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The quadratic model has these terms: Y = (b0) + (bi)(xi) + (bij)(xi)(xj) + (bii)(xi)(xi), i and j indexing the factors, 1 to k. These are called second order polynomials because the sum of the exponents is, at most, 2. Industrial processes are mimicked by quadratic models remarkably well. You could do a lifetime of industrial process improvements and never need anything more complicated.
However, every now and then a process comes along for which this quadratic model does not interpolate well enough to suit your needs. In these cases, you can extend your model to higher order terms to get more flexibility to meet your data.
For three factors, you might add the term (x1)(x2)(x3). You would do this when you think that perhaps the (x1)(x2) interaction changes when you change the level of (x3) at which you are working. If you have four factors, there are four of these terms you can add:
(x1)(x2)(x3), (x1)(x2)(x4), (x1)(x3)(x4), (x2)(x3)(x4).
They are called third order interactions. They are "cubic terms"; their sum of exponents is 3.
Also, you might add for three factors:
(x1)(x1)(x2), (x1)(x1)(x3), (x1)(x2)(x2), (x2)(x2)(x3), (x1)(x3)(x3), (x2)(x3)(x3) and (x1)(x1)(x1), (x2)(x2)(x2) and (x3)(x3)(x3).
And there is one more cubic term in 3 factors: (x1)(x2)(x3). You can add that term to your quadratic model and any one, several, or all of the cubic terms. Some experimenters will add only the terms with 1 for exponents; for example, (x1)(x2)(x3)(x4), (x1)(x2)(x5)(x7)(x8), etc.
Some might play an important part in getting good interpolation; maybe many or all; maybe none. In biotechnology you are much more likely to need these higher order terms than in other fields. In growing human tissue from amino acids in the blood, there is a 21-factor interaction.
If your company expands the budget to fit higher order terms in the model for your processes, you will have a better chance of excelling over your competitors. The things that will put you ahead are:
The models discussed here are all of the form that has one b in each term and it appears as a premultiplier. These models are easy to fit to data and they work well in real applications. There are more complex models ("nonlinear") that may well mimic your data better. If you wish to use a nonlinear model, you will need some special skills and software.
Sometimes a model will look complex at first, but with a transformation it can be seen to be simple. Look at this law from Newton: F = ma. The factors are the force, F, that you have control over and the mass, m, you select for your experiment. So the "law" (model) would be written in our conventional sequence: a = F/m. Here the factors F and m are not entering in separate terms as in Y = (b0) + (b1)(x1) + (b2)(x2). If we take the log of both sides, we have:
log a = log F - log m
in which, when proper units are used for a, F and m, the b coefficients are:
(b0) = 0, (b1) = 1, and (b2) = -1.
Thus, F=ma can be seen as a model that did not need (x1)(x2) nor (x1)(x1), nor (x2)(x2). However, keep an open mind; additional terms might be needed someday even in this model ("law") if demands for precision of prediction go up a great deal. For now, F=ma is good enough.
Will you ever need a quartic (4th order) model? Probably not. But, in 1994 a client of mine said that he expected that his then current problem would need a quartic model in two factors. We prepared a design for him; there are none in texts that I know of. He collected the data and computed the coefficients using the STRATEGY(tm) software. He was correct. The fourth order terms were needed to represent the data well.
So it's up to you. If you think that higher order terms might be needed, we will prepare for you a design that will allow you to compute their coefficients. If they are not needed, you will find that out. If they are needed, then you will have a competitive advantage.