Quadratic effect

By automation one can remove the variation of the analysis time or shorten the analysis time. Although the variation of the analysis time causes half of the delay, a reduction of the analysis time is more important. This is also true if, by reducing the analysis time, the utilization factor would remain the same (and thus q) because more samples are submitted. Since p = AT / lAT, any measure to shorten the analysis time will have a quadratic effect on the absolute delay (because vv = AT / (LAT - AT)). As a consequence the benefit of duplicate analyses (detection of gross errors) and frequent recalibration should be balanced against the negative effect on the delay. 

Another reason for augmenting the two-level design with center points is that these points allow for an overall test of curvature. It is clear that with only two levels for each variable it is impossible to detect any quadratic effect of the variables. Thus, the underlying model is assumed to... 

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

If the true model contains quadratic terms then the estimate of the intercept, Pq, of the first-order model will be biased. The lack-of-fit of the first-order model due to quadratic effects can be tested by adding center points to the design. 

Mixture variables, expressing the composition of the mobile phase as fi ac-tions, have the property that they add up to one (the mixture restriction). The consequence is that no intercept can be estimated when the effects of the solvents are evaluated [10,19]. Moreover interactions and quadratic effects, such as used when the independent variables are process variables, can not be estimated independently. Mathematically it is better to use blending effects only. Interpretation of these blending effects, i.e. explicitly stating what components are responsible for the non-linear effects, is not possible. 

This electro-optical effect, commonly observed as transient changes in optical birefringence of a solution following application, removal, or reversal of a biasing electric field E(t), has been used extensively as a probe of dynamics of blopolymer solutions, notably by O Konski, and is a valuable tool because it gives information different in form, but related to, results from conventional dielectric relaxation measurements. The state of the subject to 1975 has been comprehensively presented in two review volumes edited by O Konski (25). The discussion here is confined to an outline of a response theory treatment, to be published in more detail elsewhere, of the quadratic effect. The results are more general than earlier ones obtained from rotational diffusion models and should be a useful basis for further theoretical and experimental developments. 

The value of the hyperparameter jt may be chosen by considering the prior expected number of active effects. Illustrative calculations are now given for a full second-order model with / factors, and for subsets of active effects that include linear and quadratic main effects and linear x linear interactions. Thus, the full model contains / linear effects, / quadratic effects, and ( ) linear x linear interaction effects. Prior probabilities on the subsets being active have the form of (22) and (23) above. A straightforward extension of the calculations of Bingham and Chipman (2002) yields an expected number of active effects as... 

In a companion paper (Kleijnen et al., 2003), we changed the metamodel in (1) after the screening phase, as follows. For those controllable factors found to be important by sequential bifurcation, we augmented (1) with quadratic effects to form a predictive model for optimization. For those environmental or noise factors identified by sequential bifurcation as important, we created environmental scenarios through Latin hypercube sampling for robustness analysis. 

The quadratic effect of an externally applied field on the refractive index n is described by the third-order susceptibility (- ) w,0,0) (Kerr susceptibility). The two independent components Yilzz and x ixx can be interpreted in terms of molar polarizabilities. The results for 2 symmetric molecules with only one significant component of the second-order polarizability are expressed in (113) and (114),... 

The quadratic effect of an externally applied field on the absorption coefficient is described by the imaginary part of the third-order susceptibility -o) a),0,0). influences the molar decadic absorption coefficient of the solute. The absorption coefficient in the presence of the field is a quadratic function of the applied field strength (118),... 

So the regression coefficients have been calculated for an orthogonal composition matrix and as a consequence, for the quadratic effect, we obtain the next expressions ... 

Assuming that the pressure has a quadratic effect on the exchange current density, and the temperature has an exponential effect as observed, one can use the DUD method to conduct a least squares statistical search. The correlation for the exchange current density with temperature and pressure within the experimental range was found to be ... 

Three levels of each factor at a constant level of the other factors are required for the estimation of quadratic effects. 

To summarize, the screening experiment has identified four critical inputs HB, DT, P, and MT. It also indicates that the HB DT interaction group and quadratic effects should be included in the model. We are now ready to develop a model for the process. 

The data already collected during the screening experiment can be reused. However, the response surface study also requires additional data to be collected. In order to determine which quadratic effects exists, the six additional trials given in Table 3 must be run. The resulting data is also shown. These trials were selected using the D-optimal design method... 

The results of this calculation are given in Fig. 8. We see that the wave vector has a mean value of 2.41 10 cm and changes within values of 2.34 10 cm to 2.5 10 cm" (+,- 3.5%). We see that it can have a linear effect in the sound velocity determination but a quadratic effect in sound absorption. From here, we calculate the sound velocity for all temperatures as shown in Fig. 9. The Fig. 9 provides data for the temperature dependence of the sound velocity once corrected by the refraction effects. There we see that the sound velocity of the lower phase smoothly decreases with temperature within the two-phase region, whereas that for the upper phase shows a stronger trend. In a previous work and in a different phase transition, K. V. Kovalenko et al. ... 

In some cases, a four-factor two-level full factorial design was used in optimization. Rarely, also 2 and 2 full factorial designs were applied for optimization purposes in the literature. Such designs are not recommended because of the large number of experiments required, that is, 32 and 64, respectively. The above full factorial designs examine the factors at two levels and allow only all main and interaction effects to be estimated but not quadratic effects that is, they do not allow modeling of curvature. An intermediate optimum cannot be found because curvature in the response cannot be modeled from two-level design results. 

Estimation of Modei. Two types of models can be built mechanistic and empirical models. Usually, empirical models are applied in an experimental design context (1,7). Most frequently, a second-order polynomial quadratic model is built. Such model includes an intercept, the main effect terms, the interaction effect terms, and the quadratic effect terms. Occasionally, not all possible terms are included in the model that is, the nonsignificant terms can be deleted. 

Regression leads to a model estimating the relation between the Ai x 1 response vector y, and the Ai x r model matrix X (7,17,116) (Eq. 2.27). N is the number of design experiments, and t the number of terms included in the model. For example, in Equation 2.26, the number of terms equals six, since one intercept, two main effect terms, one interaction term, and two quadratic effect terms were included. The model matrix X is obtained by adding a row of ones before the Aix (r-1) design matrix, which consists of the coded factor levels and columns of contrast coefficients, as defined by the chosen experimental design. 

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