Quick Answer: Variance Inflation Factor Result

The verified regression model predicted G3 final grade from 13 predictors: G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout and health. The model was strong overall, with R = .923, R² = .852, adjusted R² = .849, F(13, 635) = 281.518, and p < .001. The main VIF result is that all predictors stayed below the common moderate-risk threshold of VIF = 5. The highest VIF was for G2 = 4.204, followed by G1 = 4.195. Both are elevated but still acceptable under the common VIF ≤ 5 rule.

Hypothesis-style decision: The null assumption says the regression model does not have harmful multicollinearity, meaning H₀: VIF ≤ 5 and tolerance ≥ .20 for all predictors. The alternative assumption says at least one predictor has problematic multicollinearity, meaning H₁: VIF > 5 or tolerance < .20 for at least one predictor. Since the highest VIF is 4.204 and the lowest tolerance is .238, the VIF-based decision is to retain the null assumption of acceptable multicollinearity. However, G1 and G2 should still be watched because their correlation is very strong and they are the closest predictors to the threshold.

What Is Variance Inflation Factor?

Variance Inflation Factor measures how much the variance of a regression coefficient is inflated because a predictor is correlated with other predictors in the same model. When predictors overlap strongly, the model can still predict the outcome well, but individual coefficients may become unstable. This means the sign, size, standard error, and significance of a coefficient can change when correlated predictors are added or removed.

In simple words, VIF answers this question: “How much is this predictor’s coefficient variance inflated because the predictor shares information with other predictors?” A VIF close to 1 means the predictor is mostly independent of the other predictors. A VIF around 2 to 4 shows some overlap but usually not severe. A VIF above 5 is often treated as a warning. A VIF above 10 is commonly treated as serious multicollinearity.

This guide uses a regression model predicting G3 final grade. The predictors include previous grades, study time, failures, absences, age, parental education variables, travel time, family relationship, free time, going out, and health. Because G1 and G2 are previous grade variables, they are naturally correlated. The VIF output confirms this: G1 and G2 have the highest VIF values, but they still remain below 5.

VIF Formula and Tolerance Formula

The Variance Inflation Factor for a predictor is based on how well that predictor can be predicted from all other predictors. The formula is:

Here, R²_j is the R-squared value from an auxiliary regression where predictor X_j is predicted by all the other predictors. If the other predictors explain a large amount of X_j, then R²_j is high and VIF becomes high.

Tolerance is the inverse of VIF:

So a high VIF means low tolerance. A low tolerance means the predictor has little unique variance left after accounting for the other predictors.

Null and Alternative Hypothesis for Variance Inflation Factor

VIF is a diagnostic rule rather than a p-value test, but it can still be written in a clear hypothesis-style format for reporting. This is helpful when the post needs a decision statement similar to other assumption checks.

Dataset and Regression Variables Used

The worked example uses the student performance dataset. The dependent variable is G3 final grade. The predictors are the same variables entered in the verified SPSS regression output: G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout, and health.

This post focuses on multicollinearity among predictors. For broader descriptive review, use descriptive statistics, frequency distribution, five-number summary, histogram interpretation, and box plot interpretation.

Verified SPSS Output Interpretation

The SPSS regression output gives three important pieces of evidence: overall model fit, coefficient-level results, and collinearity statistics. For VIF, the most important columns are Tolerance and VIF in the Coefficients table. The regression model used G3 as the dependent variable and entered all 13 predictors using the Enter method.

SPSS Model Summary and ANOVA

SPSS Coefficients, Tolerance and VIF

SPSS VIF Decision Summary

SPSS Collinearity Diagnostics and Condition Index

SPSS also reported a collinearity diagnostics table. The largest condition index reached 74.600, and another high dimension showed condition index = 38.634. In the dimension with condition index 38.634, G1 and G2 had large variance proportions of about .87 and .89. This supports the same conclusion shown by VIF: the clearest overlap is between G1 and G2. The very high condition index dimension also involved the constant and age, which can sometimes reflect scaling or intercept-related structure rather than a simple harmful predictor pair.

Python Chart-by-Chart Interpretation

The Python charts show the VIF pattern visually. The main message is consistent across all figures: G1 and G2 have the highest VIF values because they are strongly correlated, but all predictors remain below the common VIF = 5 threshold. Therefore, the model has acceptable multicollinearity under the selected decision rule.

Python Chart 1: VIF by Predictor

This chart ranks predictors by Variance Inflation Factor. The highest values are G2 = 4.204 and G1 = 4.195. These two bars are visibly longer than all other predictors, which means previous grade variables share a large amount of information with each other and with the remaining predictors. However, both remain below the dashed VIF = 5 threshold. That means the model does not cross the common moderate multicollinearity warning line.

Medu and Fedu are the next largest predictors, with VIF values around 1.82 and 1.76. These values are not concerning, but they make sense because mother education and father education are related. All other predictors are close to VIF = 1, meaning they have relatively high unique variance. This chart supports the final decision that multicollinearity is acceptable, while still identifying G1 and G2 as the main variables to watch.

Python Chart 2: Tolerance Cross-check

Tolerance is the inverse of VIF, so this chart verifies the same result from the opposite direction. The lowest tolerance values are for G1 and G2, both around .238. The dashed reference line at .20 marks the common warning threshold, while .10 marks a serious warning threshold. Since all predictors remain above .20, the tolerance cross-check agrees with the VIF chart.

The highest tolerance is for health = .961, followed by famrel and absences. These high tolerance values show that those predictors are not strongly explained by the other predictors. The tolerance chart is useful because it avoids a common reporting mistake: saying VIF is acceptable while ignoring tolerance. In this model, both diagnostics tell the same story: no predictor violates the selected multicollinearity rule.

Python Chart 3: Predictor Correlation Matrix

The predictor correlation heatmap explains why some predictors have higher VIF values than others. The brightest off-diagonal relationship is G1 and G2, with a correlation of about .865. This is the main reason both grade predictors have VIF values around 4.20. The next strong relationship is Medu and Fedu, with a correlation of about .647, which explains their mild VIF elevation.

The heatmap also shows moderate relationships such as freetime and goout, G1/G2 with failures, and failures with age. These correlations do not create dangerous VIF values in this model, but they provide context. A VIF table alone tells which predictor is inflated; the correlation matrix helps explain why it is inflated.

Python Chart 4: Strongest Predictor Correlations

This chart isolates the strongest pairwise predictor correlations. The strongest relationship is G1 vs G2, which is above the .80 reference line. This confirms that the previous grade predictors are highly related. The second strongest relationship is Medu vs Fedu, around .65. Other relationships are below .40 and are less likely to create severe multicollinearity by themselves.

This chart is important because VIF is not only about pairwise correlations. A predictor can have high VIF because it is explained by a combination of multiple predictors, even if no single pairwise correlation is extreme. Still, the top-pair chart is useful here because the G1-G2 relationship is strong enough to explain the highest VIF values directly. The model remains acceptable, but this chart justifies the caution statement for G1 and G2.

Python Chart 5: VIF vs Tolerance

This scatter plot shows the mathematical connection between VIF and tolerance. Predictors with low tolerance have high VIF. G1 and G2 sit in the upper-left region because their tolerance is lowest and their VIF is highest. However, they are still below the horizontal VIF = 5 warning line and above the tolerance warning line.

Most other predictors cluster near VIF = 1 and tolerance close to 1. This cluster means those predictors are not strongly redundant with the rest of the model. The chart is useful in teaching because it shows that VIF and tolerance are not separate problems; they are inverse measures of the same multicollinearity concept.

Python Chart 6: Regression Coefficients

This chart shows the regression coefficients. G2 has the largest positive coefficient, which matches the SPSS output where B = .881 and Beta = .795. G1 is also positive, with B = .146. Failures has a negative coefficient, B = -.232, meaning more prior failures are associated with lower predicted G3 after controlling for the other predictors.

The chart connects coefficient interpretation with VIF. When VIF is high, coefficient size and sign can become unstable. In this model, VIF is acceptable, but G1 and G2 are still correlated enough that their coefficients should be described as adjusted effects in the presence of the other grade variable. The coefficient chart does not replace the VIF table; it helps explain why multicollinearity matters for interpretation.

Python Chart 7: VIF Decision Summary

The decision summary chart gives the final diagnostic result: all 13 predictors fall in the acceptable VIF ≤ 5 category. No predictor falls in the moderate 5–10 category, and no predictor falls in the serious > 10 category. This chart is the easiest visual for a final report because it summarizes the VIF decision clearly.

The conclusion is not that multicollinearity is absent. The conclusion is that multicollinearity is not severe under the selected threshold. G1 and G2 still deserve caution because they are close to 5, but the model does not require immediate removal of predictors based on VIF alone.

R Chart-by-Chart Validation

The R charts validate the same VIF analysis using a separate software workflow. The R charts match the Python interpretation: G1 and G2 are the highest VIF predictors, tolerance remains above .20 for all predictors, and the final decision is acceptable multicollinearity.

R Chart 1: VIF by Predictor

The R VIF chart reproduces the same ranking as Python and SPSS. G2 and G1 have the largest VIF values, both around 4.2, while all remaining predictors are below 2. This validates the conclusion that multicollinearity is concentrated mainly in the previous grade predictors, not across the entire predictor set.

R Chart 2: Tolerance Cross-check

The R tolerance chart confirms that all predictors remain above the .20 warning threshold. G1 and G2 have the lowest tolerance, but their values remain acceptable. Health, famrel and absences have high tolerance values, showing low overlap with the rest of the predictors.

R Chart 3: Predictor Correlation Matrix

The R heatmap validates the same predictor relationship structure. The strongest off-diagonal correlation is between G1 and G2. Medu and Fedu are also related, and freetime and goout show a moderate relationship. The heatmap supports the VIF story because predictor correlations explain why some variables have higher inflation than others.

R Chart 4: Strongest Predictor Correlations

The R top-correlation chart again identifies G1 vs G2 as the strongest predictor pair. The value is above .80, which is high enough to explain why the two variables have the largest VIF values. Medu vs Fedu is the next strongest pair, explaining their mild VIF elevation. The remaining pairs are below levels that usually produce major VIF concern.

R Chart 5: VIF vs Tolerance

The R scatter plot confirms the inverse VIF-tolerance relationship. G1 and G2 appear as the highest VIF and lowest tolerance predictors. The remaining predictors cluster near VIF = 1 and high tolerance. This validates the Python scatter plot and confirms that the VIF calculation is consistent across software.

R Chart 6: Regression Coefficients

The R coefficient chart confirms the same coefficient direction as SPSS and Python. G2 has the strongest positive coefficient, G1 has a smaller positive coefficient, and failures has a negative coefficient. Since VIF is acceptable, these coefficients can be reported, but the interpretation of G1 and G2 should remain cautious because they measure related academic performance stages.

R Chart 7: VIF Decision Summary

The R decision summary confirms that all predictors fall in the acceptable category. No predictor crosses the VIF = 5 or VIF = 10 threshold. This independent validation strengthens the final conclusion: the regression model has acceptable multicollinearity under the selected diagnostic rule.

SPSS, R, Python and Excel Workflows for Variance Inflation Factor

The same VIF logic can be applied in SPSS, R, Python and Excel. SPSS reports VIF directly in the regression Coefficients table. R and Python calculate VIF from auxiliary regressions. Excel can reproduce the same idea manually by regressing each predictor on all other predictors and using the VIF formula.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Variance Inflation Factor

SPSS Syntax for VIF

* Variance Inflation Factor and Tolerance in SPSS.
* Dependent variable: G3.
* Predictors: G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health.

TITLE "Variance Inflation Factor and Tolerance through SPSS collinearity diagnostics".

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
  /CRITERIA=PIN(.05) POUT(.10)
  /NOORIGIN.

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

DESCRIPTIVES VARIABLES=G1 G2 studytime failures absences age Medu Fedu traveltime famrel freetime goout health
  /STATISTICS=MEAN STDDEV MIN MAX.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Variance-Inflation-Factor-SPSS-Output.pdf".

Python Code for VIF

import pandas as pd
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"
]

# Keep numeric data only and remove missing rows
model_data = df[[dependent] + predictors].apply(pd.to_numeric, errors="coerce").dropna()

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

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

# Calculate VIF and tolerance
vif_rows = []
for i, variable in enumerate(X_const.columns):
    if variable == "const":
        continue
    vif_value = variance_inflation_factor(X_const.values, i)
    tolerance = 1 / vif_value
    vif_rows.append({
        "predictor": variable,
        "VIF": vif_value,
        "Tolerance": tolerance,
        "Decision": "Acceptable" if vif_value <= 5 else "Review"
    })

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

# Predictor correlation matrix
corr_matrix = X.corr()
print(corr_matrix)

# Top absolute pairwise correlations
pairs = []
for i, col1 in enumerate(predictors):
    for col2 in predictors[i+1:]:
        pairs.append({
            "pair": f"{col1} vs {col2}",
            "correlation": corr_matrix.loc[col1, col2],
            "abs_correlation": abs(corr_matrix.loc[col1, col2])
        })

top_pairs = pd.DataFrame(pairs).sort_values("abs_correlation", ascending=False)
print(top_pairs.head(12))

R Code for VIF

# Variance Inflation Factor 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)

# Option 1: using car package
# install.packages("car")
library(car)
vif_values <- vif(model)

vif_table <- data.frame(
  predictor = names(vif_values),
  VIF = as.numeric(vif_values),
  Tolerance = 1 / as.numeric(vif_values)
)

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

# Predictor correlations
corr_matrix <- cor(df_model[predictors], use = "complete.obs")
print(corr_matrix)

# Top absolute pairwise correlations
pair_rows <- list()
k <- 1

for (i in 1:(length(predictors)-1)) {
  for (j in (i+1):length(predictors)) {
    pair_rows[[k]] <- data.frame(
      pair = paste(predictors[i], "vs", predictors[j]),
      correlation = corr_matrix[predictors[i], predictors[j]],
      abs_correlation = abs(corr_matrix[predictors[i], predictors[j]])
    )
    k <- k + 1
  }
}

top_pairs <- do.call(rbind, pair_rows)
top_pairs <- top_pairs[order(-top_pairs$abs_correlation), ]
head(top_pairs, 12)

Excel Formulas for VIF

Step 1:
Choose one predictor as the dependent variable in an auxiliary regression.

Example:
Predict G2 from G1, studytime, failures, absences, age, Medu, Fedu, traveltime, famrel, freetime, goout, health.

Step 2:
Get the R Square value from that auxiliary regression.

Step 3:
Calculate tolerance:
=1-R_Square

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

Step 5:
Interpret:
If VIF <= 5, commonly acceptable.
If VIF is 5 to 10, moderate warning.
If VIF > 10, serious warning.

Example:
If auxiliary R Square = 0.762

Tolerance:
=1-0.762
=0.238

VIF:
=1/(1-0.762)
=4.202

APA Reporting Wording for Variance Inflation Factor

When reporting VIF, include the threshold rule, the highest VIF value, the lowest tolerance value, and the final decision. If the model includes closely related predictors such as G1 and G2, mention that their coefficients should be interpreted cautiously even when the VIF threshold is acceptable.

APA-Style VIF Report

Full Model Report Sentence

Short Reporting Sentences

• VIF decision: All predictors had VIF values below 5, so harmful multicollinearity was not detected.
• Tolerance decision: All tolerance values were above .20, supporting the same conclusion.
• Main caution: G1 and G2 had the highest VIF values because they were strongly correlated.
• Practical conclusion: The model can be reported, but coefficients for G1 and G2 should be interpreted as adjusted effects in the presence of each other.

Common Mistakes in VIF Interpretation

Variance Inflation Factor is easy to calculate, but interpretation mistakes are common. The biggest mistake is treating VIF as a model-fit statistic. VIF does not say whether the model is good or bad overall. It says whether predictors overlap too much for stable coefficient interpretation.

When to Use Variance Inflation Factor

Use Variance Inflation Factor whenever a regression model has multiple predictors and you need to know whether predictor overlap is inflating coefficient variance. VIF is especially important when the goal is interpretation rather than only prediction.

Do not use VIF as a substitute for variance, spread, or reliability measures. If you need to compare relative variability across variables, use the coefficient of variation. If you need confidence intervals for coefficients or means, use the confidence interval guide. If you need practical importance, use effect size.

Downloads and Resources for Variance Inflation Factor

The SPSS output PDF below verifies the regression model, VIF values, tolerance values, collinearity diagnostics, predictor correlation matrix and descriptive statistics used in this guide.

Related Statistical Guides

Use these related guides to connect VIF with regression assumptions, normality checks, variance checks, model diagnostics and reporting.