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Math Details¶
A D-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 criterion and the quantities it is 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 the score is computed for a given candidate design, which is what a reader needs in order to know what the number on the sheet means.
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 |
The model matrix¶
The 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:
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 criterion¶
The design is scored by the determinant of its information matrix:
and larger is better. Of two candidate designs for the same model and the same run count, the one with the larger determinant is the one selected.
The determinant is meaningful as a comparison rather than as an absolute figure. It depends on the coding, on the run count and on the number of model terms, so a determinant from one model or one run count is not comparable with a determinant from another.
Two properties of the criterion are worth stating because they explain what the selection is and is not doing.
It is a property of the design and the model only. No response data enters \(\mathbf{X}^{\prime}\mathbf{X}\). A design can therefore be selected before a single run is performed, which is the point of the method.
A singular information matrix scores zero. 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 determinant is zero and the design is not selectable. This is why the run count has to be at least the number of model terms, and why the model has to be settled before the run count is chosen:
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 D-Optimal
- 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¶
- Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press.
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
- 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.