Quick Answer: Canonical Correlation Analysis Result

The main SPSS and Python report tested the relationship between two variable sets in the student performance data. Set X contained background and study predictors: age, Medu, Fedu, traveltime, studytime and failures. Set Y contained academic grade outcomes: G1, G2 and G3. The analysis used N = 649 complete cases and produced three canonical functions because the smaller set contained three outcome variables.

The first canonical function was the main result. Its canonical correlation was r_c = .498, with squared canonical correlation r_c² = .248. This means the first pair of canonical variates shared about 24.8% variance. The full Wilks Lambda dimension test was significant, Wilks Lambda = .7287, F ≈ 11.91, p < .001. The second function was much weaker but still passed the sequential dimension test, while the third function was not significant.

What Is Canonical Correlation Analysis?

Canonical Correlation Analysis, often shortened to CCA, is a multivariate method for studying the association between two groups of variables. A simple correlation coefficient compares one variable with one variable. Canonical correlation compares one set of variables with another set of variables by creating the strongest possible paired composites from both sets.

For example, a researcher may want to know whether a block of background and study variables is related to a block of academic grade variables. Instead of running many separate correlations and trying to interpret them one by one, canonical correlation forms one linear composite from the predictor set and one linear composite from the outcome set. The correlation between those two composites is the first canonical correlation.

After the first canonical function is extracted, the method can create additional functions that are independent of the earlier functions. Each later function explains a leftover pattern. In most practical reports, the first function is the most important, and later functions are interpreted only when they are statistically significant and practically meaningful.

Canonical correlation is connected to multiple linear regression, multicollinearity checks, variance inflation factor, tolerance statistic and effect size reporting because it depends on correlations, linear combinations and shared variance.

Canonical Correlation Formula

Canonical correlation begins with two variable sets. Let X be the first set and Y be the second set. The method creates one canonical variate from each set:

The first canonical correlation is the maximum possible correlation between U and V:

In matrix form, the canonical roots are usually obtained from the eigenvalues of this matrix:

The eigenvalues of M are squared canonical correlations. Taking the square root gives each canonical correlation:

Null and Alternative Hypotheses for Canonical Correlation

The overall canonical correlation test asks whether the two variable sets are independent. Sequential dimension tests then ask whether remaining canonical functions still explain a significant relationship after earlier functions are removed.

Dataset and Variable Sets Used

The main SPSS and Python analysis used a student performance dataset with 649 complete cases. Set X contained six background and study predictors. Set Y contained three academic grades. The goal was to test whether the multivariate pattern of age, parental education, travel time, study time and previous failures was related to the multivariate pattern of grade outcomes.

Before running canonical correlation, inspect descriptive statistics, histograms, box plots, skewness and kurtosis, Mahalanobis distance and multicollinearity. Canonical correlation can become unstable when variables inside a set are highly redundant.

SPSS Output Interpretation for Canonical Correlation

The SPSS output verifies the main canonical correlation analysis. SPSS accepted 649 cases and rejected none for missing data. The dependent block contained G1, G2 and G3, while the covariate block contained age, Medu, Fedu, traveltime, studytime and failures. The overall multivariate tests were significant, including Wilks Lambda, which shows that the two variable sets are not independent.

Open the SPSS Canonical Correlation output PDF

SPSS Canonical Roots and Dimension Tests

SPSS Structure Loadings and Redundancy

Python Chart-by-Chart Interpretation

The Python charts show the canonical correlation workflow from root strength to case-level canonical variate scores, structure loadings, redundancy and the input correlation matrix. The Python report uses Set X = age, Medu, Fedu, traveltime, studytime and failures, and Set Y = G1, G2 and G3.

Python Chart 1: Canonical Correlations by Function

This chart makes the main result easy to see. Function 1 has the largest canonical correlation, about .498. Function 2 drops sharply to about .150, and Function 3 is only about .090. The first bar is therefore the only practically meaningful root in the Python/SPSS setup, even though the dimension test for Function 2 is statistically detectable.

The visual gap between Function 1 and the later functions is important. It shows that the main multivariate relationship is concentrated in the first canonical variate pair. The later functions add very little extra shared variance, so a clear report should emphasize Function 1 and treat the remaining roots as secondary checks.

Python Chart 2: First Canonical Variate Scatterplot

The scatterplot shows each student as a point using the first Set X canonical variate score on the horizontal axis and the first Set Y canonical variate score on the vertical axis. The fitted line slopes upward, matching the displayed r = .498. The cloud is not perfectly tight, but it has a clear positive direction.

This chart validates the first canonical correlation in a way that a table cannot. Most points cluster near the center, while a smaller group extends toward higher canonical scores. The upward trend means that students with higher values on the X-side composite tend to have higher values on the Y-side composite after the weights are applied. The moderate spread around the line explains why the correlation is moderate rather than extremely high.

Python Chart 3: Set X Structure Loadings

The Set X loading heatmap shows that failures is the dominant Function 1 loading at about .82. studytime and Medu both load around -.53, while Fedu is about -.46. Age and travel time have smaller positive Function 1 loadings, around .29 to .30. This pattern means the first X-side composite is not a simple “all good factors” score; it is a contrast where previous failures sit opposite study time and parental education.

Function 2 is dominated by age with a large negative loading near -.85, while Function 3 is mainly connected with traveltime around .72 and Medu around -.62. Because the later canonical correlations are small, these later loading patterns should not be overinterpreted. The strongest and most useful story remains Function 1.

Python Chart 4: Set Y Structure Loadings

The Set Y heatmap is very clear: G1, G2 and G3 all load strongly and negatively on Function 1, with values around -.96, -.95 and -.94. Because signs in canonical correlation are arbitrary, the negative sign is not the main message. The magnitude is the main message. Function 1 represents a strong general academic-performance composite.

The later functions do not show the same clean grade pattern. Function 2 has smaller values, and Function 3 is weak. This confirms that the first canonical function is the meaningful one for the grade set. In plain words, the Y-side of the first function is “overall grades.”

Python Chart 5: Redundancy Summary

The redundancy chart gives the practical interpretation that raw canonical correlations can hide. Function 1 explains about .225 of the Y set through the X set, while the X set explained by the Y set is much smaller at about .067. In percentage language, the predictor set explains about 22.5% of the grade-set variance through Function 1, while the grade set explains about 6.7% of the predictor-set variance.

Functions 2 and 3 are almost flat in the chart. This tells the reader that the later roots do not add much practical variance explanation. The best report should therefore combine the root chart and redundancy chart: Function 1 is statistically significant and meaningful, but the practical variance explained is more modest than the canonical loading table might initially suggest.

Python Chart 6: Input Correlation Matrix

The input matrix explains why the first canonical function is mostly a grade-performance relationship. G1 and G2 correlate about .86, G2 and G3 correlate about .92, and G1 and G3 correlate about .83. These strong grade-to-grade correlations make the Y set highly coherent.

The predictor-side story is more mixed. failures correlates negatively with the grade variables, around -.38 to -.39. studytime correlates positively with grades around .24 to .26. Mother and father education also show positive links with the grade variables. This matrix supports the loading interpretation: failures, studytime and parental education provide the main cross-set connection to academic performance.

R Chart-by-Chart Validation

The R report uses an expanded validation setup. Set 1 contains G1, G2, G3 and absences. Set 2 contains background, study, family, social, alcohol and health variables. Because the R variable sets differ from the Python/SPSS setup, the exact numeric roots are different. The interpretation logic is the same: identify the strongest roots, inspect the scatterplot, read structure loadings and then check redundancy.

R Chart 1: Colorful Root Strength by Function

The R root chart shows a larger first function than the Python/SPSS setup because the variable blocks are defined differently. Function 1 is about .534, Function 2 is about .243, Function 3 is about .161, and Function 4 is about .121. The sharp decline after Function 1 again shows that most of the useful relationship sits in the first root.

The R dimension tests support the first two roots but not the last two. However, the first root is still the clearest practical story. The chart’s descending pattern is exactly what a good canonical correlation report should show: interpret the first function strongly, mention the second cautiously if significant, and avoid overexplaining weak residual roots.

R Chart 2: Colorful Function 1 Scores

The R scatterplot shows the first canonical variate scores from Set 1 and Set 2. The fitted line slopes upward, and the chart states that the case scores correlate at approximately r = .534. This confirms a moderate positive multivariate relationship in the expanded setup.

The points are spread around the line, so the relationship is not deterministic. The plot shows a general upward tendency rather than perfect separation. That is why the root is meaningful but still moderate. It supports the same reporting style as the Python chart: state the root, then explain the variables driving it through structure loadings.

R Chart 3: Colorful Structure Loadings Heatmap

The R loading heatmap shows that Function 1 is strongly tied to grades on the Set 1 side: G1 = -.93, G2 = -.93 and G3 = -.94. Absences is much smaller at about .33. On the Set 2 side, failures has a strong loading around .77, while studytime is around -.52, Medu around -.47, and alcohol variables show positive moderate loadings.

This chart gives the substantive meaning of the first function. It is mainly an academic-performance dimension linked with prior failures, study time, parental education and some lifestyle indicators. The later functions show different patterns, such as absences loading strongly on Function 2 and family relationship loading strongly on Function 4, but those functions have weaker roots and should be interpreted cautiously.

R Chart 4: Colorful Function 1 Standardized Coefficients

The R coefficient chart shows which variables receive the largest standardized weights in Function 1. failures has a large positive weight, while G3 and G1 have large negative weights. Studytime and Medu also appear with negative weights. This coefficient pattern is consistent with the loading pattern: academic performance is contrasted with previous failures and related background-study characteristics.

Coefficients should be read carefully because they can change when variables are highly correlated. In this data, G1, G2 and G3 are strongly related to one another, so structure loadings are often more stable for interpretation than raw weights. The coefficient chart is useful, but it should not replace the loading heatmap.

R Chart 5: Colorful Redundancy by Function

The R redundancy chart shows that Function 1 carries almost all of the practical explanation. Set 1 redundancy for Function 1 is about .194, while Set 2 redundancy is about .044. The later functions are very small, with Function 2 only around .012 for Set 1 and .006 for Set 2.

This chart is the practical check on the root chart. A root can be statistically significant but still explain little practical variance in the opposite set. The R redundancy plot confirms that the article should focus on Function 1 and avoid presenting the weaker roots as equally important.

Excel Results Explained

The Excel workbook gives a fully worked formula-based version of canonical correlation. It uses a compact 2 by 2 setup so the matrix method can be shown transparently using Excel formulas such as CORREL, MINVERSE and MMULT. The Excel variable sets are different from the main Python/SPSS report: X set = G1 and G2, while Y set = G3 and absences.

Excel Worked Example: G1 and G2 Against G3 and Absences

The Excel result is very strong because G1 and G2 are highly connected to G3. In the workbook, G1 correlates with G3 at about .826, G2 correlates with G3 at about .919, and G1 correlates with G2 at about .865. Absences has weaker negative relationships with the grade variables, including about -.147 with G1, -.125 with G2 and -.091 with G3. This means the first Excel canonical variate is mainly a grade-to-grade relationship, not an absence-driven relationship.

The workbook also shows the exact matrix path. First, Excel builds Rxx, Ryy and Rxy using CORREL. Then it creates M = Rxx^-1 Rxy Ryy^-1 Ryx using MINVERSE and MMULT. Because the matrix is 2 by 2, the workbook uses the quadratic eigenvalue formula to calculate the two roots. Function 1 is strong and significant; Function 2 is weak and not significant.

Excel Formula Steps Used in the Workbook

SPSS, R, Python and Excel Workflows for Canonical Correlation

The same canonical-correlation logic can be reproduced in SPSS, R, Python and Excel. The key is to define the two variable sets first. If the variable sets change, the canonical correlations, roots, redundancy and loadings will also change.

Code Blocks for Canonical Correlation

SPSS Syntax for Canonical Correlation

* Canonical Correlation Analysis in SPSS.
* Set X covariates: age Medu Fedu traveltime studytime failures.
* Set Y dependent variables: G1 G2 G3.

OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Canonical_Correlation_Output.

TITLE "Canonical Correlation Analysis".

MANOVA G1 G2 G3 WITH age Medu Fedu traveltime studytime failures
  /DISCRIM ALL ALPHA(1)
  /PRINT SIGNIF(EIGEN DIMENR) PARAM(ESTIM)
  /ANALYSIS G1 G2 G3 WITH age Medu Fedu traveltime studytime failures
  /DESIGN.

CORRELATIONS
  /VARIABLES=age Medu Fedu traveltime studytime failures G1 G2 G3
  /PRINT=TWOTAIL
  /MISSING=PAIRWISE.

DESCRIPTIVES VARIABLES=age Medu Fedu traveltime studytime failures G1 G2 G3
  /STATISTICS=MEAN STDDEV MIN MAX.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Canonical-Correlation-SPSS-Output.pdf'.

Python Code for Canonical Correlation

import pandas as pd
import numpy as np
from scipy.stats import chi2

# Load data
df = pd.read_csv("dataset.csv")

x_vars = ["age", "Medu", "Fedu", "traveltime", "studytime", "failures"]
y_vars = ["G1", "G2", "G3"]

work = df[x_vars + y_vars].apply(pd.to_numeric, errors="coerce").dropna()
X = work[x_vars].to_numpy()
Y = work[y_vars].to_numpy()

# Standardize variables
Xz = (X - X.mean(axis=0)) / X.std(axis=0, ddof=1)
Yz = (Y - Y.mean(axis=0)) / Y.std(axis=0, ddof=1)

# Correlation matrices
Z = np.column_stack([Xz, Yz])
R = np.corrcoef(Z, rowvar=False)
p = len(x_vars)
q = len(y_vars)
Rxx = R[:p, :p]
Ryy = R[p:, p:]
Rxy = R[:p, p:]
Ryx = Rxy.T

# Canonical root matrix and roots
M = np.linalg.inv(Rxx) @ Rxy @ np.linalg.inv(Ryy) @ Ryx
eigenvalues = np.real(np.linalg.eigvals(M))
eigenvalues = np.sort(eigenvalues)[::-1]
eigenvalues = eigenvalues[:min(p, q)]
canonical_r = np.sqrt(np.maximum(eigenvalues, 0))

print("Canonical correlations:", canonical_r)
print("Squared canonical correlations:", canonical_r ** 2)

# Wilks Lambda sequential tests using Bartlett approximation
n = len(work)
for i in range(len(eigenvalues)):
    remaining = eigenvalues[i:]
    wilks = np.prod(1 - remaining)
    df_test = (p - i) * (q - i)
    chi_square = -(n - 1 - (p + q + 1) / 2) * np.log(wilks)
    p_value = chi2.sf(chi_square, df_test)
    print(i + 1, "to", len(eigenvalues), "Wilks", wilks,
          "Chi-square", chi_square, "df", df_test, "p", p_value)

R Code for Canonical Correlation

# Canonical Correlation Analysis in R

df <- read.csv("dataset.csv")

x_vars <- c("age", "Medu", "Fedu", "traveltime", "studytime", "failures")
y_vars <- c("G1", "G2", "G3")

work <- na.omit(df[, c(x_vars, y_vars)])
work[] <- lapply(work, as.numeric)

X <- scale(work[, x_vars])
Y <- scale(work[, y_vars])

cca <- cancor(X, Y)
print(cca$cor)
print(cca$xcoef)
print(cca$ycoef)

# Canonical scores
U <- as.matrix(X) %*% cca$xcoef
V <- as.matrix(Y) %*% cca$ycoef

# Structure loadings
x_loadings <- cor(X, U)
y_loadings <- cor(Y, V)
print(x_loadings)
print(y_loadings)

# Redundancy approximation
r2 <- cca$cor^2
x_variance_extracted <- colMeans(x_loadings^2)
y_variance_extracted <- colMeans(y_loadings^2)
x_redundancy <- x_variance_extracted * r2
y_redundancy <- y_variance_extracted * r2
print(data.frame(r = cca$cor, r2 = r2,
                 x_variance_extracted = x_variance_extracted,
                 y_variance_extracted = y_variance_extracted,
                 x_redundancy = x_redundancy,
                 y_redundancy = y_redundancy))

Excel Formulas for Canonical Correlation

Step 1: Build the within-set correlation matrices.
=CORREL(X1_range, X2_range)
=CORREL(Y1_range, Y2_range)

Step 2: Build the cross-set correlation matrix.
=CORREL(X1_range, Y1_range)
=CORREL(X1_range, Y2_range)
=CORREL(X2_range, Y1_range)
=CORREL(X2_range, Y2_range)

Step 3: Create the canonical root matrix.
=MMULT(MMULT(MINVERSE(Rxx), Rxy), MMULT(MINVERSE(Ryy), TRANSPOSE(Rxy)))

Step 4: For a 2 by 2 matrix M, calculate eigenvalues.
=((M11+M22)+SQRT((M11-M22)^2+4*M12*M21))/2
=((M11+M22)-SQRT((M11-M22)^2+4*M12*M21))/2

Step 5: Convert eigenvalues into canonical correlations.
=SQRT(eigenvalue)

Step 6: Calculate Wilks Lambda.
=PRODUCT(1-lambda_i)

Step 7: Calculate p-value from Bartlett chi-square approximation.
=CHISQ.DIST.RT(chi_square, df)

APA Reporting Wording for Canonical Correlation

When reporting Canonical Correlation Analysis, describe the two variable sets, sample size, number of functions, canonical correlations, Wilks Lambda dimension tests, main structure loadings, redundancy results and the practical interpretation. Also state clearly which functions are interpreted and which are not.

APA-Style Full Report

Short APA-Style Version

Excel Worked-Example Wording

Common Mistakes in Canonical Correlation Interpretation

When to Use Canonical Correlation

Use Canonical Correlation Analysis when you have two meaningful sets of continuous or approximately continuous variables and you want to understand the multivariate relationship between those sets. It is useful when many separate correlations would be confusing and when the research question is naturally set-based.

Good examples include study habits and grade outcomes, psychological scales and health measures, customer behavior metrics and business performance indicators, socioeconomic variables and educational outcomes, or multiple predictors and multiple responses. If you have one outcome and many predictors, multiple linear regression may be better. If you have group differences across multiple outcomes, MANOVA may be more appropriate. If you want to reduce variables within one set, principal component analysis may be the better first tool.

Downloads and Resources for Canonical Correlation

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