Two-Way ANOVA: Main Effects, Interaction, and Treatment Combinations

1. Build From the One-Way ANOVA Example

In the previous lesson, we tested whether material type affected tensile strength.

Factor A: Material Type
Levels of A: Material A, Material B, Material C
Response: Tensile Strength

The material averages looked close:

Material	Average Strength
Material A	95.00
Material B	96.50
Material C	97.00

From the one-way view, Material Type did not look very important. But that analysis ignored temperature.

2. One-Way to Two-Way Reveal Button

Click the button to reveal the second factor: Temperature.

Now the Same Material Data Is Split by Temperature

Material	Low Temperature Mean	High Temperature Mean	Material Average	Pattern
Material A	101	89	95.00	Drops at high temperature
Material B	96	97	96.50	Stable
Material C	89	105	97.00	Increases at high temperature

The material averages are close, but the temperature-specific patterns are very different. This is the interaction.

3. What Two-Way ANOVA Tests

Two-way ANOVA tests three things:

Effect	Question It Answers	F-Statistic
Factor A: Material	Does material type affect strength on average?	\(F_A = MS_A / MS_E\)
Factor B: Temperature	Does temperature affect strength on average?	\(F_B = MS_B / MS_E\)
A × B Interaction	Does the effect of material depend on temperature?	\(F_{AB} = MS_{AB} / MS_E\)

Degrees of Freedom

\[ df_A = a - 1 \] \[ df_B = b - 1 \] \[ df_{AB} = (a - 1)(b - 1) \] \[ df_E = ab(n - 1) \] \[ df_{\text{Total}} = N - 1 = abn - 1 \]

For this example, \(a = 3\), \(b = 2\), \(n = 2\), and \(N = 12\).

4. Interactive F-Distribution Graph

Use this graph to test any of the three two-way ANOVA F-statistics. The default values show the interaction test.

Numerator df

Denominator df

Alpha

Observed F

Critical F

—

Right-tail p-value

—

Decision

—

Rule

Reject if F > Critical F

5. Two-Way ANOVA Table for This Example

Source	SS	df	MS	F	p-value
Material Type, A	8.67	2	4.33	2.17	0.196
Temperature, B	8.33	1	8.33	4.17	0.087
A × B Interaction	392.67	2	196.33	98.17	< 0.001
Error	12.00	6	2.00	—	—
Total	421.67	11	—	—	—

\[ F_A = \frac{4.33}{2.00} = 2.17 \] \[ F_B = \frac{8.33}{2.00} = 4.17 \] \[ F_{AB} = \frac{196.33}{2.00} = 98.17 \]

Material Type is not significant as an average main effect, but the Material × Temperature interaction is significant.

6. Interactive Interaction Plot

Toggle the material lines on and off. If the lines are not parallel, that is evidence of interaction.

Material A Material B Material C

Material A drops from low to high temperature. Material B stays stable. Material C rises from low to high temperature. The lines are not parallel, so there is a strong interaction.

7. Main Effect vs Interaction Toggle

Use the buttons below to compare the averaged main-effect view against the interaction view.

Main Effect View: Material Averages

Material	Average Strength
Material A	95.00
Material B	96.50
Material C	97.00

In the main-effect view, the material means look close. This is why Factor A has p-value greater than 0.05.

Interaction View: Material by Temperature

Material	Low Temperature	High Temperature	Pattern
Material A	101	89	Falls sharply
Material B	96	97	Almost flat
Material C	89	105	Rises sharply

The interaction view shows that the effect of material depends on temperature. This is why a non-significant Factor A main effect can still have a significant interaction.

Memory phrase: Averaging can hide the effect; interaction reveals the pattern.

8. Treatment Combination Explorer

This section answers the question: If Factor A has 4 levels and Factor B has 3 levels, how do we know which combination has the most effect?

Click any cell in the 4 × 3 grid. The tool will show the cell mean, row mean, column mean, grand mean, and interaction effect.

A1

A2

A3

A4

Selected Combination

Click a cell to see the details.

\[ \text{Interaction Effect}_{ij} = \bar{x}_{ij} - \bar{x}_{i.} - \bar{x}_{.j} + \bar{x}_{..} \]

The highest cell mean is not always the cell that drives the interaction most. The interaction effect shows how far the cell is from what would be expected based on row and column means.

9. How to Find the Strongest Combination

Question	What to Check
Which combination gives the highest response?	Cell means
Which combinations are statistically different?	Tukey pairwise comparisons
Which combination drives the interaction most?	Interaction effects
How much is the practical difference?	Difference in means
How strong is the interaction overall?	Interaction p-value, \(SS_{AB}\), or partial eta-squared

Effect Size for Interaction

\[ \eta_p^2 = \frac{SS_{AB}}{SS_{AB} + SS_E} \]

For the two-way ANOVA example:

\[ \eta_p^2 = \frac{392.67}{392.67 + 12.00} = 0.970 \]

This means the interaction explains a very large share of the explainable variation in this example.

10. End-of-Lesson Quiz

1. If Factor A has p-value greater than 0.05 but the interaction has p-value less than 0.05, what should you do?

Ignore Factor A completely. Interpret Factor A through the interaction. Conclude that the ANOVA failed.

2. If Factor A has 4 levels and Factor B has 3 levels, how many treatment combinations are there?

7 12 43

3. What does a significant interaction mean?

The effect of one factor depends on the level of another factor. The two factors must have equal averages. The test is two-tailed.