Some important properties of a response surface design (Box and Draper)⁸:

1. Generation of a satisfactory distribution of information throughout the region of interest, $R$ .
2. “Closeness" of $\widehat{y}\left(x\right)$ to $\eta \left(x\right)$ over $R$ .
3. Good detectibility of lack of fit.
4. Insensitivity (robustness) to extreme observations and to violations of the usual normal theory assumptions.
5. Ability to perform experiments in blocks.
6. Extendibility to a higher-order design.
7. Requiring a small number of experimental runs.

1. D-optimality: Maximization of $\left|X^{\text{'}}X\right|$

${\left|X^{\text{'}}X\right|}^{-\frac{1}{2}}$ is propotional to the volume of an ellipsoidal confidence region on $\beta$ .

2. A-optimality: Minimization of $tr\left[{\left(X^{\text{'}}X\right)}^{-1}\right]$
3. E-optimality: Minimization of the largest eigenvalue of ${\left(X^{\text{'}}X\right)}^{-1}$
4. G-optimality: Minimization of $Ma{x}_R\left\{Var\left[\widehat{y}\left(x\right)\right]\right\}$
5. The above are referred to as alphabetic optimality criteria. Some authors, including Box [9], have questioned the applicability of alphabetic optimality theory to response surface experiments, since such optimal designs are sensitive to the form of the model.
6. Rotatability: A design D is rotatable if $Var\left[\widehat{y}\left(x\right)\right]$ is constant at all points that are equidistant from the design center (Box and Hunter (1957)).
1. A rotatable design has the uniform precision property (UP) if the prediction variance at $x=0$ is equal to the prediction variance at a distance $\rho =1$ from the origin. A design is rotatable if and only if the $X^{\text{'}}X$  matrix has a particular form.
2. Rotatability is a desirable property to have, especially when there is a need to optimize $\widehat{y}\left(x\right)$ over the surface of a hypersphere, as is the case in ridge analysis.
7. Orthogonality: A design is orthogonal if it can provide independent information about the effects of the various terms in the model.

Examples:

1. $3^k$ is orthogonal, but is not rotatable.
2. Central composite design can be made rotatable and either have the
3. UP property or the orthogonality property.

These are designs for the estimation of the partial derivatives of the mean response $\eta \left(x\right)$ with respect $x_1,\dots ,x_k$
Myers and Lahoda¹⁷ extended the Box-Draper integrated mean squared error criterion to find appropriate designs for the joint estimation of the partial derivatives of $\eta \left(x\right)$ .
Hader and Park¹⁸ introduced the concept of design rotatability for second-order models. Under this design criterion, $Var\left[\frac{\partial \widehat{y}\left(x\right)}{\partial x_i}\right]$ is constant at all points that are equidistant from the design center $\left(i=1,2,\dots ,k\right)$
Park¹⁹ considered “slope rotatability over all directions": 
$$\widehat{y}\left(x\right)=f^{\text{'}}\left(x\right)\widehat{\beta }.$$

$$\frac{\partial \widehat{y}}{\partial v}=v^{\text{'}}grad\left[\widehat{y}\left(x\right)\right]=v^{\text{'}}G\widehat{\beta },$$

where $v$ is a unit vector.
$$Var\left[\frac{\partial \widehat{y}\left(x\right)}{\partial v}\right]={\sigma }_{\in }^2{v}^{\text{'}}G{\left(X^{\text{'}}X\right)}^{-1}{G}^{\text{'}}v$$

$$Av{g}_{v}Var\left[\frac{\partial \widehat{y}\left(x\right)}{\partial v}\right]=c{\displaystyle \underset{s}{\int }Var\left[\frac{\partial \widehat{y}\left(x\right)}{\partial v}\right]}dA,$$

where $S$  denotes the surface of a hypersphere of unit radius, $c$  is the reciprocal of the surface of $S$ .

$$W\left(x\right)=Ave.Slop{e}^{}Var.=\frac{{\sigma }_{\in }^2}{k}tr\left[G{\left(X^{\text{'}}X\right)}^{-1}{G}^{\text{'}}\right]$$

A design $D$ is slope rotatable over all directions if $W\left(x\right)$ is constant at all points equidistant from the design center.
Park gave necessary and sufficient conditions for a design to be slope rotatable over all directions for second-order models.
$$Rotatabilit{y}^{}{\Rightarrow }^{}Slop{e}^{}Rotatabilit{y}^{}Ove{r}^{}al{l}^{}Directions$$

Related articles

• Huda and Mukerjee²⁰
• Mukerjee and Huda,²¹
• Park²²

Related articles include Hoerl,²⁴ Draper²⁵
To maximize (minimize)
$$\widehat{y}\left(x\right)={\widehat{\beta }}_0+{\displaystyle \sum _{i=1}^k{\widehat{\beta }}_i}{x}_i+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^k{\beta }_{ij}}}{x}_i{x}_j$$

$$\widehat{y}\left(x\right)={\widehat{\beta }}_0+x^{\text{'}}b+x^{\text{'}}Bx$$

subject to the constraint ${\displaystyle {\sum }_{i=1}^k{x}_i^2}=r^2$
$$h\left(x\right)=\widehat{y}\left(x\right)-\mu \left({\displaystyle \sum _{i=1}^k{x}_i^2-r^2}\right),$$

where $\mu$  is a Lagrange multiplier.
For a maximum, $\mu$ should be larger than $e_{\max }\left(B\right)$ .
For a minimum, $\mu$ should be smaller than $e_{\min }\left(B\right)$ .
By selecting several values of $\mu$ as shown above, we obtain plots of
max. $\widehat{y}$  vs. r and plots of $x_i$  vs. $r$  for $i=1,2,\mathrm{...},k$
Khuri and Myers²⁶ introduced a certain modification to the method of ridge analysis in cases in which the design is not rotatable and may even be ill conditioned. In such cases, $Var\left[\widehat{y}\left(x\right)\right]$ can vary appreciably on a hypersphere $S\left(r\right)$ centered at the origin inside a region of interest. The proposed modification optimizes $\widehat{y}\left(x\right)$ on $S\left(r\right)$ subject to maintaining certain a constraint on the size of $Var\left[\widehat{y}\left(x\right)\right]:$
$$Var\left[\widehat{y}\left(x\right)\right]={\sigma }_{\in }^2{f}^{\text{'}}\left(x\right){\left(X^{\text{'}}X\right)}^{-1}f\left(x\right)$$

$$Var\left[\widehat{y}\left(x\right)\right]={\sigma }_{\in }^2{\displaystyle \sum _{i=1}^p\frac{{\left[f^{\text{'}}\left(x\right){\omega }_i\right]}^2}{v_i}}$$

$$f^{\text{'}}\left(x\right){\omega }_i=a_i+x^{\text{'}}{\tau }_i+x^{\text{'}}{T}_{i}x,$$

where $p$ is the number of parameters in the model, ${\omega }_1,\mathrm{...},{\omega }_p$ are orthonor-mal eigenvectors of $X^{\text{'}}X$ , and $v_1,\mathrm{....},v_p$ are the corresponding eigenvalues of $X^{\text{'}}X$ . Let $v_{\min }$ be the smallest eigenvalue of $X^{\text{'}}X$  and let ${\omega }_m$ be the corresponding eigenvector. To optimize $\widehat{y}\left(x\right)$ subject to the constraints
$${\displaystyle \sum _{i=1}^k{x}_i^2}=r^2$$

$$\left|f^{\text{'}}\left(x\right){\omega }_m\right|\le q$$

where $q$ is a small positive constant, or equivalently,
$$\left|a_m+x^{\text{'}}{\tau }_m+x^{\text{'}}{T}_{m}x\right|\le q$$

A design is rotattable if and only if the $X^{\text{'}}X$ matrix has a particular form, which we denote by $X^{\text{'}}{X}_{rot}$ . For any given design $D$ , the “closeness" of the corresponding $X^{\text{'}}X$  matrix to $X^{\text{'}}{X}_{rot}$ can be determined. On this basis, a measure $\psi \left(D\right)$ , which quantifies the percent rotatability that is inherent in $D$ , can be derived. $A$  design $D$ is rotatable if and only if $\psi \left(D\right)=100$ .

For example, the $3^2$ factorial design is 93.08% rotatable.
Properties of $\psi \left(D\right)$ :

• It is invariant to a change of scale (same change for all the input variables).
• It is invariant to the addition of experimental runs at the design center (design augmentation at the center).
• It applies to any response surface model and design.
• It is not invariant to design rotation.

Draper and Pukelsheim³² introduced a similar measure that is invariant to design rotation, but it applies only to second-order models.

• Can be used to compare designs with respect to their degree of rotatability.
• Can be used to construct a design that satisfies a certain desirable criterion, such as a bias or a variance criterion, in addition to having a relatively high percent rotatability.
• Can be used to “repair" rotatability by the addition of properly chosen experimental runs to a given design.

Giovannitti-jensen and myers (1989)

They plotted maximum and minimum values of $\frac{1}{{\sigma }_{\in }^2}Var\left|\widehat{y}\left(x\right)\right|$ , on a hyperspher $S\left(r\right)$ , against $r$ .
Relevant Reference are Khuri³³ used the measure of rotatability $\psi \left(D\right)$  to derive several upper bounds on the range of $\frac{1}{{\sigma }_{\in }^2}Var\left|\widehat{y}\left(x\right)\right|$ . These bounds are easy to compute since they only require determining eigenvalues and traces associated with the matrices $X^{\text{'}}X$ and $F$ , where
$F={\left(X^{\text{'}}X\right)}^{-1}-U_{rot}^{-1}$

and $U_{rot}$  is the “rotatable" portion of $X^{\text{'}}X$ .

Park and Kim³⁴ defined a measure $Q_k\left(D\right)$ to quantify the degree of slope rotatability over axial directions¹⁸ for second-order models. It takes the value zero if and only if the design is slope rotatable. It can also be used to “repair" slope rotatability by design augmentation.

Jang and Park³⁵ considered the maximum, $Vmax\left(r\right)$ , and minimum, $Vmin\left(r\right)$ , values of $\frac{1}{{\sigma }_{\in }^2}W\left(x\right)$  over a hypersphere $S\left(r\right)$ , where

$$W\left(x\right)=\frac{{\sigma }_{\in }^2}{k}tr\left[G{\left(X^{\text{'}}X\right)}^{-1}{G}^{\text{'}}\right]$$

The quantity
$$\Delta \left(r\right)=Vmin\left(r\right)-Vmax\left(r\right)$$
is zero if and only if the design is slope rotatable over all directions. Jang and Park plotted $Vmax\left(r\right)$ and  $Vmin\left(r\right)$ against $r$ (slope variance dispersion graphs).

Khuri and vining³⁶

Specification Problem: To determine conditions on the input variables in a process that cause the mean response $\eta \left(x\right)$   to fall within specified bounds, e.g.,  ${\alpha }^{}{<}^{}\eta {\left(x\right)}^{}{<}^{}b$ , where a and b are given, with a specified degree of confidence.

Procedure

Start with an initial n-point design $D_n$
Calculate ${\widehat{\beta }}_n={\left(X_n^{\text{'}}{X}_n\right)}^{-1}{X}_n^{\text{'}}{y}_n$ $\widehat{y}\left(x\right)=f^{\text{'}}\left(x\right){\widehat{\beta }}_n$

Compute
$$y_{1n}\left(x\right)=\widehat{y}\left(x\right)-ST{D}_n{t}_{\alpha /2,n-p}$$

$$y_{2n}\left(x\right)=\widehat{y}\left(x\right)+ST{D}_n{t}_{\alpha /2,n-p},$$

Where
$$ST{D}_n={\left[M{S}_E{f}^{\text{'}}\left(x\right){\left(X_n^{\text{'}}{X}_n\right)}^{-1}f\left(x\right)\right]}^{1/2}$$

If there is a point $x_0$  such that $\alpha <y_{1n}\left(x_0\right),y_{2n}\left(x_0\right)<b$ , then $x_0$ has a probability of at least $1-\alpha$ of satisfying
$\alpha <\eta \left(x_0\right)<b,$

$x_0$  is a solution to the specification problem.
Khuri and Vining presented a sequential procedure of adding points to $D_n$ , if necessary, until a solution can be found to the inequalities
$$\alpha <y_{1n}\left(x\right),y_{2n}\left(x\right)<b$$

for a sufficiently large $n$ .

An experiment in which a number of responses can be measured for each setting of a group of input variable, $x_1,\mathrm{...},x_k$ , is called a multiresponse experiment.

When several responses are considered simultaneously, any statistical investigation of the responses should take into consideration the multi- variate nature of the data. The response variables should not be analyzed individually or independently of one another. This is particularly true when the response variables are correlated.

Traditional response surface techniques that apply to single response models are, in general, not adequate to analyze multiresponse models.

In particular, if the individual response models are linear, then

$y_i=X_i{\beta }_i+{\in }_i$       $i=1,2,\dots ,r.$

The r linear response models,

$$y_i=X_i{\beta }_i+{\in }_i,$$ $i=1,2,\mathrm{...},r,$

can be represented as a single linear multiresponse model
$y=X\Theta +\delta ,$
Where  $y={\left[y_1^{\text{'}},y_2^{\text{'}},\dots ,y_r^{\text{'}}\right]}^{\text{'}}$

X is a block-diagonal matrix $diag\left(X_1,X_2,\dots ,X_r\right)$ ,

$\Theta ={\left[{\beta }_1^{\text{'}},{\beta }_2^{\text{'}},\dots ,{\beta }_r^{\text{'}}\right]}^{\text{'}}$    and,

$$\delta ={\left[{\in }_1^{\text{'}},{\in }_2^{\text{'}},\dots ,{\in }_r^{\text{'}}\right]}^{\text{'}}$$

The variance-covariance matrix of is given by the direct (Kronecker) product ${\displaystyle \sum \otimes I_n}$

$$\widehat{\Theta }={\left[X^{\text{'}}\left({\displaystyle \sum {}^{-1}}\otimes I_n\right)X\right]}^1{X}^{\text{'}}\left({\displaystyle \sum {}^{-1}\otimes }{I}_n\right)y$$
If ${\displaystyle \sum }$ is unknown, an estimate ${\displaystyle \sum =}\left({\widehat{\sigma }}_{ij}\right)$ can be used, where

$${\widehat{\sigma }}_{ij}=\frac{1}{n}{y}_i^{\text{'}}\left[I_n-P_j\right]y_j,$$ $i,j=1,2,\mathrm{...},r,$

and
$$P_i=X_i{\left(X_i^{\text{'}}{X}_i\right)}^1{X}_i^{\text{'}}$$ $i=1,2,\dots ,r.$
Zelner⁴⁰
In particular, if $X_1=X_2=\cdots =X_r=X_0$ , then,

$${\widehat{\beta }}_i={\left(X_0^{\text{'}}{X}_0\right)}^{-1}{X}_0^{\text{'}}{y}_i,$$

Draper and Hunter,⁴¹ Fedorov):⁴² D-optimal designs. ${\displaystyle \sum }$ must be known. Wijesinha and Khuri⁴³ used an estimate of ${\displaystyle \sum }$ . Wijesinha and Khuri⁴⁴ considered designs that maximize the power of the multi variate lack fit test.
Khuri:⁴⁵ Multiresponse rotatability for linear multirespose models.
A design is multiresponse rotatable if $Var\left[\widehat{y}\left(x\right)\right]$  is constant at all points x that are equidistant from the origin, where $\widehat{y}\left(x\right)$ is the vector of r predicted responses.
This design property can be achieved if and only if the design is rotatable for a single-response model having the highest degree among all the r response models.

Khuri⁴⁶

The models

$y_i=X_i{\beta }_i+{\in }_{i,}$ $i=1,2,\dots ,r$

can be written as
$Y=X^{*}B+\in$

where
$$Y=\left[y_1:y_2:\mathrm{...}:y_r\right]$$

$$X^{*}=\left[X_1:X_2:\dots :X_r\right]$$

$$B=diag\left({\beta }_1,{\beta }_2,\mathrm{...},{\beta }_r\right)$$

$$\in =\left[{\in }_1:{\in }_2:\mathrm{...}:{\in }_r\right]$$

Assume that replicated observations are available on all the responses at some points in a region of interest.

The above model suffers from lack of fit if and only if there exists a linear combination of the responses that suffers from lack of fit.

The multivariate lack of fit test is Roy's largest-root test, $e_{max}\left(G_{2^{-1}{G}_1}\right)$

Where
$$G_1=Y^{\text{'}}\left[I_n-X{(X^{\text{'}}X)}^{-1}{X}^{\text{'}}-K\right]Y$$

$$G_2=Y^{\text{'}}KY$$

$$K=diag\left(K_1,K_2,\dots ,K_m,0\right)$$

$K_i=I_{v_i}-\frac{1}{v_i}{J}_{v_i}$ $i=1,2,\dots ,m$

( $v_i$ repeated observations are taken at the $i$ th design point $\left(i=1,2,\dots ,m\right)$ ).
Large values of the test statistic are significant.
Note: The test may be significant even though the response models do not individually exhibit a significant lack of fit.

Richert, et al.⁴⁷ investigated the effects of heating temperature $\left(x_1\right)$ , pH level $\left(x_2\right)$ , redox potential $\left(x_3\right)$ , sodium oxalate $\left(x_4\right)$ , and sodium lauryl sulfate $\left(x_5\right)$  on foaming properties of whey protein concentrates. The responses are $y_1$ =whipping time

$y_2$ =maximum overrun

$y_3$ =percent soluble protein.

A second-order model was assumed for each response.