Main Effects & Interaction Plots
Key Takeaways
- A main-effects plot shows average response at each level of a factor; a steep line suggests a large main effect, a flat line little average change across levels.
- An interaction plot displays one factor’s levels on the x-axis with separate lines for another factor’s levels; nonparallel lines indicate interaction.
- When a strong interaction exists, you cannot set Factor A from its main-effect average alone—you must choose the A×B combination that optimizes Y.
- Parallel lines mean effects are approximately additive; crossing or strongly diverging lines mean the best level of A depends on B.
- CSSGB BoK V.A.2 is Apply level: interpret described main-effect and interaction plots, revise conclusions when interaction changes the story, and recommend factor settings from the joint pattern.
Main Effects & Interaction Plots
Quick Answer: Main-effects plots graph the average response at each level of a factor. Interaction plots show how the effect of one factor changes across levels of another. CSSGB BoK V.A.2 (Apply) expects you to read these plots, spot nonparallel (interaction) patterns, and change recommended settings when interaction makes a main-effect-only story wrong.
From Terms to Pictures
Once a factorial or fractional factorial is run, software (or hand averages) produces cell means—the average Y for each treatment. Plots turn those averages into decisions:
- Which factors move Y the most on average? → Main-effects plot
- Does the best level of A depend on B? → Interaction plot
Apply-level skill is interpretation under exam descriptions (“lines cross,” “nearly parallel,” “A high is best only when B is low”), not drawing software UI.
Main-Effects Plots
How to read one
- For factor A, compute the average of all runs at A low and the average of all runs at A high (averaging across the other factors’ levels in a balanced design).
- Plot level on the horizontal axis and average Y on the vertical axis; connect the two points with a line (for two-level factors).
- Repeat for each factor, often as a panel of small plots.
| Plot appearance | Interpretation |
|---|---|
| Steep slope | Large main effect—changing that factor moves average Y a lot |
| Flat / shallow slope | Small main effect over the levels studied |
| Upward slope (low → high) | Higher level increases average Y |
| Downward slope | Higher level decreases average Y |
Direction depends on the goal. If Y is defect rate, you want the level that lowers the line. If Y is strength, you want the level that raises the line. Always state the desired direction before declaring “high is better.”
Worked scenario — main effects only (additive process)
Two-level factors: Catalyst (Type 1 / Type 2) and Mix time (5 min / 10 min). Response: conversion (%)—higher is better. Cell means:
| Mix 5 min | Mix 10 min | |
|---|---|---|
| Catalyst 1 | 72 | 78 |
| Catalyst 2 | 80 | 86 |
Main-effect averages:
- Catalyst 1: $(72+78)/2 = 75$; Catalyst 2: $(80+86)/2 = 83$ → main effect of catalyst ≈ +8 points going to Type 2
- Mix 5: $(72+80)/2 = 76$; Mix 10: $(78+86)/2 = 82$ → main effect of mix ≈ +6 points going to 10 min
Main-effects plot description: Both lines slope up; catalyst slope is steeper than mix time.
Conclusion if lines stay parallel (check interaction next): Prefer Catalyst 2 and 10 min for highest average conversion. The estimated average at that corner is 86%, matching the table.
Interaction Plots
Construction (conceptual)
For factors A and B:
- Put A’s levels on the x-axis
- Draw one line for each level of B
- Plot the average Y for each A×B combination
Parallel vs. nonparallel
| Pattern | Meaning | Decision impact |
|---|---|---|
| Approximately parallel lines | Little or no interaction; effects are roughly additive | Main-effects story is usually safe; pick best level of A and best level of B separately |
| Nonparallel lines (converge, diverge, or cross) | Interaction present: effect of A depends on B | Must choose a treatment combination, not independent “best A” + “best B” from main effects alone |
| Crossing lines | Strong interaction; ranking of A levels reverses depending on B | Main-effect averages can be misleading—the “winner” on average may be wrong for a specific B |
Visual rule taught for CSSGB: Parallel ≈ no interaction; nonparallel ≈ interaction. Exact statistical significance still depends on error and sample size, but exam items emphasize the geometric reading and the setting implication.
Worked scenario — interaction changes the conclusion
Factors: Mold temperature (Low / High) and Hold pressure (Low / High). Response: warp (mm)—lower is better.
Cell means:
| Pressure Low | Pressure High | |
|---|---|---|
| Temp Low | 0.40 | 0.22 |
| Temp High | 0.18 | 0.35 |
Main-effect averages (warp):
- Temp Low: $(0.40+0.22)/2 = 0.31$; Temp High: $(0.18+0.35)/2 = 0.265$ → main effect slightly favors High temp
- Pressure Low: $(0.40+0.18)/2 = 0.29$; Pressure High: $(0.22+0.35)/2 = 0.285$ → main effects of pressure look almost flat
Main-effects-only trap: “Use high temperature; pressure barely matters.”
Interaction plot description: Put temperature on the x-axis. The Pressure Low line goes from 0.40 (temp low) down to 0.18 (temp high). The Pressure High line goes from 0.22 (temp low) up to 0.35 (temp high). The lines are strongly nonparallel and effectively cross in ranking: at low pressure, high temp is best; at high pressure, low temp is best.
Correct Apply-level conclusion: There is a Temp × Pressure interaction. The best combination for minimum warp is High temp + Low pressure (0.18 mm). The worst is Low temp + Low pressure (0.40). You must not set pressure from the nearly flat main-effect plot alone, and you must not assume high temp is always best if the plant is locked into high pressure for other reasons—under high pressure, low temp gives lower warp (0.22 vs. 0.35).
Worked scenario — parallel lines (additive)
Factors: Belt speed and Dryer zone (A/B). Response: moisture %—lower better.
Cell means: Speed low / Zone A = 8.0; Speed low / Zone B = 6.0; Speed high / Zone A = 7.0; Speed high / Zone B = 5.0.
Interaction plot: Zone B line sits 2 points below Zone A at both speeds; both lines slope down by 1 point from low to high speed → parallel. Interaction negligible. Best settings: High speed + Zone B (5.0%). Main-effects and interaction plots agree.
Decision Framework for CSSGB Items
- State the goal (maximize or minimize Y).
- Scan main-effects plots for large slopes and preferred levels.
- Scan interaction plots for nonparallel or crossing lines.
- If interaction is clear, open the cell means (or the plot corners) and pick the best combination.
- If lines are parallel, combine preferred levels from main effects.
- Validate with confirmation runs at the chosen settings before Control-phase lock-in.
| Situation | Safe recommendation style |
|---|---|
| Large main effects, parallel interactions | “Set A high, B low…” from main effects |
| Crossing interaction | “Use A high only when B is low; if B must be high, use A low…” |
| Flat main effect, strong interaction | “Factor looks unimportant on average but is critical with its partner—optimize the pair” |
| Flat everything | Factors/levels studied may be weak, range too narrow, or noise too large—revisit factor choice or MSA |
Linking Plots Back to DOE Hygiene
Plot beauty does not fix a bad design. If runs were not randomized, a “main effect” may be a Monday-vs-Friday artifact. If replication is missing, you cannot tell a steep line from noise. If the response measurement is weak, both main-effect and interaction plots amplify gage error. Interpretation assumes the matrix was run with the V.A.1 disciplines.
Common Exam Traps
- Picking settings from main effects only while the stem describes crossing lines
- Treating “higher on the plot” as always better without checking whether Y is a defect metric
- Confusing interaction (effect of A depends on B) with correlation from passive data
- Calling lines “interaction” when they are parallel but at different heights (that is a main effect of the second factor, not interaction)
- Ignoring practical constraints (a corner of the design may be best statistically but unsafe or un-runnable—flag constraint, then optimize inside the feasible region)
Bottom line for V.A.2: Apply main-effects plots for average factor influence; use interaction plots to see whether those averages lie. When lines are nonparallel, combination thinking replaces one-factor slogans—and that is exactly what separates DOE from OFAT in the Improve phase.
A main-effects plot for factor A is nearly flat, but the A×B interaction plot shows two clearly nonparallel lines that cross: at B low, A high gives much better Y (higher is better); at B high, A low is better. What is the best Apply-level action?
An interaction plot of temperature (x-axis) with separate lines for low and high humidity shows two nearly parallel downward slopes for defect rate (lower is better); the high-humidity line sits entirely above the low-humidity line. What does this pattern indicate?