Hi,
Asking for help with setting up a DOE.
Z and X are components of a mixture, each individual factor ranging from 0.2 to 0.8 and together summing to 1. There are three different types of X, X1 X2 and X3. X2 and X3 can be used alone or combined together in a ratio, however, they cannot be combined with X1. I tried to summarise this in the excel sheet below.
Thanks in advance!
Since the design seems to always involve pairs of factors, like Z with X (X1 or X2+X3) and X2 with X3, I'm not even sure you need mixture factors or a mixture design.
With the constraint you have, you could define 2 numerical continuous factors, ratio Z/X and ratio X2/X3, and a categorical factor "X choice" with two levels : X1 and X2+X3.
Adding a constraint with the "Disallowed Combinations filter" like "If X1 is chosen, then ratio X2/X3 can't be >0" could lead you to a design respecting your constraints :
Factors choice == "X1" & "Ratio X2-X3"n >= 0.0001
(And if ratio X2/X3 = 0 and factor choice is set on X2+X3, that means you only have X3 and Z in your formulation, with the ratio between the two defined by the factor ratio Z/X).
This might clearly not be a clean design and a nice way to proceed further (don't skip the questions from my 1st post please), but just wanted to highlight that even if you have mixture factors, depending on the settings of your study you might end up with a design with non-mixture factors.
Hi @NWolfe,
Thanks for the reply.
As factors in a mixture design are not independent from each other (mixture components sum up to 1), this constraint introduces correlations among the parameter estimates. This correlation inflates the standard errors of the parameter estimates, which in turn reduces the power and make it more difficult to screen and assess statistical significance of mixture factors.
There are some designs that can help (D-optimal mixture designs, ABCD designs), but you will still have difficulties in assessing statistical significance for the parameters in the model. You can still assess relative practical importance of your factors by looking and comparing the parameters estimates.
There might be several options to create an optimization mixture design with your constraints:
- Model-based Mixture design : Start with a Custom (D-optimal) Design, add the mixture factors with their ranges, and specify in the model main effects and some 2-factors interactions : none between X1 and X2 or X1 and X3 if you want X1 and X2/X3 to not be mixed in any formulation experiment. This type of design will look like this (example with 20 runs here, you can change the number of runs, and you can of course round up the extremely low value like 4,88e-15 to 0) :
Here is the script to generate the design :
DOE(
Custom Design,
{Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0.2, 1, "Z", 0 ),
Add Factor( Mixture, 0, 0.8, "X1", 0 ), Add Factor( Mixture, 0, 0.8, "X2", 0 ),
Add Factor( Mixture, 0, 0.8, "X3", 0 ), Set Random Seed( 550059023 ),
Number of Starts( 4436 ), Add Term( {1, 1} ), Add Term( {2, 1} ),
Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {3, 1}, {4, 1} ),
Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {1, 1}, {4, 1} ), Add Term( {1, 1}, {3, 1}, {4, 1} ),
Set Sample Size( 20 ), Optimality Criterion( "Make D-Optimal Design" ),
Simulate Responses( 0 ), Save X Matrix( 0 ), Make Design,
Set Run Order( Randomize ), Make Table}
)- Model-agnostic Mixture design : Since you mention optimization, another possible option could be to consider Space-filling Mixture design with your constraints. This type of design does not assume any model before the analysis, so you have some flexibility about the modeling, from "standard" regression models (with different complexities) to Machine Learning models like SVM, Gaussian Process, Neural Networks... If this option is interesting for you and you would like to have an example for your case, I can also propose a possible Space-filling design.
I hope this first answer will help you,