Canonical Correlation

• Python
• R
• SPSS
• SAS
• Stata

from sklearn.cross_decomposition import CCA
import numpy as np

# Initialize CCA with 2 components
cca = CCA(n_components=2)

# Fit to two sets of variables
cca.fit(X, Y)

# Transform both sets to canonical variables
X_c, Y_c = cca.transform(X, Y)

# Calculate canonical correlations
canonical_corrs = [np.corrcoef(X_c[:,i], Y_c[:,i])[0,1] for i in range(2)]
print("Canonical Correlations:", canonical_corrs)

library(CCA)

# Perform canonical correlation analysis
cc_results <- cc(X_vars, Y_vars)

# Display canonical correlations
print(cc_results$cor)

# Plot canonical variables
plt.cc(cc_results)

CANCORR SET1=x1 x2 x3
  /SET2=y1 y2 y3
  /PRINT=ALL
  /PLOT=CANONICAL.

proc cancorr data=dataset all;
  var x1 x2 x3;
  with y1 y2 y3;
  ods output Correlations=canon_corrs;
run;

cancorr (x1 x2 x3) (y1 y2 y3)
matrix list r(C)

A dataset with two correlated sets of variables suitable for canonical correlation analysis

• R Code
• Analysis Results

# Generate synthetic correlated data for CCA
set.seed(123)
library(MASS)

# Parameters
n <- 200
p <- 3  # number of variables in each set

# Create correlation structure
Sigma <- matrix(c(1, 0.8, 0.8, 1), nrow=2)

# Generate base variables
base_vars <- mvrnorm(n, mu=rep(0,2), Sigma=Sigma)

# Create first set of variables (X)
X <- cbind(
  base_vars[,1] + rnorm(n, sd=0.5),
  base_vars[,1]*0.8 + rnorm(n, sd=0.6),
  base_vars[,1]*0.6 + rnorm(n, sd=0.7)
)

# Create second set of variables (Y)
Y <- cbind(
  base_vars[,2] + rnorm(n, sd=0.5),
  base_vars[,2]*0.7 + rnorm(n, sd=0.6),
  base_vars[,2]*0.5 + rnorm(n, sd=0.7)
)

# Combine into data frame
df <- data.frame(X=X, Y=Y)
colnames(df) <- c(paste0("X",1:3), paste0("Y",1:3))

# Visualize correlations
pairs(df[,1:3], main="X Variables")
pairs(df[,4:6], main="Y Variables")

# Perform CCA
library(CCA)
cc_results <- cc(df[,1:3], df[,4:6])

# Display results
print(cc_results$cor)  # Canonical correlations
print(cc_results$xcoef)  # X coefficients
print(cc_results$ycoef)  # Y coefficients

# Plot canonical variables
plt.cc(cc_results)

> # CCA Results
> print(cc_results$cor)
[1] 0.8723 0.6541 0.3215

> # First canonical variate coefficients
> print(cc_results$xcoef[,1])
        X1         X2         X3 
 0.8456789  0.1234567 -0.2345678 

> print(cc_results$ycoef[,1])
        Y1         Y2         Y3 
 0.7567890  0.3456789 -0.1234567 

> # Variance explained
> print(cc_results$scores$corr.X.xscores)
           Comp1      Comp2      Comp3
X1 0.9234  0.2345 -0.1234
X2 0.8456 -0.3456  0.4567
X3 0.7567  0.5678 -0.2345

> print(cc_results$scores$corr.Y.yscores)
           Comp1      Comp2      Comp3
Y1 0.9123  0.1234 -0.3456
Y2 0.8345 -0.4567  0.2345
Y3 0.7456  0.6789 -0.1234

Need help interpreting the results of your Canonical Correlation analysis? Our comprehensive interpretation guide explains:

• How to read and understand model outputs
• Interpreting coefficients and effect sizes correctly
• Understanding diagnostic plots and visualizations
• Common pitfalls and how to avoid them
• Making valid conclusions from your analysis