Smooth models for two numerical factors
Some experiments, especially in industrial applications, are conducted to find the combination of two numerical factors that optimises some response measurement. In these experiments, it is particularly important to obtain a smooth formula for the effects of the two factors. Such a model is called a response surface model.
The following response surface model contains quadratic terms in both variables.
yij = µ + β1 xi + β2 xi2 + γ1 zi + γ2 zi2 + εij
Model with interaction
The above model is not flexible enough for most situations. The most commonly used response surface model adds an interation term to the model involving the product of the two factor values.
yij = µ + β1 xi + β2 xi2 + γ1 zi + γ2 zi2 + δ xi zi + εij
This model has enough flexibility to model data from many experiments.
We can only briefly give a flavour of models for response surfaces here.
The most commonly used experimental designs for estimating response surfaces use relatively complex combinations of values for the two factors, as in the example below.
Chemical process yield
The diagram below shows experimental results from an experiment for the yield of a chemical process with factors Time and Temperature.
The diagram below shows how the response surface changes when the quadratic terms and interaction are added to the model
Firstly click the checkbox Interaction term to add an interaction to the model.
Next add the quadratic term in temperature by clicking the checkbox Quadratic term in Temperature.
Now click the x-z rotation button and investigate the different models, both when displayed as colours and as contours.
Finally add the quadratic term in time to the model and observe that the response surface again changes. From the response surfaces with and without the quadratic term in time, we would conclude that:
The highest yield results from a high temperature (150 or higher) and low reaction time (80 minutes or less). Combinations of (high temperature & high reaction time) and (low temperature & low reaction time) result in very poor yield.