Analysis strategies for constrained mixture and mixture process experiments using jmp pro 14

Analysis Strategies for Constrained Mixture and Mixture
Process Experiments Using JMP Pro 14
Dr. Philip J. Ramsey
Principal Lecturer in Statistics
University of New Hampshire
Durham, NH, USA
philip.ramsey@unh.edu
North Haven Group
Owner and Chief Consultant
Brookline, New Hampshire, USA
pjrstats@gmail.com
Discovery March 2019 1

Introduction
Mixture experiments have a history dating back well over 200 years;
e.g., George Pearson, 1784, Observations and Experiments for
investigating the chymical history.
A small sub-discipline of experimental design: a few key researchers.
John A. Cornell (1941–2016): Mr.
Mixtures
Henri Scheffe (1907–1977):
Pioneered modern mixture designs
(published key paper 1958)
Greg F. Piepel: Student
of John and prolific
researcher and author
on mixture designs.

Introduction
Mixture experiments consist of experiments where the experimental
factors are components of a formulation or recipe.
The components can be either a liquid, a gas, or even a solid.
The impact of the mixture components on the responses of interest
depends only on the proportions of the components and not on
the amounts.
If there is a dependency on the amount of the components present,
then classical mixture designs are inappropriate – more on this
later.
Because the experimental factors are components of a formulation,
there are inherent constraints on the settings of these factors.

Introduction
Factorial experimental designs allow independent settings for each of
the experimental factors - no explicit constraints exist on the
mutual factor settings.
Mixture experimental designs impose constraints on the possible
settings for each factor since they are components of a formulation.
Mixture factors can not be independently manipulated.
Mixture experimental designs, with q factors, impose the constraints
and for i=1,…,q1 2 1.0qX X X    0 1.0i
X 

Introduction
A type of important effect in mixture experiments is referred to as
Nonlinear Blending, or NLB .
NLB effects represent the departure of the observed response from
what one would predict the response to be as a linear, additive
combination of pure component blends.
NLB is common, e.g., mixtures of water and alcohol vs boiling point
of the mixture.
NLB typically has two forms: Synergistic where the response is
higher than expected and Antagonistic where the response is lower
than expected from the pure component blend responses.
Large NLB effects of second and third order are commonly
encountered in mixture experiments.

Introduction
Example of synergistic NLB for a two component mixture experiment
on blends of sweeteners glucose and fructose.
Linear Blending
Response Line.
Observed
Response Nonlinear
Blending Effect

Introduction
Additional constraints on mixture factor settings are common.
Regional or inequality constraints (upper and lower bounds)
Example, one would not have cake mixes with 100% eggs or 100%
flour, etc., a proportion of some or all ingredients must be present in
every mixture trial performed.
Another type of constraint often exists in which bounds are placed on a
linear combination of some of the components in the mixture.
Typically, linear constraints involve 2 and sometimes 3 components
of the mixture and have the general form.
, , 1.0i jX X a i j a   
i
i
, 0, 1.0, X 1.0, 1, ,i i ii iLower X pper iL qU U     

Introduction
When regional constraints exist, it is common to rescale the mixture
components to help mitigate the multicollinearity issues.
Basically, the component settings are stretched to better fit the
encompassing simplex region and reduce multicollinearity.
A common rescaling method is referred to as L-pseudocomponent
scaling (can be used with both upper and lower constraints)
Note, the scaling method of Cornell and Gorman (JQT, 2002) is
more effective but is not implemented in JMP.
* *
1
1
1.0, 1.0, 1 , i=1, ,q
= or (1 )
1
i i i i
i i
q
i i i i
i i i i iq
i
i
i
L X X L L
X L X L
X X X L L
L
L 

    
 
   

 



Analyzing Mixture Experiments
Mixture designs are really a type of response surface design and as
such the primary goal of a mixture experiment is to develop a model
that satisfactorily predicts future performance of the response(s).
The model is subsequently used to characterize the behavior of the
response(s) over the constrained mixture space – the models are
commonly used for optimization and sensitivity analysis.
Selecting satisfactory predictive models require that we divide the data
into a training set to fit the model and a validation set, not used to
fit the model, that assesses the prediction capability.
Predictive models are selected that have smallest prediction error on
the validation set – should predict future performance the best.
Unfortunately, in designed experiments rarely is there sufficient data
to form a training and validation set.
So how does one select a best predictive model in DOE?

The model selection problem is particularly difficult in mixtures.
 The existence of constraints results in a great deal of
multicollinearity.
 The multicollinearity results in correlations among the
estimated effects, especially NLB effects; standard errors of the
estimates are generally quite large.
 Traditional model selection techniques using p-values are not
useful for mixtures; p-values are distorted by even small
amounts of multicollinearity and are misleading.
 The pure component terms must be forced into all models.
 Mixture models have 0 intercept – sometimes a problem.
 The designs are typically very efficient and even supersaturated
depending upon the mixture model of interest.

Today’s powerful computers, with double precision, and modern
statistical software such as JMP provide us with better capabilities to
analyze and build predictive models for mixtures.
We will investigate and illustrate a couple methods (many more are
possible, users should try an array of methods).
 Traditional Forward selection using the pseudo factor method
of Miller (1992) – interesting and unappreciated approach.
 Traditional Forward selection and Pruned Forward using
fractionally weighted bootstrapping and auto-validation
(Gotwalt and Ramsey, 2018).
The Generalized Regression platform in JMP Pro makes it
straightforward to implement these methods for mixtures.
Note, I do not recommend All Possible Models for mixtures in part
due to the need to force the pure component terms in every model

Miller’s Pseudo Factor Method
Calculate the largest possible number of effects p that could occur in a
mixture model; includes cross products, polynomials, etc.
Create an additional set of p random factors or pseudo factors – Miller
used random uniformly distributed factors.
Define the largest possible model in the mixture factors and then add
the additional set of pseudo factors.
Under the assumption that all effects are 0, the probability of a mixture
effect or pseudo factor entering the model is the same, hence the use
of p pseudo factors – Miller never explained his rationale.
Use Forward Selection with say AICc or BIC as the objective function
and stop the selection process when either the first pseudo factor
enters the model or the fitting criterion is minimized.
For large mixture problems, this approach can become unwieldy.

Example (due to Montgomery): In semiconductor manufacture,
silicon wafers are acid etched prior to a metalization step in the
process.
An experiment has been performed by research scientists to find the
optimum composition of etchant – the composition yielding the
highest etching rate.
The etchant is composed of three acids:
A = nitric,
B = hydrochloric, and
C = Phosphoric.
A simplex centroid design was used and the largest possible model
contains 7 effects: 3 pure component terms, 3 two-way NLB terms,
and 1 three-way NLB term.

Below is the design with pseudo factors added and the response.
In Fit Model define a mixture response surface model and add the
three-way NLB term, then add the 7 pseudo factors.
We will use the Generalized Regression platform to do the forward
selection analysis; if you do not have JMP Pro use the Stepwise
platform.

Below is a screenshot of the Fit Model launch dialog for the analysis.

To the right is a screenshot of the Gen
Reg Model Launch configuration.
Of course other options are possible.
Important to force in the pure
component terms!
I do not recommend enforcing
heredity in predictive model
building whether a mixture model or
not; can lead to over fitting.
This is controversial.
Here we use AICc as the objective
function for model selection; again
you can use other options.

The algorithm stops after two steps, the three pure component effects
the A*B NLB term and the A*B*C NLB terms define the model.

Using the model selection slider (red vertical line) we manually force a
third selection step. Notice a pseudo factor enters the model, so we
stop at two steps.

For reference below is a screenshot of the same analysis performed in
the Stepwise platform; the results are identical to Gen Reg.

FWB and Autovalidation
Gotwalt and Ramsey (2018 Discovery Frankfurt, Cary) propose a new
method of model selection based upon fractionally weighted
bootstrapping and autovalidation; slides on JMP Communities.
The method also makes use of a pseudo or null factor.
Recall, a problem in building predictive models in DOE is the lack of a
validation set to assess prediction capability.
Autovalidation uses a copy of the original data as a validation set;
sounds crazy but give it a chance.
The key is that the observations in the original or training set and
validation set (copy) are differentially weighted such that the
original or training set is anticorrelated with the validation copy set.
Thousands of bootstrap repetitions of model selection are performed
and on each bootstrap trial new weights are generated randomly.

FWB and Autovalidation
The method is straightforward.
 In the Fit Model launch dialog define the largest model of
interest and then add a null or pseudo factor to the model.
 Use Generalized Regression, pick the desired model selection
algorithm, force the pure components into the model, do not
enforce heredity (same as Miller’s method).
 Perform the model selection, then use the Simulate function in
JMP Pro to perform thousands of fractionally weighted
bootstrap repetitions of the model selection.
 Gen Reg keeps track of which effects enter the model on each
repetition.
 All effects entering the model with higher frequency then the
null factor are considered potential model terms.

Case Study: The Waste Glass Experiment
Redgate, Piepel, et. al., 1992, discuss a 10 component (oxides), highly
constrained mixture experiment to study a vitrification process to
transform high level radioactive waste to a borosilicate glass form
for subsequent permanent sequester.
The response of interest for this case study is Viscosity @1150oC.
The design consisted of 81 trials (68 unique settings and 13 replicate
trials). The authors do not discuss how the design was generated;
most likely an early form of D optimal design.
Both regional and linear (multicomponent) constraints exist.
The design was created to estimate pure component effects and two-
way NLB effects, however assuming some amount of effect
sparsity, higher order effects could be considered.
For q = 10 components the full 2nd order Scheffe model has 55 terms.

The oxide components and regional constraints are given in the table
below.
Oxide Lower Bound Upper Bound
SiO2 0.42 0.57
B2O3 0.05 0.20
Na2O 0.05 0.20
Li2O 0.01 0.07
CaO 0.00 0.10
MgO 0.00 0.08
Fe2O3 0.02 0.15
Al2O3 0.00 0.15
ZrO2 0.00 0.13
Remainder 0.01 0.10

The linear constraints involving the oxides are as follows:
The response limits for Viscosity are [1.5, 12].
We begin by analyzing the full second order Scheffe model using
autovalidation and the Generalized Regression platform.
In the original paper the authors used Miller’s method (55 pseudo
factors added to the model) as well as traditional Stepwise selection
without pseudo factors.
2 2 3
2 3 2 3 2
2 3 2
2
3 0.0
0.08
Remainder 0.21
0.14
0.18
SiO Al O
MgO CaO
Fe O Al O ZrO
Al O ZrO
MgO CaO ZrO
 
 
   
 
  

The traditonal 2nd order Scheffe model (Scheffe, 1958) has the form
Notice, that no pure quadratic terms are included. It can be shown
(Smith, 2005) that a pure quadratic mixture term can be re-expressed
as combination of a linear term and string of cross product terms.
Therefore, if we add all of the pure quadratics to the mixture model,
the model matrix is singular; Scheffe elected to drop the quadratics.
However, a valid mixture model can contain a combination of cross
product terms and pure quadratic terms, just not all of them, and
actually fit better.
Redgate and Piepel included cross products and pure quadratics in
their model selection strategy in order to improve the fit.
1
q
ijk i i ij i ji
i j
Y X X X 

  

The quadratic Scheffe model had considerable lack of fit, as did the
models selected by Redgate and Piepel, so we next tried fitting
Scheffe special cubic models.
The original design was not generated to fit special cubic terms,
however using a forward selection strategy and assuming some
effect sparsity it is possible to fit special cubic models.
The special cubic model in 10 components has n = 166 terms apart
from the pure component terms. Obvioiusly the design is
supersatured for this model.
The autovalidation method was repeated, but this time using a special
cubic model in the ten components.
1
q
ijk i i ij i j ijk i j k ijki
i j i j k
Y X X X X X X   
  
      

Quick diversion: How might we display the results of the simulation?
One nice approach is odds ratios. If pi is the proportion of times the ith
effect entered the model during the simulation, then pi/(1-pi) is the
estimated odds of entering the model.
Under a null hypothesis that all effects are independent and inactive p0
is the probability of entering the model purely at random and
p0/(1-p0) is the odds of randomly entering the model.
The odds ratio for the ith effect is then
As an alternative the proportion for
the null factor could be used in the
denominator.
ORs have nice statistical properties that could be exploited for further
inference.
 
 
0
0
1
1
i
i
i
p
p
OR
p
p




Below is a screenshot of the simulation results ordered descending by
odds ratio of entering the model vs the null hypothesis.

To the right is a screenshot of the final
selected special cubic model.
Although some lack of fit still exists,
overall the model fits much better than
the quadratic model.
It is common to have some level of lack of
fit in mixture models that cannot be
completely accommodated.
At this point it is wise to consult with
subject matter experts as to whether the
amount of LOF is of concern.
Overall, the LOF does not appear large
and may not be of concern.

Given there were 13 replicate runs, it is possible that some of them
could be used as a validation set for model selection and the
remainder as the training set.
There were also some extra degrees of freedom in the experiment that
could also be used for validation – this is not the usual case.
To investigate the approach, a randomly selected subset of 10 runs
were used as a validation set (mostly from the replicate runs) and the
remainder for training.
The special cubic model was defined and Generalized Regression
platform used for model selection.
As before pure component terms are forced into the model and
Forward selection was used as the model selection algorithm.

To the right is a screenshot of the
final fitted model.
Overall this model has the best fit
with no significant LOF.
However, the autovalidation
models had 13 df for the LOF
test while this model has 6 due to
the use of some trials for the
validation set.
The autovalidation model and true
validation model are actually
similar.

Below is a comparison of the two models, they are not identical but
similar and both fit the data quite well.

Mixture Process Factor Experiments
A much underutilized class of experimental designs for process
development and improvement are designs with combinations of
mixture and process (factorial) factors.
In general, formulations developed under very controlled process
conditions (often only one setting) do not work well in practice.
The reason is that the blending behavior of the mixture factors changes
as a function of the settings of the process factors.
In order to develop robust and optimal processes it is important to
simultaneously experiment with both the mixture and process
factors.
Such experimental designs are commonly called mixture process
factor designs.

Mixture Process Factor Experiments
Mixture process factor designs have a long history, however the
classical designs were often infeasibly large.
With modern optimal designs (Custom Design) mixture process factor
experiments can be generated that are reasonable in size.
The most common type of mixture process factor experiment is the
mixture amount experiment, where the total amount of the mixture
used is a process factor.
We will illustrate with a case study.
Mixture process factors still cannot have an intercept and are a bit
tricky to create in JMP, but not so hard with a little practice.
Historically, mixture process factor experiments have been very
difficult to analyze for reasons given earlier and often badly over fit
models were selected.

Mixture-Process Factor Designs
A special notation is often used to write MPV models in a compact
form. Each coefficient has a subscript defining the mixture factors
and a superscript defining the process factors that are associated
with that coefficient.
Here is a full MPV model for three mixture factors Xi and two process
factors Z. Note the 2nd order terms in Zj are often necessary.
1 2 3 12 13 23 123
1 2 3 12 13 23 123
1 2 3 12 13 23 123
1 2
0 0 0 0 0 0 0
1 2 3 1 2 1 3 2 3 1 2 3
1 1 1 1 1 1 1
1 2 3 1 2 1 3 2 3 1 2 3 1
2 2 2 2 2 2 2
1 2 3 1 2 1 3 2 3 1 2 3 2
12 12
1 2
iY X X X X X X X X X X X X
X X X X X X X X X X X X Z
X X
      
      
      
 
       
        
        
  3 12 13 23 123
1 2 3 12 13 23 123
1 2 3 12 13 23 123
12 12 12 12 12
3 1 2 1 3 2 3 1 2 3 12
11 11 11 11 11 11 11 2
1 2 3 1 2 1 3 2 3 1 2 3 1
22 22 22 22 22 22 22 2
1 2 3 1 2 1 3 2 3 1 2 3 2
X X X X X X X X X X Z
    
      
      
      
        
       

Mixture-Process Factor Case Study
Chardon (1989) discusses the development of a finishing material for
cotton-polyester fabrics. We will use this example (file
MixtureProcessFabric.JMP) to demonstrate the analysis of a MPV
experiment.
The finisher is comprised of three fabric softeners and a catalyzed
resin.
We will designate the three mixture components as A, B, and C (the
softeners).
There are two process factors. D is the level of the catalyzed resin
used in the finisher, and E is the total amount of softener in the
finisher.
Note factor E is a total amount factor for the three mixture factors.

The response is hydrophobicity of the fabric (water repellent
measure), which the researchers would like to maximize.
Due to heteroscedasticity the authors recommended analyzing the log
of hydrophobicity.
The researchers ran a simplex centroid mixture design at each of 7
settings of the process factors.
The process factor design is known as an equiradial (Doelhert)
response surface design, which is hexagonal in shape.
There are a total of 49 runs in the MPV and the full MPV model has
42 potential terms to estimate – similar to our example given
earlier.
48Discovery March 2019

We will use autovalidation and Gen Reg to analyze the experiment.
Below is the full model, with 42 terms:
1 2 3 12 13 23 123
1 2 3 12 13 23 123
1 2 3 12 13 23 123
1 2 3 12 13 23 123
0 0 0 0 0 0 0
i
D D D D D D D
E E E E E E E
DE DE DE DE DE DE DE
Y A B C AB AC BC ABC
A B C AB AC BC ABC D
A B C AB AC BC ABC E
A B C AB AC BC ABC
      
      
      
      
       
        
        
      
2 2 2 2 2 2 2
1 2 3 12 13 23 123
2 2 2 2 2 2 2
1 2 3 12 13 23 123
2
2
D D D D D D D
E E E E E E E
DE
A B C AB AC BC ABC D
A B C AB AC BC ABC E
      
      
  
       
 
      
 

Below is a plot of the effects by Odds Ratio for the autovalidation
results. We will fit a model containing the pure components and the
effects > null factor.

To the right is a screenshot of fitted
model based on the autovalidation
results.
The fit appears quite good.
The published model is much larger and does
not fit as well; see below.

The goal of the experiment was to find settings of the factors that
maximizes hydrophobicity. Using the autovalidation model and the
Prediction Profiler we find the optimized settings – quite different
from the original published recommendation.

Summary and Conclusions
Traditionally building predictive models from mixture and mixture
process DOE data has been limited by the lack of validation trials to
control over fitting.
Typically a DOE budget and time does not make it feasible to perform a
separate set of validation trials.
Autovalidation and Fractionally Weighted Bootstrapping are two new
viable techniques that enable predictive modeling from DOE data
without running a set of validation trials.
The techniques use the original training data and bootstrapping concepts
to form validation sets.
Autovalidation and FWB should be considered a part of mixture DOE
analysis where the goal is to build predictive models.

Bibliography
Cornell, J.A. (2002). Experiments with Mixtures 3rd Ed. John Wiley
& Sons.
Cornell, J.A., and Gorman, J.W. (2002). “Two New Mixture Models:
Living With Collinearity but Removing Its Influence.” Journal of
Quality Technology, Vol 35, No 1.
Miller, A.J. (2002). Subset Selection in Regression, 2nd Edition.
Chapman Hall/CRC, 235 pp.
Redgate, P.E., Piepel, G.F., and Hrma, P. (1992). “Second-Order
Model Selection in Mixture Experiments.” ASA, Proceedings of
the Section on Physical and Engineering Sciences.
Scheffe, Henri (1958). “Experiments with Mixtures.” Journal of the
Royal Statistical Society, Ser. B, 20, 344-360.

Bibliography
Smith, W.F. (2005). “Experimental Design for Formulation.” ASA-
SIAM Series on Statistics and Applied Probability, ASA,
Alexandria, VA, USA.
Wu, Y., Boos, D., and Stefanski, L. (2007), “Controlling Variable
Selection by the Addition of Pseudovariables”, Journal of the
American Statistical Association.

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