Model estimation, response surface designs

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]. 

The response surface designs require at least three levels for each variable, in order to be able to detect and model curvature in the response. The model is very often an empirical second-order or quadratic one (see Eq. (6.2)). The coefficients in the second-order model are estimated using multiple regression and they allow to predict... 

The number of experiments N in response surface designs is larger than the number of b-coefficients that needs to be estimated. The obtained model then can be used to predict the response for given experimental conditions. It should be emphasized that only predictions within the examined experimental domain are recommended. Extrapolations should be avoided because the model may not be correct anymore and the prediction error will increase (7). However, most frequently the model is used to determine the optimum, and this is selected from the graphical representation (Figure 2.18), rather than using the model for predictive purposes. 

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 ... 

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. 

The design must enable estimation of the first-order effects, preferably free from interference by the interactions between factors other variables. It should also allow testing for the fit of the model and, in particular, for the existence of curvature of the response surface (center points). Two-level factorial designs may be used for this (shown earlier). 

The parameters in the response surface model will allow for an evaluation of each variable if the estimated value of each model parameter is independent of the estimated value of other model parameters. (For certain experimental designs, this is not possible to attain, but the estimates are as independent as possible.)... 

From this conclusion follows, that a factorial design can be used to fit a response surface model to account for main effects and interaction effects. In the concluding section of this chapter is discussed how the properties of the model matrix X influence the quality of the estimated parameters in multiple regression. It is shown that factorial design have optimum qualities. 

The reasoning above is general and applies to all least squares estimations. In the example below is shown how these criteria are fulfilled when a two-level factorial design is used to estimate the parameters of a response surface model. 

If the variables do not have any influence whatsoever on the response, the response surface is completely flat and the true value of all BpS and Byzs is zero. The estimated effects in such cases would be nothing but different average summations of the experimental error. If we have randomized the order of execution of the experiments in the design, and have done all what we can do to avoid systematic errors, the set of estimated model parameters, [ft, b2,...b, would be a... 

Assume that you have run experiments by a factorial design (with Np runs) with a view to assessing the significance of the experimental variables fi om estimates, hj, of the coefficients in a linear response surface model. Assume also that you have made Nq repeated runs of one experiment to obtain an estimate of the experimental error standard deviation. From the average response, J, in repeated runs, an estimate of the experimental error standard deviation, Sq, with (Nq - 1) degrees of freedom is obtained as... 

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