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.
실행 가능한 후보에서 목적별 설계를 비교하세요
표준설계가 들어가지 않는 영역에서 제약·run 수·D/I 기준을 바꾸며 추정가능성과 평균 예측분산을 확인합니다.
처음이라면: 무엇을 눌러야 하나요?
- 1. 질문을 먼저 읽기Lab 제목에서 이번에 비교할 한 가지를 확인합니다.
- 2. 조건 하나만 바꾸기처음에는 n, 효과, 산포 같은 입력 중 하나만 바꾸십시오.
- 3. 새 합성 표본 누르기새 합성 데이터가 만들어집니다. 같은 조건도 표본에 따라 달라질 수 있습니다.
- 4. 그림과 계산 결과 비교하기바꾸기 전후 무엇이 움직이고 무엇이 그대로인지 한 문장으로 적어보십시오.
막히면 초기화로 돌아가 기본 결과를 본 뒤 조건 하나만 바꾸십시오. 이 Lab은 정답 판정기가 아니라 패턴 관찰 도구입니다.
같은 설정의 합성 관측
계산 결과
Lab 선택기는 deterministic-greedy-teaching-approximation입니다. JMP의 최적화 구현을 재현하거나 보편적 최적성을 보증하지 않으며, 실제 설계는 여러 시작값과 diagnostics를 확인해야 합니다.
교육용 synthetic model · bjs-advanced-design-sequence-v1. 한 행은 별도 표시가 없는 한 하나의 독립 simulation 또는 설계 run입니다. 실제 연구·품질·규제 판단에는 사용할 수 없습니다.
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
모형·후보점·run 수·optimality 기준의 계약을 확인합니다.
어느 영역의 예측이 약한지 평균 하나 밖의 모양을 봅니다.
estimability·power·상관을 목적별로 검토합니다.
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.