Detecting Multicollinearity with VIF – Python

Multicollinearity is a common problem in regression analysis, where two or more independent variables are highly correlated. This issue can lead to inaccurate predictions and misinterpretation of the model, as it inflates the standard errors of regression coefficients, causing unreliable estimates. To effectively detect multicollinearity, the Variance Inflation Factor (VIF) is widely used.

VIF quantifies the level of multicollinearity by measuring how much the variance of a regression coefficient increases due to the correlation between predictor variables. A high VIF value signals that multicollinearity is present, affecting the stability of the model. In this article, we will explore how to detect multicollinearity using the Variance Inflation Factor (VIF).

Understanding the Variance Inflation Factor (VIF) Formula

In the Variance Inflation Factor (VIF) method, we assess the degree of multicollinearity by selecting each feature and regressing it against all other features in the model. This process calculates how much the variance of a regression coefficient is inflated due to the correlations between independent variables.

The VIF for each feature is given by the following formula:[Tex] VIF=\frac{1}{1-R^2} [/Tex]

Where, R-squared is the coefficient of determination in linear regression. Its value lies between 0 and 1 from the linear regression of one feature against the others. R-squared measures the proportion of variance in the dependent variable that is predictable from the independent variables, and it ranges between 0 and 1. A higher R-squared value suggests a stronger relationship between the feature and other predictors, which results in a higher VIF. If R-squared is close to 1, this indicates a high degree of multicollinearity, as the feature can be largely explained by the other variables in the model.

As we see from the formula, greater the value of R-squared, greater is the VIF. Hence, greater VIF denotes greater correlation. This is in agreement with the fact that a higher R-squared value denotes a stronger collinearity. Generally, a VIF above 5 indicates a high multicollinearity.

By understanding the VIF formula, we can accurately detect multicollinearity in our regression models and take the necessary steps to address it. The interpretation of VIF values is key to ensuring that multicollinearity does not affect the reliability of your regression analysis.

Detect Multicollinearity Using VIF in Python

To detect multicollinearity in regression analysis, we can implement the Variance Inflation Factor (VIF) using the statsmodels library. The statsmodels package provides a function named variance_inflation_factor() for calculating VIF that helps calculate the VIF for each feature in the dataset, indicating the presence of multicollinearity.

Syntax : statsmodels.stats.outliers_influence.variance_inflation_factor(exog, exog_idx)

Parameters :

• exog : an array containing features on which linear regression is performed.
• exog_idx : index of the additional feature whose influence on the other features is to be measured.

The dataset : Let’s walk through an example using a dataset that contains information about 500 individuals, including their height, weight, gender, and Body Mass Index (BMI). In this case, the dependent variable is Index, and the other variables serve as independent features. Our goal is to check for multicollinearity among these features using VIF. Let us see an example to implement the method on this dataset.

import pandas as pd 

# the dataset
data = pd.read_csv('BMI.csv')

# printing first few rows
print(data.head())

Output :

Gender Height Weight Index
0 Male 174 96 4
1 Male 189 87 2
2 Female 185 110 4
3 Female 195 104 3
4 Male 149 61 3

Approach :

• Each of the feature indices are passed to variance_inflation_factor() to find the corresponding VIF.
• These values are stored in the form of a Pandas DataFrame.

 from statsmodels.stats.outliers_influence import variance_inflation_factor

# creating dummies for gender
data['Gender'] = data['Gender'].map({'Male':0, 'Female':1})

# the independent variables set
X = data[['Gender', 'Height', 'Weight']]

# VIF dataframe
vif_data = pd.DataFrame()
vif_data["feature"] = X.columns

# calculating VIF for each feature
vif_data["VIF"] = [variance_inflation_factor(X.values, i)
                          for i in range(len(X.columns))]

print(vif_data)

Output :

feature VIF
0 Gender 2.028864
1 Height 11.623103
2 Weight 10.688377

• As we can see, height and weight have very high values of VIF, indicating that these two variables are highly correlated.
• This is expected as the height of a person does influence their weight. Hence, considering these two features together leads to a model with high multicollinearity.

By implementing VIF using statsmodels, you can efficiently assess the presence of multicollinearity in your regression models, leading to more robust and reliable analyses.

Addressing Multicollinearity: What to Do If VIF Is High

When conducting regression analysis, encountering high Variance Inflation Factor (VIF) values indicates the presence of multicollinearity, which can compromise the reliability of your model. Here are several effective strategies to address high VIF values and improve model performance:

1. Removing Highly Correlated Features

One of the most straightforward approaches to dealing with multicollinearity is to remove one or more of the highly correlated features from your dataset. This can be accomplished by:

• Analyzing the correlation matrix to identify pairs of features with high correlation coefficients (generally above 0.7 or 0.8).
• Dropping the feature that is less important for your analysis or has a higher VIF value.

By removing features with high multicollinearity, you can reduce redundancy and make your regression model more interpretable, ultimately leading to more reliable coefficient estimates.

2. Combining Variables or Using Dimensionality Reduction Techniques

Another effective strategy for addressing multicollinearity is to combine correlated features into a single variable. This can involve:

• Creating composite scores by averaging or summing related features. For example, if height and weight are highly correlated, you might create a new feature representing a Body Mass Index (BMI).
• Utilizing dimensionality reduction techniques like Principal Component Analysis (PCA). PCA transforms correlated variables into a set of uncorrelated components, which can capture most of the variance in the original dataset while eliminating multicollinearity issues. The principal components can then be used as features in your regression model.

By incorporating regularization techniques into your regression analysis, you can enhance model performance and interpretability while addressing the challenges posed by multicollinearity.

Conclusion

Detecting multicollinearity using VIF allows data scientists and statisticians to address collinearity issues and enhance the performance of their regression models. Understanding and correcting multicollinearity in regression is crucial for improving model accuracy, especially in fields like econometrics, where variable relationships play a key role.