Which of these factors actually move the process — and do any of them depend on each other?
Run every combination of two-level factors and estimate every effect at once — including the interactions that changing one factor at a time is structurally unable to see.
Design of Experiments (2ᵏ Factorial) · improve · Green Belt
Use this when
- Several candidate factors and no proof which ones matter
- A suspicion that the right setting for one factor depends on another
- Before a one-factor-at-a-time trial spends its runs finding a false optimum
Two-level factorial experiment
sandbox5 effects distinguishable from noise — screening judgement, no error estimate exists
5 of 15 effects are distinguishable from noise (Temp, Conc, TempConc, Stir, TempStir), judged against Lenth's margin of 6.748 — a screening criterion, since an unreplicated design has no error estimate. The interactions TempConc, TempStir mean the factors involved cannot be set independently: the right setting for one depends on the other. The simple-effects table below shows each combination, and the recommendation has to be a PAIR of settings, not two separate ones. This is also exactly what one-factor-at-a-time experimentation cannot see. Before acting on this: run the winning combination again as a confirmation. An effect estimate is a prediction, and the confirmation run is where it meets the process.
- Design
- 2^4, 1× replicated
- Method
- Lenth's method
- Lenth PSE
- 2.625
- Margin of error
- 6.748
| Effect | Estimate | vs margin 6.75 | Verdict |
|---|---|---|---|
| Temp | 21.625 | |21.625| | Distinguishable |
| Pressure | 3.125 | |3.125| | Noise-sized |
| TempPressure | 0.125 | |0.125| | Noise-sized |
| Conc | 9.875 | |9.875| | Distinguishable |
| TempConc | -18.125 | |18.125| | Distinguishable |
| PressureConc | 2.375 | |2.375| | Noise-sized |
| TempPressureConc | 1.875 | |1.875| | Noise-sized |
| Stir | 14.625 | |14.625| | Distinguishable |
| TempStir | 16.625 | |16.625| | Distinguishable |
| PressureStir | -0.375 | |0.375| | Noise-sized |
| TempPressureStir | 4.125 | |4.125| | Noise-sized |
| ConcStir | -1.125 | |1.125| | Noise-sized |
| TempConcStir | -1.625 | |1.625| | Noise-sized |
| PressureConcStir | -2.625 | |2.625| | Noise-sized |
| TempPressureConcStir | 1.375 | |1.375| | Noise-sized |
The interactions, decomposed — why one setting at a time is not a recommendation
Effect of Temp is 39.750 with Conc low, and 3.500 with Conc high.
| Conc low | Conc high | |
|---|---|---|
| Temp low | 45.25 | 73.25 |
| Temp high | 85.00 | 76.75 |
Effect of Temp is 5.000 with Stir low, and 38.250 with Stir high.
| Stir low | Stir high | |
|---|---|---|
| Temp low | 60.25 | 58.25 |
| Temp high | 65.25 | 96.50 |
This design is unreplicated, so there is no estimate of experimental error — every degree of freedom went into effects. Significance here comes from Lenth's method, which builds a noise floor from the smaller contrasts themselves. It is a screening judgement, good at finding the effects worth a confirmation run; it is not an F test, and a borderline effect deserves the replicated follow-up rather than a verdict.
Temp and Conc interact: the effect of Temp is 39.750 with Conc low and 3.500 with Conc high. The main effects of Temp and Conc average those two answers and describe neither; any recommendation has to name both settings together.
Temp and Stir interact: the effect of Temp is 5.000 with Stir low and 38.250 with Stir high. The main effects of Temp and Stir average those two answers and describe neither; any recommendation has to name both settings together.
| # | Run | Temp | Pressure | Conc | Stir |
|---|---|---|---|---|---|
| 1 | (1) | − | − | − | − |
| 2 | a | + | − | − | − |
| 3 | b | − | + | − | − |
| 4 | ab | + | + | − | − |
| 5 | c | − | − | + | − |
| 6 | ac | + | − | + | − |
| 7 | bc | − | + | + | − |
| 8 | abc | + | + | + | − |
| 9 | d | − | − | − | + |
| 10 | ad | + | − | − | + |
| 11 | bd | − | + | − | + |
| 12 | abd | + | + | − | + |
| 13 | cd | − | − | + | + |
| 14 | acd | + | − | + | + |
| 15 | bcd | − | + | + | + |
| 16 | abcd | + | + | + | + |
Replicates on the same line, separated by commas.
Load a published experiment
How this is calculated
- Effects are contrasts: the average response with the effect’s factors high minus the average with them low — effect = contrast / (n·2k−1), SS = contrast² / (n·2k). The contrasts are orthogonal, which is what lets 2k runs estimate 2k−1 effects independently.
- Replicated designs test each effect with F on 1 and 2k(n−1) degrees of freedom against the pooled error.
- Unreplicated designs have no error estimate — every degree of freedom went into effects. Significance uses Lenth’s pseudo standard error: PSE = 1.5 × median of the |effects| below 2.5 × (1.5 × median|effect|), with margin t(1−α/2, m/3) × PSE. It is a screening criterion, and the tool labels it as one rather than presenting it as an F test.
- The half-normal plot ranks |effects| against half-normal quantiles. Effects that are pure noise fall on a line through the origin; real ones peel away above it.
- Simple effects decompose each significant interaction, because the main effects it involves are averages of two different answers and cannot be quoted alone.
Source: Montgomery, D.C., Design and Analysis of Experiments 8e, Ch. 6 (Examples 6.1 and 6.2 are the presets, and the verification fixtures); Lenth, R.V. (1989), Technometrics 31(4), 469–473; Daniel, C. (1959) on the half-normal plot.
How this is calculated
Full 2^k factorial in standard (Yates) order. Each effect is an orthogonal contrast: effect = contrast/(n*2^(k-1)), SS = contrast^2/(n*2^k). Replicated designs test each effect by F on 1 and 2^k(n-1) degrees of freedom against the pooled error. Unreplicated designs have no error estimate — every degree of freedom went into effects — so significance uses Lenth's method: s0 = 1.5 x median|effect|, PSE = 1.5 x median of the |effects| below 2.5*s0, margin of error = t(1-alpha/2, m/3) x PSE, stated as a screening judgement rather than presented as an F test. The half-normal plot ranks |effects| against half-normal quantiles with the noise line drawn at the effect standard error. Every significant two-factor interaction is decomposed into simple effects with its cell means, because the main effects it involves are averages of two different answers and cannot be quoted alone.
Source: Montgomery, D.C., Design and Analysis of Experiments 8e, Ch. 6 — Examples 6.1 (plasma etch, replicated 2^3) and 6.2 (filtration, unreplicated 2^4) are the presets and the verification fixtures; Lenth, R.V. (1989), Quick and Easy Analysis of Unreplicated Factorials, Technometrics 31(4), 469-473; Daniel, C. (1959) on the half-normal plot.
Learn the method
- Change everything at once, carefullyGreen Belt · 15 min · free