What Is One-Way ANOVA?

ANOVA stands for Analysis of Variance. The one-way ANOVA tests whether the means of three or more independent groups differ significantly. It is the extension of the independent samples t-test to more than two groups.

Despite its name, ANOVA actually compares means by analyzing variance — specifically, it compares the variability between groups to the variability within groups. If the between-group variability is substantially larger than the within-group variability, there is evidence that at least one group mean differs from the others.

The test produces an F-ratio:

$$F = \frac{\text{Variance between groups}}{\text{Variance within groups}} = \frac{MS_{between}}{MS_{within}}$$

If the null hypothesis is true (all group means are equal), the F-ratio should be close to 1.0. The further $F$ is above 1, the stronger the evidence against $H_0$.

When to Use It

Use a one-way ANOVA when:

• You have one continuous dependent variable (e.g., test scores, reaction time, weight loss)
• You have one categorical independent variable (the factor) with three or more levels (groups)
• The groups are independent (different participants in each group)

Examples:

• Comparing patient satisfaction scores across four hospital departments
• Comparing crop yield for three fertilizer types
• Comparing exam performance across five study methods

Why not just run multiple t-tests? Running all pairwise t-tests inflates the Type I error rate. With 3 groups, you would need 3 comparisons; with 5 groups, 10 comparisons. At $\alpha = .05$, the probability of at least one false positive with $k$ comparisons is:

$$1 - (1 - .05)^k$$

For 3 comparisons: $1 - 0.95^3 = .143$ (14.3% false positive rate instead of 5%). ANOVA controls this by testing all groups simultaneously with a single omnibus test.

Assumptions

1. Independence of observations. Participants are randomly and independently assigned to groups. Each person appears in only one group.

2. Normality. The dependent variable is approximately normally distributed within each group. ANOVA is robust to moderate violations when sample sizes are roughly equal and each group has $n \geq 20$.

3. Homogeneity of variances. The population variances are equal across all groups. Test this with Levene's test. If violated, use Welch's ANOVA or the Brown-Forsythe test as alternatives.

Rule of thumb: If the largest group standard deviation is no more than twice the smallest, the assumption is reasonably satisfied.

Formula

Decomposing Variance

The total variability in the data is partitioned into two sources:

$$SS_{total} = SS_{between} + SS_{within}$$

Between-groups sum of squares (variability due to group differences):

$$SS_{between} = \sum_{j=1}^{k} n_j (\bar{X}_j - \bar{X}_{grand})^2$$

Within-groups sum of squares (variability due to individual differences within groups):

$$SS_{within} = \sum_{j=1}^{k} \sum_{i=1}^{n_j} (X_{ij} - \bar{X}_j)^2$$

Mean Squares

$$MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{SS_{between}}{k - 1}$$

$$MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{SS_{within}}{N - k}$$

Where $k$ is the number of groups and $N$ is the total sample size.

F-Ratio

$$F = \frac{MS_{between}}{MS_{within}}$$

The F-statistic follows an $F$-distribution with $df_1 = k - 1$ and $df_2 = N - k$.

Effect Size: Eta-Squared

$$\eta^2 = \frac{SS_{between}}{SS_{total}}$$

This tells you the proportion of total variance explained by the group variable.

Partial eta-squared ($\eta_p^2$) is often reported in factorial designs and equals $\eta^2$ in a one-way ANOVA. Be aware that SPSS reports partial $\eta^2$ by default.

Omega-squared ($\omega^2$) is a less biased alternative:

$$\omega^2 = \frac{SS_{between} - (k-1) \cdot MS_{within}}{SS_{total} + MS_{within}}$$

Worked Example

Scenario: A marketing researcher tests three advertising strategies (humor, emotional, and informational) to see which produces the highest purchase intention ratings (scale of 1-10). Each group has 8 participants.

Step 1: Calculate group means and grand mean.

• $\bar{X}_1$ (Humor) $= 54 / 8 = 6.75$
• $\bar{X}_2$ (Emotional) $= 62 / 8 = 7.75$
• $\bar{X}_3$ (Informational) $= 38 / 8 = 4.75$
• $\bar{X}_{grand} = 154 / 24 = 6.42$

Step 2: Calculate $SS_{between}$.

$$SS_{between} = 8(6.75 - 6.42)^2 + 8(7.75 - 6.42)^2 + 8(4.75 - 6.42)^2$$

$$= 8(0.11) + 8(1.77) + 8(2.79) = 0.87 + 14.15 + 22.35 = 37.37$$

Step 3: Calculate $SS_{within}$.

For each group, sum the squared deviations from the group mean:

• Humor: $(7-6.75)^2 + (6-6.75)^2 + \cdots + (7-6.75)^2 = 5.50$
• Emotional: $(8-7.75)^2 + (9-7.75)^2 + \cdots + (6-7.75)^2 = 7.50$
• Informational: $(5-4.75)^2 + (4-4.75)^2 + \cdots + (5-4.75)^2 = 7.50$

$$SS_{within} = 5.50 + 7.50 + 7.50 = 20.50$$

Step 4: Calculate mean squares.

$$MS_{between} = \frac{37.37}{3 - 1} = \frac{37.37}{2} = 18.69$$

$$MS_{within} = \frac{20.50}{24 - 3} = \frac{20.50}{21} = 0.976$$

Step 5: Calculate the F-ratio.

$$F = \frac{18.69}{0.976} = 19.15$$

Step 6: Determine the p-value.

With $df_1 = 2$ and $df_2 = 21$, the critical value of $F$ at $\alpha = .05$ is approximately 3.47. Our $F = 19.15$ far exceeds this, so $p < .001$.

Step 7: Calculate effect size.

$$SS_{total} = SS_{between} + SS_{within} = 37.37 + 20.50 = 57.87$$

$$\eta^2 = \frac{37.37}{57.87} = .65$$

This is a very large effect — advertising strategy explains 65% of the variance in purchase intention.

Post-Hoc Tests

A significant F-test tells you that at least one group mean differs, but not which groups differ. Post-hoc tests identify specific pairwise differences while controlling the family-wise error rate.

Tukey's HSD (Honestly Significant Difference)

The most commonly used post-hoc test. Compares all possible pairs of means while maintaining the overall $\alpha = .05$.

$$HSD = q_{\alpha, k, df_{within}} \times \sqrt{\frac{MS_{within}}{n}}$$

Where $q$ is the studentized range statistic.

Bonferroni Correction

Divides $\alpha$ by the number of comparisons. More conservative than Tukey for many comparisons but works with unequal sample sizes.

$$\alpha_{adjusted} = \frac{\alpha}{m}$$

Where $m$ is the number of pairwise comparisons: $m = \frac{k(k-1)}{2}$.

Which Post-Hoc Test to Use?

Interpretation

For our example, $F(2, 21) = 19.15$, $p < .001$, $\eta^2 = .65$.

This means:

• The omnibus test is significant. At least one advertising strategy produces different purchase intention ratings than the others.
• The effect is large. Advertising strategy explains 65% of the variance in purchase intention.
• Post-hoc tests are needed to determine which specific groups differ.

Tukey HSD post-hoc tests would likely reveal:

• Emotional > Informational ($p < .001$)
• Humor > Informational ($p < .01$)
• Emotional vs. Humor ($p$ may or may not be significant — the difference is 1.0 point)

Common Mistakes

1. Running multiple t-tests instead of ANOVA. This inflates the family-wise Type I error rate. Use ANOVA first, then post-hoc tests if significant.

2. Skipping post-hoc tests after a significant F. The ANOVA F-test only tells you something differs — you need post-hoc tests to learn what differs.

3. Running post-hoc tests after a non-significant F. If the omnibus F is not significant, do not go fishing for pairwise differences.

4. Ignoring unequal variances. If Levene's test is significant, use Welch's ANOVA with Games-Howell post-hoc tests instead of the standard ANOVA with Tukey.

5. Reporting $\eta^2$ as partial $\eta^2$ (or vice versa). In a one-way ANOVA these are identical, but in factorial designs they differ. Be explicit about which you report.

6. Confusing Cohen's $d$ and Cohen's $f$. For ANOVA, use Cohen's $f$ for power analysis:

$$f = \sqrt{\frac{\eta^2}{1 - \eta^2}}$$

Where $f = 0.10$ (small), $f = 0.25$ (medium), $f = 0.40$ (large).

How to Run It