Mixture designs are a special case of polynomial regression with no intercept term. The intercept is undefined in the mixture design because the point where all terms are zero does not exist. If n-1 terms are zero, then the nth term must be at its maximum value.

The problem is that standard statistical methods assume that the independent variables are uncorrelated, or try to develop techniques to avoid/mitigate the effects of such correlation. The mixture design requires that some set of the independent variables are correlated. Moreover that you can in some way force them to be perfectly correlated so that knowing the value of n-1 of them enables you to calculate the value of the nth independent variable with perfect accuracy. More complex designs allow for process variables which are not part of the mixture.

The intuition about cubic interaction seems to be impossible to get.

B

Here is how I understand the scheffe cubic “interaction” term(s).  I included a couple of diagrams that I find useful when thinking about mixture effects.

Technically, there are no interaction terms in a scheffe mixture model.  However, there are terms that look like interaction terms.  For example, in a 2-component mixture the term that looks like an interaction term, AB for the quadratic model and AB(A-B) for the cubic model.  The term “interaction” applies to factors that can be varied independently, but because the “factors” of a mixture cannot be varied independently they are called “components” and the “interaction” effect called nonlinear blending in mixture designs.

Nonlinear blending occurs when the response deviates either above or below what would be predicted by a linear relationship.  The quadratic blending pic illustrates a response that is greater than what would be expected if the response were linear.  An example of a nonlinear blending response is the melting point of ordinary lead/tin solder – lead melts at 327C, tin at 232C, but a blend (specifically 0.62 tin and 0.38 lead) melts at 183.  If the melting point was linear, the blend would melt at 268.  Further, if the nonlinear blending is considered a good thing it is called a synergism and, if not a good thing it is called an antagonism.  The cubic diagram shows what nonlinear blending looks like in a cubic response - thus the AB(A-B) term.

From a geometric perspective each term in the model contributes to the shape of the response surface.

As an aside, I typically run many of my experiments using a mixture model because 1) mixture effects are common in biology and, 2) mixture effects cannot be detected otherwise.

What bothers me in Randall's explanation is that it seems that the people doing mixture designs have repurposed the term "nonlinear." In more standard regression the options are linear versus nonlinear. The difference is if the first derivative with respect to the parameters includes one or more of the parameters. So y=b0+b1x1+b2x2 is linear because dy/dbo = 1, dy/b1 = x, and dy/b2 = x2. The example comes from the SAS user manual. With this definition, mixture designs are linear regression (thought it is the subcategory polynomial regression which does not yield a straight line). The interaction terms all are allowing for waves (or departures from linearity when used in a geometric sense, as Randall showed.).

thanks Mr Randall and Mr Timothy for your kind response. Mr. Tomthy, i agree with you input variables should be independent varaibales, In mixture design quanitity of one factor depends upon another factor, but they are impacting independently on response variables, am i right?  Mr Randall,  i am unable to understand this term AB(A-B) in scheffe cubic interaction. could you please share some literature on it.

Yes and no. If they were truly independent then all interaction terms would not be significant. AB(A-B) is an interaction term. There are more of them in mixture designs with more variables. So DF(D-F) is present if there are mixture variables A through F. You should also look at process variables.

There is no specific meaning to the AB(A-B) interaction term. As Randall pointed out the main effects are straight lines. If a unit increase in factor A results in a unit change in the response, then you will have no interaction terms. If there is any curvature in the response, then you will have significant interaction terms. The AB interaction term is depicted in Randall's top picture. The AB(A-B) interaction is the lower figure.

Please be aware that the number of treatments increase as you model higher order interaction terms. If you run an experiment designed for a quadratic interaction you cannot change your mind at the end and try to model cubic interactions. You can go the other way, and end up with a quadratic model from a design optimized for a cubic model. I always favored running experiments designed for higher order models because it gave me the most options. I had a three component mixture, so I used the points of the triangle, two points on each edge, a centroid, and three treatments midway between the centroid and the point. This was more than Design Expert (http://www.statease.com/dx9.html) wanted me to use, but I had a very non-traditional application. I had a mixture design in log-space. Number of droplets multiplied by droplet size multiplied by toxicant/volume equaled the total applied dose. I kept total dose constant. This becomes a mixture design after a log transformation. One has to also place some bounds by selecting a maximum droplet size or a maximum dilution. An infinitely dilute toxicant will require an infinitely large droplet. The problem is solved by selecting a maximum droplet size based on what could reasonably be encountered in an agricultural spray application.

The AB(A-B) term is the binary cubic mixture term.  A 2-component cubic mixture model has 4 model terms – linear (A and B), quadratic (AB), and cubic (AB(A-B).  A 3-component cubic mixture model has 10 terms – linear (3), quadratic (3 – AB, AC, BC), binary cubic (3 – AB(A-B), AC(A-C), BC(B-C)), and trinary cubic (ABC).  All these terms together represent all the ways the 3 components can twist and turn.  So, for example, AB(A-B) captures cubic blending between 2 components, A and B – regardless of the number of components in the mixture.

To see where the cubic term AB(A-B) exerts its effect and its relationship to the other two cubic terms AC(A-C) and BC(B-C), I made a graph and labeled the 3-component mixture design space with the various terms.

How might the cubic term AB(A-B) actually look with “real” data?  Thanks to Mark Anderson at Statease where I first learned of this explanation.  It almost certainly will not look like the graph (sine wave type) I posted previously.  I constructed a second graph to illustrate what happens in a 2-component mixture.  With real data the polynomial will first define the slope, this is the linear component; it will next define curvature, this is the quadratic component; and lastly, it will define any additional asymmetry, this is the AB(A-B) cubic component.  The cubic term can capture both positive and negative curvature (i.e., the sine wave) if it is present.    Each additional term therefore explains something over and above the previous terms.  The AB(A-B) is able to capture the small amount of asymmetry at point 0.75:0.25, which the quadratic cannot quite capture; this is what the cubic term does.  As Mark puts it, "it (the cubic term) creates a far more subtle "shaping" of the surface ..."

Randal,

I think your original reply was easier to grasp if the person has no prior background. Of course your new answer is more technically accurate. One could also simply say that the linear term has no inflection point, the quadratic term has one inflection point, and the cubic has two inflection points. Dots 2 and 4 in your bottom graph are the infection points for the cubic model, while dot 3 is the inflection point for the quadratic.

Maybe a different answer: The AB(A-B) interaction term has no specific meaning in the same sense as something like the exponential decay model N(t)=Noe-rt where r is the rate at which elements decay. There is no specific meaning to the (AB(A-B) term. In my example the "droplet size*number of droplets*(droplet size - number of droplets)" doesn't have a specific meaning, is just allows the response surface to flex more along the droplet size by droplet number axis. In my experiments I was interested in two things:

1) Were there significant cubic interaction terms? If there was even one, then I was stuck with the cubic model and all the work that implied.

2) In the entire response surface, where were there high and low points. The paradigm was that this was a basically linear relationship: smaller droplets increase efficacy. I showed that it was not linear.

Timothy,

Your point about a term, or an entire model, having no inherent meaning is a very good one, and one that we always need to remember.  The meaning of these types of models is strictly empirical, and as you explain, how it shapes the response surface.  The potential value ultimately is, "can the model predict?"  Thanks for bringing this into the conversation.