library(afex)
data(Loblolly)
dat <- as.data.frame(Loblolly)
dat$Seed <- factor(as.character(dat$Seed)) # subject
dat$age <- factor(dat$age) # repeated within-subject factor ("Day")
afmod <- afex::aov_car(
height ~ age + Error(Seed / age), # mirrors: y ~ Day + Error(Subject/Day)
data = dat,
anova_table = list(correction = "none", # "GG" or "HF" for sphericity correction
es = "ges") # generalized eta-squared
)
as.data.frame(afmod$anova_table) # num Df, den Df, MSE, F, ges and p for age
Computational Provenance & Reproducibility Record repeated-measures one-way ANOVA · 1.0.0 · DOI 10.5281/zenodo.23013600
This Computational Provenance Record documents the statistical computing environment, software dependencies, computational provenance, and bibliographic references associated with the RAISINS Repeated-Measures One-Way ANOVA module. It is intended to support computational reproducibility and software transparency. Detailed statistical methodology, mathematical derivations, and user guidance are provided separately in the official module documentation.
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1 Module Metadata
| Parameter | Specification |
|---|---|
| Module | Repeated-Measures One-Way ANOVA |
| Module Version | 1.0.0 |
| DOI | 10.5281/zenodo.23013600 |
| Document Type | Computational workflow |
| Statistical Engine | R |
| R Version | 4.5.2 |
| Reproducibility | Execution Environment: Posit Connect · GCR · renv-locked |
2 Statistical Dependency Manifest
| Package | Version | Repository | Core Statistical Functions |
|---|---|---|---|
| afex | 1.5-1 |
CRAN | aov_car() |
| car | 3.1-5 |
CRAN | Anova() (sphericity tests and GG / HF epsilon, via afex) |
| emmeans | 2.0.3 |
CRAN | emmeans(), pairs() |
| multcomp | 1.4-30 |
CRAN | cld() |
| moments | 0.14.1 |
CRAN | skewness(), kurtosis() |
| nortest | 1.0-4 |
CRAN | ad.test() |
| factoextra | 2.1.0 |
CRAN | get_eigenvalue() |
| scales | 1.4.0 |
CRAN | rescale() |
| gtools | 3.9.5 |
CRAN | mixedsort(), mixedorder() |
| stats | 4.5.2 |
Base R | aov(), lm(), qt(), qtukey(), prcomp(), shapiro.test(), ks.test() |
3 Statistical Function Registry
| Analytical Role | Primary Function(s) |
|---|---|
| Repeated-measures ANOVA model fit | afex::aov_car() |
| Mauchly’s test of sphericity | summary(afex_model$Anova)$sphericity.tests (car) |
| Sphericity correction (GG / HF) | afex::aov_car(anova_table = list(correction = ...)) |
| Effect size | Generalized eta-squared, afex::aov_car(anova_table = list(es = "ges")) |
| Complete stratified ANOVA table (SS, MSS) | stats::aov(y ~ Day + Error(Subject/Day)) |
| Estimated marginal means | emmeans::emmeans() |
| Pairwise comparisons | emmeans::pairs() |
| Compact letter display (grouping) | multcomp::cld() |
| Critical difference (LSD / Tukey HSD) | stats::qt(), stats::qtukey() |
| Descriptive statistics | mean(), sd(), moments::skewness(), moments::kurtosis() |
| Residual normality (QQ plot) | stats::lm(), stats::shapiro.test(), nortest::ad.test(), stats::ks.test() |
| PCA-based index score | stats::prcomp(), factoextra::get_eigenvalue(), scales::rescale() |
| Result ordering | gtools::mixedsort(), gtools::mixedorder() |
4 Default Methods & Parameters
| Analysis | Step / Parameter | Default Method / Value |
|---|---|---|
| Repeated-Measures One-Way ANOVA ( afex::aov_car()) |
Model | Fitted separately for each selected response as response ~ Day + Error(Subject/Day), where Day is the single within-subject (repeated) factor and Subject is the unit measured at every time point. There is no between-subject factor, so no interaction term |
| Sphericity test | Mauchly's test for the Day effect, reported for every response; sphericity is flagged as violated when p < α. Not reported when Day has only two levels (sphericity then holds automatically) |
|
| Sphericity correction | Default none; optional Greenhouse–Geisser (GG) or Huynh–Feldt (HF) correction of the numerator and denominator degrees of freedom. A Huynh–Feldt epsilon above 1 is set to 1 |
|
| Effect size | Generalized eta-squared (es = "ges") |
|
| Multiple Comparison |
Test | Default LSD (adjust = "none"); TUKEY optional (adjust = "tukey"). Applied to the estimated marginal means of Day |
| Significance level α | Default 0.05; 0.01 optional |
|
| Letter Grouping ( multcomp::cld()) |
Method | cld(…, Letters = letters, adjust = test, alpha = α, decreasing = TRUE), so the letter a goes to the highest mean. Computed when Day has ≤ 30 levels |
| Display rule | Letters are shown only when the Day p-value ≤ α; otherwise the grouping is left blank |
|
| Critical Difference (CD) |
LSD | qt(1 − α/2, df) × SE(d), computed for each pair |
| Tukey (HSD) | qtukey(1 − α, nmeans, df) × SE(d) / √2, computed for each pair |
|
| Residual Normality |
QQ plot tests | Residuals of lm(response ~ Day + Subject); Shapiro–Wilk by default (3 ≤ n ≤ 5000), Anderson–Darling and Kolmogorov–Smirnov optional |
| Transformation | Options | Log10 (shifted by −min + 1 when any value ≤ 0); square root (√(x + 0.5) when zeros are present); arcsine asin(√p) for proportions, with 0 and 1 replaced by 1/(4n) and 1 − 1/(4n) |
| Reporting | The analysis is run on the transformed values; the table shows original-scale means with transformed means in parentheses | |
| PCA Index | Method | prcomp(center = TRUE, scale = TRUE) on the Day means of the selected responses (at least three); PC1 and PC2 scores rescaled to 0–1 |
| Rounding | Displayed decimals | Controlled by the "Digits after decimal" input (default 2, range 1–4) |
5 R Code for Key Analytical Steps
The code blocks below demonstrate the computation behind each reported result using the built-in Loblolly dataset from the datasets package. The height of each of 14 loblolly pine trees (Seed) was measured at the same six ages (3, 5, 10, 15, 20 and 25 years). This gives a balanced one-way repeated-measures design that mirrors the module’s structure: Seed is the subject, age is the within-subject (repeated) factor, the “Day” of the module, and height is the response.
5.1 Repeated-Measures One-Way ANOVA
5.2 Mauchly’s Test & Sphericity Correction
s <- summary(afmod$Anova)
s$sphericity.tests # Mauchly's W and p-value
s$pval.adjustments # GG and HF epsilon with corrected p-values
# Refit with the Greenhouse-Geisser correction
afmod_gg <- afex::aov_car(height ~ age + Error(Seed / age), data = dat,
anova_table = list(correction = "GG", es = "ges"))
as.data.frame(afmod_gg$anova_table)5.3 Complete ANOVA Table
# Sums of squares and mean squares for age and its within-subject error
summary(aov(height ~ age + Error(Seed / age), data = dat))5.4 Estimated Marginal Means & Letter Grouping
library(emmeans)
alpha <- 0.05
adjust_method <- "none" # "none" = LSD (default) ; "tukey" = TUKEY
emm <- emmeans(afmod, "age")
summary(emm) # emmean and SE per age
multcomp::cld(emm, Letters = letters, adjust = adjust_method,
alpha = alpha, decreasing = TRUE)5.5 Pairwise Comparisons & Critical Difference
pw <- as.data.frame(pairs(emm, adjust = adjust_method))
pw # contrast, estimate, SE, df, t.ratio, p.value
# --- LSD (adjust = "none") ---
pw$CD_LSD <- qt(1 - alpha / 2, pw$df) * pw$SE
# --- Tukey (adjust = "tukey") ---
n_means <- nrow(as.data.frame(emm))
pw$HSD <- qtukey(1 - alpha, nmeans = n_means, df = pw$df) * pw$SE / sqrt(2)
pw5.6 Residual Normality
res <- residuals(lm(height ~ age + Seed, data = dat))
shapiro.test(res)
nortest::ad.test(res)Explore the entire Repeated-Measures One-Way ANOVA module in preview mode using our demo datasets. To submit suggestions or report a workflow issue, please use the discussion section below, or visit the official RAISINS website.
6 RAISINS Native Statistical Framework
RAISINS uses R for all its statistical computations. Every package used to generate major results is listed and demonstrated with examples, so results can be reproduced independently. These results are then organized and formatted on the RAISINS website along with visualisation to make them easier to use and interpret. RAISINS also has its own custom-built statistical tools for managing workflows, validating results, and generating reports. Details of these are not fully covered here, they’re shared with outside researchers only on request, and are subject to licensing terms.
7 Package References
R Core Team. (2025). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
Singmann, H., Bolker, B., Westfall, J., Aust, F., & Ben-Shachar, M. S. (2025). afex: Analysis of Factorial Experiments (R package version 1.5-1). https://doi.org/10.32614/CRAN.package.afex
Fox, J., & Weisberg, S. (2026). car: Companion to Applied Regression (R package version 3.1-5). https://doi.org/10.32614/CRAN.package.car
Lenth, R. V. (2026). emmeans: Estimated Marginal Means, aka Least-Squares Means (R package version 2.0.3). https://doi.org/10.32614/CRAN.package.emmeans
Hothorn, T., Bretz, F., & Westfall, P. (2026). multcomp: Simultaneous Inference in General Parametric Models (R package version 1.4-30). https://doi.org/10.32614/CRAN.package.multcomp
Komsta, L., & Novomestky, F. (2022). moments: Moments, Cumulants, Skewness, Kurtosis and Related Tests (R package version 0.14.1). https://doi.org/10.32614/CRAN.package.moments
Gross, J., & Ligges, U. (2015). nortest: Tests for Normality (R package version 1.0-4). https://doi.org/10.32614/CRAN.package.nortest
Kassambara, A., & Mundt, F. (2020). factoextra: Extract and Visualize the Results of Multivariate Data Analyses (R package version 2.1.0). https://doi.org/10.32614/CRAN.package.factoextra
Wickham, H., Pedersen, T. L., & Seidel, D. (2025). scales: Scale Functions for Visualization (R package version 1.4.0). https://doi.org/10.32614/CRAN.package.scales
Warnes, G. R., Bolker, B., & Lumley, T. (2023). gtools: Various R Programming Tools (R package version 3.9.5). https://doi.org/10.32614/CRAN.package.gtools
Allaire, J. J., Xie, Y., Dervieux, C., McPherson, J., Luraschi, J., Ushey, K., Atkins, A., Wickham, H., Cheng, J., Chang, W., & Iannone, R. (2026). rmarkdown: Dynamic Documents for R (R package version 2.31). https://github.com/rstudio/rmarkdown