[I had to correct computer confusion to get this through. Sorry for
the delay]

Three thoughts:

1) The variables are ingredients. That means that the amounts of the
variables (ingredients) are constrained - the sum must = 1, or 100%.
I think you should be looking very closely at mixture designs. At
least, to explain why they need not be used.

2) In many cases, the effect of an ingredient is not (more or less)
linear, especially near a 'boundary' near 0. In cases where you wish
to remove the ingredient (as inclusions & contaminants in a metal
alloy) you can't get to zero anyway. It may be valuable to consider
the concentration on a logarithmic scale, or through an arcsin
transformation. This will effectively 'stretch' the scale near 0,
putting more emphasis on those low concentrations.

3) An afterthought. Is your response linear? I once had a case
where the response function was a function of one component
'strength' so that the optimum was to go to 0. But I was concerned
with variation in plating thickness, and the minimum variation was
obtained at 0 thickness. Right; we'll set up that one right away.

I'm not sure this computer account is prepared to get general
distribution, so it may go only to you. Feel free to post generally
if you like.

Jay
On Sep 17, 2008, at 10:49:57 PM, Pradipta Sarkar wrote:

> I have a real life problem .... would appreciate any help you can
> provide.
>
> Let us say we have k variables (which are ingredients of a product,
> k is about 20). We want to study some technical response Y as a
> function of the k variables X_1, X_2, ..., X_k. The variable X_i
> can take value between 0 and a_i
>
> So we can create a response surface design using some optimality
> criteria such as D-Optimality or I-Optimality. Ater running the
> experiment and collecting data we can fit the model and then find
> the combination of X_1, X_2, ..., X_k that maximizes the response
> Y. For this specific application (involving this particular
> product) usually the maximum is attained at a point where several
> of the X_1, X_2, ..., X_k variables are 0 which means it is on the
> boundary of the design space.
>
> We know that the confidence interval of prediction is very wide at
> the boundary (no matter what optimality criteria we use), so the
> confidence interval on the maximum reponse is likely to be very wide.
>
> So here is my question: Is there a way to design an experiment so
> that "it has lower prediction variance at the boundary"? My
> intuition says that while designing the experiment put more points
> at the corners and sides of the design space and less inside the
> design space. Is that a good strategy or can you suggest another
> better one? Won't that destroy some desired properties of DOE?
> What will I lose? Is there a standard way to handle such design
> of experiments?
>
> Remember that the X variables are ingredients, so the minimum value
> is 0(cannot be negative).
>
> Thanks,
>
> Pradipta.
>
>
>
>
>
>
>
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Jay Warner
Principal Scientist
Warner Consulting, Inc.
4444 North Green Bay Road
Racine, WI 53404-1216
USA

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