data(ToothGrowth)
tg <- ToothGrowth
tg$treatment <- factor(tg$supp) # treatment factor
tg$covar <- tg$dose # continuous covariate
# One-way ANCOVA: main effects only (common slope assumed)
fit <- aov(len ~ treatment + covar, data = tg)
# Type III ANCOVA table, as reported (Intercept / Treatment / Covariate / Error)
tab <- car::Anova(fit, type = "III")
tab$MS <- tab$`Sum Sq` / tab$Df # mean squares
tab
Computational Provenance & Reproducibility Record ANCOVA in CRD · 2.0.0 · DOI ⟨VERIFY — Zenodo DOI not yet minted⟩
This Computational Provenance Record documents the statistical computing environment, software dependencies, computational provenance, and bibliographic references associated with the RAISINS ANCOVA (CRD). 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 | ANCOVA (CRD) |
| Module Version | 2.0.0 |
| DOI | ⟨VERIFY — Zenodo DOI not yet minted⟩ |
| 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 |
|---|---|---|---|
| car | 3.1-3 |
CRAN | Anova() (Type III ANCOVA table) |
| emmeans | 1.11.2 |
CRAN | emmeans(), pairs() |
| multcomp | 1.4-28 |
CRAN | cld() (compact letter display) |
| multcompView | 0.1-10 |
CRAN | letter-grouping backend for cld() |
| effectsize | 1.0.1 |
CRAN | eta_squared() |
| broom | 1.0.8 |
CRAN | tidy() |
3 Statistical Function Registry
| Analytical Role | Primary Function(s) |
|---|---|
| ANCOVA model fitting | stats::aov(response ~ treatment + covar) |
| Covariate-adjusted ANCOVA table (Type III SS) | car::Anova(..., type = "III") |
| Adjusted (estimated marginal) means | emmeans::emmeans(model, "treatment") |
| Pairwise comparison of adjusted means | emmeans::pairs() |
| Multiple-comparison letter groupings | multcomp::cld() (via multcompView) |
4 Default Methods & Parameters
| Analysis | Step / Parameter | Default Method / Value |
|---|---|---|
| ANCOVA Model ( aov() +car::Anova()) |
Model formula | One-way ANCOVA: response ~ treatment + covar, a single continuous covariate, main-effects only. No treatment×covariate interaction term is fitted. |
| Sums of squares | Type III, via car::Anova(model, type = "III") |
|
| Significance flags | Significance level α selectable (choices 0.05 and 0.01, default 0.05). Stars from the p-value: ** when p ≤ 0.01, * when p ≤ 0.05, otherwise NS |
|
| Adjusted Means ( emmeans()) |
Estimate | Estimated marginal means for treatment at the mean of the covariate, reported with their standard errors (SE) |
| Multiple Comparisons ( pairs(),cld()) |
Test | Choice of LSD (default) or Tukey's HSD. LSD uses pairs(emm, adjust = padj); Tukey uses adjust = "tukey" |
| p-adjustment (LSD only) | User-selectable: none (default), bonferroni, holm, or BH. Tukey uses its own "tukey" adjustment |
|
| Critical difference | CD for each treatment pair computed as SE(d) × |t ratio| | |
| Letter groupings | Compact letter display via multcomp::cld(emm, Letters = letters, alpha = α); letters are shown only when the Treatment effect is significant (p ≤ α), otherwise the grouping column is left blank |
|
| Transformations (response variables only) |
Logarithmic | log10(x); if any value ≤ 0, an offset is applied as log10(x − min(x) + 1) |
| Square root | sqrt(x); if any value = 0, sqrt(x + 0.5); negative values are rejected with a warning |
|
| Arcsine | asin(sqrt(p)) for proportions in [0, 1]; boundary values corrected by 1/(4n) (0 → 1/(4n), 1 → 1 − 1/(4n)); values outside [0, 1] are rejected |
5 R Code for Key Analytical Steps
The code blocks below demonstrate the exact computation behind each reported result. The ANCOVA and adjusted-means steps use the ToothGrowth dataset from the base datasets package (supp as the treatment, dose as the continuous covariate, len as the response); the multivariate step uses iris (Species as the treatment) to illustrate the several-trait case that the index requires.
5.1 Covariate-adjusted ANCOVA table
5.2 Adjusted means and multiple comparisons
library(emmeans)
library(multcomp)
alpha <- 0.05
emm <- emmeans(fit, "treatment") # estimated marginal (adjusted) means
summary(emm) # adjusted means + SE
# Pairwise comparisons (LSD-style default: adjust = "none")
pw <- as.data.frame(pairs(emm, adjust = "none"))
pw$CD <- pw$SE * abs(pw$t.ratio) # critical difference, per pair
pw
# Compact letter display (groupings); shown only when Treatment is significant
cld(emm, Letters = letters, adjust = "none", alpha = alpha)
# Tukey's HSD alternative:
# pairs(emm, adjust = "tukey"); cld(emm, Letters = letters, adjust = "tukey", alpha = alpha)Explore the entire ANCOVA in CRD 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/
Fox, J., & Weisberg, S. (2019). car: Companion to Applied Regression (R package version 3.1-3). https://doi.org/10.32614/CRAN.package.car
Lenth, R. V. (2025). emmeans: Estimated Marginal Means, aka Least-Squares Means (R package version 1.11.2). https://doi.org/10.32614/CRAN.package.emmeans
Hothorn, T., Bretz, F., & Westfall, P. (2008). multcomp: Simultaneous Inference in General Parametric Models (R package version 1.4-28). https://doi.org/10.32614/CRAN.package.multcomp
Graves, S., Piepho, H.-P., & Selzer, L. (2024). multcompView: Visualizations of Paired Comparisons (R package version 0.1-10). https://doi.org/10.32614/CRAN.package.multcompView
Ben-Shachar, M. S., Lüdecke, D., & Makowski, D. (2020). effectsize: Estimation of Effect Size Indices and Standardized Parameters (R package version 1.0.1). https://doi.org/10.32614/CRAN.package.effectsize
Robinson, D., Hayes, A., & Couch, S. (2025). broom: Convert Statistical Objects into Tidy Tibbles (R package version 1.0.8). https://doi.org/10.32614/CRAN.package.broom
Kassambara, A., & Mundt, F. (2020). factoextra: Extract and Visualize the Results of Multivariate Data Analyses (R package version 1.0.7). 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
Schloerke, B., Cook, D., Larmarange, J., Briatte, F., Marbach, M., Thoen, E., Elberg, A., & Crowley, J. (2024). GGally: Extension to ‘ggplot2’ (R package version 2.2.1). https://doi.org/10.32614/CRAN.package.GGally
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