Response surface modeling of the mean and standard deviation

1 Response surface modeling of the mean and standard deviation 

In Section 2.2 it was shown that response surface methodology can be applied to enable a researcher to model the effect of multiple quantitative variables on a response with a low-degree polynomial. Frequently, response surface techniques have focused on the mean response as the only response of interest. However, by regarding the variation in the response as an additional response of interest, the researcher can investigate how to achieve a mean response that is on target with minimum variation. In particular, if a researcher replicates each design point in an experiment, then an estimate of the standard deviation at each point can be calculated and used to model the effect of the variables on the variability of the response. 

Now if each of the design points in the central composite design is replicated five times, so that the complete design has 75 runs, then at each design point we can calculate the average response and the standard deviation of the response. The analysis techniques associated with response surface methodology can then be applied to fit separate models to 

The advantages of using the response surface approach to study both the mean and the variability are that it is easy to apply, no new methods of analysis are required, and the standard analysis methods can be used to bring insight to bear on the dual objective of the mean response and the variability. Some of these methods of analysis are considered in Section 2.3.2. 

As was mentioned above, a disadvantage of this approach is that the variation that is measured at each replicated design point will be from many sources, including sources of environmental variation, and it will be impossible to attribute the variation to a particular source. Another disadvantage of this approach is that it assumes that the variation experienced at the design points during the course of the experiment is similar to that experienced in practice in the real world. Frequently an experiment will be well-controlled and so the variation experienced will be considerably less than that normally encountered. 

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