Quick Answer: Multicollinearity Check Result

The worked regression model predicts G3 final grade using multiple predictors. The main Multicollinearity Check uses pairwise correlations, VIF, tolerance, condition index, and variance decomposition proportions. The strongest predictor relationship is between G1 and G2, with a correlation of approximately .865. Their VIF values are also the largest: G1 VIF ≈ 4.195 and G2 VIF ≈ 4.204. Their tolerance values are approximately .238, which is low but still above the common strict cutoff of .10.

The overall decision is that the model shows moderate multicollinearity concern mainly between G1 and G2, but not severe multicollinearity under the common VIF > 10 or tolerance < .10 rules. However, the condition index and variance decomposition output should still be reviewed because the condition index can reveal near-dependency patterns that VIF alone may not fully explain.

What Is Multicollinearity?

Multicollinearity happens when two or more predictor variables in a regression model contain overlapping information. In simple words, predictors are too strongly related to each other. This makes it harder for the regression model to separate the unique effect of each predictor.

For example, G1 and G2 are both previous grade measures. It is expected that students who performed well in G1 also tend to perform well in G2. Because the two predictors are highly related, their unique regression coefficients can become less stable. The model can still predict G3 very well, but interpretation of each individual grade predictor becomes more delicate.

Multicollinearity Check is part of regression diagnostics. It should be used with other diagnostic tools such as the Ramsey RESET test, Goldfeld-Quandt test, Q-Q plot normality check, P-P plot normality check, descriptive statistics, and effect size.

Multicollinearity Check Formula: VIF and Tolerance

The two most common numeric indicators in a Multicollinearity Check are VIF and tolerance. VIF stands for Variance Inflation Factor. Tolerance is the inverse of VIF.

Here, R_j² is the R-squared value obtained when predictor j is regressed on all the other predictors. If a predictor can be strongly predicted by the other predictors, its R-squared is high and its VIF becomes large.

A low tolerance means the predictor shares a large amount of variance with the other predictors. A high VIF means the standard error of the coefficient is inflated because of predictor overlap.

VIF, Tolerance and Condition Index Thresholds

There is no single universal cutoff for Multicollinearity Check, but applied regression reports commonly use practical rules. The strictness depends on the field, sample size, model purpose and whether the goal is prediction or coefficient interpretation.

Diagnostic Hypotheses for Multicollinearity Check

Multicollinearity Check is usually a diagnostic decision rather than a single traditional p-value test. However, a hypothesis-style statement makes the reporting clearer.

Dataset and Regression Variables Used

The worked Multicollinearity Check uses the student performance dataset. The dependent variable is G3 final grade. The predictors include previous grades, study habits, failures, absences, age, parental education, travel time, family relationship, free time, going out and health. The goal is to check whether these predictors overlap too strongly before interpreting the regression coefficients.

Before checking multicollinearity, it is useful to understand each variable with descriptive statistics, frequency distribution, histogram interpretation, box plot interpretation, and five-number summary.

SPSS Output Interpretation for Multicollinearity Check

The SPSS output PDF verifies the Multicollinearity Check using collinearity statistics and diagnostics. SPSS reports tolerance and VIF in the coefficients table, and condition index with variance proportions in the collinearity diagnostics table.

SPSS Model Summary

SPSS VIF and Tolerance Table

SPSS Pairwise Correlation Findings

The top pairwise correlation in the SPSS-supported workflow is G1 with G2 = .865. This is the strongest predictor-predictor relationship and explains why both G1 and G2 have the largest VIF values. Another notable relationship is Medu with Fedu = .647, which explains their moderate but acceptable VIF values.

SPSS Condition Index and Variance Decomposition Interpretation

SPSS collinearity diagnostics can show a high condition index even when VIF values do not exceed 10. In the related output, the maximum condition index reached about 74.600. A high condition index should be interpreted with the variance decomposition proportions. If two or more predictors have high variance proportions on the same high-index dimension, that dimension may indicate a dependency structure.

SPSS Final Decision

Python Chart-by-Chart Interpretation

The Python charts provide the complete visual workflow for a Multicollinearity Check. They show predictor correlations, VIF values, tolerance values, condition index, top pairwise correlations, VIF-tolerance relationship and variance decomposition proportions.

Python Chart 1: Predictor Correlation Heatmap

This heatmap gives the first visual warning for multicollinearity. Strong color intensity indicates stronger predictor-predictor relationships. The most important relationship is between G1 and G2, which are strongly correlated because both measure previous academic performance. This pair is the most likely source of coefficient interpretation difficulty.

The heatmap is useful because it shows the structure of predictor overlap before any regression-specific statistics are calculated. However, pairwise correlations are only the first step. A predictor may have acceptable pairwise correlations but still be collinear with a combination of other predictors. Therefore, the heatmap must be followed by VIF, tolerance and condition index checks.

Python Chart 2: VIF by Predictor

The VIF chart is the main numeric diagnostic for multicollinearity. G1 and G2 have the largest VIF values, around 4.2. These values are below the common severe cutoff of 10 and below the more cautious cutoff of 5, but they are clearly higher than the other predictors. This means G1 and G2 share substantial information.

The correct interpretation is moderate collinearity, not severe collinearity. The model can remain useful, especially for prediction, but individual coefficient interpretation for G1 and G2 should be careful because the model must separate two closely related prior-grade predictors.

Python Chart 3: Tolerance by Predictor

The tolerance chart is the inverse view of the VIF chart. Lower tolerance means greater predictor overlap. G1 and G2 have the lowest tolerance values, approximately .238. This means about 23.8% of their variance remains unique after accounting for the other predictors, while the rest is shared with the predictor set.

Since the values are above .10, the result does not meet the common severe collinearity rule. However, the low relative tolerance compared with other predictors supports the same conclusion as the VIF chart: G1 and G2 deserve special attention in reporting.

Python Chart 4: Condition Index Plot

The condition index plot checks whether the predictor matrix has dimensions that are nearly dependent. A high condition index can indicate that some predictors form a near-linear dependency. This diagnostic is helpful because it can reveal collinearity patterns that are not obvious from VIF alone.

If the condition index is high, the next step is to examine variance decomposition proportions. A high condition index is most concerning when two or more predictors have high variance proportions on the same high-index dimension. Therefore, this chart should be interpreted together with the variance decomposition heatmap.

Python Chart 5: Top Pairwise Correlations

This chart ranks the strongest predictor-predictor correlations. It confirms that G1 and G2 are the dominant pairwise relationship. It also shows other moderate relationships, such as Medu and Fedu, but these do not produce severe VIF values.

The chart is useful for explaining why some predictors have higher VIF values than others. It also helps readers understand whether the collinearity is caused by one obvious predictor pair or by a broader pattern across many variables.

Python Chart 6: VIF vs Tolerance

This chart shows the mathematical relationship between VIF and tolerance. As VIF increases, tolerance decreases. G1 and G2 appear in the higher-VIF, lower-tolerance region, which confirms that both diagnostics identify the same predictors as the main collinearity concern.

This chart is helpful for teaching because many students treat VIF and tolerance as unrelated. They are actually two versions of the same diagnostic relationship. Reporting both makes the result easier to understand, especially in SPSS output where both columns appear together.

Python Chart 7: Variance Decomposition Heatmap

The variance decomposition heatmap completes the condition-index diagnosis. It shows which predictors load heavily on the same diagnostic dimensions. A high condition index becomes more concerning when multiple predictors have large variance proportions on that same dimension.

This chart is especially important for advanced reporting. VIF tells which individual predictors are inflated. Condition index and variance decomposition show whether the predictor set contains near-dependency structures. If G1 and G2 share high variance proportions on a high-index dimension, that supports the conclusion of moderate collinearity involving previous-grade predictors.

R Chart-by-Chart Validation

The R charts validate the Python and SPSS Multicollinearity Check using a separate workflow. The same diagnostic pattern appears again: G1 and G2 dominate the correlation and VIF results, tolerance is lowest for the same variables, and the condition index plus variance decomposition help confirm the structure.

R Chart 1: Predictor Correlation Heatmap

The R heatmap confirms the Python correlation pattern. G1 and G2 are the strongest predictor pair. This validates the first stage of the multicollinearity diagnosis and supports the need for VIF and tolerance checks.

R Chart 2: VIF by Predictor

The R VIF chart validates that G1 and G2 have the highest VIF values while the remaining predictors are much lower. This supports the conclusion that collinearity is concentrated in the previous-grade predictors rather than spread severely across the entire model.

R Chart 3: Tolerance by Predictor

The R tolerance chart confirms the inverse pattern of VIF. G1 and G2 have the lowest tolerance values, but not below the severe .10 cutoff. This reinforces the interpretation of moderate rather than severe multicollinearity.

R Chart 4: Condition Index Plot

The R condition index chart validates the condition-diagnostics stage. It shows whether the predictor matrix has high-index dimensions that deserve review. This is important because VIF alone may not fully explain dependency patterns in a multi-predictor model.

R Chart 5: Top Pairwise Correlations

The R top-correlation chart confirms that the strongest pairwise relationship is between G1 and G2. This chart is useful for communicating the main source of multicollinearity in a simple visual form.

R Chart 6: VIF vs Tolerance

The R VIF-versus-tolerance chart confirms the same inverse diagnostic relationship shown by Python. It helps readers understand that high VIF and low tolerance are two views of the same collinearity problem.

R Chart 7: Variance Decomposition Heatmap

The R variance decomposition heatmap validates the advanced collinearity interpretation. It shows which predictors contribute to high condition-index dimensions. This chart is the final confirmation step when reporting a complete multicollinearity diagnosis.

SPSS, R, Python and Excel Workflows for Multicollinearity Check

A complete Multicollinearity Check can be completed in SPSS, R, Python and Excel. The workflow is similar in all tools: fit the regression model, calculate pairwise correlations, compute VIF and tolerance, examine condition index and variance decomposition, then decide whether predictors should be retained, combined, transformed or removed.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Multicollinearity Check

SPSS Syntax for Multicollinearity Check

* Multicollinearity Check in SPSS.
* Dependent variable: G3.
* Predictors: G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health.

TITLE "Multicollinearity Check: VIF, Tolerance and Collinearity Diagnostics".

CORRELATIONS
  /VARIABLES=G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /STATISTICS COEFF OUTS R ANOVA COLLIN TOL CHANGE
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Multicollinearity-Check-SPSS-Output.pdf".

Python Code for Multicollinearity Check

import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor

df = pd.read_csv("dataset.csv")

dependent = "G3"
predictors = [
    "G1", "G2", "studytime", "failures", "absences", "age",
    "Medu", "Fedu", "traveltime", "famrel", "freetime", "goout", "health"
]

model_data = df[[dependent] + predictors].apply(pd.to_numeric, errors="coerce").dropna()

X = model_data[predictors]
X_const = sm.add_constant(X)

# Correlation matrix
corr_matrix = X.corr()
print(corr_matrix)

# VIF and tolerance
rows = []
for i, col in enumerate(X_const.columns):
    if col == "const":
        continue
    vif_value = variance_inflation_factor(X_const.values, i)
    rows.append({
        "predictor": col,
        "VIF": vif_value,
        "Tolerance": 1 / vif_value
    })

vif_table = pd.DataFrame(rows).sort_values("VIF", ascending=False)
print(vif_table)

# Fit regression model
model = sm.OLS(model_data[dependent], X_const).fit()
print(model.summary())

# Condition index
X_scaled = (X - X.mean()) / X.std(ddof=1)
xtx = np.dot(X_scaled.T, X_scaled)
eigenvalues, eigenvectors = np.linalg.eig(xtx)
eigenvalues = np.real(eigenvalues)

condition_index = np.sqrt(eigenvalues.max() / eigenvalues)
condition_table = pd.DataFrame({
    "dimension": np.arange(1, len(condition_index) + 1),
    "eigenvalue": eigenvalues,
    "condition_index": condition_index
}).sort_values("condition_index")

print(condition_table)

R Code for Multicollinearity Check

# Multicollinearity Check in R

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

predictors <- c(
  "G1", "G2", "studytime", "failures", "absences", "age",
  "Medu", "Fedu", "traveltime", "famrel", "freetime", "goout", "health"
)

vars_needed <- c("G3", predictors)
df_model <- df[vars_needed]
df_model[] <- lapply(df_model, as.numeric)
df_model <- na.omit(df_model)

model <- lm(G3 ~ G1 + G2 + studytime + failures + absences + age +
              Medu + Fedu + traveltime + famrel + freetime + goout + health,
            data = df_model)

summary(model)

# install.packages("car")
library(car)

vif_values <- vif(model)
tolerance_values <- 1 / vif_values

vif_table <- data.frame(
  predictor = names(vif_values),
  VIF = as.numeric(vif_values),
  Tolerance = as.numeric(tolerance_values)
)

print(vif_table[order(-vif_table$VIF), ])

# Correlation matrix
corr_matrix <- cor(df_model[predictors], use = "pairwise.complete.obs")
print(corr_matrix)

# Condition index
X <- scale(df_model[predictors])
xtx <- t(X) %*% X
eig <- eigen(xtx)

condition_index <- sqrt(max(eig$values) / eig$values)

condition_table <- data.frame(
  dimension = 1:length(condition_index),
  eigenvalue = eig$values,
  condition_index = condition_index
)

print(condition_table)

Excel Formula Block for Multicollinearity Check

Step 1:
Create a correlation matrix:
Data > Data Analysis > Correlation

Step 2:
For each predictor, run an auxiliary regression:
Example: Regress G1 on all other predictors.

Step 3:
Record the R-squared from each auxiliary regression.

Step 4:
Calculate VIF:
=1/(1-R_squared)

Step 5:
Calculate tolerance:
=1/VIF

Step 6:
Flag severe VIF:
=IF(VIF>10,"Severe multicollinearity","Acceptable")

Step 7:
Flag low tolerance:
=IF(Tolerance<0.1,"Severe low tolerance","Acceptable")

Step 8:
Interpret:
High VIF and low tolerance indicate predictor overlap.
Review pairwise correlations and theory before removing variables.

APA Reporting Wording for Multicollinearity Check

APA reporting for a Multicollinearity Check should include the diagnostic method, the cutoff rule, the highest VIF, the lowest tolerance, and whether any predictor showed problematic overlap. If condition index and variance decomposition were reviewed, mention them as supporting diagnostics.

APA-Style Full Report

Short APA-Style Version

If Severe Multicollinearity Is Found

Common Mistakes in Multicollinearity Check

When to Use Multicollinearity Check

Use a Multicollinearity Check whenever you fit a multiple regression model with more than one predictor. It is especially important when predictors measure similar concepts, when coefficients look unstable, when signs are unexpected, when standard errors are large, or when a model includes many related variables.

Use this guide with related regression diagnostics such as the Ramsey RESET test, Goldfeld-Quandt test, Levene test, Brown-Forsythe test, Q-Q plot normality check, and P-P plot normality check.

Downloads and Resources for Multicollinearity Check

The SPSS output PDF below verifies the Multicollinearity Check workflow used in this guide. Use it as the supporting output file for VIF, tolerance, condition index, variance decomposition, predictor correlations and regression diagnostic reporting.

Related Statistical Guides

Use these related guides to connect Multicollinearity Check with regression diagnostics, assumption tests, descriptive statistics, transformations and reporting.