data(mtcars)
X <- mtcars[, c("mpg", "hp", "qsec")] # performance set
Y <- mtcars[, c("wt", "disp", "drat")] # body / weight set
cca <- CCA::cc(X, Y)
rho <- cca$cor # canonical correlations
rho2 <- rho^2 # r-squared (shared variance) per function
Computational Provenance & Reproducibility Record Canonical Correlation Analysis · 2.0.0 · DOI 10.5281/zenodo.21882155
This Computational Provenance Record documents the statistical computing environment, software dependencies, computational provenance, and bibliographic references associated with the RAISINS Canonical Correlation Analysis module. It is intended to support computational reproducibility and software transparency. Detailed statistical methodology, mathematical derivations, and user guidance are provided separately in the official module documentation.
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1 Module Metadata
| Parameter | Specification |
|---|---|
| Module | Canonical Correlation Analysis |
| Module Version | 2.0.0 |
| DOI | 10.5281/zenodo.21882155 |
| Document Type | Computational workflow |
| Statistical Engine | R |
| R Version | 4.5.2 |
| Reproducibility | Execution Environment: Posit Connect · GCR · renv-locked |
2 Statistical Dependency Manifest
| Package | Version | Repository | Core Statistical Functions |
|---|---|---|---|
| CCA | 1.2.2 |
CRAN | cc(), comput() |
| CCP | 1.2 |
CRAN | p.asym() |
| stats | 4.5.2 |
Base R | cor(), na.omit() |
3 Statistical Function Registry
| Analytical Role | Primary Function(s) |
|---|---|
| Canonical correlation extraction | CCA::cc() |
| Raw canonical coefficients, structure loadings & cross-loadings | CCA::comput() |
| Significance testing (Wilks’ Lambda) | CCP::p.asym(tstat = "Wilks") |
| Canonical variate scores (for plots) | CCA::comput()$xscores, $yscores |
| Structure correlations (biplot / network plots) | stats::cor() of each set against the joint canonical scores |
4 Default Methods & Parameters
| Analysis | Step / Parameter | Default Method / Value |
|---|---|---|
| Canonical Correlation |
Method | Canonical correlation between an X-set and a Y-set of numeric variables via CCA::cc(); rows with missing values in any selected column are removed listwise (na.omit()) before fitting |
| Number of canonical functions | min(p, q), where p and q are the number of variables in the X-set and Y-set respectively | |
| Variable requirements | At least 2 numeric variables in each of the X-set and Y-set; non-numeric columns in a selected set are dropped before fitting | |
| Significance Testing |
Test | Wilks' Lambda, sequential test of function k through min(p,q), via CCP::p.asym(tstat = "Wilks"), using Rao's F-approximation |
| Significance level | User-selectable via a "Level of significance (α)" control; α = 0.05 (default) or 0.01, applied uniformly to all functions | |
| Coefficients & Loadings |
Raw canonical coefficients | Standardized coefficients of the linear combinations that define each canonical variate (cc()$xcoef, cc()$ycoef) |
| Structure loadings & cross-loadings | Pearson correlations between each original variable and the canonical variates of its own set (structure loadings) and of the other set (cross-loadings), via comput() |
|
| Practical Retention Rule |
Retained for interpretation | A canonical function is flagged "Retained" when its Wilks' Lambda p-value is below the selected significance level α (see above) and its r² (squared canonical correlation) ≥ 0.10. A function can be statistically significant yet not retained if it explains too little shared variance to be practically meaningful |
| Loading Importance |
Threshold | A structure or cross-loading with |value| > 0.3 is treated as an important/dominant contributor to that canonical function |
| Plots | Canonical scores, scree, biplot | Canonical variate scores from comput()$xscores/$yscores; the biplot's joint observation position is the average of the X-side and Y-side scores on the chosen pair of dimensions |
| Loadings heatmap, network plot | Structure/cross-loading matrices rendered directly; the network plot additionally filters variables by a user-adjustable minimum |loading| threshold |
5 R Code for Key Analytical Steps
The code blocks below demonstrate the exact computation behind each reported result, using the built-in mtcars dataset from the datasets package, split into two illustrative variable sets: an engine/performance set and a body/weight set.
5.1 Canonical Correlation
5.2 Significance Testing (Wilks’ Lambda)
n <- nrow(X); p <- ncol(X); q <- ncol(Y)
wilks <- CCP::p.asym(rho, n, p, q, tstat = "Wilks")
# wilks$stat, $approx (F), $df1, $df2, $p.value - one row per function,
# sequentially testing function k through min(p,q)
alpha <- 0.05 # user-selectable in the app: 0.05 (default) or 0.01
retained <- (wilks$p.value < alpha) & (rho2 >= 0.10) # practical retention rule5.3 Raw Canonical Coefficients, Loadings and Cross-Loadings
comp <- CCA::comput(X, Y, cca)
cca$xcoef # raw canonical coefficients, X-set
cca$ycoef # raw canonical coefficients, Y-set
comp$corr.X.xscores # structure loadings: X-set vs its own canonical variate (U)
comp$corr.Y.yscores # structure loadings: Y-set vs its own canonical variate (V)
comp$corr.X.yscores # cross-loadings: X-set vs the Y-set's canonical variate (V)
comp$corr.Y.xscores # cross-loadings: Y-set vs the X-set's canonical variate (U)5.4 Canonical Variate Scores (for Plots)
U <- comp$xscores # per-observation scores on each X-side canonical variate
V <- comp$yscores # per-observation scores on each Y-side canonical variate
# the canonical scores plot pairs U and V per function;
# the biplot's joint observation position, for a chosen pair of dimensions
# dim_x, dim_y, is the average of the corresponding U and V columns
joint <- (U[, c("Dim1", "Dim2")] + V[, c("Dim1", "Dim2")]) / 2 # illustrativeExplore the entire Canonical Correlation Analysis 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/
González, I., Déjean, S., Martin, P. G. P., & Baccini, A. (2008). CCA: An R Package to Extend Canonical Correlation Analysis. Journal of Statistical Software, 23(12), 1-14. https://doi.org/10.18637/jss.v023.i12
Menzel, U. (2012). CCP: Significance Tests for Canonical Correlation Analysis (R package version 1.2). https://CRAN.R-project.org/package=CCP
Hotelling, H. (1936). Relations Between Two Sets of Variates. Biometrika, 28(3/4), 321-377. https://doi.org/10.2307/2333955