topic Re: DOE Mixtures with disallowed combinations of components in Discussions

DOE Mixtures with disallowed combinations of components

Asier — Thu, 08 Jun 2023 20:58:37 GMT

Hi,

I have read one of the documents in the site (7 Component Mixture Design with Additional Constraints)

I have a similar problem; I have one mixture component (A) for which i have three different types (A1, A2 and A3). I have similar constraints to what include in your paper (min/max % of A as a whole). However i also want to evaluate mixtures with

* only A1 or A2 or A3 within total limits min/max for A

* a combination of A1+A2 or A1+A3 or A2+A3 within total min/max for A

Is there a way to put these constraints in JMP? I have found that the disallowed combination filter does not work for continous mixture factors so i dont really know how to do this.

I suppose a way around is to do 3 different designs (A1+A2) and (A1+A3) and (A2+A3) but i was wondering if there a way to do only 1 design with those constraints.

Any suggestions?

Re: DOE Mixtures with disallowed combinations of components

cwillden — Tue, 05 Mar 2019 18:47:16 GMT

You should be able to apply a linear constraint to mixture designs (e.g. to make it so that the 3 "A" components cannot exceed 70% of the total mixture, set A1 + A2 + A3 < 0.7). You would just need to specify all 3 components. It gets a bit trickier if you want to make it so only 2 of the 3 are used in any mixture. One idea would be to only specify 2 generic "A" components and then include a categorical factor that specifies *which* substances are A1 and A2. Here's what I mean:

Re: DOE Mixtures with disallowed combinations of components

tom_donnelly — Tue, 05 Mar 2019 21:06:06 GMT

Asier,

In reading your requirements on constraints for A1, A2, and A3

"* only A1 or A2 or A3 within total limits min/max for A" and

"* a combination of A1+A2 or A1+A3 or A2+A3 within total min/max for A"

I found that the original ranges and constraint in my example

A1: 0 to 0.36

A2: 0 to 0.36

A3: 0 to 0.36

with constraint 0.2 <= A1 + A2 + A3 <= 0.36

yielded a design (supporting 2nd order model, with 56 trials) that contains trials with only A1, A2, or A3, as well as trials with the three pairwise sums of A1 + A2, A2+ A3, and A1 + A3, i.e. the case that A1, A2, or A3 = 0.

Separate constraints on the pairwise sums were not required. The key I believe is allowing zero as the low limit for each components range. That may not be a guarantee that all three pairwise sums show up, but it worked. I hope this helps and that you can get your design to work.

Thanks for following my example and for your question.

Good luck,

Tom

Re: DOE Mixtures with disallowed combinations of components

Asier — Thu, 07 Mar 2019 15:22:54 GMT

Hi Cameron,

Thank you for your reply!

I saw a similar reply to yours in a different discussion. So I had a go at it in JMP. I put a few screens shots below to

clarify (factors, constraints, model and design).

The dummy variables (1&2) are only there as you said for the combination of component A (I want only two types of A at the same time in the mixture).

Then, i have also defined a "allowed combination of A" categorical variable where i have the 3 possible combinations of component A (A1+A2, A1+A3, A2+A3).

I have selected only main factors just for the sake of the exercise.

I get a design and yes, in this case, with the categorical value I can see which 2 A components I will put in the mixture. However, this design is not really telling me anything about how I have to combine the different quantities for A (1,2 or 3) for each experiment. For example, for run 1. I will only use A1 and A2 to a total of 33% but i dont get any information about how much of A1 or A2 I should put in that experiment.

Do you have any suggestion?

Thanks

Re: DOE Mixtures with disallowed combinations of components

cwillden — Thu, 07 Mar 2019 15:45:47 GMT

The intention in my suggestion was that Dummy 1 and Dummy 2 would correspond respectively to the first and second "A" component in the categorical factor Combination. So, row 1 indicates 0% A1 and 33% A2, row 4 is 36% A1 and 0% A2. Since you only have main effects, you're not going to get blends of Dummy 1 and Dummy 2 in your example. Add an interaction between Dummy1 and Dummy2 and you should see some blends.

Re: DOE Mixtures with disallowed combinations of components

Asier — Thu, 07 Mar 2019 16:03:37 GMT

Hello Tom,

Thank you for your reply!

Indeed, I can use your example as guidance (and I did). I put below, some of the screenshots to help the discussion.

In this case, i have 1 ingredient A with 3 different subtypes (A1, A2 and A3). If possible I would like to evaluate combinations of maximum 2 A ingredients (A1+A2, A1+A3, A2+A3) and obviously mixtures with only one type of A.

The total sum of ingredients in this case is only 44.34% (the other ingredients are not changed).Something similar to another of your examples.

I only put main factors in the model. I just wanted to know what I would get. In this case, for some reason, I only get mixtures with only 1 A ingredient (out of the 3 A ingredients that i have) . Do you know why we get this type of design?

Another question that I have (and this is more fundamental i believe) is about the correlations. From the correlations map, I would understand that this DOE is not great. Is that right? What can we do in general in this type of situations?

Similary, with the Power, I have the feeling that this is not great either? (Although I have to admit that I dont fully grasp the concept of power :(

Again, what can we do in this type of situations?

Thank you again for any help!

Re: DOE Mixtures with disallowed combinations of components

Mark_Bailey — Thu, 07 Mar 2019 16:33:29 GMT

The correlation of the estimates and the power are related. The correlation inflates the variance of the estimates. The inflated variance decreases the power.

This behavior (high correlation of estimates, low power) is characteristic of mixture designs because of the ever-present constraint that the components must sum to 1. This behavior is exacerbated by restricting the range of the component (not full 0-1 range) and by adding further constraints among the components.

Re: DOE Mixtures with disallowed combinations of components

tom_donnelly — Thu, 07 Mar 2019 16:46:07 GMT

Thanks Mark for responding to the power and correlation questions.
As for the design not having two As, I believe part of the problem is the ranges of all the factors particularly the non-A factors may be overly constraining the design space. The last four factors min and max ranges sum to 0.06 and 0.1604 respectively. When added to the 0.5566 proportion being held constant, the total range available for As is 0.283 to 0.3834, AND the max of the As is 0.36 making the actual sum of the mins of the last four factors 0.0834 AND when the minimum A of 0.33 is used the max sum of the last four is 0.1134. What we have is a design in the sum of the As or in the sum of the other 4 factors is only 0.03 wide. In my example highly constrained design I had at 2 "slack" variables ranging from 0 to 1 even though I knew we will never hit those levels. But these wide ranges give the As some room to have a broader range than the current situation we have here. I will try to work out a design and respond later today.
Tom

Re: DOE Mixtures with disallowed combinations of components

tom_donnelly — Thu, 07 Mar 2019 18:01:52 GMT

I think the key to getting two of the As in the trials is including in the model the 3 interaction terms among the As.

Here's my resulting design with the 10 term (vs. 7 term) model showing lots of pairwise combinations and no three-way combinations of As.

I hope this delevers what you want. Thanks for the good discussion.

Re: DOE Mixtures with disallowed combinations of components

Asier — Fri, 08 Mar 2019 12:55:21 GMT

Hi Mark,
Thank you for your reply.
I am doing this design for a colleague with no experience in DOE. I was suspecting as well that the experimental ranges are too small and I passed that comment already to "question" the reasons for that. I have to wait to see if the ranges for the factors can be increased
My first concern was about how accurately we can control the weight of the ingredients within such a small range, and secondly about the real effect (and the measurement) of changing some factors by so little.
I see from your reply that we also have mathematical consequences from the slow ranges and the linear constraints. Thanks you for the information!
Asier

Re: DOE Mixtures with disallowed combinations of components

Mark_Bailey — Fri, 08 Mar 2019 13:41:26 GMT

I often find that experimenters limit factor ranges for no good reason. The choice is guided by what they think they know and not by the requirements of the regression analysis ahead. "I think that the best level is 12% so I will test 10% to 15%." Not a good idea. That idea comes from a testing mentality, not an experimenting mentality. The range should always be as wide as is realistically possible in order to produce the largest effects. (I am fully aware that there are often physical limitations on the ranges. That limitation is not what I am talking about.) That way will maximize power (without necessarilky increasing the number of runs), minimize the relative standard error of parameter estimates and predicted response, and narrow the confidence intervals. Most informative.

Why narrow factor ranges? Why not widen them? This lesson is stubbornly ignored. It is a mentality thing.

(Note that this is my personal opinion.)

Re: DOE Mixtures with disallowed combinations of components

Asier — Fri, 08 Mar 2019 14:09:46 GMT

Hi Cameron,

Thank you again for your reply!

So Dummy1 is component A1 and Dummy2 is component A2

Then from that design table i get

A1 A2 A3 Combination

Run1 33% 0% 0% A1+A2

Run4 36% 0% 0% A1+A2

Run7 0% 35.64% 0% A2+A3

Run14 35.64% 0% 0% A1+A3

etc..

But when&how do I get to put some component A3 then?

in Run7 and A1 (0%) and

in Run14 as A2 (0%)?

Is this what you mean? or do I get this wrong?

Re: DOE Mixtures with disallowed combinations of components

Asier — Fri, 08 Mar 2019 14:18:21 GMT

Hi Mark,
I agree. I think in this case the limitations from the factors are just coming from what you said (I want 12% and I will test between 10% and 15%). That is what I told my colleague already (let’s hope we can move away from that) :smiling_face_with_smiling_eyes:
At the same time, the Power concept is not really clear to me… so I cannot really “reason” this topic well with her…. if you can elaborate further on that (or give a good starting reference on your website, I think some people will find it useful). I had a look at a video from Tom and that helped already (and probably after a few “encounters” with this topic, it will become clearer… if you have a good basic reference (with some examples, please!) you will make my weekend!
Thanks again

Re: DOE Mixtures with disallowed combinations of components

Mark_Bailey — Fri, 08 Mar 2019 14:32:09 GMT

Power is the probability that you will decide that a real effect is significant or that you won't make a type II error. That is, the alternative hypothesis is true (there is a real effect). For example, in a regression analysis such as the one used to fit the linear model to the experimental data, the null hypothesis is that a parameter is 0 (no effect). The alternative hypothesis is that a parameter is not 0. We could use a t-test to decide. The t-ratio is the (estimate - hypothesized value) / (standard error of the estimate). By widening the factor range, you produce a larger effect (and a larger estimate). That change, in turn, produces a larger numerator in the t-ratio. A larger t-ratio will have a smaller p-value. You are more likely to decide that the real effect is significant.

On the other hand, if you arbitrarily narrow the factor range, the change is in the opposite direction. You will produce a smaller effect that leads to a smaller numerator and, therefore, a smaller t-ratio with a higher p-value. Now it is more likely that the decision will be that the effect is not significant.

Caution: even if you can get them to widen the range now, once they know where the factor should be set, the mentality will return and they will want to narrow the range when this factor is in a future experiment. You never narrow the factor range.

Re: DOE Mixtures with disallowed combinations of components

cwillden — Fri, 08 Mar 2019 15:23:00 GMT

No, dummy 1 and dummy 2 are whatever components are in the row for Combination.
In row 2, Combination is A2+A3. That means Dummy1 will indicate how much A2 for that row, and Dummy2 will indicate how much A3. In that row, A1 is 0% because Combination indicates it is not part of the mixture, A2 = 0% (Dummy1), and A3 = 33% (Dummy2).

For row 7, it’s the same thing except now A3 = 35.6% and A1 and A2 are 0%.

In short, if Combination = A1+A2 then Dummy1 is A1 and Dummy2 is A2, if Combination = A1+A3 then Dummy1 is A1 and Dummy2 is A3, and if Combination is A2+A3 then Dummy1 is A2 and Dummy2 is A3.

Re: DOE Mixtures with disallowed combinations of components

Asier — Fri, 08 Mar 2019 16:06:24 GMT

Hi Tom,

Thank you again for your reply.

I did include the interaction between the A components and I get something similar to your design. I attach the views below.

Would you say that this is a good "design" or do you think as well that we should make more effort to enlarge the ranges for the factors to have a more robust experimental program?

Thanks

asier

Re: DOE Mixtures with disallowed combinations of components

tom_donnelly — Fri, 08 Mar 2019 16:55:33 GMT

Asier,

Happy to see you can produce a similar design satisfying the desire to have binary blends of the As.

As for is it a good design? You never stated and I never asked what is(are) your goal(s)? If it is to be able to optimize the process, then why not include all interactions to make a potentially more predictive model? If it is to do screening to find out better performing levels or whether a component makes any difference over the proposed ranges, then I would say "yes, it is good." As you have already discussed with Mark, expanding ranges (being bold) helps to increase the size of the effect relative to the noise in the process and therefore increases power. Good luck experimenting.

Tom

Re: DOE Mixtures with disallowed combinations of components

Asier — Mon, 11 Mar 2019 15:02:21 GMT

Hi Tom,
Thank again for your reply.
I think the final objective would be to optimize the process (however, my colleague for the time being is not asking for that, just for "an experimental program to evaluate some improvements". This is probably one of the reason of the small ranges for the factors (because they already have an idea of "what" the want to replace and by "how much"... although they don't really know the impact on performance.
>From the factors, they want to see if they can replace some of A1 by A2 or A3 (or potentially even a A2+A3 mixture). Then, they also want to see if they can replace some of FL by FS. And also see if we can decrease the content of P and F. I would say this is more optimization than screening (Would you agree? Although a good outcome may be that A2 or A3 are not as good as A1 (which would be more screening that optimization then?).
I did not put all the interactions from the beginning because when I did, I was getting very few runs with only two A components (8 out of 56 runs). So, I thought that this was defeating my purpose. However, I was also thinking that for a proper optimization I would need the interactions between all factors.... And that is why I was trying to see if we can define these constraint somehow into the problem.
The reply from Cameron could be a potential solution for the constraint (only 2 A's). However, I am not sure of the mathematical implications of such a solution (I will not be able to fill directly the "runs table" with the results and somehow I am not sure how I will have to fill in the results and interpretate the results... what do you think?
Then, I am still curious about the implications of the little variation of the factors (the low power, etc) and getting little "value" from the experiments... so that is why I wonder how we can improve the design (if possible).
Thanks again!