Created:December 24, 2025
Last Updated:December 24, 2025
This calculator performs comprehensive Canonical Correlation Analysis (CCA), a multivariate statistical method used to identify and measure the associations between two sets of variables. CCA finds linear combinations of variables in each set that have maximum correlation with each other.
What You'll Get:
- Canonical Correlations: Correlation coefficients between canonical variates
- Canonical Coefficients: Weights for creating canonical variates from original variables
- Canonical Loadings: Structure coefficients showing relationships between original variables and canonical variates
- Significance Tests: Wilks' Lambda test for each canonical function
- Redundancy Analysis: Proportion of variance in one set explained by the other set
- Visualizations: Heatmaps and bar charts showing canonical loadings and correlations
- APA-Formatted Report: Professional statistical reporting ready for publication
Pro Tip: CCA is useful when you want to understand relationships between two groups of variables (e.g., academic performance vs. psychological factors, physiological measures vs. behavioral outcomes). Each set should contain at least 2 variables. Larger sample sizes (n ≥ 20 per variable) provide more reliable results.
Ready to explore multivariate relationships? (academic performance and psychological factors) to see CCA in action, or upload your own data to discover relationships between your variable sets.
Calculator
1. Load Your Data
2. Define Variable Sets
Important: Assign variables to Set 1 and Set 2 based on your research question. Each variable can only belong to one set. Each set should contain at least 2 variables for meaningful analysis.
Select variables for the first set (e.g., academic performance measures)
Select variables for the second set (e.g., psychological factors)
3. Analysis Options
Related Calculators
Learn More
Definition
Canonical Correlation Analysis (CCA) is a multivariate statistical technique that identifies and quantifies the associations between two sets of variables. It finds pairs of linear combinations (canonical variates) from each set that have maximum correlation with each other, allowing researchers to understand complex relationships between multiple predictors and multiple outcomes.
When to Use CCA
Use Canonical Correlation Analysis when you want to:
- Explore multivariate relationships: Understand how two sets of variables relate to each other
- Reduce dimensionality: Summarize complex relationships using fewer canonical dimensions
- Test theoretical models: Examine relationships between constructs measured by multiple indicators
- Predict multiple outcomes: Analyze how multiple predictors relate to multiple dependent variables
- Compare variable sets: Determine which variables contribute most to relationships between sets
Understanding CCA Output
- Canonical Correlations: Range from 0 to 1, with higher values indicating stronger relationships between variable sets
- Canonical Coefficients: Weights used to create canonical variates (similar to regression coefficients)
- Canonical Loadings: Correlations between original variables and canonical variates (easier to interpret than coefficients)
- Wilks' Lambda: Tests statistical significance; values closer to 0 indicate stronger relationships (p < .05 suggests significance)
Interpreting CCA Results
Canonical Correlations
- ≥ 0.70: Strong relationship
- 0.40 - 0.70: Moderate relationship
- < 0.40: Weak relationship
- First canonical correlation is always the largest; subsequent ones are progressively smaller
Canonical Loadings
- |r| ≥ 0.45: Variable substantially contributes to the canonical variate
- |r| 0.30 - 0.45: Moderate contribution
- |r| < 0.30: Minimal contribution
- Loadings are easier to interpret than canonical coefficients
Significance Testing
- Wilks' Lambda: Values closer to 0 indicate stronger relationships
- p < .05: Canonical function is statistically significant
- Test each canonical function sequentially; stop when non-significant
Redundancy Analysis
- Indicates proportion of variance in one set explained by the other set
- ≥ 10%: Meaningful redundancy
- Low redundancy suggests sets share little common variance
Assumptions and Considerations
Key Assumptions:
- Linear relationships between variables
- Multivariate normality (especially for significance tests)
- Homoscedasticity of residuals
- No extreme multicollinearity within sets
Sample Size Requirements:
Minimum n ≥ 20 per variable in the larger set. For reliable results, aim for n ≥ 10 × (total number of variables). Smaller samples may produce unstable estimates.
Note on Interpretation:
Focus on the first 1-3 canonical functions, as later functions often explain trivial variance and may not be practically meaningful even if statistically significant.
Example Code
R Code (CCA & CCP packages)
R
library(CCA)
library(CCP)
library(tidyverse)
# Academic performance and psychological factors data
df <- tibble(
math = c(85, 78, 92, 88, 76, 90, 82, 87, 91, 79, 84, 89, 77, 86, 93),
reading = c(88, 82, 90, 85, 79, 92, 84, 89, 93, 81, 86, 91, 80, 87, 95),
writing = c(82, 76, 88, 84, 74, 89, 80, 85, 90, 77, 83, 87, 75, 84, 91),
motivation = c(7.5, 6.8, 8.2, 7.8, 6.5, 8.5, 7.2, 7.9, 8.8, 6.9, 7.6, 8.3, 6.7, 7.7, 8.9),
anxiety = c(3.2, 4.1, 2.5, 3.0, 4.5, 2.3, 3.8, 2.9, 2.1, 4.2, 3.3, 2.6, 4.3, 3.1, 2.0),
self_efficacy = c(8.1, 7.2, 8.9, 8.3, 7.0, 9.0, 7.8, 8.5, 9.2, 7.3, 8.0, 8.7, 7.1, 8.2, 9.3)
)
# Define variable sets
# Set 1: Academic Performance (math, reading, writing)
X <- df |> select(math, reading, writing)
# Set 2: Psychological Factors (motivation, anxiety, self_efficacy)
Y <- df |> select(motivation, anxiety, self_efficacy)
# Perform Canonical Correlation Analysis
cc_result <- cc(as.matrix(X), as.matrix(Y))
# Display canonical correlations
print("Canonical Correlations:")
print(cc_result$cor)
# Canonical coefficients (standardized)
print("Canonical Coefficients for Set 1 (X):")
print(cc_result$xcoef)
print("Canonical Coefficients for Set 2 (Y):")
print(cc_result$ycoef)
# Calculate canonical loadings (structure coefficients)
X_std <- scale(X)
Y_std <- scale(Y)
# Canonical variates
U <- X_std %*% cc_result$xcoef
V <- Y_std %*% cc_result$ycoef
# Loadings (correlations between original variables and canonical variates)
loadings_X <- cor(X_std, U)
loadings_Y <- cor(Y_std, V)
print("Canonical Loadings for Set 1 (X):")
print(loadings_X)
print("Canonical Loadings for Set 2 (Y):")
print(loadings_Y)
# Test significance using Wilks' Lambda
n <- nrow(df)
p <- ncol(X)
q <- ncol(Y)
# Use CCP package for significance testing
sig_test <- p.asym(cc_result$cor, n, p, q)
print("Significance Tests:")
print(sig_test)
# Redundancy analysis
redundancy_X <- colSums(loadings_X^2) / p * cc_result$cor^2
redundancy_Y <- colSums(loadings_Y^2) / q * cc_result$cor^2
print("Redundancy (variance in X explained by Y's canonical variates):")
print(redundancy_X)
print("Redundancy (variance in Y explained by X's canonical variates):")
print(redundancy_Y)Python Code (scikit-learn)
Python
import pandas as pd
import numpy as np
from sklearn.cross_decomposition import CCA
from scipy import stats
import matplotlib.pyplot as plt
# Academic performance and psychological factors data
data = pd.DataFrame({
'math': [85, 78, 92, 88, 76, 90, 82, 87, 91, 79, 84, 89, 77, 86, 93],
'reading': [88, 82, 90, 85, 79, 92, 84, 89, 93, 81, 86, 91, 80, 87, 95],
'writing': [82, 76, 88, 84, 74, 89, 80, 85, 90, 77, 83, 87, 75, 84, 91],
'motivation': [7.5, 6.8, 8.2, 7.8, 6.5, 8.5, 7.2, 7.9, 8.8, 6.9, 7.6, 8.3, 6.7, 7.7, 8.9],
'anxiety': [3.2, 4.1, 2.5, 3.0, 4.5, 2.3, 3.8, 2.9, 2.1, 4.2, 3.3, 2.6, 4.3, 3.1, 2.0],
'self_efficacy': [8.1, 7.2, 8.9, 8.3, 7.0, 9.0, 7.8, 8.5, 9.2, 7.3, 8.0, 8.7, 7.1, 8.2, 9.3]
})
# Define variable sets
# Set 1: Academic Performance
X = data[['math', 'reading', 'writing']].values
# Set 2: Psychological Factors
Y = data[['motivation', 'anxiety', 'self_efficacy']].values
# Standardize the data
from sklearn.preprocessing import StandardScaler
scaler_X = StandardScaler()
scaler_Y = StandardScaler()
X_std = scaler_X.fit_transform(X)
Y_std = scaler_Y.fit_transform(Y)
# Perform Canonical Correlation Analysis
n_components = min(X.shape[1], Y.shape[1])
cca = CCA(n_components=n_components)
cca.fit(X_std, Y_std)
# Transform to canonical variates
X_c, Y_c = cca.transform(X_std, Y_std)
# Calculate canonical correlations
canonical_correlations = [np.corrcoef(X_c[:, i], Y_c[:, i])[0, 1]
for i in range(n_components)]
print("Canonical Correlations:")
print(canonical_correlations)
# Canonical coefficients
print("\nCanonical Coefficients for Set 1 (X):")
print(cca.x_weights_)
print("\nCanonical Coefficients for Set 2 (Y):")
print(cca.y_weights_)
# Calculate canonical loadings (structure coefficients)
loadings_X = np.corrcoef(X_std.T, X_c.T)[:X.shape[1], X.shape[1]:]
loadings_Y = np.corrcoef(Y_std.T, Y_c.T)[:Y.shape[1], Y.shape[1]:]
print("\nCanonical Loadings for Set 1 (X):")
print(loadings_X)
print("\nCanonical Loadings for Set 2 (Y):")
print(loadings_Y)
# Wilks' Lambda test for significance
def wilks_lambda_test(canonical_corrs, n, p, q):
"""
Test significance of canonical correlations using Wilks' Lambda
n: sample size
p: number of variables in set 1
q: number of variables in set 2
"""
m = min(p, q)
lambda_vals = []
for k in range(m):
# Product of (1 - r^2) for remaining correlations
lambda_k = np.prod([1 - r**2 for r in canonical_corrs[k:]])
lambda_vals.append(lambda_k)
# Chi-square approximation
df = (p - k) * (q - k)
chi_sq = -(n - 1 - (p + q + 1) / 2) * np.log(lambda_k)
p_value = 1 - stats.chi2.cdf(chi_sq, df)
print(f"\nFunction {k+1}:")
print(f" Wilks' Lambda: {lambda_k:.4f}")
print(f" Chi-square: {chi_sq:.4f}")
print(f" df: {df}")
print(f" p-value: {p_value:.4f}")
print("\nSignificance Tests:")
wilks_lambda_test(canonical_correlations, len(data), X.shape[1], Y.shape[1])Verification
Created:December 24, 2025
Last Updated:December 24, 2025
This calculator performs comprehensive Canonical Correlation Analysis (CCA), a multivariate statistical method used to identify and measure the associations between two sets of variables. CCA finds linear combinations of variables in each set that have maximum correlation with each other.
What You'll Get:
- Canonical Correlations: Correlation coefficients between canonical variates
- Canonical Coefficients: Weights for creating canonical variates from original variables
- Canonical Loadings: Structure coefficients showing relationships between original variables and canonical variates
- Significance Tests: Wilks' Lambda test for each canonical function
- Redundancy Analysis: Proportion of variance in one set explained by the other set
- Visualizations: Heatmaps and bar charts showing canonical loadings and correlations
- APA-Formatted Report: Professional statistical reporting ready for publication
Pro Tip: CCA is useful when you want to understand relationships between two groups of variables (e.g., academic performance vs. psychological factors, physiological measures vs. behavioral outcomes). Each set should contain at least 2 variables. Larger sample sizes (n ≥ 20 per variable) provide more reliable results.
Ready to explore multivariate relationships? (academic performance and psychological factors) to see CCA in action, or upload your own data to discover relationships between your variable sets.
Calculator
1. Load Your Data
2. Define Variable Sets
Important: Assign variables to Set 1 and Set 2 based on your research question. Each variable can only belong to one set. Each set should contain at least 2 variables for meaningful analysis.
Select variables for the first set (e.g., academic performance measures)
Select variables for the second set (e.g., psychological factors)
3. Analysis Options
Related Calculators
Learn More
Definition
Canonical Correlation Analysis (CCA) is a multivariate statistical technique that identifies and quantifies the associations between two sets of variables. It finds pairs of linear combinations (canonical variates) from each set that have maximum correlation with each other, allowing researchers to understand complex relationships between multiple predictors and multiple outcomes.
When to Use CCA
Use Canonical Correlation Analysis when you want to:
- Explore multivariate relationships: Understand how two sets of variables relate to each other
- Reduce dimensionality: Summarize complex relationships using fewer canonical dimensions
- Test theoretical models: Examine relationships between constructs measured by multiple indicators
- Predict multiple outcomes: Analyze how multiple predictors relate to multiple dependent variables
- Compare variable sets: Determine which variables contribute most to relationships between sets
Understanding CCA Output
- Canonical Correlations: Range from 0 to 1, with higher values indicating stronger relationships between variable sets
- Canonical Coefficients: Weights used to create canonical variates (similar to regression coefficients)
- Canonical Loadings: Correlations between original variables and canonical variates (easier to interpret than coefficients)
- Wilks' Lambda: Tests statistical significance; values closer to 0 indicate stronger relationships (p < .05 suggests significance)
Interpreting CCA Results
Canonical Correlations
- ≥ 0.70: Strong relationship
- 0.40 - 0.70: Moderate relationship
- < 0.40: Weak relationship
- First canonical correlation is always the largest; subsequent ones are progressively smaller
Canonical Loadings
- |r| ≥ 0.45: Variable substantially contributes to the canonical variate
- |r| 0.30 - 0.45: Moderate contribution
- |r| < 0.30: Minimal contribution
- Loadings are easier to interpret than canonical coefficients
Significance Testing
- Wilks' Lambda: Values closer to 0 indicate stronger relationships
- p < .05: Canonical function is statistically significant
- Test each canonical function sequentially; stop when non-significant
Redundancy Analysis
- Indicates proportion of variance in one set explained by the other set
- ≥ 10%: Meaningful redundancy
- Low redundancy suggests sets share little common variance
Assumptions and Considerations
Key Assumptions:
- Linear relationships between variables
- Multivariate normality (especially for significance tests)
- Homoscedasticity of residuals
- No extreme multicollinearity within sets
Sample Size Requirements:
Minimum n ≥ 20 per variable in the larger set. For reliable results, aim for n ≥ 10 × (total number of variables). Smaller samples may produce unstable estimates.
Note on Interpretation:
Focus on the first 1-3 canonical functions, as later functions often explain trivial variance and may not be practically meaningful even if statistically significant.
Example Code
R Code (CCA & CCP packages)
R
library(CCA)
library(CCP)
library(tidyverse)
# Academic performance and psychological factors data
df <- tibble(
math = c(85, 78, 92, 88, 76, 90, 82, 87, 91, 79, 84, 89, 77, 86, 93),
reading = c(88, 82, 90, 85, 79, 92, 84, 89, 93, 81, 86, 91, 80, 87, 95),
writing = c(82, 76, 88, 84, 74, 89, 80, 85, 90, 77, 83, 87, 75, 84, 91),
motivation = c(7.5, 6.8, 8.2, 7.8, 6.5, 8.5, 7.2, 7.9, 8.8, 6.9, 7.6, 8.3, 6.7, 7.7, 8.9),
anxiety = c(3.2, 4.1, 2.5, 3.0, 4.5, 2.3, 3.8, 2.9, 2.1, 4.2, 3.3, 2.6, 4.3, 3.1, 2.0),
self_efficacy = c(8.1, 7.2, 8.9, 8.3, 7.0, 9.0, 7.8, 8.5, 9.2, 7.3, 8.0, 8.7, 7.1, 8.2, 9.3)
)
# Define variable sets
# Set 1: Academic Performance (math, reading, writing)
X <- df |> select(math, reading, writing)
# Set 2: Psychological Factors (motivation, anxiety, self_efficacy)
Y <- df |> select(motivation, anxiety, self_efficacy)
# Perform Canonical Correlation Analysis
cc_result <- cc(as.matrix(X), as.matrix(Y))
# Display canonical correlations
print("Canonical Correlations:")
print(cc_result$cor)
# Canonical coefficients (standardized)
print("Canonical Coefficients for Set 1 (X):")
print(cc_result$xcoef)
print("Canonical Coefficients for Set 2 (Y):")
print(cc_result$ycoef)
# Calculate canonical loadings (structure coefficients)
X_std <- scale(X)
Y_std <- scale(Y)
# Canonical variates
U <- X_std %*% cc_result$xcoef
V <- Y_std %*% cc_result$ycoef
# Loadings (correlations between original variables and canonical variates)
loadings_X <- cor(X_std, U)
loadings_Y <- cor(Y_std, V)
print("Canonical Loadings for Set 1 (X):")
print(loadings_X)
print("Canonical Loadings for Set 2 (Y):")
print(loadings_Y)
# Test significance using Wilks' Lambda
n <- nrow(df)
p <- ncol(X)
q <- ncol(Y)
# Use CCP package for significance testing
sig_test <- p.asym(cc_result$cor, n, p, q)
print("Significance Tests:")
print(sig_test)
# Redundancy analysis
redundancy_X <- colSums(loadings_X^2) / p * cc_result$cor^2
redundancy_Y <- colSums(loadings_Y^2) / q * cc_result$cor^2
print("Redundancy (variance in X explained by Y's canonical variates):")
print(redundancy_X)
print("Redundancy (variance in Y explained by X's canonical variates):")
print(redundancy_Y)Python Code (scikit-learn)
Python
import pandas as pd
import numpy as np
from sklearn.cross_decomposition import CCA
from scipy import stats
import matplotlib.pyplot as plt
# Academic performance and psychological factors data
data = pd.DataFrame({
'math': [85, 78, 92, 88, 76, 90, 82, 87, 91, 79, 84, 89, 77, 86, 93],
'reading': [88, 82, 90, 85, 79, 92, 84, 89, 93, 81, 86, 91, 80, 87, 95],
'writing': [82, 76, 88, 84, 74, 89, 80, 85, 90, 77, 83, 87, 75, 84, 91],
'motivation': [7.5, 6.8, 8.2, 7.8, 6.5, 8.5, 7.2, 7.9, 8.8, 6.9, 7.6, 8.3, 6.7, 7.7, 8.9],
'anxiety': [3.2, 4.1, 2.5, 3.0, 4.5, 2.3, 3.8, 2.9, 2.1, 4.2, 3.3, 2.6, 4.3, 3.1, 2.0],
'self_efficacy': [8.1, 7.2, 8.9, 8.3, 7.0, 9.0, 7.8, 8.5, 9.2, 7.3, 8.0, 8.7, 7.1, 8.2, 9.3]
})
# Define variable sets
# Set 1: Academic Performance
X = data[['math', 'reading', 'writing']].values
# Set 2: Psychological Factors
Y = data[['motivation', 'anxiety', 'self_efficacy']].values
# Standardize the data
from sklearn.preprocessing import StandardScaler
scaler_X = StandardScaler()
scaler_Y = StandardScaler()
X_std = scaler_X.fit_transform(X)
Y_std = scaler_Y.fit_transform(Y)
# Perform Canonical Correlation Analysis
n_components = min(X.shape[1], Y.shape[1])
cca = CCA(n_components=n_components)
cca.fit(X_std, Y_std)
# Transform to canonical variates
X_c, Y_c = cca.transform(X_std, Y_std)
# Calculate canonical correlations
canonical_correlations = [np.corrcoef(X_c[:, i], Y_c[:, i])[0, 1]
for i in range(n_components)]
print("Canonical Correlations:")
print(canonical_correlations)
# Canonical coefficients
print("\nCanonical Coefficients for Set 1 (X):")
print(cca.x_weights_)
print("\nCanonical Coefficients for Set 2 (Y):")
print(cca.y_weights_)
# Calculate canonical loadings (structure coefficients)
loadings_X = np.corrcoef(X_std.T, X_c.T)[:X.shape[1], X.shape[1]:]
loadings_Y = np.corrcoef(Y_std.T, Y_c.T)[:Y.shape[1], Y.shape[1]:]
print("\nCanonical Loadings for Set 1 (X):")
print(loadings_X)
print("\nCanonical Loadings for Set 2 (Y):")
print(loadings_Y)
# Wilks' Lambda test for significance
def wilks_lambda_test(canonical_corrs, n, p, q):
"""
Test significance of canonical correlations using Wilks' Lambda
n: sample size
p: number of variables in set 1
q: number of variables in set 2
"""
m = min(p, q)
lambda_vals = []
for k in range(m):
# Product of (1 - r^2) for remaining correlations
lambda_k = np.prod([1 - r**2 for r in canonical_corrs[k:]])
lambda_vals.append(lambda_k)
# Chi-square approximation
df = (p - k) * (q - k)
chi_sq = -(n - 1 - (p + q + 1) / 2) * np.log(lambda_k)
p_value = 1 - stats.chi2.cdf(chi_sq, df)
print(f"\nFunction {k+1}:")
print(f" Wilks' Lambda: {lambda_k:.4f}")
print(f" Chi-square: {chi_sq:.4f}")
print(f" df: {df}")
print(f" p-value: {p_value:.4f}")
print("\nSignificance Tests:")
wilks_lambda_test(canonical_correlations, len(data), X.shape[1], Y.shape[1])