One of the axial points of the CCD is outside the equipment range, and the high-high combination of the two factors is unsafe. Forcibly running or arbitrarily deleting points from the standard design can destroy the information structure of the second-order model.
The question for this section is:
When a standard design cannot be executed, which points should be selected?
The candidate set defines the boundary of feasible experiments
Custom Design first creates a set of conditions that are actually feasible to run. It can create a dense grid of candidate points over a continuous range, or it can use discrete candidates, such as a list of manufacturable recipes.
When creating candidates, first incorporate the following:
- Physical minimum and maximum of the equipment
- Combinations prohibited for safety and stability
- Mathematical constraints such as total amount or compositional sum
- Runs already used or controls that must be included
- Hard-to-change factors and block structure
Areas omitted from the candidate set cannot be evaluated by the optimizer. Conversely, including infeasible points can produce a mathematically attractive design that fails in practice.
The model must be determined first for the information to have meaning.
Even with the same candidate set, the points required by a main-effect model and a full quadratic model are different. The design algorithm evaluates the information in the specified model matrix X.
Information = XᵀX
The model terms are columns, and the runs are rows. If a column is a combination of other columns, its coefficient cannot be estimated independently. This is the estimability problem. Even if the number of runs is greater than the number of terms, a poor arrangement can lead to rank deficiency.
D-optimal and I-optimal have different objectives.
D-optimal
By maximizing det(XᵀX), it minimizes the volume of the coefficients' joint confidence region. It is a natural choice when the objective is overall estimation precision for the specified model coefficients.
I-optimal
It places points in a direction that minimizes the average prediction variance over the candidate region. It is useful when prediction within the region is the objective.
The two criteria are not simply different rankings of the same question. A high D does not necessarily mean that the prediction variance is low at all locations, and a low I does not necessarily mean that the power for a specific coefficient is sufficient.
| Decision aspect | What D criterion asks | What I criterion asks |
|---|---|---|
| primary objective | How precisely can the coefficients be estimated together? | How precisely can the candidate region be predicted on average? |
| Point placement tendency | Extreme/structural points that separate the model columns | Points that evenly cover the prediction region |
| Additional verification | Power, alias, prediction variance profile | Location of maximum variance, coefficient precision, power |
If any of the following change – candidate set, prior model, number of runs, block, or optimality criterion – a different design may result. Check multiple random starts and Design Diagnostics, and do not interpret the algorithm name as scientific superiority.
The number of runs should be more than the minimum required
A 2-factor full quadratic has 6 coefficients. Even if 6 runs produce rank 6, there is very little room to evaluate residual degrees of freedom, pure error, and lack of fit. Plan the run count with independent replicates, center points, blocks, and the possibility of missing runs in mind.
Power depends on the assumed effect and noise. Forcing all terms to have the same power target may increase the number of runs, so distinguish between primary terms and nuisance terms.
Prediction Variance Profile shows what lies beyond the average
Do not stop after examining only I-optimality's average prediction variance. Use a profile or map to see where variance is high and low across the candidate region. If the boundary of an important operating region is weak, add information at that location.
Design Evaluation should include at least the following:
- model rank and estimability
- correlation of estimates or alias structure
- location-specific shape of relative/absolute prediction variance
- expected power of primary effects
- stability of design quality with different starting values
In-Silico Lab: Change Constraints and Objectives
- Observe whether the upper-right candidate disappears in
no-high-high. - Swap D and I and compare the selected points and the average prediction variance.
- Change the number of runs to 8, 10, and 12 and check the rank of the 6-term model.
- Explain why the standard cube corner cannot be used in the ellipse constraint.
Compare designs by purpose across viable candidates
In areas where standard design does not apply, the constraints, number of runs, and D/I standards are changed and the estimability and average prediction variance are checked.
If this is your first time: What should I press?
- 1. Read the question firstIn the Lab title, check the one thing you will compare this time.
- 2. Change just one conditionInitially, change only one of the inputs: n, effect, or spread.
- 3. new composite specimen pressureNew synthetic data is created. The same conditions may vary depending on the sample.
- 4. Pictures and calculation results CompareWrite in one sentence what moves and what stays the same before and after the change.
If it gets stuckresetGo back to see the default results and change just one condition. This Lab is not a correct answer tester but a pattern observation tool.
Synthetic observations of the same settings
calculation result
The Lab selector is deterministic-greedy-teaching-approximation. It neither reproduces JMP's optimization implementation nor guarantees universal optimality; real designs require multiple starting points and diagnostic checks.
educational synthetic modelbjs-advanced-design-sequence-v1. One row is one independent simulation or design run unless otherwise indicated. It cannot be used for actual research, quality, or regulatory decisions.
The Lab is a deterministic greedy teaching approximation that selects one point at a time from candidate points at 0.5 intervals. It does not replicate JMP's Custom Design algorithm or global optimality. The objective is to transparently view the dependence on candidates, models, and criteria.
Use JMP Design Evaluation as a design receipt
Confirm contracts based on model, candidate points, number of runs, and optimization standards.
We look at the shape of one area outside the average to see which areas have weak predictions.
Estimability, power, and correlation are examined for each purpose.
In the Custom Design evaluation, verify that the factor range, disallowed combinations, model terms, number of runs, and criterion are consistent with the plan. Preserve the prediction variance profile, estimability, power, and correlation before execution. After obtaining the results and changing the model, note which terms the initial design was created for.
Example of Result Statement
After excluding the prohibited high-high combination, we used 21 candidate points and a prespecified 6-term full quadratic model. We evaluated a 10-run D-optimal design from several starting values, and all terms were estimable. The operating region's prediction variance profile and primary-term power were reviewed together. This optimality is limited to the specified candidates, model, and run count.
Concluding the Section
- The candidate set defines the feasible region.
- Optimal designs can only be evaluated if a prior model exists.
- D and I prioritize coefficient information and mean prediction information, respectively.
- Do not merely meet the minimum rank; allow for residual degrees of freedom, pure error, and missing runs.
- Optimal is a property within the specified candidate points, model, runs, and criterion.
In the next section, we will examine DSD, which connects screening of many continuous factors with curvature exploration in a three-level structure.
Supplemental Materials
- NIST/SEMATECH · Choosing an Experimental Design
- JMP Help · Prediction Variance Profile
- JMP Help · Example of a Custom Design Construction
This article and the Lab's design selection are educational approximations and should not be used as evidence for actual research, process, quality, or regulatory decisions.