Intervals for response surfaces

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. 

For response surface modelling, scaled variables, Xj, will be used intstead of the "natural" variable, u. The range of variation of each continuous variable, Uj, in the experimental domain will be linearly transformed into a variation of Xj centered around zero and usually spanning the interval [-1 < Xj < +1]. (For discrete variables on two levels, each level will be arbitrarily assigned the value — 1 or +1). The scaling is done in the following way ... 

Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

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). 

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. 

One type of simultaneous approach, termed an exhaustive grid search, requires the collection of a very large number of data points throughout the experimental set of conditions at regular intervals in order to map the response surface. Response surfaces are usually complex (complexity will increase with the number of solutes and the number of variables to be optimized), and thus require a large number of data points for accurate mapping. 

The presented block diagrams link the factor-fixing accuracy, range of response change and response-surface curvature with the width of factor-variation interval. When selecting a factor variation interval one should, if possible, account for the number of factor variation levels in the experimental domain. Depending on the number of these levels, are the experiment range and optimization efficiency. 

In accord with recommendations in sect. 2.1.3 for the choice of the variation interval, one must take into consideration accuracy of fixing the factor, information on response surface curvature and range of change of the optimization parameter. 

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. 

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

If we choose a = /k, the star points will be placed farther from the center point as the number of factors increases. This choice should be made — if it is made at all — with much care, because we run the risk of leaving too much of the intermediate region uninvestigated. With nine factors, for example, a would be equal to 3. The experiments would teU us nothing about the behavior of the response surface in the 1-3 intervals along each axis. 

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