Response predicted by the model

Although it is beyond the scope of this presentation, it can be shown that if the model yj. = 0 + r, is a true representation of the behavior of the system, then the three sui.. s of squares SS and divided by the associated degrees of freedom (2, 1, and 1 respectively for this example) will all provide unbiased estimates of and there will not be significant differences among these estimates. If y, = 0 + r, is not the true model, the parameter estimate will still be a good estimate of the purely experimental uncertainty, (the estimate of purely experimental uncertainty is independent of any model - see Sections 5.5 and 5.6). The parameter estimate however, will be inflated because it now includes a non-random contribution from a nonzero difference between the mean of the observed replicate responses, y, and the responses predicted by the model, y, (see Equation 6.13). The less likely it is that y, - 0 + r, is the true model, the more biased and therefore larger should be the term Si f compared to 5. ... 

By using UCL and assuming that the model accurately reflects the dose-response relationship at low doses, there is only a five percent chance that the true response is higher than the response predicted by the model. 

Another possibility to evaluate the model is by performing a residual analysis (1,7,17,116). Here, the experimental response and the response predicted by the model are compared for each experimental design point. Large residuals or tendencies in the residuals indicate that the model is not adequate and should be revised. 

The method for estimating parameters from Monte Carlo simulation, described in mathematical detail by Reilly and Duever (in preparation), uses a Bayesian approach to establish the posterior distribution for the parameters based on a Monte Carlo model. The numerical nature of the solution requires that the posterior distribution be handled in discretised form as an array in computer storage using the method of Reilly 2). The stochastic nature of Monte Carlo methods implies that output responses are predicted by the model with some amount of uncertainty for which the term "shimmer" as suggested by Andres (D.B. Chambers, SENES Consultants Limited, personal communication, 1985) has been adopted. The model for the uth of n experiments can be expressed by... 

From the value predicted by the model (y,) to the mean of replicate responses (if any) at the same factor level (y,). This distance is a measure of the lack of fit of the model to the data if the model does a good job of predicting the response, this distance should be small. 

From the value predicted by the model (y,) to the response itself (y,). This distance corresponds to the already familiar residual. ... 

A response surface model of the effects of HA protein concentration (gliadin, the wheat prolamin), HA polyphenol concentration (tannic acid, TA), alcohol, and pH on the amount of haze formed was constructed using a buffer model system (Siebert et al., 1996a). Figure 2.12 shows the effects of protein and polyphenol on haze predicted by the model at fixed levels of pH and alcohol. The model indicates that as protein increases at fixed polyphenol levels, the haze rises to a point and then starts to decline. Similarly, when polyphenol increases at a fixed protein level, the haze increases to a maximum and then declines. 

We see from Fig. 7.7 that the experimental values of show the same trends across the series as that predicted by the model calculation. We will find later in subsequent sections that this structure in the densities of states is responsible for the ferromagnetism in bcc iron and the structural trend from hep -> bcc - hep - fee across the nonmagnetic 4d and 5d series. 

Another non - parametric approach is deconvolution by discrete Fourier transformation with built - in windowing. The samples obtained in pharmacokinetic applications are, however, usually short with non - equidistant sample time points. Therefore, a variety of parametric deconvolution methods have been proposed (refs. 20, 21, 26, 28). In these methods an input of known form depending on unknown parameters is assumed, and the model response predicted by the convolution integral (5.66) is fitted to the data. 

This has important implications with respect to the shape of the voltammetric response predicted by the different models. Thus, the MHC model has been proven, theoretically and experimentally, to be unable to fit the voltammetric response of redox systems that show BV transfer coefficients significantly different from 0.5 [30]. In such cases, as well as in the analysis of surface-confined redox systems, the use of the asymmetric Marcus-Hush theory has been recommended [35] which considers that the force constants for the redox species can be different leading to Gibbs energy curves of different curvatures. 

For the most frequently used low-dose models, the multi-stage and one-hit, there is an inherent mathematical uncertainty in the extrapolation from high to low doses that arises from the limited number of data points and the limited number of animals tested at each dose (Crump et al., 1976). The statistical term confidence limits is used to describe the degree of confidence that the estimated response from a particular dose is not likely to differ by more than a specified amount from the response that would be predicted by the model if much more data were available. EPA and other agencies generally use the 95 percent upper confidence limit (UCL) of the dose-response data to estimate stochastic responses at low doses. 

The statistics reported for the tit are the number of compounds used in the model (n), the squared multiple correlation coefficient (R2), the cross-validated multiple correlation coefficient (R2Cv) the standard error of the fit (s), and the F statistic. The squared multiple correlation coefficient can take values between 0 (no fit at all) and 1 (a perfect fit) and when multiplied by 100 gives the percentage of variance in the response explained by the model (here 83%). This equation is actually quite a good fit to the data as can be seen by the plot of predicted against observed values shown in Figure 7.6. 

Table 14 shows results obtained for every formula development according to MODDE 4.0 software. The collected experimental data were fitted by a multilinear regression (MLR) model with which several responses can be dealt with simultaneously to provide an overview of how all the factors affect all the responses. The responses of the model, R2 and Q2 values, were over 0.99 and 0.93 for tm% and 0.98 and 0.89 for /30%, respectively, implying that the data fitted well with the model. Here, R2 is the fraction of the variation of the response that can be modeled and Q2 is the fraction of the variation of the response that can be predicted by the model. The relationship between a response y and the variables xh xh... can be described by the polynomial ... 

A response surface model which has been determined Ity regression to experimental data can be used to predict the response for any given settings of the experimental variables. The presence of a random experimental error is, however, transmitted into the model and gives a probability distribution of the model parameters. Hence, the precision of the predictions by the model will depend on the precision of the parameters of the response surface model. The error variance of a predicted response, V(yi), for a given setting, Xj = [Xjj,. .. x ] of the experimental variables is determined by the variance function, d, introduced in Chapter 1. 

A consensus modeling can provide for each molecule the standard deviation of the responses predicted by the selected models, which is a measure of the convergence of all the selected models toward a unique response. 

Validation of Optimized Conditions. Once the relationship between the experimental parameters and the response has been modeled and the optimum conditions predicted, experiments should be performed to verify that the response is in fact the desired one. Most commonly, the resolution among the peaks should meet a quantitative requirement. Another method of verification is to compare the predicted response (dehned by the model-predicted optimal conditions) to the actual experimental response. In the case of Nielsen et al., the experimental response fell within the conhdence intervals of the predicted response, and therefore, the model used to optimize the separation of fungal metabolites was a success (66). In the case of the MEKC separation of anionic metal complexes by Breadmore et al., in which the model predicted the electrophoretic mobility of each complex, the model-predicted separation was overlaid with an actual separation, shown in Figure 5.4. Inspection of the coinciding peaks shows that the prediction was, in fact, accurate. Once the separation is deemed optimized, validation of criteria by figures of merit such as precision, dynamic range, selectivity, limit of detection, limit of quantitation, and robustness (see Table 5.1) are typically performed to ensure reproducible and secure results (34). 

Utilization and uptake of the pesticide was rapid within the first 5 days. At day five utilization and uptake decline as the substrate became limited. The population dropped in response to the loss of the carbon supply. Soulas (26) had developed a theoretical model that predicted a short lag phase and a rapid increase and subsequent decline in biomass size in systems adapted for growth on a particular pesticide. Our data conforms to the curves predicted by the model. However, the model indicated complete conversion of the applied pesticide. This was not the case in the enhanced soils. Buildup of the metabolite may have decreased the activity of the population. This was not considered in Soulas (26) model. The estimated 5.14 x 10 cells gr soil found at day 15 implies that the degradation ability is not widespread and is harbored in a distinct subsection of the population. 

Analysis of the data of reference 8 by the Box-Cox method shows that the experimental variance cannot be considered constant within the domain and the most adequate value of X is zero. This corresponds to a logarithmic transformation of the data, which was the one used by the authors. The coefficients of the model, (with the logarithmic transformation) are given in table 10.11 (column A). The response surface predicted by the model is shown in figure 10.8 (dotted line contours). 

Abstract The Canadian Nuclear Safety Commission (CNSC) used the finite element code FRACON to perform blind predictions of the FEBEX heater experiment. The FRACON code numerically solves the extended equations of Biot s poro-elasticity. The rock was assumed to be linearly elastic, however, the poro-elastic coefficients of variably saturated bentonite were expressed as functions of net stress and void ratio using the state surface equation obtained from suction-controlled oedometer tests. In this paper, we will summarize our approach and predictive results for the Thermo-Hydro-Mechanical response of the bentonite. It is shown that the model correctly predicts drying of the bentonite near the heaters and re-saturation near the rock interface. The evolution of temperature and the heater thermal output were reasonably well predicted by the model. The trends in the total stresses developed in the bentonite were also correctly predicted, however the absolute values were underestimated probably due to the neglect of pore pressure build-up in the rock mass. 

Fig. 7A.2. Responses predicted by the quadratic model plotted against the observed responses.

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