Scheffe model

For the regression analysis of a mixture design of this type, the NOCONSTANT regression command in MINITAB was used. Because of the constraint that the sum of all components must equal unity, the resultant models are in the form of Scheffe polynomials(13), in which the constant term is included in the other coefficients. However, the calculation of correlation coefficients and F values given by MINITAB are not correct for this situation. Therefore, these values had to be calculated in a separate program. Again, the computer made these repetitive and Involved calculations easily. The correct equations are shown below (13) ... 

Any component level change must be compensated by changes of the remaining component levels to maintain the unity constraint. To describe the relationships between the response and the component levels, polynomial models of special forms are fit to the data. The Scheffe model (1 ), expressed in quadratic canonical form as... 

The results of three-component mixture designs are often presented as response surfaces over the triangular mixture space as shown in Figure 12.34. The Scheffe model parameters are seen to be equivalent to the responses at the vertexes. 

A quadratic equation was found to fit the crushing strength values the best compared to other Scheffe models [31]. The calculated coefficients of the quadratic model equation (21) are presented in Table 4.4. 

Cornell, J.A. The tting of Scheffe-type models for estimating solubilities of multisolvent systems,... 

VNfells, M.L., Tong, W.-Q., Campbell, J.W., IV, McSorley, E.O., and Emptage, M.R. Afour component study for estimating solubilities of a poorly soluble compound in multisolvent systems using a scheffe-type model,Drug Dev. Ind. Pharm., 22, 881-889, 1996. 

Li, A., Jang, K., Scheff, P., 2003. Application of EPA CMB8.2 model source apportionment of Sediment PAHs in Lake Calumet, Chicago. Environ. Sci. Technol. 37(13), 2598-2965. 

Any apparently more general relationship involving an intercept term and pure quadratic terms can by use of (5-15) be shown to be equivalent to (5-19) in the mixture context.) Relationships of the type of (5-19) are often called Scheffe models, after the first author to treat them in the statistical literature. Other more complicated equation forms are also useful in some applications, but we will not present them in this chapter. The interested reader is again referred to Cornell25,26 for more information on forms that have been found to be tractable and effective. 

Designs in this case, primarily attributed to Scheffe, are derived very simply. That shown in Fig. 7 for three eomponents is suitable for first-, second-, and partial third-order models. The latter is the central composite design and is quite eommonly used. Test points for eheeking model fit are also shown. 

Mixture models (such as those of Scheffe) are still useful, especially when there are three or more such excipients with fairly large ranges of variation. In solid formulations, this is often the case for diluents (or fillers) and also for the polymers or waxes incorporated into controlled-release tablets to form a matrix through which the drug diffuses slowly out when immersed in aqueous fluid, i.e., in the gastrointestinal tract. 

Selected blends of styrene-acrylonitrile copolymer (30 to 55%), a styrene-butadiene copolymer grafted with styrene and acrylonitrile (45 to 70%), and a coal-tar pitch (0 to 25%), were prepared. Physical properties of the experimental blends were determined and statistical techniques were used to develop empirical equations relating these properties to blend composition. Scheff canonical polynominal models and response surfaces provided a thorough understanding of the mixture system. These models were used to determine the amount of coal-tar pitch that could be incorporated into ABS compounds that would still meet ASTM requirements for various pipe-material designations. ... 

Empirical models were used to relate the changes in composition to resultant changes in properties. The form of the model used is the special cubic which was developed by Scheffe (2). 

The polynomial models, first described by Scheffe (4) and traditionally used for studying mixtures, are of two kinds ... 

The first group of models corresponds to Scheffe s centroid designs, the second to the Scheffe lattice designs. In fact, the models and the designs are only distinct from the third-order onwards and centroid and lattice designs are identical for the first and second degree. 

The Scheffe models (whether canonical polynomial or reduced models) can be used for the analysis of both simplex and non-simplex designs. 

These designs, which are saturated, allow the corresponding reduced models to be determined directly, and are optimal for the Scheffe reduced models. The points for the 4 component factor space are shown in figure 9.8. 

The experimental regions we have treated, when in the form of a simplex, have had the same shape as the total factor space. Thus they have been equilateral triangles, regular tetrahedra, etc. However this is not a necessary limitation. Any point in an irregular simplex, like the simplex of figure 9.17a, may be defined in terms of a mixture of the pseudocomponents at eaeh of the vertices. A response may then be analysed within that simplex by means of the Scheffe models and designs. This approach was used by the authors of references (7) and (8). 

Equation 9.8 suggests the use of a 2 factorial design to study the effect of the temperature. Equation 9.9 would require a first-order Scheffe design at each temperature (simplex vertices). In fact two independent measurements of solubility were carried out at each point. Also unreplicated test points were set up at the midpoints of the binary mixtures (points 7-12) that would allow use of a more complex model, if necessary. The resulting design is given in table 9.14. 

Experiments were carried out at each vertex of the space (table 10.2, experiments 1-4). It thus resembles a 2 factorial design. Analysis would normally be carried out using one of two models. There is the first-order Scheffe model (see chapter 9) ... 

In the case of the solubility problem, figure 10.6, it can be seen that 5 midpoints of edges have been added, as well as a point close to the overall centre of gravity. There are 11 experiments to determine a second-order Scheffe model... 

First-order, second-order, or reduced third-order Scheffe models may be used. Alternatively, it may be postulated that certain components have more pronounced effects than others on the curvature of the response. "Interaction terms" between certain components might thus be omitted. In the model which follows, it is assumed that neither the 2 polymers, nor the 2 diluents exhibit synergism. The second-order terms and P34X 4are left out. 

Instead of the 12 element model of equation 10.4 or the full second-order Scheffe model, we may postulate a model where the total polymer is to be considered as one component, X, but where the proportion of permeable acrylic polymer is taken as an independent coded variable Zj. This approach is possible because there are only 2 components in the class. The full second-order model is ... 

The optimal design of 9 experiments for this model is obtained by multiplying the second-order Scheff design for the polymers (3 points) by the first-order Scheffe design for the diluents (3 points). It is saturated. 

Another way of treating the above problem, where the major component "polymer" is fixed and has only 2 minor components, is to define a Scheffe model for the remaining components and treat the relative proportions of polymers A and B as an independent "process" variable. It may be transformed to a coded variable Z,. Thus when all the polymer is "A", z, = -1, when all of it is "B", Zj = -i-l, and for a 50 50 mixture, Zj = 0. The models and designs are those of mixture/process models and designs. For example, the second-order model is ... 

Scheffe canonical First-order Scheff6 canonical polynomial model SchefK canonical polynomial model... 

The closure constraint has to be taken into account also in modelling results of mixture experiments. The closure means that the columns of the model matrix are linearly dependent making the matrix singular. One way to overcome this problem is to make the model using only N-1 variables, because we need to know only the values of N-1 variables, and the value of the N th variable is one minus the sum of the others. However, this may make the interpretation of the model coefficients quite difficult. Another alternative is to use the so-called Scheffe polynomials, i.e. polynomials without the intercept and the quadratic terms. It can be shown that Scheffe polynomials of N variables represent the same model as an ordinary polynomial of N-1 variables, naturally with different values for the polynomial coefficients. For example the quadratic polynomial of two... 

Ravikumar R, Eugaccia I, Scheff SW, Geddes JW, Srinivasan C, Toborek M (2005) Nicotine attenuates morphological deficits in a contusion model of spinal cord injury. J Neurotrauma 22 240-251... 

Rizzo MJ, Scheff PA (2007) Utilizing the chemical mass balance and positive matrix factorization models to determine influential species and examine possible rotations in receptor modeling results. Atmos Environ 41 6986-6998... 

To obtain the estimates of the model coefficients allowing the best forecast quality in the experimental domain studied, Scheffe propo.ses an experimental design that he calls simple.x lattice design. In the case of q components and for a polynomial of degree m, the corresponding simplex lattice design is noted q, /n. The coordinates of each point are multiples of /m and such that ... 

---