Model estimation, response surface

Canonical analysis achieves this geometric interpretation of the response surface by transforming the estimated polynomial model into a simpler form. The origin of the factor space is first translated to the stationary point of the estimated response surface, the point at which the partial derivatives of the response with respect to all of the factors are simultaneously equal to zero (see Section 10.5). The new factor... 

SSC describes the squared deviations from the mean which should be accounted for by the regression parameters associated with the experimental variables. If the response surface should be totally flat (no slopes, no twists, no curvatures), the average response would be representative for all experiments, i.e. the estimated response surface model would be... 

When such time effect is present, the design responses are corrected relative to the (nominal) experiment performed at the beginning of the experimental design (Eq. 2.10) (Figure 2.16) (5,16,106). These corrected responses are then used to estimate the factor effects from screening designs or to build the model from response surface designs (see further). From both the estimated effects and the model coefficients then the time effect has been removed ... 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

In reference 69, results were analyzed by drawing response surfaces. However, the data set only allows obtaining flat or twisted surfaces because the factors were only examined at two levels. Curvature cannot be modeled. An alternative is to calculate main and interaction effects with Equation (3), and to interpret the estimated effects statistically, for instance, with error estimates from negligible effects (Equation (8)) or from the algorithm of Dong (Equations (9), (12), and (13)). Eor the error estimation from negligible effects, not only two-factor interactions but also three- and four-factor interactions could be used to calculate (SE)e. 

If the factor combinations are chosen far apart, the variances and covariances of the parameter estimates will be smaller, and the probability of bracketing the optimal pH will be greater. However, the assumed second-order model might not be as good an approximation to the true response surface over such a large domain of the factor as it would be over a smaller domain. 

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 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. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

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. 

It can be seen that both V(y ) (equation (27)) and E(y ) (equation (28)) are essentially response surface models. From an experiment, estimates of y, D, cTg, Pq, and p can be derived. Suppose, also that the elements of V are known, or can be estimated. Then the search for a choice of design variables that yields a response that is robust to the environmental variation and close to target will involve an examination of these two response surfaces. At this point, the scientist might proceed by following... 

Our research [5,7] showed the value of the so-called indirect optimization designs [8]. The exploration of the response surfaces and the contoured curves enabled us to observe the significant number of combinations giving an optimal point. From the mathematical models, the precise experimental conditions of an optimal point could be estimated and confirmed for the major response (AUC). 

Values for all variation levels are shown in Table 2.154. Select FUFE 23 as a basic design of experiment. Determine the linear regression model from experimental outcomes, Table 2.155. Assume that the obtained linear model is inadequate and that there is curvature of the response surface. To check these assumptions, additional design points were done in the experimental center so that their average is y0=0.1097 (y0—estimate of free member in linear regression, i.e. y0 — 30). Since h0 — y0 = 3 is the measure... 

It should be said that q-responses of pure components makes determination of regression coefficients of linear model possible, while q-internal and central points serve to estimate the nonlinearity of the response surface. It is useful to include in the mentioned designs of experiments q-points of "null effects" in this form ... 

The method of response-surface modeling provides a framework for addressing the above problems and provides accurate estimates of the real coefficients, fi. The basic steps of RSM methodology are... 

Chowdhury and Fard (2001) presented a method for estimating dispersion effects from robust design experiments with right censored data. Kim and Lin (2002) proposed a method to determine optimal design factor settings that take account of both location and dispersion effects when there are multiple responses. They based their approach on response surface models for location and dispersion of each response variable. 

This issue is addressed, for instance, in the discussion of model coefficient aliasing in books on response surface analysis such as Myers and Montgomery (2002). It is clear that this aliasing can introduce serious problems into decision processes based upon the realized estimates of model coefficients. 

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