One-Way ANOVA in R: Compare Group Means With aov()

One-way ANOVA in R tests whether the means of three or more groups are equal. Use aov(y ~ group, data = df) to fit the model and summary() for the F-statistic and p-value. Pair with TukeyHSD() for pairwise comparisons.

fit <- aov(mpg ~ factor(cyl), data = mtcars)    # fit
summary(fit)                                    # F-stat + p-value
TukeyHSD(fit)                                   # pairwise post-hoc
oneway.test(mpg ~ cyl, data = mtcars)           # Welch (unequal var)
shapiro.test(residuals(fit))                    # check normality
bartlett.test(mpg ~ factor(cyl), data = mtcars) # check equal variance
DescTools::EtaSq(fit)                           # effect size (eta squared)

What one-way ANOVA does in one sentence

One-way ANOVA tests the null hypothesis that all group means are equal by comparing the variance BETWEEN groups to the variance WITHIN groups. A large F-statistic means between-group variance dominates; a small F-statistic means groups are indistinguishable.

A significant ANOVA tells you SOMEWHERE in the groups, means differ. It does NOT tell you which pair. For that, run a post-hoc test like Tukey's HSD.

Syntax

aov(y ~ group, data = df) fits the model. summary() produces the ANOVA table.

The F-statistic (39.70) and p-value (4.98e-09) tell you the groups differ.

Five common patterns

1. Basic ANOVA + summary

The F-statistic compares between- to within-group variance. The p-value tests "all group means equal".

2. Tukey HSD for pairwise differences

Each row shows the difference between two groups, its CI, and family-adjusted p-value. All three pairs differ here.

3. Welch's one-way (unequal variances)

oneway.test() does NOT assume equal variances. Use it when Bartlett's or Levene's test rejects equal variance.

4. Check assumptions

Both assumptions pass here. ANOVA is appropriate.

5. Effect size (eta-squared)

Eta-squared is the proportion of total variance explained by the grouping. Convention: 0.01 small, 0.06 medium, 0.14+ large. Here 73% of mpg variance is explained by cyl, a very large effect.

ANOVA assumptions

Practical workflow for one-way ANOVA

ANOVA is rarely a one-line analysis in real research. The full workflow has a sequence: prepare data, check assumptions, fit, test, post-hoc, interpret, report.

Start by inspecting your groups visually with a boxplot or violin plot. This reveals outliers, asymmetry, and obvious group differences before any test runs. Next, check the assumptions: residual normality with shapiro.test(residuals(fit)) and equal variances with bartlett.test() or Levene's test. If either fails badly, switch tools (Welch's ANOVA for unequal variances, Kruskal-Wallis for non-normality).

Once assumptions hold, fit aov() and read summary(). The F-statistic and its p-value answer "are any group means different?" If yes, run TukeyHSD() to identify which pairs differ. Compute eta-squared as the effect-size measure: significance with a tiny effect is rarely useful in practice.

When reporting, include all the moving parts: F(df1, df2) = F-value, p = p-value, eta-squared = effect_size, plus the post-hoc comparisons (which pairs differ and by how much). A standalone p-value is incomplete. Pair the result with a visualization (boxplot or means-with-error-bars chart) so readers can see what the test detected.

In notebook environments, save the model object so you can re-extract residuals, run additional post-hocs, or compute marginal means with emmeans later. Re-fitting the model each time is wasteful and error-prone.

Common pitfalls

Pitfall 1: forgetting factor() on a numeric group variable. Without it, aov(y ~ group) treats group as continuous and fits a linear regression. Always factor() if the grouping variable is numeric.

Pitfall 2: skipping post-hoc tests. A significant ANOVA only says "at least one pair differs". Without TukeyHSD() or pairwise.t.test(), you do not know WHERE.

Pitfall 3: running multiple t-tests instead of ANOVA. Pairwise t-tests inflate the false-positive rate. Use ANOVA + Tukey HSD (or another adjusted post-hoc) for proper multiple-comparison control.

Try it yourself

Related tests

After mastering one-way ANOVA, look at:

• kruskal.test(): non-parametric alternative for non-normal data
• oneway.test(): Welch's ANOVA for unequal variances
• aov(y ~ a * b): two-way ANOVA for factorial designs
• aov(y ~ group + Error(subject)): repeated-measures ANOVA
• lm(y ~ group): equivalent to one-way ANOVA via regression
• emmeans::emmeans(): estimated marginal means with custom contrasts

For sample size planning, pwr::pwr.anova.test() computes required N per group for a target power.

FAQ

How do I run one-way ANOVA in R?

aov(y ~ group, data = df) followed by summary(). Wrap numeric group variables in factor(): aov(y ~ factor(group)). Pair with TukeyHSD() for post-hoc pairwise tests.

What is the difference between aov and lm in R?

Both fit the same linear model. lm() returns regression coefficients; aov() returns the ANOVA decomposition (sums of squares). They are mathematically equivalent for one-way designs. Use aov() when you want the ANOVA table; lm() when you want coefficient interpretations.

How do I do post-hoc tests after ANOVA in R?

TukeyHSD(fit) is the standard. It computes all pairwise comparisons with family-wise error correction. Alternatives: pairwise.t.test(y, group, p.adjust.method = "bonferroni"), emmeans::pairs(emmeans(fit, ~ group)).

What if my data violate the equal variance assumption?

Use oneway.test(y ~ group, data = df), which is Welch's ANOVA (does not assume equal variances). Or transform the data (log, sqrt) to stabilize variance, then re-run aov.

How do I report ANOVA results?

Standard format: F(df_between, df_within) = F-value, p = p-value, eta-squared = effect_size. Example: "F(2, 29) = 39.70, p < .001, eta^2 = 0.73". Always include effect size and post-hoc results.