Quality engineers often learn two-way ANOVA in a statistics course and factorial design of experiments (DOE) in a Six Sigma course, with different notation, different vocabulary, and different software menus. It's easy to come away thinking they're different methods.
They're not. A two-factor factorial experiment is a two-way ANOVA. The DOE framework adds a way of thinking about which levels to choose, how to run the experiment, and how to express the results as effects and a prediction equation. The underlying arithmetic is the same.
This article shows the correspondence on one dataset, explains what each framework emphasizes, and covers where they diverge.
The vocabulary map
Two-way ANOVA | Factorial DOE |
|---|---|
Factor A, factor B | Factor A, factor B (often coded −1, +1) |
Levels | Levels (low, high) |
Cells | Runs or treatment combinations |
Replicates per cell | Replicates |
Main effect (via level means) | Main effect (high average minus low average) |
Interaction (via cell means) | Interaction effect (AB) |
Error mean square | Residual or pure error |
ANOVA table | ANOVA table plus coefficients and a model |
Interaction plot | Interaction plot |
The main difference in language: ANOVA describes effects in terms of means at each level, while DOE describes them as differences between levels, which become coefficients in a regression model.
One dataset, both ways
A packaging line seals pouches. Two factors are tested in a 2 × 2 factorial with four replicates each:
A: seal temperature — low (−1), high (+1)
B: seal pressure — low (−1), high (+1)
The response is seal strength (N).
Cell means:
Pressure low | Pressure high | |
|---|---|---|
Temperature low | 41.40 | 45.00 |
Temperature high | 46.90 | 56.45 |
Grand mean: 47.44 N. Within-cell SD: about 0.61 N.
The two-way ANOVA view
Source | SS | df | MS | F | p |
|---|---|---|---|---|---|
Temperature | 287.30 | 1 | 287.30 | 761.07 | < 0.0001 |
Pressure | 172.92 | 1 | 172.92 | 458.07 | < 0.0001 |
Temperature × Pressure | 35.40 | 1 | 35.40 | 93.78 | < 0.0001 |
Error | 4.53 | 12 | 0.378 | ||
Total | 500.16 | 15 |
All three terms are significant. The temperature effect depends on pressure.
The factorial DOE view
DOE expresses each effect as the change in the average response when a factor moves from its low to its high level.
Main effect of temperature: (average at high temp) − (average at low temp) = (46.90 + 56.45) / 2 − (41.40 + 45.00) / 2 = 51.675 − 43.20 = 8.475 N
Main effect of pressure: (45.00 + 56.45) / 2 − (41.40 + 46.90) / 2 = 50.725 − 44.15 = 6.575 N
Interaction effect AB: half the difference between the temperature effect at high pressure and the temperature effect at low pressure = [(56.45 − 45.00) − (46.90 − 41.40)] / 2 = (11.45 − 5.50) / 2 = 2.975 N
The link between them
For a 2^k factorial with N total runs, each effect's sum of squares is:
SS = N × (effect)² / 4
Temperature: 16 × 8.475² / 4 = 287.30 ✓
Pressure: 16 × 6.575² / 4 = 172.92 ✓
Interaction: 16 × 2.975² / 4 = 35.40 ✓
These are exactly the sums of squares in the ANOVA table. The two methods computed the same quantities by different routes.
The prediction equation
DOE goes one step further and writes the result as a regression model in coded units:
ŷ = 47.44 + 4.24·A + 3.29·B + 1.49·AB
where A and B are −1 or +1. Each coefficient is half the corresponding effect, because moving from −1 to +1 is two units.
Plug in A = +1, B = +1: 47.44 + 4.24 + 3.29 + 1.49 = 56.46, which matches the high-high cell mean (56.45, to rounding).
The t-statistic for each coefficient, squared, equals its F in the ANOVA table. For temperature: the standard error of an effect is 2√(MS_E / N) = 2√(0.378 / 16) = 0.307, so t = 8.475 / 0.307 = 27.6, and t² = 761. ✓
What each framework emphasizes
Two-way ANOVA emphasizes
Testing hypotheses. Is there a main effect? Is there an interaction?
Categorical factors with any number of levels. Three suppliers, four machines, five operators.
Comparing levels. Which suppliers differ? Post hoc comparisons and cell-mean tables.
Observational or existing data, where the levels weren't chosen to optimize anything.
Factorial DOE emphasizes
Estimating effect sizes and building a predictive model.
Choosing levels deliberately to span the range of interest, usually at two levels per factor.
Running the experiment properly: randomized run order, replication, blocking, center points.
Optimization: using the model to find the settings that maximize or minimize the response.
Scaling to more factors: fractional factorials, screening designs, response surfaces.
Where they diverge
More than two levels
Two-way ANOVA handles any number of levels naturally. Two-level DOE does not; with three or more levels per factor, the effects can no longer each be summarized by one number, and the DOE framework moves to general full factorials (which are analyzed with ANOVA anyway) or to response surface methods.
Numeric factors and curvature
DOE treats numeric factors (temperature, pressure) as continuous and fits a linear model across the range. ANOVA treats every factor as categorical, even when it's numeric. If you have three temperatures, ANOVA asks whether they differ; a DOE analysis could ask whether the relationship is linear or curved. With two levels, curvature is invisible; DOE adds center points to detect it.
More than two factors
A DOE framework scales cleanly to three, four, or more factors, with all their interactions, and fractional designs that run a fraction of the combinations. Two-way ANOVA is by definition two factors; beyond that, it becomes a general factorial ANOVA.
Random factors
ANOVA has well-developed methods for random factors (operators, batches, days treated as samples from a population), such as the variance-component analysis used in gage R&R. DOE typically treats factors as fixed.
Which to use
If you chose the levels to optimize a process and have two levels per factor: think in DOE terms. Report effects, the prediction equation, and an interaction plot.
If the factors are categorical choices (suppliers, machines, materials) or have more than two levels: think in ANOVA terms. Report the ANOVA table, cell means, and post hoc comparisons.
Either way, the ANOVA table is the common foundation, and the significance tests are identical.
Running the analysis
For a two-factor experiment, the ANOVA table is the shared starting point regardless of which framework you're using. A two-way ANOVA calculator will return the sums of squares and F-tests; for a 2 × 2 design, you can convert them to DOE effects directly with effect = √(4·SS / N), taking the sign from the cell means. For the seal data, that gives back 8.475, 6.575, and 2.975.
Summary
A two-factor factorial experiment and a two-way ANOVA are the same analysis. The ANOVA computes sums of squares and F-tests from level means and cell means; the DOE framework computes the same information as effects (high average minus low average) and expresses them as a prediction equation, with SS = N × effect² / 4 for two-level designs. ANOVA is better suited to categorical factors and multi-level comparisons; DOE adds deliberate level selection, run-order discipline, effect estimation, and a path to optimization and to more factors.