Confidence intervals for response surfaces

In Section 6.1, the concept of confidence intervals of parameter estimates was presented. In this section, we consider a general approach to the estimation of confidence intervals for parameter estimates and response surfaces based on models that have been shown to be adequate (i.e., the lack of fit is not highly significant, either in a statistical or in a practical sense). 

If a model does not show a serious lack of fit, then 

The variances of the parameter estimates can be used to set confidence intervals that would include the true value of the parameter a certain percentage of the time. In general, the confidence interval for a parameter p, based on is given by 

Although the derivation is beyond the scope of this presentation, it can be shown that the estimated variance of predicting a single new value of response at a given point in factor space, is equal to the purely experimental uncertainty variance, plus the variance of estimating the mean response at that point, 5, 0 that is. 

For a given experimental design (such as that of Equation 11.15), the variance of predicting the mean response at a point in factor space is 

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If the model is adequate but still not perfectly correct, then the estimate sJiq based on 5 (Equation 11.70) will be too low because it does not take into account the lack of fit of the model. To partially compensate for the possibility of a slight lack of fit between the model and the data, it is customary to use to estimate in setting confidence intervals for response surfaces. 

It is to be stressed that if the model is grossly incorrect, it is of little practical use to estimate confidence intervals for response surfaces. 

Confidence intervals for single-factor response surfaces were discussed in Section 11.6. The equations developed for estimating different types of confidence intervals (Equations 11.76, 11.79, 11.80, and 11.81) are entirely general and can be used for multi-factor response surfaces as well. 

A (100 - a) confidence interval for the true value, of the response surface parameter is obtained as... 

Figure 11 shows the plots of the semi-amplitude of the 95% confidence interval for each of the two responses. These plots, derived from the leverage plot, are anyway much more easily interpretable as they give a direct idea of the uncertainty of the predicted value. So, we can immediately see that for the first response the uncertainty is between about 2 and about 3, while for the second respmise it is in the range of 0.1 to 0.2. It is very important to note that the cmifidence interval changes according to the position in the experimental domain, and that the shape of this surface depends on the distribution of the experiments in the experimental domain. Again, what is important is which experiments are performed, not how many. 

These and most other equations developed by statisticians assume that the experimental error is the same over the entire response surface there is no satisfactory agreement for how to incorporate heteroscedastic errors. Note that there are several different equations in the literature according to the specific aims of the confidence interval calculations, but for brevity we introduce only two which can be generally applied to most situations. 

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