 terms to estimate, and if you add one centre point, you can still estimate one of the quadratic effect with it without changing the number of runs. For the same reasons explained before, if you want more centre points (2+) in your design, then your minimum number of runs will increase, as the centre points will "take the place" of runs used for estimation (and differentiation) of quadratic effects, and since they can only be used to estimate one quadratic effect, you'll need other uns to estimate separately the other quadratic effects from your RSM model.

Hope the follow-up explanation clarify the minimum runs calculation,

[#6]

Thank you very much for your answer. I now fully understand the earlier question regarding the addition of center points. I would like to ask one more question: if a blocking factor is included, and the model either incorporates RSM or not, along with added replicates, how should the minimum number of runs be calculated?

[#7]

Hi @CompositeCamel5, sure.

In a "Custom design" JMP displays "Number of Replicate Runs", meaning the number of run(s) to replicate. Replicate run is an independent repetition of one experiment in the design. JMP choose the location of replicate runs based on the design optimality criterion chosen, to improve estimation of effects and/or to reduce the variance prediction : Design Generation

As replicate runs do not enable to estimate more effects in the design (they are only repeating a run that contributes to effect estimation), they can increase the minimum number of runs if you do not have enough independant runs to estimate all the effects assumed in the model.

In the example of a RSM Custom design for 3 continuous factor and 1 blocking factor with 6 runs/block, the minimum number of runs was 11 with 11 terms to estimate, so since you do not have any degree of freedom left to include a replicate run, the minimum number of runs will increase to 12.

Hope this clarify the situation,

[#8]

Thank you so much for your answer!!!

What I’d really like to explore now is the relationship among **replicated run count**, **minimum required number of trials**, and **suggested (default) number of trials** in experimental designs that include a **blocking factor**.

For example, you mentioned a design with three continuous factors and a block factor where each block contains 6 runs. As I incrementally increase the total number of trials (i.e., add replications), I notice that the **minimum required number of trials** sometimes changes and sometimes stays the same.

I’d like to understand the underlying pattern or rule governing this behavior. Could you provide a clearer, more illustrative example to explain this?

Thank you!

I’d like to understand the underlying pattern or rule behind this behavior. Could you provide a clearer, more illustrative example to explain this?

Thank you!

[#9]

Hi @SampleOwl447,

If you're interested in the recommended number of runs, you can check the following topics :

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

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

Custom Design: Default Number of Runs 

The documentation can also help you understand how this default number of runs is calculated : Design Generation 

If you read the previous answers, you can get the general understanding of the calculation :

1. Count the number of terms you need to estimate based on your assumed model. Remember that k-levels categorical factors or blockign factors creating k blocks will each require k-1 terms to estimate. Don't forget to count the intercept.
2. Depending on your design choice in terms of centre points and/or replicate runs :
   • Centre points can be added without any change on the runs number only if you have some spare experiments not used for terms estimation, and/or if you use centre point(s) for the estimation of only one quadratic effect (without change on the number of runs). If you add more centre points than the number of degree of freedom you have, then the number of runs will increase.
   • Replicate runs do not enable to estimate different terms than the term the original run is estimating. So they can be added as long as you have some degree of freedom / spare runs left, but else, they will increase the number of runs.

In the specific example mentioned :

1. You have 11 terms to estimate, and the minimum number of runs is 11 in these settings. So you don't have any degree of freedom left to include replicate runs. If you add one (or two), then the minimum number of runs will increase to 12. Note that for two replicate runs, you will not have enough degrees of freedom (10 independant runs + 2 replicate runs for the estimation of 11 terms, so it will create a Singularity in your model, as effects can be all estimated independantly).
2. If you increase the number of replicate runs to 3, the minimum number of runs increase to 18. You have a big change happening here because :
   • You still have 11 terms to estimate and you want 3 replicate runs, so your minimum number of runs should be 14. However, 14 runs can't be shared in 2 blocks of 6 runs, so you need to add an extra block (3rd one).
   • By adding a third block, you now have 12 terms to estimate (+1 because of the new level of the block requiring a new block effect estimate), so 12 runs + 3 replicate runs = 15 runs. Since you specify 6 runs/block, JMP will increase the minimum number of runs to 18 (3 blocks of 6 runs). You still have some degrees of freedom left if you want to add additional replicate runs or centre points.

Hope this clarifies the calculation,