# --- self-contained balanced 3-factor factorial RBD (a=2, b=2, c=3, r=4 blocks) ---
set.seed(1)
d <- expand.grid(
FactorC = factor(paste0("c", 1:3)),
FactorB = factor(paste0("b", 1:2)),
FactorA = factor(paste0("a", 1:2)),
Block = factor(paste0("R", 1:4))
)
d$Yield <- rnorm(nrow(d), mean = 50, sd = 5)
# --- ANOVA: three-factor factorial in RBD, Type II sums of squares ---
fit <- lm(Yield ~ Block + FactorA + FactorB + FactorC +
FactorA:FactorB + FactorA:FactorC + FactorB:FactorC +
FactorA:FactorB:FactorC, data = d)
aov_tab <- car::Anova(fit, type = "II")
aov_tab
# --- error term for post-hoc (last row = Residuals) ---
DFerror <- aov_tab["Residuals", "Df"]
MSerror <- aov_tab["Residuals", "Sum Sq"] / DFerror
# --- post-hoc: Fisher's LSD on Factor A (critical difference + letter grouping) ---
lsd_A <- agricolae::LSD.test(d$Yield, d$FactorA,
DFerror = DFerror, MSerror = MSerror, alpha = 0.05)
lsd_A$statistics # includes CD (critical difference), CV, MSE
lsd_A$groups # mean-separation letters
# --- effect size: Cohen's f (partial) = sqrt(SS_effect / SS_error) ---
# index the ANOVA table by [term, "Sum Sq"] so the row labels are kept
# (extracting the column with [, "Sum Sq"] would drop the names -> NA on lookup)
sqrt(aov_tab["FactorA", "Sum Sq"] / aov_tab["Residuals", "Sum Sq"])
Computational Provenance & Reproducibility Record 3FRBD · 2.0.0 · DOI 10.5281/zenodo.22209966
This Computational Provenance Record documents the statistical computing environment, software dependencies, computational provenance, and bibliographic references associated with the RAISINS Three-Factor Factorial RBD module. It is intended to support computational reproducibility and software transparency. Detailed statistical methodology, and user guidance are provided separately in the official module documentation.
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1 Module Metadata
| Parameter | Specification |
|---|---|
| Module | Three-Factor Factorial RBD (3FRBD) |
| Module Version | 2.0.0 |
| DOI | 10.5281/zenodo.22209966 |
| Document Type | Computational workflow |
| Statistical Engine | R |
| R Version | 4.5.2 |
| Reproducibility | GCP · renv-locked |
2 Statistical Dependency Manifest
| Package | Version | Repository | Core Statistical Functions |
|---|---|---|---|
| car | 3.1-5 |
CRAN | Anova (Type II sums of squares) |
| agricolae | 1.3-7 |
CRAN | LSD.test, HSD.test, duncan.test |
| effectsize | 1.0.3 |
CRAN | eta_squared |
| phia | 0.3-2 |
CRAN | interactionMeans |
| nortest | 1.0-4 |
CRAN | ad.test |
| moments | 0.14.1 |
CRAN | skewness, kurtosis |
| broom | 1.0.13 |
CRAN | tidy |
| gtools | 3.9.5 |
CRAN | mixedsort |
Base R (stats) |
4.5.2 |
Base R | lm, manova, prcomp, cor, aov |
3 Statistical Function Registry
| Analytical Role | Primary Function(s) |
|---|---|
| Analysis of variance (three-factor factorial in RBD) | lm, car::Anova |
| Mean separation / post-hoc | agricolae::LSD.test, agricolae::HSD.test, agricolae::duncan.test |
| Effect size (univariate) | Cohen’s f = sqrt(SS_effect / SS_error) (base R) |
| Multivariate analysis of variance (MANOVA) | manova, broom::tidy, effectsize::eta_squared |
| Principal component analysis | prcomp |
| Interaction (adjusted) means | phia::interactionMeans |
| Normality assessment | nortest::ad.test |
| Descriptive statistics | mean, sd, moments::skewness, moments::kurtosis |
| Correlation | cor |
| Ordered factor-level handling | gtools::mixedsort |
4 Default Methods & Parameters
| Analysis | Step / Parameter | Default Method / Value |
|---|---|---|
| Analysis of Variance ( lm() + car::Anova()) |
Model | Block + A + B + C + A:B + A:C + B:C + A:B:C |
| Sum-of-squares type | Type II (type = "II") |
|
| Mean Separation (Post-hoc) ( agricolae::LSD.test() / HSD.test() / duncan.test()) |
Default method | Fisher's LSD (options: Tukey's HSD, Duncan's DMRT) |
| Significance level (α) | 0.05 (option: 0.01) | |
| Error term | Residual mean square & df from the ANOVA table | |
| Effect Size (univariate) | Cohen's f (partial) | √(SSeffect / SSerror) |
| MANOVA ( manova()) |
Test statistic | Pillai's trace |
| Effect size | Partial η² (effectsize::eta_squared()) |
5 R Code for Key Analytical Steps
The code blocks below demonstrate the exact computation behind each reported result using a small self-contained balanced three-factor factorial RBD constructed inline.
5.1 Three-Factor Factorial ANOVA & Mean Separation
5.2 Multivariate Summaries (MANOVA & PCA)
# a second response variable for a multivariate illustration
d$Yield2 <- rnorm(nrow(d), mean = 30, sd = 4)
# --- MANOVA across the response matrix, Pillai's trace ---
man <- manova(cbind(Yield, Yield2) ~ FactorA + FactorB + FactorC +
FactorA:FactorB + FactorA:FactorC + FactorB:FactorC +
FactorA:FactorB:FactorC, data = d)
summary(man, test = "Pillai")
effectsize::eta_squared(man) # partial eta-squared per source
# --- PCA on standardised treatment-combination means ---
mc <- aggregate(cbind(Yield, Yield2) ~ FactorA + FactorB + FactorC,
data = d, FUN = mean)
pca <- prcomp(mc[, c("Yield", "Yield2")], center = TRUE, scale. = TRUE)
summary(pca)Explore the entire Three-Factor Factorial RBD 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). An R Companion to Applied Regression (3rd ed.) [car: Companion to Applied Regression, R package version 3.1-5]. Sage. https://CRAN.R-project.org/package=car
de Mendiburu, F. (2023). agricolae: Statistical Procedures for Agricultural Research (R package version 1.3-7). https://CRAN.R-project.org/package=agricolae
Ben-Shachar, M. S., Lüdecke, D., & Makowski, D. (2020). effectsize: Estimation of Effect Size Indices and Standardized Parameters (R package version 1.0.3). https://CRAN.R-project.org/package=effectsize
De Rosario-Martinez, H. (2015). phia: Post-Hoc Interaction Analysis (R package version 0.3-2). https://CRAN.R-project.org/package=phia
Gross, J., & Ligges, U. (2015). nortest: Tests for Normality (R package version 1.0-4). https://CRAN.R-project.org/package=nortest
Komsta, L., & Novomestky, F. (2022). moments: Moments, Cumulants, Skewness, Kurtosis and Related Tests (R package version 0.14.1). https://CRAN.R-project.org/package=moments
Robinson, D., Hayes, A., & Couch, S. (2025). broom: Convert Statistical Objects into Tidy Tibbles (R package version 1.0.13). https://CRAN.R-project.org/package=broom
Warnes, G. R., Bolker, B., & Lumley, T. (2023). gtools: Various R Programming Tools (R package version 3.9.5). https://CRAN.R-project.org/package=gtools