Tensions in Practice: A small screening design in tension with separating effects¶
A toy experiment with three two-level factors
Factors A, B and C each take levels −1 and +1. A four-run design keeps only combinations where A = B × C. On those runs, the candidate response laws Y = 10 + A and Y = 10 + B × C give identical predictions. Running all eight combinations separates the two laws because A and B × C disagree on the added rows. The reduced design costs fewer runs, but it cannot decide which of those terms produced a response.
Screen with fewer runs
Use four controlled combinations when runs are expensive.
Separate rival effects
Distinguish an A contribution from a B-by-C interaction.
Why these aims pull against each other
The fractional design makes two analysis columns identical. Separation requires more combinations or an explicit assumption about one of the effects.
Choose an arrangement to see what changes and what remains difficult.
Row labels list A, B and C; − and + mean −1 and +1. The eight candidate runs stay fixed. Yes marks the chosen design; the A and B × C columns expose equality on the fraction and disagreement on added rows.
What this choice protects
What it costs
When it fits
Compare the arrangements
Run the selected half
Use the four Yes rows satisfying A = B × C.
| A | B × C | Run? | |
|---|---|---|---|
| − − − | −1 | 1 | No |
| − − + | −1 | −1 | Yes |
| − + − | −1 | −1 | Yes |
| − + + | −1 | 1 | No |
| + − − | 1 | 1 | Yes |
| + − + | 1 | −1 | No |
| + + − | 1 | −1 | No |
| + + + | 1 | 1 | Yes |
- What it protects
- Only four experimental runs are required.
- What it costs
- A and B × C are aliased: their separate coefficients cannot be recovered from these runs alone.
- When it fits
- Fits screening when the alias is acceptable and any neglect of the interaction is explicitly justified.
Illustration note: This is an exact equality of design columns, not a noisy failure to detect an effect. No interaction is silently assumed zero.
Run the full design
Add the four previously omitted combinations.
| A | B × C | Run? | |
|---|---|---|---|
| − − − | −1 | 1 | Yes |
| − − + | −1 | −1 | Yes |
| − + − | −1 | −1 | Yes |
| − + + | −1 | 1 | Yes |
| + − − | 1 | 1 | Yes |
| + − + | 1 | −1 | Yes |
| + + − | 1 | −1 | Yes |
| + + + | 1 | 1 | Yes |
- What it protects
- The A and B × C columns now differ, separating the stated response laws.
- What it costs
- Eight runs are needed rather than four.
- When it fits
- Fits when distinguishing this interaction from A matters and all combinations are feasible.
Illustration note: The table establishes algebraic separation. With real noise, precision and replication still matter.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- The row label gives A, B, C signs in that order. The two candidate laws are invented, not observed outcomes.
- A full design without replication does not by itself establish a noise model or precise effect estimates.
- Controlled factor settings and the selected alias are declared; this is not a claim about arbitrary observational records.
Source entries
Factorial Design
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Interaction discovery versus run-count growth
The tension is how aggressively to fractionate: too little fractionation leaves the design infeasible; too much fractionation creates "aliasing" where main effects are confounded with two-way interactions, potentially producing misleading results.
The source operation
(1) A factorial design varies two or more factors simultaneously at multiple levels within a single integrated experiment, so that every combination of factor levels (or a carefully balanced subset) is observed, rather than studying one factor at a time while holding others fixed.