Test for curvature

When points are added to the center of the 2 design, then the test for curvature actually tests the following hypotheses ... 

Although the Rockwell test is intended to be used on flat parallel-sided specimens, its use can be extended to rounded surfaces by using a curvature correction factor. Compound surfaces such as gear teeth can be tested but the results must be corrected for curvature. 

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... 

The method of caustics has also been used to study the formation of cracks and crazes formed by exposure of PMMA to solvents (259). ISO 4599 (260) has been developed to better control the application of stress using a jig having the curve of the arc of a circle for shaping the specimen and maintaining a set curvature during exposure to the agent. After a predetermined time the specimen is tested for tensile or flexural properties and compared to preexposure test values. ISO 4600 (261) uses the technique of impressing an oversized ball or pin into a hole drilled in the specimen to apply a strain. 

In this model, the regression coefficients of the pure quadratic terms (the fijj) are not estimable because the typical screening design has all factors at only two levels. However, the experimenter should be alert to the possibility that the second-order model is required. By adding center points to the basic 2f factorial design we can obtain a formal test for second-order curvature, that is, a test of the null hypothesis... 

Whereas the curvature types for truncation are complementary, the above two (a,b) maps cannot yet be compared directly, since in a direct comparison of these maps, identical, and not complementary, a and b values occur for the two molecules. However, the complementarity of density thresholds and curvatures can be taken into account by a simple transformation by inverting the (a,b) parameter map of molecule M2 centrally with respect to the point (ao,0), and by comparing the centrally inverted (a,b) map of M2 to the original (a,b) map of M ]. This transformation ensures that domain types, density thresholds, and curvature parameters are matched properly, as required by the pairing scheme (6.77) - (6.79). For example, the locally convex domains of MIDCO G(ao-a, M ) relative to the reference curvature b are tested for shape complementarity against the locally concave domains of MIDCO G(ao+a, M2) relative to a reference curvature - b. 

The design must enable estimation of the first-order effects, preferably free from interference by the interactions between factors other variables. It should also allow testing for the fit of the model and, in particular, for the existence of curvature of the response surface (center points). Two-level factorial designs may be used for this (shown earlier). 

The Saam-Cole approach has several approximations, among which are the neglect of the sohd-adsorbate interaction and curvature effects on the adsorbate chemical potential, and curvature effects on surface tension in symmetrical and asymmetrical states, while modeling the multilayer region. Here, a more accurate version of the above approach has been introduced and tested for explaining the reversibihty of adsorption in MCM-41. For fluid molecules inside a cyhndrical pore of radius R, the incremental potential function has been expressed as [4,6,7]... 

P-value associated with a test for the significance of each factor. P-values less than 0.05 indicate a strong effect. Values less than 0.10 (or perhaps 0.15) indicate factors that we should continue to track and use in our model building. Note that the table does not list the interaction effects. This is because for nearly all the responses none of the interactions were significant (at a = 0.1). Only the amount of E at 12 hours had significant interactions, with X1 X2, X1 X3, and curvature all having significant effects. 

Evidently, the locally convex domains of the MIDCO G(Ki,a0—a ) of fragment F, relative to the reference curvature b are tested for shape complementarity against the locally concave domains of the MIDCO G(K2,a0 + a ) of fragment F2 relative to a reference curvature —b. This is precisely what is required for both curvature and density threshold. 

If P2 - P = 200 MPa and T = 298 K, we need a precision of 3 per cent in the rate measurements to achieve 8(AT ) = 0.5 cm moT. It is some help that the slope is derived not from just two points but from a least squares fit to several points, but the slope is heavily influenced by the extreme values. Overall, a 3 per cent precision is not an easy goal if it is to represent a 95 or 99 per cent confidence interval and imply conscientious tests for systematic errors at that level. This error analysis shows why the pressure range cannot be reduced to much less than 200 MPa and still yield the desired precision. One would prefer a bigger range, were it not for the fact that the curvature, or uncertainty about the curvature, in the plots makes a large range less useful. 

To demonstrate this we use the simple example that was introduced in chapter 4, that of solubility in a mixed surfactant system. The treatment is in two stages, the first being a intuitive rather than mathematical demonstration of testing for lack of fit and curvature of a response surface. Then, in section Il.B, we will carry out a more detailed, statistical analysis of the same process, showing how prediction confidence limits are calculated and the use of ANOVA in validating a model. 

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