Two-Way Analysis of Variance (ANOVA): Concepts, Procedure, and Example

Analysis of Variance (ANOVA)

Introduction to ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare means across multiple groups. It helps determine whether observed differences among group means are statistically significant. In this section, we focus on Two-Way ANOVA, which examines the effects of two categorical factors on a quantitative response variable.

Two-Way ANOVA

Key Concept

Two-way analysis of variance is used when data are partitioned into categories according to two factors. The procedure involves:

• Testing for an interaction between the two factors.

• Testing for an effect from the row factor.

• Testing for an effect from the column factor.

Example Dataset: Femur Force in Car Crash Tests

Consider measured forces on femurs in car crash tests, categorized by vehicle size and femur side (left/right). The data are organized in a table with two factors, each cell containing five values.

	Small	Midsize	Large	SUV
Left Femur	1.6, 1.4, 0.5, 0.2	0.4, 0.7, 1.0, 0.7, 0.5	1.6, 1.8, 0.3, 1.3, 1.1	0.4, 0.4, 0.6, 0.2, 0.2
Right Femur	2.8, 1.0, 0.3, 0.3, 0.2	0.6, 0.8, 1.3, 0.5, 1.1	1.5, 1.7, 0.2, 0.9, 0.6	0.7, 0.7, 0.3, 0.0, 0.2

Additional info: The table above is reconstructed from the provided notes and images.

Definition: Interaction

Interaction between two factors occurs if the effect of one factor changes for different categories of the other factor.

• Interaction Effect: The impact of one factor depends on the level of the other factor.

• No Interaction Effect: The effect of one factor is consistent across levels of the other factor.

Exploring Data with Means and Interaction Graphs

To explore possible interactions, calculate the mean for each cell and plot the results. Variation in cell means may suggest interaction.

	Small	Midsize	Large	SUV
Left Femur	0.82	0.68	1.02	0.36
Right Femur	0.92	0.86	0.96	0.96

Interpretation of Interaction Graphs:

• Interaction Effect: Suggested when line segments connecting cell means are far from parallel.

• No Interaction Effect: If line segments are approximately parallel, there is likely no interaction effect.

Objectives of Two-Way ANOVA

With data categorized by two factors, Two-Way ANOVA is used to:

1. Test for an interaction effect between the row and column factors.

2. Test for an effect from the row factor.

3. Test for an effect from the column factor.

Requirements for Two-Way ANOVA

• Normality: Each cell's sample values come from a population with an approximately normal distribution.

• Variation: Populations have the same variance ($\sigma^2$) or standard deviation ($\sigma$).

• Sampling: Samples are simple random samples of quantitative data.

• Independence: Samples are independent of each other.

• Two-Way Categorization: Sample values are categorized in two ways.

• Balanced Design: All cells have the same number of sample values.

Procedure for Two-Way ANOVA

Step 1: Test for Interaction Effect

Begin by testing the null hypothesis that there is no interaction between the two factors. The test statistic is:

$F = \frac{\text{MS(interaction)}}{\text{MS(error)}}$

• Reject $H_0$: If the P-value is small ($\leq 0.05$), reject the null hypothesis. Conclude there is an interaction effect.

• Fail to Reject $H_0$: If the P-value is large ($> 0.05$), fail to reject the null hypothesis. Conclude there is no interaction effect.

Step 2: Test for Row and Column Effects

If there is an interaction effect, stop the analysis; do not proceed with further tests. If there is no interaction effect, test for effects from the row and column factors:

• Row Factor: Test $H_0$: No effect from the row factor. Use $F = \frac{\text{MS(row)}}{\text{MS(error)}}$.

• Column Factor: Test $H_0$: No effect from the column factor. Use $F = \frac{\text{MS(column)}}{\text{MS(error)}}$.

For both tests:

• Reject $H_0$: If the P-value is small ($\leq 0.05$), conclude there is an effect from the factor.

• Fail to Reject $H_0$: If the P-value is large ($> 0.05$), conclude there is no effect from the factor.

Flowchart of Two-Way ANOVA Procedure

The procedure follows this sequence:

1. Test for interaction effect.

2. If no interaction, test for row effect.

3. If no interaction, test for column effect.

Additional info: Flowchart omitted for brevity; see textbook for visual representation.

Example: Femur Impact in Car Crash Tests

Problem Statement

Given crash force measurements on left and right femurs in car crash tests, use two-way ANOVA to test for:

• Interaction effect

• Effect from femur side (row factor)

• Effect from vehicle size category (column factor)

Significance level: 0.05

Requirement Check

• Most cells have approximately normal distributions (checked via normal quantile plots).

• Cell variances differ, but the test is robust to departures from equal variances.

• Samples are simple random samples and independent.

• Data are categorized by femur side and vehicle size.

• All cells have five sample values (balanced design).

Step 1: Interaction Effect

Calculate the test statistic:

$F = \frac{\text{MS(interaction)}}{\text{MS(error)}} = \frac{0.18966667}{0.489875} = 0.387$

The corresponding P-value is 0.763. Since $0.763 > 0.05$, fail to reject the null hypothesis. Conclusion: No interaction effect.

Step 2: Row Factor

Calculate the test statistic:

$F = \frac{\text{MS(leg side)}}{\text{MS(error)}} = \frac{0.441}{0.489875} = 0.900$

P-value is 0.3498. Since $0.3498 > 0.05$, fail to reject the null hypothesis. Conclusion: No effect from femur side.

Step 2: Column Factor

Calculate the test statistic:

$F = \frac{\text{MS(size)}}{\text{MS(error)}} = \frac{0.2096667}{0.489875} = 0.428$

P-value is 0.7343. Since $0.7343 > 0.05$, fail to reject the null hypothesis. Conclusion: No effect from vehicle size category.

Final Interpretation

Based on the sample data, crash force measurements on the femur are not affected by an interaction between femur side and vehicle size category, nor by either factor individually.

Summary Table: Two-Way ANOVA Steps and Decisions

Test	Test Statistic (F)	P-value	Decision
Interaction Effect	0.387	0.763	Fail to Reject $H_0$
Row Factor (Femur Side)	0.900	0.3498	Fail to Reject $H_0$
Column Factor (Vehicle Size)	0.428	0.7343	Fail to Reject $H_0$

Key Terms

• Two-Way ANOVA: Statistical test for effects of two categorical factors and their interaction.

• Interaction: When the effect of one factor depends on the level of another factor.

• Mean Square (MS): Estimate of variance used in F-tests.

• F-test: Statistical test comparing variances to assess significance.

Applications

• Used in experimental designs with two categorical independent variables.

• Common in fields such as biology, engineering, psychology, and social sciences.