Adequate linear model

Figure 2.37 Block diagram of search for an optimum for an adequate linear model... 

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. 

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 ... 

When a blank appears, it has to be estimated from a sufficiently large number of blank measurements and the measured values must be corrected in this respect. To ensure the adequateness of the SA calibration model, p >2 additions should be carried out. Only in the case when it is definitely known that the linear model holds true, then one single addition (ft times repeated) may be carried out. In general, linearity can be tested according to Eqs. (6.49)-(6.51). 

Simulation methods have been proved to be useful in the study of many different molecular systems, in particular in the case of flexible polymers chains [ 14]. According to the variety of structures and the theoretical difficulties inherent to branched structures, simulation work is a very powerful tool in the study of this type of polymer, and can be applied to the general problems outHned above. Sometimes, this utility is manifested even for behaviors which can be explained with simple theoretical treatments in the case of linear chains. Thus, the description of the theta state of a star chain cannot be performed through the use of the simple Gaussian model. The adequate simulation model and method depend strongly on the particular problem investigated. Some cases require a realistic representation of the atoms in the molecular models [10]. Other cases, however, only require simplified coarse-grained models, where some real mon-... 

The quadratic model is an improvement on the linear model since it gives information about the curvature of the function and contains a stationary point. However, the model is still unbounded and it is a good approximation to fix) only in some region around xc. The region where we can trust the model to represent fix) adequately is called the trust region. Usually it is impossible to specify this region in detail and for convenience we assume that it has the shape of a hypersphere s <, h where h is the trust... 

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. 

With nonlinear models, which are aimed at mathematical modeling or adequate description of the optimum region and that as a rule have numerous regression coefficients, rejection of insignificant regression coefficients is not so important as in the phase of linear modeling. For second-order models, an estimate of lack of fit or inadequacy of the model is of particular importance. 

It is much more complicated to draw conclusions after an unsuccessful application of the method of movement to optimum along the gradient. Drawing conclusions depends much more on whether the optimum is close by, far away or its position is unknown, and whether the linear model is adequate. Typical cases of such situations are demonstrated in Fig. 2.46. 

Optimum area far away-linear model adequate... 

If the optimum area is far away and the linear model adequate, there exist good reasons for the method of steepest ascent to be successful. A possible explanation for a failure in applying the gradient method may lie in the form of the response surface with one extreme. The response surface may in reality have the form shown Fig. 2.47. 

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. 

The residuals e, = (y - yt) should also fit a normal distribution, i.e. they should not correlate with xt. An easy way to check this is to plot et versus x . If the residuals do not scatter randomly around zero, the linear model may not be adequate for the data. An indication of a wrongly specified model may be the occurrence of auto-correlated residuals, which can be checked by the DURBIN-WATSON test [MAGER, 1982],... 

Rigorous nonlinear models must be used in analyzing batch reactors because of the changing process parameters. Continuous reactors operate around some steady-state level, so linear models are sometime adequate for establishing controller tuning constants. 

The simplest model that adequately describes the concentration-response relationship should be used, e.g., a linear model is simpler than a quadratic model. At the completion of the validation, evaluation of different regression models must be performed. Justification for using a quadratic regression equation must be documented. 

It was assumed that a linear model augmented with the above-mentioned interaction terms would give an adequate description of the variation of the observed yield, y, i.e. 

The linear-effect model defines this relationship adequately as long as biophase drug concentrations/ [B], are substantially less than is the EC50. However/ the decision to use an Emax rather than a linear model is usually determined by the available data rather than by theoretical considerations. For example/ in one study of QT interval prolongation by an anti-arrhythmic drug/ a linear-effect model was satisfactory for analyzing the response of four patients but an Emax model was required to analyze the exaggerated response of a fifth patient (15). 

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