What it does

One-way analysis of variance asks whether the means of three or more independent groups differ more than measurement variability would explain. It generalises the independent-samples t-test; running several t-tests instead inflates the family-wise error rate.

Typical uses

• Comparing three or more formulations, batches or suppliers for hardness, assay, friability or disintegration time
• Effect of a process factor with several levels (compression force, drying temperature, mixing time) on a quality attribute
• Percentage dissolved at one time point across several formulations

How to read the output

• ANOVA table: F = MS_between/MS_within; p < α means at least one mean differs, without saying which.
• Tukey HSD: all pairwise comparisons with the family-wise error held at α. With unequal group sizes the Tukey-Kramer standard error is used. A confidence interval that excludes zero marks a significant pair.
• η² and ω²: share of total variance explained by group membership; ω² is the less biased population estimate.
• Levene: tests equality of variances (mean-centred, as on the t-test page). If it is significant the F test and Tukey lose reliability; consider a Welch ANOVA or Games-Howell in dedicated software.

Prerequisites

• Independent observations; each group at least two, preferably five or more values
• Approximately normal residuals and similar variances; ANOVA is fairly robust to moderate departures when group sizes are similar
• Continuous response. Counts, percentages near 0 or 100 and ordinal scores may need transformation or a non-parametric test (Kruskal-Wallis)

References: Levene H. (1960) in Contributions to Probability and Statistics, Stanford Univ. Press; Tukey J.W. (1949) Biometrics 5:99–114, doi:10.2307/3001913; Kramer C.Y. (1956) Biometrics 12:307–310, doi:10.2307/3001469. Results are compared with SciPy/statsmodels in the unit tests.