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Celeus

One-way ANOVA

Shown in the app as one_way_anova. Part of the method reference, under Comparing groups.

When to use

3+ independent groups, continuous outcome, equal variances.

What you provide

  • Outcome: one numeric column.
  • Group: one categorical column.

Options

Missing data

  • Complete case (listwise deletion)
  • Mean imputation (single)
  • Median imputation (single)

Multiple comparisons

  • Not adjustable for this method. Where it makes post-hoc comparisons, the procedure applies its own correction.

Confidence level

Every interval in the result is reported at the level you choose:

  • 90%
  • 95%
  • 99%

Alternatives that take the same inputs:

Other methods under Comparing groups:

Guides that discuss this method

How to read the output

These methods compare one outcome across three or more independent groups, for example crop yield under four irrigation schedules, or test scores across several schools.

The overall test. The p-value answers one question: is there any difference among the groups? A small p-value does not say which groups differ or by how much. One-way ANOVA assumes equal variances across groups; Welch’s ANOVA does not; the Kruskal-Wallis test works on ranks and does not assume normality.

Effect size. For the two ANOVAs, eta-squared is the proportion of the outcome’s variation that lies between the groups, from 0 to 1. For Kruskal-Wallis, epsilon-squared plays the same role on ranks. Both come with a two-sided confidence interval; the epsilon-squared interval is a seeded bootstrap, so the same data always give the same interval. The reproducible script recomputes the effect size and its interval too.

Groups coded as numbers. A group column stored as numbers (1, 2, 3) is still treated as a set of labels, never as a measurement. When missing values are filled with a mean or median, a missing group code is not filled, because the mean of two codes is not a group; that row is left out instead.

Which groups differ. With three or more groups, Celeus adds pairwise comparisons, each corrected for making several comparisons at once: Tukey’s honestly significant difference after one-way ANOVA, the Games-Howell procedure after Welch’s ANOVA (it does not assume equal variances either), and Dunn’s test after Kruskal-Wallis, where you choose the correction (Holm by default). Read each pair’s adjusted p-value and its effect size together; a pair can differ reliably by a trivial amount.

Options. Every interval of a run (the effect size, the Tukey intervals and each pair’s effect size) uses one confidence level: 90%, 95% (the default) or 99%. The overall test has no one-sided version: it asks whether any group differs, which has no direction.

Sources: Tukey JW (1949), Biometrics 5(2):99-114, doi:10.2307/3001913. Games PA, Howell JF (1976), Journal of Educational Statistics 1(2):113-125, doi:10.3102/10769986001002113. Dunn OJ (1964), Technometrics 6(3):241-252, doi:10.1080/00401706.1964.10490181.

Worked examples

Celeus's result on a published or illustrative dataset, next to an independent computation and, where one exists, a published value. The data themselves are not reproduced here.

datasets::PlantGrowth

  • Source: Published dataset distributed with an R package
  • Licence: base R (GPL)
  • Reference: Dobson (1983), An Introduction to Statistical Modelling (the data); independent computation
QuantityCeleusIndependent computationPublished valueAgrees
Test statistic4.846094.84609-Yes
Degrees of freedom2,272,27-Yes
p-value0.015910.01591-Yes
Effect size0.2641480.264148-Yes

datasets::InsectSprays (spray coded 1-6)

  • Source: Published dataset distributed with an R package
  • Licence: base R (GPL)
  • Reference: Beall (1942), Biometrika 32(3/4):243-262 (the data); independent computation
QuantityCeleusIndependent computationPublished valueAgrees
Test statistic34.702334.7023-Yes
Degrees of freedom5,665,66-Yes
p-value3.18258e-173.18258e-17-Yes
Effect size0.7244390.724439-Yes
Effect size confidence interval0.601653, 0.7949890.601653, 0.794989-Yes

datasets::chickwts

  • Source: Published dataset distributed with an R package
  • Licence: base R (GPL)
  • Reference: Anonymous (1948), Biometrika 35 (the data); independent computation
QuantityCeleusIndependent computationPublished valueAgrees
Test statistic15.364815.3648-Yes
p-value5.93642e-105.93642e-10-Yes
Adjusted p-value for each pair of groups3.07020e-08, 2.10015e-04, 3.32458e-01, 8.36531e-03, 9.99890e-01, 1.41333e-01, 1.06209e-04, 4.21665e-03, 1.21989e-08, 1.27696e-01, 7.93285e-01, 8.84323e-05, 7.39136e-01, 2.20696e-01, 3.88452e-033.07020e-08, 2.10015e-04, 3.32458e-01, 8.36531e-03, 9.99890e-01, 1.41333e-01, 1.06209e-04, 4.21665e-03, 1.21989e-08, 1.27696e-01, 7.93285e-01, 8.84323e-05, 7.39136e-01, 2.20696e-01, 3.88452e-03-Yes
Lower simultaneous confidence limit, first pair of groups-225.614-225.614-Yes
Upper simultaneous confidence limit, first pair of groups-101.153-101.153-Yes

datasets::airquality (Ozone by Month)

  • Source: Published dataset distributed with an R package
  • Licence: base R (GPL)
  • Reference: Chambers et al. (1983), Graphical Methods for Data Analysis (the data); independent computation
QuantityCeleusIndependent computationPublished valueAgrees
Test statistic8.930898.93089-Yes
Degrees of freedom4,1484,148-Yes
p-value1.73129e-061.73129e-06-Yes
Number of observations153153-Yes
Effect size0.1944420.194442-Yes

datasets::PlantGrowth (first five plants per group)

  • Source: Published dataset distributed with an R package
  • Licence: base R (GPL)
  • Reference: Dobson (1983), An Introduction to Statistical Modelling (the data); independent computation
QuantityCeleusIndependent computationPublished valueAgrees
Test statistic2.425962.42596-Yes
Degrees of freedom2,122,12-Yes
p-value0.1303740.130374-Yes
Effect size0.2879150.287915-Yes
Effect size confidence interval0.000000, 0.5897240.000000, 0.589724-Yes

Validation report

References

  1. Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh.