2.3 How to design a DOE ?

2.3.1 An optimally wrong choice

Proposition 2.3.1   Since $n\ge p$, matrix $S\doteq\raisebox{0.5 em}{\normalfont\textsf{t}}{A} . A$ is invertible if and only if the rank of the lines (each describing a point of measure) is at least equal to the number of columns (i.e. the number of parameters to estimate).

Remark 2.3.2   Apart from the case $\det S=0$ (not enough measures, or points of measure not spanning the whole factorial space $\Omega$), the design "a factor at a time" gives a design as bad as possible. Better avoid such a design !

2.3.2 Several algorithms

Algorithm 2.3.3 (only random)   Choose the number $n$ of measures. It must be at least the number of parameters to be determined, plus a "remainder" to determine the quality factor. Then build $\omega$ by $n$ random choices among the elements of $0..\left(\char93 \Omega-1\right)$ (repetitions are allowed, provided that $\omega$ spans $\Omega$). Finally, use Proposition 2.1.2 to decode the numbers obtained.

Definition 2.3.4   A design $\omega_{1}$ is better than another design $\omega_{2}$ of same size $n$ when eigenvalues of matrix $S_{1}$ are greater than eigenvalues of matrix $S_{2}$. This criterion is related to the transmission of errors from $b$ towards $f^{*}$. A quite equivalent criterion is to require an increasing determinant and a decreasing sum of the absolute non-diagonal elements.

Algorithm 2.3.5 (iterated random)   Choose several DOE using zal:only-random and retain the best (using Definition 2.3.4).

Algorithm 2.3.6 (quasi gradient)   Choose a DOE $\omega$ using zal:only-random. Then randomly select an element $\epsilon$ of $\omega$ and replace it successively by an element of $\Omega\setminus\omega$. Then replace $\epsilon$ by its best substitute selected according to Definition 2.3.4. Iterate. When no improvement is obtained, be not so random by choosing non yet tryied $\omega$.

Exercise 2.3.7   For each of the situations where there is an explicit method to obtain the optimal design, verify that zal:quasi-gradient is not that far from optimal.

2.3.3 Full design $2^{m}$

Definition 2.3.8   A full $2^{m}$ design is conducing once each measure of $\Omega$ when there are $m$ two-levels factors.

Algorithm 2.3.9   A full binary design can be generated using the binary writing of all integers between 0 and $2^{m}-1$ :

$$\left(z_{1}, \cdots, z_{m}\right)\quad\mathrm{such that}\quad\sum_{k=1}^{k=m}z_{i} 2^{i-1}\in\left[0..2^{m}-1\right]$$

Here the factors ared coded using variables $z_{i}\in\left\{ 0,1\right\}$. When using factors coded using $x_{i}\in\left\{ -1, +1\right\}$, the following must be used :

$$\left(x_{1}, \cdots, x_{m}\right)\quad\mathrm{such that}\qua... ...{1}{2} \sum_{k=1}^{k=m}\left(x_{i}+1\right) 2^{i-1}\in\left[0..2^{m}-1\right]$$

Of course, the order of achieving the measure must be randomized in order to, "in the average" eliminate the influence of external factors (ambient temperature, derive of equipment, etc).

Theorem 2.3.10   When the $m$ factors are actually binary, a full $2^{m}$ design lead to a model that accounts for all interactions between factors.

Proof. Let $X_{j}$ be the variables obtained by making all possible products of variables $x_{i}\in\left\{ +1,-1\right\}$ describing the factors. It is convenient to sort the $X_{j}$ by their full degree, then lexically : $X_{0}$ is $1$ and $X_{i}=x_{i}$ when $1\leq i\leq m$. Since any square is the neutral $1=X_{0}$, an obvious map exists between $\left\{ X_{j}\right\}$ and the power set of $\left\{ x_{i}\right\}$. By expansion in series, any function can be described as a linear (finite) combination of the $X_{i}$.

It remains to prove that the corresponding $S$ is invertible. An obvious computation shows that in fact $S=2^{m}I$ : the matrix is not only invertible, but orthogonal and this design is optimal in terms of propagation of errors. $\qedsymbol$

Remark 2.3.11   Theorem 2.3.10 states nothing when the factors are in fact continuous and the levels are arbitrarily chosen.

2.3.4 Fractional $2^{m-k}$ design

Definition 2.3.12   A fractional $2^{m-k}$ design is a design with $2^{m-k}$ measure points that addresses $m$ binary factors. The measure points have to be chosen in order to obtain a full factorial design when restricting the factors to any $\left(m-k\right)$-tuple of factors. Of course, the ability to highlight the interactions between factors is significantly reduced.

Example 2.3.13   A full $2^{4}$ factorial design $\Omega_{x}$ contains $16$ measure points, concerning four factors $x_{i}\in\left\{ -1,+1\right\}$. Defining $y_{i}=x_{i}$ ($i\leq4$) and $y_{5}=x_{1} x_{2} x_{3} x_{4}$, one obtains a fractional $2^{5-1}$ design $\Omega_{y}$. Indeed we can use $x_{1}=y_{2} y_{3} y_{4} y_{5}$ to see that, restricted to $y_{2}, y_{3}, y_{4}, y_{5}$, the design $\Omega_{y}$ is a full design for these four factors. The $16$ measures can be used to determine the coefficients of a second order model, that requires a constant, 5 linear terms and $\mathrm{binom}\left(5,2\right)=10$ cross terms.

When comparing the $\Omega_{y}$ design with the full $2^{4}$ design ( $\Omega_{x}$), the coefficient associated with $y_{2}y_{3}$ in $\Omega_{y}$ is identical to the coefficient associated with $x_{2}x_{3}$ in $\Omega_{x}$, while the coefficient associated with $y_{1} y_{5}$ in $\Omega_{y}$ is identical to the coefficient associated with $x_{2} x_{3} x_{4}$ in $\Omega_{x}$. On the other hand, when comparing $\Omega_{y}$ with the $2^{5}$ full design, the coefficient of $y_{2} y_{3}$ in $\Omega_{y}$ appears to be shared between the $y_{2} y_{3}$ and the $y_{1} y_{4} y_{5}$ terms of the full $2^{5}$ design. This phaenomenom is called aliasing.

Algorithm 2.3.14   The best design for a fractional choice is obtained by considering the aliasing and "keeping alone" the most probable interactions.

Exercise 2.3.15   Determine the number of coefficients required by a second order model adressing $6$ binary factors. Characterize a fractional design to obtain these coefficients.

Exercise 2.3.16   Same question with $8$ factors.