Kitstead Labs
DOE interaction
DOE Interaction Effects: Why One Factor's Effect Depends on Another
A longer bake may have a different effect at a higher temperature. Averaging across temperatures can hide that difference. This guide follows a complete replicated 2×2 example through cell means, an interaction plot and a nested-model test, then separates inference from descriptive screening of observed conditions. No unmeasured recipe or production optimum is inferred.
Quick answer
An interaction means one factor’s effect depends on the level of another. Compare an additive model with one containing A:B, using replication to estimate residual variation. Nonparallel lines are descriptive evidence; the partial F-test and its assumptions determine the statistical conclusion.
Main effects can hide condition-dependent behavior
A main effect averages across the other factor. An A:B interaction asks whether A's effect changes with B. In a two-factor experiment, report the conditional changes as well as any average: an average improvement can conceal a much smaller effect at one setting.
Here A is bake time (30 or 60 seconds), B is bake temperature (150 or 170 °C), and Y is a synthetic post-bake thickness response in nm. These labels provide an educational scenario, not a validated bake recipe. A and B are treated as categorical factors at the observed settings; the model does not establish what happens between or beyond them.
Compare additive and interaction models
The additive model Y ~ A + B assigns the same time effect at both temperatures. The full model Y ~ A + B + A:B permits different effects. Fit both to exactly the same usable observations and compare their residual sums of squares (SSE).
F = (ΔSS / Δdf) / (SSEfull / dffull)
Partial η² = ΔSS / (ΔSS + SSEfull)
Δdf is the number of additional model parameters; dffull is the full model residual degrees of freedom. F tests whether the added interaction terms improve fit under the model assumptions. Partial eta-squared describes the share of interaction-plus-residual variation associated with interaction after the main effects; it is not a yield gain or causal percentage.
Worked example: a complete replicated synthetic 2×2
Assume independent randomized experimental units, normal residual errors with a common variance and no omitted blocking effect. There are three usable independent replicates in each of all four design cells. No missing combination is imputed.
| Time A | Temperature B | Replicates | N | Cell mean |
|---|---|---|---|---|
| 30 s | 150 °C | 99, 100, 101 | 3 | 100 |
| 60 s | 150 °C | 103, 104, 105 | 3 | 104 |
| 30 s | 170 °C | 101, 102, 103 | 3 | 102 |
| 60 s | 170 °C | 113, 114, 115 | 3 | 114 |
At 150 °C, increasing time changes the mean by 104−100 = 4 nm. At 170 °C, it changes it by 114−102 = 12 nm. The difference of these differences is 8 nm. The average time effect is 8 nm, which conceals the separate 4 and 12 nm effects.
Fit both models to all 12 values
For this balanced dataset the grand mean is 105 nm. Time marginal means are 101 and 109 nm; temperature marginal means are 102 and 108 nm. The additive fitted cell values are 98, 106, 104 and 112 nm in the table's order (time mean + temperature mean − grand mean). Full-model fitted values are the four observed cell means.
Each cell has within-cell squared residuals 1 + 0 + 1 = 2 nm², so SSEfull = 8 nm². The additive cell means miss each observed cell mean by 2 nm, adding 4 cells × 3 replicates × 2² = 48 nm². Thus SSEadditive = 56 nm².
| Model | Parameters | Residual df | SSE (nm²) |
|---|---|---|---|
| Additive A + B | 3 | 9 | 56 |
| Full A + B + A:B | 4 | 8 | 8 |
F(1, 8) = (48 / 1) / (8 / 8) = 48
p ≈ 0.000121040
Partial η² = 48 / (48 + 8) = 0.857143
Under the stated assumptions, this is evidence against an additive-only response at these settings. The effect of time depends on temperature in the synthetic example. Partial η² ≈ 0.857 is a conditional effect-size description; it does not mean a process is 85.7% improved. These invented replicates demonstrate arithmetic, not qualification evidence.
Read interaction plots without overclaiming
Nonparallel lines suggest a pattern worth testing, but sampling noise also changes slopes. Crossing lines are neither necessary nor sufficient for statistical significance. Confidence depends on replication, residual variation and the model, not line geometry alone. Failure to reject an interaction test does not prove exact additivity.
For unequal cell counts, fit the nested models to the individual observations. The balanced shortcuts used above do not generally transfer by simply averaging cell means. Examine residuals and the assignment design before trusting nominal p-values; repeated sites on the same wafer are not independent experimental replicates.
Missing cells and unavailable interaction results
The Product 02 v0.3.0 review-candidate interaction requires two distinct factors, 2–12 observed levels per factor and at least two usable observations in every cell of the full A×B cross. Unequal cell counts are supported. A missing cell or insufficient usable replication makes this interaction unavailable; removing invalid measurements can change eligibility.
INTERACTION UNAVAILABLE does not mean no interaction. It means the requested inference cannot be provided for those inputs. Do not fill a missing cell with a guessed mean or recommend an untested combination from the plot. The candidate plot displays up to six Factor B series even when selected statistics may remain available for more levels.
Observed-condition decisions are a separate question
The candidate DECISION workflow screens observed conditions using configured response-mean limits, Minimum N and goals: MINIMIZE, MAXIMIZE, CLOSEST TO TARGET or SCREEN ONLY. Individual OOS warnings are reported separately. A mean inside its screen does not imply that each individual measurement is within specification.
For example, a hypothetical inclusive mean screen of 101–110 nm and Minimum N = 3 retains the observed cells with means 102 and 104 nm in this dataset. It excludes the means 100 and 114 nm. This is descriptive filtering under an invented requirement, not another significance test or a recommendation to adopt either condition. An individual-value specification must be checked separately.
For multiple responses, a non-dominated candidate has no other eligible observed condition that is at least as good on all configured objectives and better on at least one. Trade-offs can leave several candidates. NON-DOMINATED ≠ statistically best or production-qualified. These comparisons use descriptive observed means, without confidence-adjusted winner selection.
A partial observed design may still support DECISION screening while the full-cross INTERACTION test is unavailable. DETAILS supplies the complete factor-level mapping for auditing. Neither workflow invents measurements for unobserved conditions.
Product status and related analysis
The Semiconductor DOE & Split-Lot Analyzer v0.3.0 remains a Windows-first review candidate, not commercially released. Its commercial gate is pending; current review verification is limited to Windows Excel LTSC 2024, with Mac deferred and untested. The calculations here were independently audited and are educational, not evidence of a public product launch.
Read semiconductor split-lot analysis for marginal two-level and multi-level comparisons. A significant marginal factor comparison does not replace the A:B test, and neither removes confounding created by the experiment's design.
Sources and further reading
- NIST: Two-way ANOVA — factorial design and interaction decomposition.
- NIST: Interaction effects — interaction plots and confounding in an experimental example.
- SciPy: F distribution — upper-tail probabilities used in the independent numerical audit.