Resolution V Screen with Confounded Curvature
A coatings plant screens five factors for adhesion strength (target ≥ 4.5 N/mm, baseline mean 3.9 N/mm): A oven temperature, B line speed, C coat weight, D die gap and E ionizer voltage, using a 2^(5−1) design with generator E = ABCD. The 16 runs were randomized within two days; three centre points were added but, because of scheduling, were all run last, immediately after a planned maintenance and with a new batch of coating. Analysis shows A (p = 0.002) and B (p = 0.004) positive, A×B negative (p = 0.011), no other term below 0.05, and R² = 0.91. The centre-point mean is 4.62 N/mm against a factorial mean of 3.90 N/mm, with curvature p = 0.003. No confirmation runs have been made. The team's slide reads: 'curvature confirmed, optimum is a factor setting, so run A and B at their high levels, expect 4.6, launch in three weeks.' One operator ran day one and another ran day two; no block variable was recorded.
1. Is the curvature conclusion trustworthy? Explain the design structure and the confounding you can identify from the information given.
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Design structure first. With E = ABCD the defining relation is I = ABCDE, so this is a resolution V design: main effects are clear of two-factor interactions and two-factor interactions are clear of each other. That part of the analysis is sound - A, B and A×B can be read at face value, and no two-factor interaction is being mistaken for a main effect.
Curvature is a different question. Centre points estimate pure quadratic curvature only if they are treated the same way as the factorial runs. Here all three were run consecutively at the end of day two, after a maintenance intervention and on a different coating batch. The p = 0.003 is therefore confounded with at least three alternative explanations: a step change from maintenance (new nozzle/cleaning), a batch-to-batch raw-material difference in the coating, and an operator or day effect (day one and day two had different operators, and no block variable was recorded, so this effect is not even estimable).
What this means in practice: the centre-point test answers 'is the mean of these three runs different from the mean of the sixteen?' - and the answer is almost certainly yes. It does not answer 'is there curvature in the response surface?'
Verdict: do not accept the slide. The correct next step is cheap: re-run centre points interleaved into normal production (two per block, several blocks, ideally with each operator and at least two coating batches). If the gap persists with centre points embedded in the factorial blocks, curvature is real.
2. If curvature is real, why is 'set A and B to their high levels' unsafe, and what design or analysis would you authorise next?
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Why the corner setting is unsafe. A first-order model with A and B both positive has its maximum at the corner of the experimental region, which is exactly why the team wants to go there. But significant curvature means the response is not planar: the surface is bending, so the true optimum is either inside the region or beyond it. Setting both factors high risks overshooting into a lower-adhesion zone, and it extrapolates outside the tested region with no data - the one thing a screening design cannot license.
Design next - augment, do not restart:
- Keep the 16 factorial runs and add axial points at ±1.68 (α = 2^(5/4)) plus new centre points, turning the design into a rotatable central composite. That is ten new runs plus centre points - cheap compared with a fresh screen.
- Block by day/batch and record operator, so maintenance and batch effects are estimable rather than lurking in the residual.
- Analyse as a second-order model (main effects, two-factor interactions, pure quadratic terms), then map the response surface and locate the stationary point; test whether it is a maximum and whether it falls inside the region.
- If a full CCD is impractical, run steepest ascent along the A+B direction in short, cheap increments until the response stops improving, then place the RSM design around that new region.
Do not put AB's negative sign and the centre-point lift into one linear model and predict 4.6 N/mm - those two statements are not compatible in a first-order fit.
3. Design the confirmation and launch plan you would require before the process change is released, including acceptance criteria.
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Pre-commit the prediction. From the augmented model, state the predicted mean and 95% prediction interval at the proposed setting, in writing, before running anything. A predicted 4.62 with a prediction interval reaching 4.2 is a different commercial proposition from 4.62 ± 0.1.
Confirmation runs:
- Minimum five runs at the proposed setting, spread across different days, both operators and at least two coating batches - the same conditions that confounded the centre points, so the confirmation also tests those effects.
- Run them randomized against three baseline-setting runs, not in a single block at the end of a shift; that repeats the original mistake.
- Acceptance criteria agreed in advance: mean adhesion ≥ 4.5 N/mm and the lower bound of the 95% CI ≥ 4.5, plus every individual result above a defined floor (e.g. 4.0 N/mm), plus no increase in the current defect rate on the adhesion test.
Launch controls:
- Verify the settings are actually controllable in production - record the achieved A and B values for the first two weeks, not the setpoints.
- Add the new settings to the control plan and FMEA, with a reaction plan for adhesion results trending down.
- Hold the three-week launch promise to the confirmation result; if the lower confidence bound is below 4.5, extend the study rather than launch and hope.
One sentence for the sponsor: curvature plus a corner optimum means we have not yet found the optimum, only the edge of what we have tested.