Inadequate linear model

If the optimum region is close by, the research by this model ends and we switch to constructing the design of experiments for the second-order model. Fig. 2.38 shows the block diagram of searching for an optimum for an inadequate linear model. 

Figure 2.38 Block diagram of search for optimum for an inadequate linear model...

Movement to optimum by an inadequate linear model is also possible in cases when doing the mentioned eight trials is not acceptable. The values of linear regression coefficients are considerably above the values of those for interactions, the more so since linear effects are not aliased/confounded with interaction effects. Although the movement to optimum by an inadequate linear model is mathematically incorrect, it may be accepted in practice with an adequate risk. Note that when trying to optimize a process one should aspire towards both the smallest possible interaction effects and approximate or symmetrical linear coefficients. In problems of interpolation models, the situation is exactly the opposite since it insists on interaction effects, which may be significant. 

If the experimental objective is to obtain an interpolation model, an adequate linear model is the solution. In the case of an inadequate linear model, one of the following activities is undertaken indusion of interaction effects into the model, upgrading the design, transformation of variables, change of variation intervals. 

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

As with an inadequate linear model, it is possible to switch to a higher order or a third-order model. Realization, processing of experimental results and analysis and interpretation are very complicated for third-order designs, which makes such a suggestion not efficient enough [16]. 

Screening designs are mainly used in the intial exploratory phase to identify the most important variables governing the system performance. Once all the important parameters have been identified and it is anticipated that the linear model in Eqn (2) is inadequate to model the experimental data, then second-order polynomials are commonly used to extend the linear model. These models take the form of Eqn (3), where (3j are the coefficients for the squared terms in the model and 3-way and higher-order interactions are excluded. 

Linearity. Whether the chosen linear model is adequate can be seen from the residuals ey over the x values. In Fig. 6.8a the deviations scatter randomly around the zero fine indicating that the model is suitable. On the other hand, in Fig. 6.8b it can be seen that the errors show systematic deviations and even in the given case where the deviations alternate in the real way, it is indicated that the linear model is inadequate and a nonlinear model must be chosen. The hypothesis of linearity can be tested ... 

The complete linear models were determined by regression (Table 9). The R2 values for In Y2, T4, In Y5 In Y6 and mainly for Y3 are quite low, indicating that the linear model is inadequate for describing the situation for these variables and that a quadratic model could better fit the data. Nevertheless, Y6 from experiment 12 is a good value, as well as the other responses in this trial and may produce the optimum. 

If a linear model is inadequate it means that the response surface is not approximated to the plane. Apart from Fisher s criterion, which is there to judge the lack of fit of a regression model, inadequacy may also be recognized in this way ... 

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... 

In case linear model has been inadequate, we turn back to block diagram in Fig. 2.39 to establish reasons for inadequacy. 

Assume this situation the basic design half-replica, linear model, is inadequate, the method of steepest ascent proved to be inefficient, the optimum area is close by. The system response is the product yield. Maximal possible yield is 100%. The best yield in realizing half-replica is 80%. Trial error is 1%. 

Optimum area far away-linear model inadequate... 

The linear regression model is inadequate with 95% confidence. Since the linear model is neither symmetrical nor adequate and since the application of the method of steepest ascent would lead to a one-factor optimization (b2 is by far the greatest), a new FRFE 24 1 has been designed with doubled variation intervals for X3 X3 and X4. 

To draw conclusions about the obtained regression model it is necessary to interpret the outcomes or to translate the model into the researcher s language. Since a linear model is inadequate, one must when interpreting the model analyze the even the factor interactions. Linear regression coefficients show by their values approximately the same effect of all factors on response. Their signs are identical so that the response value increases with an increase in factor value. The response or coefficient of the separation value in the last trial reaches the value from the reference... 

The obtained linear model proved to be inadequate, which means that the response surface is curved. The method of steepest ascent is applied to linear members of the regression mode], Table 2.242. 

When simple univariate or multivariate linear models are inadequate, higher-order models can be pursued. For example, in the case of only one instrument response (wavelength), Equation 5.8... 

In practice, augmentation can be performed after the experimenter has completed a full factorial design and found a linear model to be inadequate. A possible reason is that the true response function may be second order. Instead of starting a completely new set of experiments, we can use the results of the previous design and perform an additional set of measurements at points having one or more zero coordinates. All of the data collected can be used to tit a second-order model. 

In Figures 12.10 and 12.11, we illustrate a comparison of simulated (curve-fitted) versus predicted BTCs for the BC-II column. Clearly, regardless of whether a kinetic or equilibrium model was used, the predictions overestimated the extent of sorption and resulted in much-delayed BTCs. These predictions were obtained with independently derived parameters from the BC-I column for the equilibrium linear model as well as the kinetic model. It is obvious that the independently measured parameters for the high concentration were inadequate in describing BTC for the low concentration. Therefore, the reactivity or retention of S04 during transport is concentration dependent. 

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