Unveiling Multicollinearity: The Power of Variance Inflation Factor (VIF) in Regression Analysis

Introduction:

In the world of statistical modeling, understanding the relationships between variables is crucial for drawing meaningful insights from data. Regression analysis, a widely used method, helps researchers unravel these connections and predict outcomes based on different predictors. However, when multiple predictor variables are highly correlated, a phenomenon known as multicollinearity can wreak havoc on the regression model, leading to unstable coefficients and diminished interpretability.

Enter the Variance Inflation Factor, or VIF, a powerful tool in the arsenal of statisticians and data analysts. VIF is a statistical metric that plays a fundamental role in regression analysis, offering a clear lens through which to detect and address multicollinearity among predictor variables. In this blog, we will delve into the intricacies of VIF, its significance in regression analysis, and how it enables us to navigate the complexities of multicollinearity to build more robust and accurate models.

Defining VIF and its Significance

VIF, short for Variance Inflation Factor, is a statistical measure used to assess multicollinearity among predictor variables in a regression model. Multicollinearity arises when two or more predictor variables exhibit strong correlations with each other, posing a significant challenge to the regression analysis process. The consequences of multicollinearity include inflated standard errors, imprecise coefficient estimates, and compromised predictive accuracy.

The significance of VIF lies in its ability to quantify the degree of multicollinearity within the model. By providing a numerical value for each predictor variable, VIF helps researchers identify the variables that contribute most to the multicollinearity problem, facilitating effective strategies to mitigate its adverse effects. Through this analysis, VIF empowers us to build regression models that not only perform better but also yield more interpretable and reliable results.

Alright, imagine you have some colorful building blocks, and you want to build a tower with them. Each block represents something different, like how tall a person is, how much they eat, and how much they play outside.

Now, VIF is like a way to check if some of these blocks are too similar to each other. It’s like when two blocks look almost the same and are almost the same size. When this happens, it becomes a little tricky to figure out how important each block is in making the tower tall because they are kind of doing similar things.

If we have these very similar blocks in our tower, it can make our measurements less accurate and make it harder to predict how tall the tower will be. So, we use VIF to look at these blocks and see if they are too similar, causing problems when we want to know exactly how much each one affects the tower’s height.

If the blocks are too similar, it’s like they get mixed up, and we can’t be sure how much each one is contributing to the tower’s height. That’s why we use VIF to spot these similar blocks and make sure our tower stays sturdy and accurate.

Example: Let’s say we have two blocks — one for how much a kid eats and another for how much they exercise. If these blocks are very similar, it might be hard to tell which one has a bigger effect on a kid’s height. But if they are different from each other, we can clearly see that eating more helps the kid grow taller, and exercising also adds a little bit to their height. That’s what VIF helps us with — making sure our blocks are not too similar so we can build our tower of knowledge in a strong and reliable way.

The Purpose of VIF in Detecting Multicollinearity

The primary purpose of VIF is to detect multicollinearity among predictor variables in a regression model. It accomplishes this by measuring the inflation of the variance of the estimated coefficients due to the presence of correlated predictors. In simple terms, VIF gauges how much the precision of a predictor’s coefficient estimate is compromised by multicollinearity.

To calculate the VIF for a specific predictor variable, it is regressed against all the other predictor variables in the model. The resulting VIF value indicates how much larger the variance of the coefficient estimate for that predictor becomes compared to what it would be in the absence of multicollinearity. Consequently, a high VIF value suggests a high degree of multicollinearity, warranting closer inspection and corrective measures.

let’s use a fun example to understand how VIF works!

Imagine we are trying to predict a person’s height (how tall they are) based on two factors: how much milk they drink every day and how many hours they sleep each night. We collected data from many people, and we want to build a regression model to see how these factors affect their height.

Now, let’s see how VIF helps us check if there is any problem with using these two factors together in our model:

Step 1: First, we build a regression model with height as the outcome and milk consumption and sleep duration as predictors.

Step 2: Now, for each predictor (milk consumption and sleep duration), we calculate its VIF value. The VIF value tells us how much the precision of a predictor’s coefficient estimate is affected by multicollinearity with other predictors.

Step 3: Let’s say we find that the VIF for milk consumption is 4, and the VIF for sleep duration is 5.

Interpretation: The VIF values indicate the level of multicollinearity in the model. Generally, a VIF value greater than 5 or 10 suggests a problem with multicollinearity. In our example, both milk consumption and sleep duration have VIF values greater than 5, so there might be an issue.

Step 4: To understand what’s going on, let’s think about why VIF values are high. It’s because milk consumption and sleep duration are closely related to each other. For example, if someone drinks a lot of milk, they might also sleep more because milk can help you sleep better. This relationship makes it difficult for our model to separate the effects of each factor on height accurately.

Step 5: We need to decide what to do next. Since we identified high VIF values, we might consider removing one of the predictors from our model. For example, we could remove milk consumption and only use sleep duration as the predictor for height.

Step 6: After removing one predictor, we check the VIF values again to make sure there’s no more problematic multicollinearity in the model. If the VIF values are low (ideally below 5), it means we have reduced the multicollinearity issue, and our model can now more accurately predict a person’s height based on sleep duration.

That’s how VIF works! It helps us spot when some predictors in our model are too similar, making it hard to get accurate results. By removing or adjusting those predictors, we can build a better and more reliable regression model.

Understanding Multicollinearity

Multicollinearity refers to a statistical phenomenon in regression analysis where two or more predictor variables in a model are highly correlated with each other. In other words, multicollinearity occurs when there is a linear relationship between independent variables, leading to redundancy and overlap in the information they provide. This correlation among predictors can cause several implications and challenges in the regression modeling process.

Implications of Multicollinearity in Regression Models:

1. Unreliable Coefficient Estimates: Multicollinearity makes it challenging for the regression model to accurately estimate the individual effects of each predictor variable. As predictors become more and more correlated, it becomes difficult for the model to distinguish their unique contributions to the dependent variable. Consequently, the coefficients’ estimates become unstable and highly sensitive to changes in the data.
2. Inflated Standard Errors: With multicollinearity, the standard errors of the coefficient estimates increase significantly. Larger standard errors imply reduced precision in estimating the true relationship between predictors and the outcome variable. As a result, it becomes harder to determine whether the coefficients are statistically significant or simply the result of random fluctuations in the data.
3. Ambiguous Interpretation: Multicollinearity can lead to misleading interpretations of the regression results. Even though the overall model might be statistically significant, it becomes challenging to identify which specific predictors are driving the observed effects. The lack of clarity in interpretation can hinder the ability to make informed decisions based on the model’s findings.
4. Overfitting: When multicollinearity is present, the model may capture noise or random variations in the data rather than meaningful relationships. This overfitting can lead to a model that performs well on the training data but generalizes poorly to new, unseen data, reducing the model’s predictive accuracy.
5. Difficulty in Identifying Important Predictors: When highly correlated predictors are included in the model, it becomes challenging to discern which variables truly have a significant impact on the dependent variable. Multicollinearity can obscure the identification of important predictors, making it difficult to focus on the most influential factors.

Why Multicollinearity Can Be Problematic for Regression Analysis:

Multicollinearity poses several significant problems for regression analysis, which can undermine the reliability and usefulness of the model. Some of the key reasons why multicollinearity is problematic include:

a. Violation of Assumptions: Multicollinearity violates the assumption of independence among predictor variables, which is a fundamental assumption in regression analysis. This violation can lead to incorrect conclusions and predictions.

b. Loss of Interpretability: Multicollinearity hinders the ability to interpret the relationships between individual predictor variables and the outcome variable. Researchers may struggle to isolate the unique effects of each predictor, making it challenging to understand the true impact of the variables.

c. Model Instability: Multicollinearity causes the model to be highly sensitive to small changes in the data, leading to unstable and unpredictable coefficient estimates. This instability reduces the model’s reliability and undermines its ability to generalize to new data.

d. Diminished Statistical Power: With inflated standard errors and imprecise coefficient estimates, the statistical power of the model is reduced. As a result, the model may fail to detect significant effects that do exist in the data.

Relation between Correlation and Multicollinearity

Correlation and multicollinearity are both concepts related to the relationships between predictor variables in regression analysis, but they serve different purposes and have distinct implications:

1. Correlation:

• Definition: Correlation measures the strength and direction of the linear relationship between two variables. It is a statistical metric that ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.
• Purpose: Correlation helps identify how two variables move together or in opposite directions. It provides insights into the direction and strength of the association between two variables.
• Implications: Correlation is useful for understanding bivariate relationships between two variables. However, it does not directly indicate the presence of multicollinearity among multiple predictors in a regression model.

2. Multicollinearity:

• Definition: Multicollinearity refers to the high correlation among two or more predictor variables in a regression model. It occurs when the predictors are highly linearly dependent on each other, leading to redundancy and instability in the model.
• Purpose: The main purpose of identifying multicollinearity is to assess how the predictors collectively impact the response variable. It helps understand whether there are strong interconnections among predictors that may impact the validity and interpretability of the regression results.
• Implications: Multicollinearity can cause several issues in regression analysis, such as unstable coefficient estimates, inflated standard errors, and difficulty in interpreting the individual effects of predictors. It does not directly measure the strength of the relationships between individual pairs of predictors, as correlation does.

Calculation of VIF

The Variance Inflation Factor (VIF) is a metric used to quantify the extent of multicollinearity for each predictor variable in a regression model. The VIF for a particular predictor variable is calculated by regressing that variable against all the other predictor variables in the model. The formula for calculating the VIF for variable ‘i’ is as follows:

VIF(i) = 1 / (1 — R²(i))

where:

• VIF(i): Represents the Variance Inflation Factor for the ith predictor variable.
• R²(i): Denotes the coefficient of determination for the regression of the ith predictor variable against all other predictor variables.

To calculate the VIF, follow these steps:

Step 1: Choose one predictor variable as the dependent variable and regress it against all the other predictor variables in the model.

Step 2: Calculate the coefficient of determination (R²) for this regression. R² represents the proportion of the variance in the dependent variable that can be explained by the independent variables.

Step 3: Calculate the VIF using the formula mentioned above. VIF(i) represents how much the variance of the coefficient estimate for variable ‘i’ is inflated due to multicollinearity.

Step 4: Repeat the above steps for each predictor variable in the model to obtain the VIF values for all the variables.

Interpreting VIF Values and Detecting Multicollinearity

The VIF values provide essential insights into the presence and severity of multicollinearity in a regression model. A VIF value of 1 indicates no multicollinearity, implying that the predictor variable is completely independent of the other variables in the model. As a general rule of thumb, a VIF value greater than 1 suggests some degree of correlation, but the real concern arises when the VIF value exceeds a certain threshold, commonly set at 5 or 10.

Higher VIF values indicate a higher degree of multicollinearity associated with the respective predictor variable. For example:

• A VIF value of 1: No multicollinearity (perfectly independent variable).
• A VIF value between 1 and 5: Low to moderate multicollinearity (no significant concern).
• A VIF value between 5 and 10: Moderate to high multicollinearity (considered problematic).
• A VIF value greater than 10: High multicollinearity (severe concern, requires action).

VIF values greater than 10 indicate a high level of multicollinearity, and in such cases, the regression model’s coefficient estimates become highly unstable and less reliable. It becomes challenging to interpret the individual effects of the predictor variables, and the standard errors of the coefficient estimates are inflated, reducing the model’s statistical power.

When multicollinearity is detected, it is essential to address the issue by employing appropriate strategies such as removing one or more of the highly correlated variables, combining variables into composite indices, or applying dimensionality reduction techniques like Principal Component Analysis (PCA). By reducing multicollinearity, the regression model becomes more robust and interpretable, allowing for better insights and improved predictive accuracy.

Dealing with Multicollinearity

1. Variable Selection:

• Advantages: Variable selection involves choosing a subset of predictor variables from the original set to include in the model. This method simplifies the model, making it easier to interpret and reducing the risk of multicollinearity. Selecting only the most relevant predictors can also improve the model’s predictive accuracy and reduce overfitting.
• Disadvantages: The main challenge with variable selection is deciding which variables to keep and which to exclude. Arbitrary selection of variables may lead to biased results or the exclusion of potentially important predictors. Additionally, this method may not fully address the underlying correlation issues between the remaining predictors.

2. Data Transformation:

• Advantages: Data transformation techniques like centering, scaling, or standardizing the variables can help reduce multicollinearity. Centering involves subtracting the mean from each variable, while scaling standardizes variables to have a mean of 0 and a standard deviation of 1. These transformations can help remove the scale-related multicollinearity.
• Disadvantages: While data transformation can be effective in some cases, it may not entirely eliminate multicollinearity, especially when the underlying relationships between predictors remain highly correlated. Additionally, the interpretation of the coefficients in the transformed model may become less intuitive for practitioners.

3. Combining Correlated Predictors:

• Advantages: Combining correlated predictors into composite indices or summary variables can be a powerful approach to reduce multicollinearity. For example, if two predictors measure similar constructs, combining them into a single composite variable can effectively capture the shared variance and reduce redundancy.
• Disadvantages: While combining correlated predictors can address multicollinearity, it may come at the cost of losing the individual interpretability of the original variables. The new composite variable may be less meaningful or challenging to interpret in the context of the research question.

4. Dimensionality Reduction Techniques:

• Advantages: Dimensionality reduction methods, such as Principal Component Analysis (PCA) or Factor Analysis, transform the original predictors into a set of uncorrelated components. These components capture the most significant variability in the data while reducing multicollinearity.
• Disadvantages: While dimensionality reduction effectively tackles multicollinearity, it may lead to a loss of interpretability since the new components may not directly correspond to the original predictors. Explaining the relationships between the components and the outcome variable can be more challenging for stakeholders.

5. Ridge Regression and Lasso Regression:

• Advantages: Ridge regression and Lasso regression are regularized regression techniques that can handle multicollinearity by penalizing large coefficient estimates. These methods shrink the coefficients toward zero, effectively reducing multicollinearity’s impact while preserving all variables in the model.
• Disadvantages: Ridge regression and Lasso regression may require tuning hyperparameters, and the results might depend on the chosen penalty terms. Additionally, interpreting the coefficients in these models can be more complex compared to standard linear regression.

Practical example to demonstrate how VIF is calculated and its interpretation

Example Application: Detecting and Addressing Multicollinearity using VIF

Let’s consider a practical example of predicting house prices based on two predictor variables: “Square Footage” (sq. ft.) and “Number of Bedrooms.” We will create a simulated dataset for illustration purposes.

Step 1: Dataset Creation

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

# Step 1: Create a Simulated Dataset
data = {
    "House": [1, 2, 3, 4, 5, 6],
    "Square Footage (sq. ft.)": [1500, 1800, 1600, 1400, 1900, 1700],
    "Number of Bedrooms": [3, 4, 3, 2, 4, 3],
    "Price (in $1000s)": [250, 280, 260, 220, 300, 270]
}

dataset = pd.DataFrame(data)

Step 2: Perform the Regression Analysis

X = dataset[['Square Footage (sq. ft.)', 'Number of Bedrooms']]
X = sm.add_constant(X)  # Add a constant term for the intercept
y = dataset['Price (in $1000s)']

model = sm.OLS(y, X).fit()

Step 3: Calculate the Variance Inflation Factor (VIF)

vif = pd.DataFrame()
vif["Variable"] = X.columns
vif["VIF"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]

Step 4: Interpretation of VIF Values

print(vif)

VIF

Now, let’s understand the specific values in the table:

1. “Square Footage (sq. ft.)” has a VIF of approximately 6.76.
2. “Number of Bedrooms” also has a VIF of approximately 6.76.

Interpretation: Both variables have VIF values around 6.76. Typically, a VIF value above 5 or 10 is considered a sign of multicollinearity. So, in this case, both “Square Footage (sq. ft.)” and “Number of Bedrooms” might be showing some multicollinearity.

To resolve this issue, one option could be to remove one of the variables from the regression model. By doing so, we can reduce the multicollinearity and make the model more reliable for predicting the outcome variable.

Step 5: Regression Results

print(model.summary())

Model Summary

Best Practices for Using VIF in Regression Analysis:

1. Calculate VIF for All Predictor Variables: Calculate VIF values for all predictor variables in the model, not just for a selected few. This ensures a comprehensive assessment of multicollinearity and helps identify potential issues with any variable combinations.
2. Set a Reasonable VIF Threshold: While there is no strict rule for the acceptable VIF threshold, it is generally recommended to consider a VIF value of 5 or 10 as an indication of multicollinearity. However, the threshold may vary depending on the field of study and the specific context of the analysis. Use domain knowledge and interpretability considerations to determine an appropriate threshold.
3. Interpret VIF Values with Context: The absolute VIF values alone may not provide a complete picture. Consider the context of the problem and the level of multicollinearity in the predictors. A slightly elevated VIF may not be a cause for concern if the model’s overall performance and interpretability are satisfactory.
4. Address High VIF Values Proactively: If high VIF values are detected, investigate the underlying reasons for multicollinearity and take appropriate action. Use techniques such as variable selection, data transformation, combining correlated predictors, or dimensionality reduction to mitigate multicollinearity.
5. Educate Stakeholders on VIF Interpretation: Communicate the concept of VIF and its implications to stakeholders. Make sure they understand the impact of multicollinearity on the model’s reliability and how addressing it can lead to more robust and interpretable results.

Potential Pitfalls and Common Mistakes to Avoid:

1. Overlooking Higher-Order Multicollinearity: VIF primarily assesses pairwise correlations between predictors. Be aware that multicollinearity can also arise from higher-order relationships among three or more predictors, which VIF might not capture. Consider using condition numbers or eigenvalues for a broader assessment.
2. Incorrect Interpretation of VIF Threshold: Setting an arbitrary VIF threshold without considering the specific context and complexity of the model can lead to incorrect decisions. Evaluate the impact of multicollinearity on the regression model’s performance and consider the trade-offs between model simplicity and interpretability.
3. Blindly Removing Variables Based on VIF: If a variable is removed solely based on its high VIF value, it might lead to omitting important predictors and biasing the results. Instead, consider the relevance and theoretical significance of the variables before removing them from the model.
4. Ignoring the Impact of Data Transformation: Data transformation can help mitigate multicollinearity, but it may alter the interpretation of the coefficients. Be mindful of the transformations applied and their implications for the model’s interpretability.
5. Neglecting the Model’s Purpose: Addressing multicollinearity is essential, but it should not overshadow the primary purpose of the regression analysis. Keep the research question and the model’s predictive accuracy in focus while handling multicollinearity.
6. Using VIF in Non-Linear Models: VIF is primarily designed for linear regression models. Its application in non-linear models like logistic regression or time series analysis may not be appropriate.

Real-Life Examples multicollinearity caused issues in regression analysis, and how VIF helped resolve them

Case Study 1: Housing Market Analysis

Scenario: A real estate agency aims to predict house prices based on various predictor variables such as square footage, number of bedrooms, number of bathrooms, and proximity to amenities. They perform a linear regression analysis to build the predictive model.

Issue: During the analysis, they find that the coefficient estimates for both square footage and number of bedrooms are highly unstable, and their standard errors are inflated. Additionally, the p-values for these variables are not significant, making it difficult to interpret their effects on house prices.

Resolution with VIF: To identify the cause of the instability, they calculate the VIF values for all predictor variables. They discover that both square footage and number of bedrooms have high VIF values (e.g., VIF > 10), indicating severe multicollinearity between these variables.

Action Taken: With the help of VIF, the team decides to address the multicollinearity issue by combining the square footage and number of bedrooms into a single composite variable that represents the overall size of the house. They create a new predictor called “House Size Index” by taking a weighted average of square footage and number of bedrooms. This combination effectively reduces the multicollinearity between the two variables.

Result: After re-running the regression analysis with the new composite variable, the coefficient estimates become stable, and their standard errors are reduced. The p-values for the predictors are now significant, allowing the team to interpret their effects on house prices with more confidence. The improved model provides more reliable predictions for house prices in the market.

Case Study 2: Marketing Campaign Analysis

Scenario: A marketing firm conducts a regression analysis to understand the factors influencing the success of a marketing campaign. They consider variables such as ad spending, social media engagement, website traffic, and customer demographics.

Issue: When interpreting the regression results, they notice that some predictor variables, such as ad spending and social media engagement, have high VIF values, suggesting multicollinearity.

Resolution with VIF: To assess the extent of multicollinearity, the marketing firm calculates VIF values for all predictor variables. They find that ad spending and social media engagement are highly correlated with each other.

Action Taken: To deal with the multicollinearity, the team decides to retain the variable that has more theoretical significance in the context of the marketing campaign. They choose to keep ad spending, as it aligns better with their campaign strategy, and remove social media engagement from the model.

Result: After removing the highly correlated predictor, the marketing firm re-evaluates the regression model. The coefficient estimates become more stable, and the standard errors are reduced. The model’s interpretability improves, allowing the firm to focus on the impact of ad spending on the campaign’s success. This helps them make more informed decisions about allocating resources for future marketing campaigns.

These real-life examples demonstrate how VIF can be a valuable tool in identifying and resolving multicollinearity issues in regression analysis. By addressing multicollinearity through appropriate strategies guided by VIF, analysts can build more reliable and interpretable models, leading to better insights and more effective decision-making.

Some Important Interview Questions you can expect from VIF

Here are some interview questions related to Variance Inflation Factor (VIF) and their answers:

Question: What is Variance Inflation Factor (VIF), and what does it measure in regression analysis?

Answer: Variance Inflation Factor (VIF) is a statistical measure used to detect multicollinearity among predictor variables in a regression model. It quantifies how much the variance of the estimated coefficients for each predictor variable is inflated due to the presence of correlated predictors. In simple terms, VIF measures the extent to which the precision of a predictor’s coefficient estimate is compromised by multicollinearity.

Question: How is VIF calculated for a predictor variable in a regression model?

Answer: To calculate the VIF for a specific predictor variable, you perform a linear regression of that variable against all other predictor variables in the model. The formula for VIF is VIF(i) = 1 / (1 — R²(i)), where VIF(i) is the VIF for the ith predictor, and R²(i) is the coefficient of determination for the regression of the ith predictor against all other predictors.

Question: What does a VIF value greater than 1 imply? How do you interpret high VIF values?

Answer: A VIF value greater than 1 indicates some level of correlation between the predictor variable and the other predictors in the model. Generally, a VIF greater than 5 or 10 is considered indicative of moderate to high multicollinearity. High VIF values suggest that the predictor’s coefficient estimate is less precise due to its strong correlation with other predictors, which can lead to instability in the regression model.

Question: How do you identify multicollinearity using VIF? What VIF threshold do you consider concerning?

Answer: To identify multicollinearity, calculate the VIF values for all predictor variables in the model. Look for VIF values above a certain threshold, typically set at 5 or 10, as an indication of problematic multicollinearity. However, the threshold may vary based on the specific context and research field, and interpretation should be guided by the model’s complexity and the importance of predictor relationships.

Question: What are some common strategies to address multicollinearity identified through VIF?

Answer: There are several strategies to address multicollinearity:

• Variable Selection: Choose a subset of predictors based on theoretical relevance and domain knowledge.
• Data Transformation: Apply centering, scaling, or standardization to reduce scale-related multicollinearity.
• Combining Correlated Predictors: Create composite variables or indices to capture shared variance among highly correlated predictors.
• Dimensionality Reduction Techniques: Use methods like Principal Component Analysis (PCA) to transform predictors into uncorrelated components.
• Regularized Regression: Employ techniques like Ridge Regression or Lasso Regression to penalize large coefficient estimates and mitigate multicollinearity’s impact.

Question: How does dealing with multicollinearity through VIF affect the regression model’s reliability and interpretability?

Answer: Addressing multicollinearity using VIF can significantly improve the regression model’s reliability and interpretability. By reducing multicollinearity, the model’s coefficient estimates become more stable and reliable, leading to more accurate predictions. Moreover, the model becomes easier to interpret as it allows for clearer identification of the individual effects of each predictor variable on the response variable.

Question: What is the relationship between VIF and R-squared?

The relationship between Variance Inflation Factor (VIF) and R-squared (R²) is essential in understanding multicollinearity’s impact on the regression model.

1. VIF and R-squared: VIF and R-squared are related because VIF is derived from R-squared values. Specifically, VIF is calculated based on the R-squared obtained from regressing each predictor variable against all other predictor variables in the model.
2. VIF Calculation: To calculate the VIF for a particular predictor variable, you perform a linear regression of that variable against all other predictors. The R-squared from this regression represents the proportion of the variance in that predictor explained by the other predictors in the model. The formula for VIF is given as:

VIF(i) = 1 / (1 — R²(i))

where VIF(i) is the Variance Inflation Factor for the ith predictor variable, and R²(i) is the coefficient of determination from the regression of the ith predictor against all other predictors.

Interpretation: The VIF quantifies how much the variance of the coefficient estimate for a specific predictor is inflated due to multicollinearity with other predictors. A VIF value greater than 1 indicates some level of correlation between the predictor and the other predictors in the model. A VIF of 1 means no multicollinearity (i.e., the predictor is perfectly independent), and higher VIF values indicate increasing multicollinearity.

Relationship with R-squared: As VIF is derived from R-squared, an increase in the R-squared value from the regression of a predictor variable indicates that the predictor is well-explained by the other predictors in the model. Consequently, a higher R-squared leads to a smaller denominator in the VIF formula, resulting in a smaller VIF value.

Interpretation of VIF and R-squared in the Context of Multicollinearity: When assessing multicollinearity, both high R-squared values and high VIF values are indicators of concern. A high R-squared means that the predictor is well-predicted by the other predictors, which may lead to multicollinearity issues. Consequently, the VIF value will be larger as the R-squared increases, implying that multicollinearity inflates the variance of the predictor’s coefficient estimate.