Second-order polynomial model

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

Anticipating a later section on canonical analysis of second-order polynomial models, we will show that the first-order term can be made to equal zero if we code the model using the stationary point as the center of the symmetrical design. For this new system of coding, c, = 10 2/3 and (see Section 8.5). 

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Efficiency of full second-order polynomial models fit to data from central composite designs without replication. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

To find the coordinates of the stationary point, we first differentiate the full second-order polynomial model with respect to each of the factors and set each derivative equal to zero. For two-factor models we obtain... 

The corresponding matrix least squares treatment for the full second-order polynomial model proceeds as follows. 

Write full second-order polynomial models for 1, 2, 3, 4, and 5 factors. 

Show that the full two-factor second-order polynomial model may be written y, - Po + Do d sd where Dq = [x, Xj,]. Show that this may be extended to full three-factor second-order polynomial models. 

Confidence Intervals for Full Second-Order Polynomial Models... 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

Figure 13.1 Sums of squares and degrees of freedom tree for a two-factor full second-order polynomial model fitted to a central composite design with a total of four center point replicates.

Add one design point to a two-factor star design to generate a design that is sufficient to fit a full second-order polynomial model ( y, = Po + PiJCj,- + P Tj, + Pn- ii + P22 i + Pi2 iA2i + "ii)- Hint see Figure 13.11. 

Trivedi et al. utilized Sorby s experimental data for water-ethanol-propylene glycol and Ltted to a complete second order polynomial model and performed a stepwise regression to arrive at following equation where andy represent fractions of ethanol and propylene glycol, respectively ... 

Here t is time in years. This equation provides a perfect fit for the sales figures for EPO as a function of time in years. Once again, t = 0 here indicates the base year, 1992, for which the sales numbers were first available. For practical or predictive purposes Eq. (2) may be better to use than Eq. (3) since it is a power-law type of equation and has some physical basis, even though Eq. (3) provides more accurate estimates than Eq. (2). Second-order polynomial fits can of course be applied to a wide variety of sales curves for different drugs with considerable accuracy. In other words, if one were to attempt to predict future sales of EPO, one should (a) obviously use caution, (b) extrapolate only for a short period (since one cannot predict market forces), and (c) use the powerlaw model as compared to the second-order polynomial model (Eq. 3). 

The analysis of variance (ANOVA) indicated that the second-order polynomial model (above) was statistically significant and adequate to represent the actual relationship between the response (percent weight conversion) and the significant variables, with very small p-value (0.0001) and a satisfactory coefficient of determination (R2 = 0.955). 

The extremely versatile full second-order polynomial model in Equation 3.32 can also be fitted when at least three-level factorial designs are used and 3 experiments are run. Alternatively, a central composite design may be used effectively (see Figure... 

The most useful empirical models are the second-order polynomial models and occasionally a first- or third-order polynomial. The measurement scales of the responses or factors may be transformed to another metric, such as a logarithmic scale, or... 

Second-order polynomial models are versatile enough to describe most relationships between factors and responses. These models consist of an intercept term, first-order and second-order terms for each factor, and two-factor interaction terms for each combination of two factors. The second-order model is shown below for two factors and for multiple factors. 

In summary, RSM is a useful technique for finding the optimum conditions for one or more responses over up to about five factors. The types of experimental designs often used for RSM are the CCD and Box-Behnken designs. The response surface can be well-described by a second-order polynomial model, and thus can be used to readily find the optimum conditions for a single response or to perform a tradeoff analysis among two or more responses. 

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