What Is a Main Effects and Interaction Plot in Factorial Design (and How Do I Read It)?

If you run a factorial experiment and only look at the individual factor averages, you can miss the most important insight: whether one factor’s effect changes depending on the level of another factor. That hidden behavior is an interaction, and the standard way to see it is with a main effects plot and an interaction plot.

What It Is

A main effects plot shows the average response at each level of a single factor, with all other factors averaged out. An interaction plot shows the average response for every combination of two factors, connecting the points by level of the second factor.

Together, these plots are the first diagnostic step in a factorial design of experiments (DOE), as described in Montgomery’s Design and Analysis of Experiments. They tell you which factors matter and whether their effects are additive or conditional.

How It Works / Formula and Steps

For a two‑level factorial design (2^k), the main effect of factor A is:

```
Effect(A) = (Average response at A = high) – (Average response at A = low)
```

The interaction effect between A and B is half the difference of the simple effects:

```
Effect(A×B) = ½ × [ (Effect of A at B = high) – (Effect of A at B = low) ]
```

To build the plots:

1. Main effects plot: For each factor, compute the mean response at each level. Plot the two points and connect them with a line.
2. Interaction plot: For each pair of factors, compute the mean response for all four combinations (A low/high × B low/high). Plot the four points, connecting the two points at B = low with one line and the two points at B = high with another line.
3. Interpretation:

- Parallel lines → no interaction (effects are additive).
- Crossing or non‑parallel lines → interaction exists.
- A steep main‑effect line → large factor effect.

A Worked Illustrative Example

Example data (illustrative only). Suppose you run a 2² design on a chemical yield with factors Temperature (T) and Catalyst (C). The average yields are:

T | C | Yield

• Low | Low | 60
• High | Low | 80
• Low | High | 70
• High | High | 90

Main effects (averaging over the other factor):

• Effect of T = (80 + 90)/2 – (60 + 70)/2 = 85 – 65 = 20
• Effect of C = (70 + 90)/2 – (60 + 80)/2 = 80 – 70 = 10

Interaction effect:

• Effect of T at C = Low: 80 – 60 = 20
• Effect of T at C = High: 90 – 70 = 20
• Interaction = ½ × (20 – 20) = 0

The interaction plot would show two parallel lines (both slopes = 20), confirming no interaction. The main effects plot would show T rising steeply and C rising moderately.

If instead the yields were 60, 80, 70, 60, then:

• Effect of T at C = Low: 20; at C = High: –10
• Interaction = ½ × (20 – (–10)) = 15

The interaction plot would show crossing lines — a clear sign that the effect of temperature reverses when catalyst is high.

Common Pitfalls

• Reading a single main effect without checking interactions. If an interaction is large, the main effect alone is misleading.
• Using line plots for factors with more than two levels. The plot is still valid, but the “parallel lines” rule only strictly applies to two‑level factors.
• Confusing statistical significance with practical size. The plot shows magnitude; use ANOVA to test significance.
• Forgetting that interaction plots only show two factors at a time. In a 2^k design with k > 2, higher‑order interactions may exist — check the interaction matrix as well.

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To quickly generate main effects and interaction plots from your own DOE data, use the free tool at https://www.6sq.com/tools/me_interact/ — paste your averages and see the plots instantly.