One-Way Analysis of Variance

Overview

One-way analysis of variance, abbreviated one-way ANOVA, compares the means of one numerical outcome across three or more independent groups defined by one categorical factor.

Examples include comparing:

• mean functional score across three rehabilitation programmes;
• mean haemoglobin concentration across four disease-severity groups;
• mean hospital stay across several surgical techniques; and
• mean laboratory yield across multiple preparation methods.

The overall ANOVA asks whether all population means are equal. If the overall null is rejected, the analysis does not identify automatically which groups differ. Planned contrasts or multiplicity-adjusted pairwise comparisons are then required.

ANOVA is a linear model. The F statistic compares variation explained by group membership with residual variation within groups.

Why this matters

Performing several ordinary two-sample t-tests across three or more groups inflates the probability of at least one false-positive conclusion.

For three groups there are three pairwise comparisons. For five groups there are ten. The analysis should therefore distinguish:

• the overall group question;
• prespecified scientific contrasts;
• exploratory pairwise comparisons;
• and multiplicity control.

A significant overall F test means that at least one population mean differs from another. It does not imply that every pair differs.

When to use this method

Use one-way ANOVA when:

• the outcome is numerical;
• one categorical factor defines three or more independent groups;
• one outcome value is observed per analytical unit;
• mean comparisons answer the question;
• and the linear-model assumptions are sufficiently reasonable.

Use another method when:

• observations are paired, repeated or clustered;
• two factors are central;
• baseline adjustment is required;
• the outcome is binary, ordinal or count;
• or a rank-based distributional contrast is the estimand.

Research question and variables

The outcome is physical-function score.

The explanatory factor is programme with three levels.

The population means are:

• $\mu_U$: usual care;
• $\mu_S$: standard rehabilitation;
• $\mu_E$: enhanced rehabilitation.

The overall alternative is that at least one population mean differs.

The effect should also be described using:

• group mean differences;
• simultaneous confidence intervals;
• and an overall effect-size measure.

Data structure

Each participant appears once. Programme should be a factor with a prespecified reference and display order.

Statistical assumptions

Moderate non-normality is often less concerning in balanced groups of reasonable size than severe outliers or dependence.

If variances differ materially, Welch’s one-way test can be used with oneway.test(..., var.equal = FALSE) [2]. If covariate adjustment is required, use linear regression or ANCOVA rather than a simple one-way procedure.

How the method works

The total variability is partitioned into:

• between-group sum of squares;
• within-group or residual sum of squares.

A large F value indicates that group mean differences are large relative to within-group variation.

Effect-size measures

Eta squared is:

Omega squared adjusts partly for positive small-sample bias and is a useful supplementary ANOVA effect-size measure [6]:

These proportions do not replace pairwise differences in clinical units.

Worked example

Reproducible R analysis

Base R fits the balanced model with aov() and constructs Tukey family-wise intervals with TukeyHSD() [4], [5].

rehab <- data.frame(
  programme = factor(
    rep(c("Usual care", "Standard rehabilitation", "Enhanced rehabilitation"), each = 24),
    levels = c("Usual care", "Standard rehabilitation", "Enhanced rehabilitation")
  ),
  function_score = c(
    81, 61, 68, 71, 66, 66, 80, 67, 80, 69, 71, 76,
    78, 67, 77, 62, 77, 81, 63, 78, 53, 87, 75, 73,
    82, 68, 79, 72, 72, 75, 75, 79, 71, 79, 82, 73,
    83, 88, 80, 83, 74, 79, 83, 73, 74, 94, 76, 68,
    81, 84, 81, 95, 69, 104, 82, 90, 79, 82, 86, 90,
    81, 88, 88, 86, 87, 65, 83, 79, 92, 74, 70, 101
  )
)

# `aov()` fits the balanced one-way model [4]
fit <- aov(function_score ~ programme, data = rehab)

summary(fit)
# `TukeyHSD()` gives family-wise simultaneous intervals [5]
TukeyHSD(fit, "programme", conf.level = 0.95)

R ANOVA output

             Df Sum Sq Mean Sq F value   Pr(>F)
programme     2 1754.9   877.4   13.98 7.98e-06
Residuals    69 4329.8    62.8

R Tukey output

Standard rehabilitation-Usual care
 diff = 5.6250, lwr = 0.1476, upr = 11.1024, p adj = 0.0428

Enhanced rehabilitation-Usual care
 diff = 12.0833, lwr = 6.6059, upr = 17.5608, p adj = 0.000004

Enhanced rehabilitation-Standard rehabilitation
 diff = 6.4583, lwr = 0.9809, upr = 11.9358, p adj = 0.0168

Cleaned output and checks

aov_table <- summary(fit)[[1]]
ss_between <- aov_table["programme", "Sum Sq"]
ss_residual <- aov_table["Residuals", "Sum Sq"]
ms_residual <- aov_table["Residuals", "Mean Sq"]

eta_squared <- ss_between / (ss_between + ss_residual)
omega_squared <- (
  ss_between - 2 * ms_residual
) / (
  ss_between + ss_residual + ms_residual
)

aggregate(
  function_score ~ programme,
  data = rehab,
  FUN = function(x) c(n = length(x), mean = mean(x), sd = sd(x))
)

data.frame(
  F = aov_table["programme", "F value"],
  df1 = 2,
  df2 = 69,
  p_value = aov_table["programme", "Pr(>F)"],
  eta_squared = eta_squared,
  omega_squared = omega_squared
)

Interpreting the output

Presenting the results

Overall one-way ANOVA: $F(2,69)=13.98$, $p<0.001$, $\omega^2=0.27$.

Visualising the results

Suggested reporting

oneway.test(
  function_score ~ programme,
  data = rehab,
  var.equal = FALSE
)

Common mistakes

Knowledge check

Key take-home messages

References

1. Fisher RA. Statistical methods for research workers. Edinburgh: Oliver and Boyd; 1925.
2. Welch BL. On the comparison of several mean values: an alternative approach. Biometrika. 1951;38(3–4):330–336. https://doi.org/10.1093/biomet/38.3-4.330
3. Tukey JW. Comparing individual means in the analysis of variance. Biometrics. 1949;5(2):99–114. https://doi.org/10.2307/3001913
4. R Core Team. Fit an analysis of variance model. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/aov.html
5. R Core Team. Compute Tukey honest significant differences. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://search.r-project.org/R/refmans/stats/html/TukeyHSD.html
6. Lakens D. Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front Psychol. 2013;4:863. https://doi.org/10.3389/fpsyg.2013.00863