12.2 - Combined Array Design

The combined array design approach treats all the variables the same, no matter they are controllable or noise. These models are capable of modeling the main effects of controllable and noise factors and also their interactions. To illustrate, consider a case with two controllable and one noise factor. Equation 12.1 in the textbook gives a first-order model:

\(y = \beta _{0}+\beta _{1}x _{1}+\beta _{2}x _{2}+\beta _{12}x _{1}x _{2}+\gamma _{1}z_{1}+\delta _{11}x _{1}z_{1}+\delta _{21}x _{2}z_{1}+\varepsilon \)

where the \(\beta_i\) are coefficients of controllable factors, \(\beta_{12}\) is the coefficient of the interaction of controllable factors, \(\gamma_1\) is the coefficient of the noise factor and \(\delta_{ij}\) are the coefficients of interaction between controllable and noise factors. As can be seen, the response model approach puts all of the variables, no matter they are controllable or noise, in a single experimental design. There exist some assumptions which are mentioned as follows:

  • \(\epsilon\) is a random variable with mean zero and variance \(\sigma^2\)
  • Noise factors are random variables (although controllable in the experiment) with mean zero and variance \(\sigma ^{2}_{z} \)
  • If there exist several noise factors their covariance is zero

Under these general assumptions, we will find the mean and variance for the given example, as following:

\(E(y) = \beta _{0}+\beta _{1}x _{1}+\beta _{2}x _{2}+\beta _{12}x _{1}x _{2} \)


\(Var(y) = \sigma ^{2}_{z}\left (\gamma _{1}+\delta _{11}x_{1}+\delta _{21}x_{2} \right )^{2}+\sigma ^{2}\)

Notice that although the variance model involves only controllable variables it also considers the interaction regression coefficients between the controllable and noise factors.

Finally, as before, we perform the optimization using any dual response approach like overlaid contours, desirability functions or etc. (Example 12.1 from the textbook is a good example of overlaid contour plots approach).

From the design point of view, using any resolution V (or higher) design for the two-level factor designs is efficient. Because these designs allow any main effect or two-factor interaction to be estimated separately, assuming that three and higher factor interactions are negligible.