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Math Details

An optimal design is not built from a stored pattern the way the other designs in this section are. It is selected: the product chooses a set of runs that scores best on a stated criterion. This page gives the three criteria and the quantities they are computed from.

What this page does not cover

How the product searches for the best set of runs is not documented. This page states what is being optimized and how a given candidate design is scored, which is what a reader needs in order to know what the numbers on the Results page mean.

Notation

Term	Description
\(N\)	the number of runs in the design
\(p\)	the number of terms in the model, including the constant
\(\mathbf{X}\)	the model matrix, \(N\) rows by \(p\) columns
\(\mathbf{X}^{\prime}\mathbf{X}\)	the information matrix, \(p\) by \(p\)
\(\lvert \mathbf{X}^{\prime}\mathbf{X} \rvert\)	its determinant
\(\operatorname{tr}(\mathbf{A})\)	the trace of \(\mathbf{A}\), the sum of its diagonal entries
\(\mathcal{R}\)	the design region

The model matrix

Every criterion is a property of the model, not of the factors alone, so the model has to be fixed before any design can be scored. The model is the constant, the main effects, and whichever interactions and higher order terms are selected.

\(\mathbf{X}\) has one row per run and one column per model term. A row is built from that run's coded factor settings: the constant column holds \(1\), a main effect column holds the factor's coded value, and an interaction column holds the product of the coded values of the factors in it. This is the same matrix the fit uses, so a design's score and its fit are computed from the same construction.

Every quantitative factor is put on a common scale before the regression runs. One expression does it, whichever coding is in force, using a centre \(m_c\) and a scale \(s_c\) that the coding chooses:

\[ x_{\text{coded}} = \frac{x - m_c}{s_c} \]

A categorical factor is never coded this way. Its coded value is \(1\), and the level it sits at selects which coefficient is used instead.

An interaction's coded value is the product of the coded values of the factors in it.

The three criteria

The wizard optimizes one of three. All three are properties of \(\mathbf{X}\) and the model alone.

D

\[ D = \left\lvert \mathbf{X}^{\prime}\mathbf{X} \right\rvert \]

and larger is better. Maximizing the determinant minimizes the volume of the joint confidence region for the vector of regression coefficients, which is why D is the estimation-oriented criterion and the usual choice for first-order and screening designs.

A

\[ A = \operatorname{tr}\!\left( \left( \mathbf{X}^{\prime}\mathbf{X} \right)^{-1} \right) \]

and smaller is better. That trace is the sum of the variances of the regression coefficients, so A controls those variances one at a time rather than jointly, which is the difference between it and D.

Strictly the sum of the variances is that trace multiplied by the error variance. The criterion is stated here without that factor, following Montgomery, because a constant multiplier does not change which design is best.

I

\[ I = \frac{1}{\operatorname{vol}(\mathcal{R})} \int_{\mathcal{R}} \operatorname{var}\!\left( \hat{y}(\mathbf{x}) \right) \, d\mathbf{x} \]

and smaller is better. It is the prediction variance integrated over the design region and divided by the volume of that region, so it is an average over the whole region rather than a figure evaluated at the design points. That makes I the prediction-oriented criterion, and the usual choice for second-order designs fitted to find an optimum.

The three are not interchangeable

A design that is best on one criterion is generally not best on another, and the gap is not symmetric. Jones and Goos measured this on a protein-extraction example: the D-optimal design reached 85.3 percent of the I-optimal design on the D criterion, while the D-optimal design reached only 60.2 percent of the I-optimal design on the I criterion.

This is why the Results page reports every measure for every design, whichever one you optimized.

What the Results page reports

Column	What it is
D-efficiency (%)	the design's standing on the D criterion, as a percentage. Higher is better
A-efficiency (%)	its standing on the A criterion, as a percentage. Higher is better
G-efficiency (%)	its standing on the G criterion, which concerns the largest prediction variance over the region rather than the average. Higher is better
I (average prediction variance)	the value of \(I\) above, reported directly rather than as a percentage. Lower is better
Max VIF	the largest variance inflation factor over the model columns

The condition number is not reported. The published bands for interpreting it belong to the condition number of \(\mathbf{X}\) computed after each column is scaled to unit length. That is not the same quantity as the condition number of the coded columns as they stand, and the two part company as soon as a design has center points, quadratic terms or effect-coded categorical factors. The column was removed rather than print a figure with bands that may not belong to it.

Two properties worth stating

No response data enters any criterion. All three are computed from \(\mathbf{X}\) alone, so a design can be selected before a single run is performed, which is the point of the method.

A singular information matrix cannot be scored. If the chosen runs cannot support the chosen model, because there are fewer runs than model terms or because two model columns coincide over the selected runs, the design is singular. The wizard reports that rather than a number, and neither the alias matrix nor the power figures are available for it. This is why the run count must be at least the number of model terms:

\[ N \ge p \]

What is chosen and what is fixed

The run count is an input: you state how many runs the design should have, and the selection then picks that many. It is not derived from the factor count the way a factorial's run count is, which is the reason this design can hit a run count no factorial or Plackett Burman array offers.

Replicates are applied after the selection, by repeating the chosen design, so they multiply the row count and do not change which runs were chosen.

Shared Math Details used here

This design uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.

Shared concept	Used here for	Reference
Design coding	the coded values that fill the model matrix	Design Coding
Prediction equation	the model whose terms are the columns of \(\mathbf{X}\)	Prediction Equation

See Also

• Create Optimal Design
• Create Custom Design Math Details, which takes runs as supplied rather than selecting them
• Two Level Factorial Designs Math Details, where the run count follows from the factor count

References

1. Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press.
2. Gilmour, S. G., and Trinca, L. A. (2012). Optimum design of experiments for statistical inference. Journal of the Royal Statistical Society, Series C (Applied Statistics), 61(3), 345 to 401.
3. Jones, B., and Goos, P. (2012). I-optimal versus D-optimal split-plot response surface designs. University of Antwerp, Faculty of Applied Economics, Research Paper 2012-002.
4. Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
5. Myers, R. H., Montgomery, D. C., and Anderson-Cook, C. M. (2016). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 4th edition. Wiley.