Long page
descriptions

Chapter 4   Response Surfaces

4.1   Quadratic model for two factors

4.1.1   Quadratic model for two factors

Nonlinear relationships between a response and two numerical factors can be modelled by adding quadratic terms and an interaction to the linear model.

4.1.2   Shape of quadratic response surface

The quadratic model defines a surface that can take various different shapes, and may have a maximum, minimum or saddle point, depending on the values of the model parameters.

4.1.3   Central composite designs

The quadratic model cannot be fitted using data from a factorial experiment in which both factors take two levels, even if augmented with centre points. A central composite design adds 'star points' to the factorial model.

4.1.4   Display of surface with contours

The quadratic model corresponds to a surface in 3 dimensions. Contours on this surface can be used to represent the surface on paper.

4.1.5   Choosing best model

An analysis of variance table can be used to test the significance of the model terms.

4.2   Quadratic model for three factors

4.2.1   Design of experiment

A factorial experiment using 2 levels for each factor must be augmented by star points to allow a quadratic model to be fitted. An alternative is to use a Box-Behnken design in which each factor only takes 3 different levels.

4.2.2   Displaying the model

A quadratic model for three factors defines a surface in 4 dimensions. It can be displayed as 'slices' through this 4-dimensional surface at fixed values of one factor.

4.2.3   Choosing the best model

The significance of the quadratic model's terms can be tested in an analysis of variance table. Care must be taken when a response surface is interpreted since deletion of insignificant terms may changes its shape.

4.3   Mixtures of ingredients

4.3.1   Mixtures of three ingredients

If an experiment has 3 factors whose values are proportions that sum to 1, the possible combinations of factor values correspond to a triangle in 3 dimensions.

4.3.2   Experimental designs for mixtures

Mixture designs try to spread design points evenly over the design space (a triangle for mixtures of 3 ingredients). The simplex-centroid and simplex-lattice designs are often used.

4.3.3   Representing designs in two dimensions

The design space for a mixture of 3 ingredients is an equilateral triangle. Each vertex represents 100% of one ingredient and the opposite face corresponds to none of that ingredient.

4.3.4   Designs for mixtures of four ingredients

When mixtures contain four ingredients, the design space can be represented as a regular tetrahedron (pyramid with 4 equilateral triangles as faces) in 3 dimensions. The simplex-centroid design and simplex-lattice designs are points on the surface and interior of this tetrahedron.

4.3.5   Modelling the response surface

The response can be modelled as a linear or quadratic function of the constituent proportions. For mixtures of 3 ingredients, the mean response can be displayed as a surface above the design space (equilateral triangle).

4.3.6   Hypothesis tests

An anova table describes the reduction in residual sum of squares corresponding to adding linear and quadratic terms to the model. It can be used to test the significance of these terms. If there are more design points than model parameters, lack of fit can also be tested in an anova table.

4.3.7   Constrained mixtures

If the experiment must be conducted with mixtures in which the proportions of some components are fixed or have minimum values, a standard mixture design for the rest of the product can be used.

4.3.8   Constraints on maximum

When there is a contraint on the maximum proportion of a component, the design region is no longer a triangle (or tetrahedron if there are 4 components).

4.3.9   Designs for constrained mixtures

In experiments for mixtures with 3 components where the maximum values of some components are specified, it is often possible to create a suitable design by hand. A computer should be used to generate the design if there are 4 or more components in the mixture.