Domain of factors

It is evident that Equation 11.71 can be used to plot the variance (or uncertainty) of predicting a single new value of response if Xq is made to vary across the domain of factor space. Such a plot of standard deviation of predicting a single new value of response as a function of pH is shown in Figure 11.7 for the experimental design of Equation 11.15, the data of Equation 11.16, and the second-order model of Equation 11.39. 

Figure 12.2 is a graphic representation of a portion of two-dimensional factor space associated with the system shown in Figure 12.1. In this illustration, the domain of factor (the horizontal axis ) lies between 0 and +10 similarly, the domain of factor X2 (the vertical axis ) lies between 0 and +10. The response axis is not shown in this representation, although it might be imagined to rise perpendicularly from the intersection of the factor axes (at jCj = 0, X2 = 0). Figure 12.2 shows the location in factor space of a single experiment at jc, = +3, X21 = +7.

Having selected the system response, we start choosing factors, levels of the factors and center point of the design (basic level or the null point). By factor we understand the controllable independent variable that corresponds to one possibility of influence on the object of research. A factor is considered defined if its name and domain of factors are determined. A factor may take several values in this field. The chosen factor values, both qualitative and quantitative, are called factor variation levels. Factor variation levels in the design of experiments are coded values. Under factor interval of variation we understand the difference between two factor levels, which in their coded form have value one. When selecting the factors one should pay attention to the conditions they must meet. 

Factor singularity means its property to directly change the status of a research subject, i.e. it is not a function of other factors and it may be fixed to any value in the domain of factors. 

Factor concordance is a property that makes it possible for all factor combinations to be realized in an experiment. This property is very important when an experiment with several simultaneous factor variations is designed. It is not a rare case where the lack of this property brings about a change in defining a research problem, excluding some factors from the experiment, or it changes the domain of factors. 

When selecting a domain of factors one should pay special attention to choosing the center point of the design (basic level or the null point). The choice of a null point is associated with selection of the initial status of the research subject to perform optimization. As optimization is connected with improvement of the subject status in comparison with the status in the null point, it is desirable that the point is in the optimum region or as close to it as possible. If the mentioned research was preceded by other experiments on the same subject, the status having the most convenient response value is taken as the null experiment. The null point is quite often the center of the domain of factors. The most important alternatives in selecting the basic and null levels are shown in Fig. 2.10. 

Finally, we call the extreme values those factors can take, without changing the physical-chemical properties of research subject, physical limits of factors, and the interval Xlrn lx-X min domain of factors. Geometric interpretation is shown in Fig. 2.14. 

Domain of factors is marked O . The figure clearly shows that intervals of factor variations are part of the domain of factors when the optimization problem is being solved. This is necessary in order to realize movement towards optimum in this domain. The experiment domain is in the same figure marked by letter E". In studies with an objective of approximation or interpolation, that is mathematical modeling, the factor-variation intervals cover the whole of the domain of factors. For a two-factor experiment the upper level of factors X and X2 corresponds to values Xlmax,-and X2max, while the lower levels have values Xlmin, X2min. Domain of factors O is in that case called intcrpolational, and E the domain of extreme experiment. 

Let us stick to response geometrical interpretation of black box with two input factors. A simple graphic system with x-y coordinates is sufficient for this. One may insert values of variation levels of one factor on one axis, and those of the other factor on another axis. Each black box status will have a corresponding point in the surface. As has been said in Sect. 2.1.3, factors are defined by their domains. This means that each factor is defined by its minimal and maximal values where it may be changed continuously or discontinuously. If the factors are concordant then those limits in the plane form a rectangle within which are the points that coincide with black box statuses. Dashed lines in Fig. 2.28 mark the limit values of the domain of factors and full lines the limits of concordant domain of factors. To present graphically the response values, we use the third axis of the coordinate system, so that the response surface has the shape given in Fig. 2.29. 

The first series of design points-trials of a basic design is preceded by numerous activities meant to select the local domain of factor space. Thereby, limits of factors space that in principle determine the limitations, technoeconomic possibilities and concrete conditions for doing the process, are being estimated. The factor space requires careful analysis of preliminary information on the scope of response change and on curvature of the response surface. 

Jones TD, Phillips WJ, Smith BJ, Bamford CA, Nayee PD, Baglin TP, Gaston JS, Baker MP. Identification and removal of a promiscuous CD4+ T cell epitope from the Cl domain of factor VIII. J Thromb Haemost 2005 3 991-1000. 

Also see color figure.) Tissue factor-factor Vila complex. The three-dimensional structure of the complex of factor Vila and tissue factor (minus the transmembrane polypeptide domain of the tissue factor) in the absence of membrane surface. It is approximately 115 A in length and has a diameter of 40-50 A. Factor Vila shows its four distinct domains the Gla domain, two EGF-like domains, and the proteinase domain. Tissue factor contacts factor VHa via the interface between the two fibronectin type Ill-like domains. All four domains of factor Vila appear to be involved in the interaction between tissue factor and factor Vila. The Gla domain of factor Vila is folded very similarly to the Gla domain of prothrombin (Gla domain of prothrombin fragment 1). Activation of factor VII can be catalyzed by thrombin, factor Xa, factor Vila, and factor Xlla—all by cleavage at Arg -Ile . Secondary structures are shown in the center diagram two views of the close interactions between TF and factor Vila are shown in the two diagrams at each side. 

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