Second-order polynomial quadratic model

Estimation of Modei. Two types of models can be built mechanistic and empirical models. Usually, empirical models are applied in an experimental design context (1,7). Most frequently, a second-order polynomial quadratic model is built. Such model includes an intercept, the main effect terms, the interaction effect terms, and the quadratic effect terms. Occasionally, not all possible terms are included in the model that is, the nonsignificant terms can be deleted. 

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... 

A linear (first order) calibration model requires five standards, a quadratic (second order) model requires six standards, and a third order polynomial calibration model requires seven standards. 

The design used is a function of the model proposed. Thus, if it is expected that the important responses vary relatively little over the domain, a first-order polynomial will be selected. This will also be the case if the experimenter wishes to perform rather a few experiments at first to check initial assumptions. He may then change to a second-order (quadratic) polynomial model. Second-order polynomials are those most commonly used for response surface modeling and process optimization for up to five variables. 

The results were modeled with a second-order polynomial equation, i.e. each response j was fitted by a quadratic model, given as follows for four factors. 

Figure 1.5 Sample residual plot. Paired (x, Y) data were simulated using the model Y = 13 + 1.25x + 0.265x2. To each Y value was added random error from a normal distribution with mean zero and standard deviation 25. The top plot is a plot ordinary residuals versus predicted values when the fitted model was a second-order polynomial, the same model as the data-generating model. The bottom plot is the same plot when the fitted model was linear model (no quadratic term). Residual plots should appear as a shotgun blast (like the top plot) with no systematic trend (like the bottom plot).

The most common nonlinear empirical model is a second order polynomial of the design variables, often called a quadratic response surface model, or simply, a quadratic model. It is a linear plus pairwise interactions model added with quadratic terms, i.e. design variables raised to power 2. For example, a quadratic model for two variables is y = 0 + b-yX-y +12X2+ bi2XiX2 + + 22XI. In general, we use the notation that b,- is the... 

Response surface morphologies were utilized to analyze the experimental data (Table 15.5) and a second order polynomial equation was fitted by multiple regression analysis [44-45]. The quality of fit for the model was estimated by the coefficients of determination (R ) and the analysis of variances. After the coefficients were examined and the model was fine-tuned, all the insignificant coefficients were omitted. As a result, the quadratic response model was fitted to the following equation ... 

These models are polynomials of the second order, i.e., they contain squared terms and binary interactions. In principle one could think of third- and higher-order polynomials, but this is rarely necessary. Ternary interactions are rarely relevant and third-order models or nonlinear models (in the statistical sense of the term nonlinear) do not often occur. Nature can in practice often be approximated, at least locally, by smooth functions such as second-order equations. Exceptions exist for example, pH often leads to sigmoid curves when the measured response is due to the cumulated response of the ionized and nonionized species of the same substance. This curve would be difficult to model over the whole experimental domain with a second-order equation. Quite often, one will not be interested in the whole domain but in a more restricted region. In that case it may be feasible to model the response using a quadratic function of the independent variable. 

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