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Apr 30, 2015 1:17 AM
(3399 views)

Hi,

I'm trying to make a DoE with three mixture factors (x1, x2, x3) which sum up to some constant, for example x1+x2+x3=50.

I've noticed that, when I make an optimal mixture design (with proportions - i.e. I suppose that x1+x2+x3=1) that the power in design evaluation window, for all coefficients is equal to 0.073.

On the other hand, If I change "Mixture Sum" parameter to 50 (i.e. x1+x2+x3=50) and then make the same design, the power for all coefficients is 1.

This is puzzling to me, since model matrix isn't changed, that is the range of all factors is translated to [0,1] interval.

Can someone explain the reason for that?

From the technical details of power calculation (DESIGN OF EXPERIMENTS GUIDE • EVALUATE DESIGNS • TECHNICAL DETAILS) I don't see the reason for a change.

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The model matrix X has changed because the design matrix D has changed. Here is a simple example for illustration. I am using Custom Design with three mixture components X1-X3. For the simple model with only the main effects and intercept, I get this design in 6 runs with its associated power:

[Missing image]

If I click Back and change the Mixture Sum to 50 and then click Make Design I get a different design matrix, which in turn is expanded into a different model matrix:

[Missing image]

The power depends on the model matrix, which is determined by the design matrix and the terms in the model. Comparing these two designs, the one that spans a smaller factor space will have less power to estimate the same sized effect (coefficient).

The same thing happens if you leave the mixture sum set to the default value of 1, but restrict the range of any of the components.

Learn it once, use it forever!

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Not sure what is going on here but my experience has told me that the Power calculations for Mixture Designs are not very useful due to the interdependence of the terms. If I change the RSME for the magnitude of the design that uses mixture sum to 50 the Power for the main effects are equal. I prefer the Fraction of the Design Space Plots to evaluate the designs and in the case that you propose they are superimposable.

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Hi LouV, thank you for your answer. I'm aware of that power of test for coefficients == 0 isn't helpful here.

Information about prediction variance and stuff like G-efficiency is in my focus also, but still I'm trying to understand how things are calculated because that is the only way I can remember what and why is important to look at

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The model matrix X has changed because the design matrix D has changed. Here is a simple example for illustration. I am using Custom Design with three mixture components X1-X3. For the simple model with only the main effects and intercept, I get this design in 6 runs with its associated power:

[Missing image]

If I click Back and change the Mixture Sum to 50 and then click Make Design I get a different design matrix, which in turn is expanded into a different model matrix:

[Missing image]

The power depends on the model matrix, which is determined by the design matrix and the terms in the model. Comparing these two designs, the one that spans a smaller factor space will have less power to estimate the same sized effect (coefficient).

The same thing happens if you leave the mixture sum set to the default value of 1, but restrict the range of any of the components.

Learn it once, use it forever!

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Hi Mark, thx for your explanation.

I figured out that the reason must be in model matrix X, but I looked at coding table (i.e. wrong table) of both designs and didn't see the difference. Now when I look at the correct table, everything is clear now (although, at this moment I don't understand why model matrix doesn't have coded values (as documentation states) instead of 0-50 ?)