What is Multicollinearity? | Causes, Effects and Detection Using VIF

Introduction

Multicollinearity might be a handful to pronounce, but it’s a topic you should be aware of in the field of data science and machine learning, especially if you’re sitting for data scientist interviews! In this article, we will understand what multicollinearity is and how it is caused. We will also try to understand why it is a problem and how we can detect and fix it.

Before diving further, it is imperative to have a basic understanding of regression and some statistical terms. For this, I highly recommend going through the below resources:

• Fundamentals of Regression Analysis (Free Course!)
• Beginner’s Guide to Linear Regression

Learning Objective

• Understand what multicollinearity is and why it is a problem in a regression model.
• Learn the causes of multicollinearity.
• Understand how to detect multicollinearity using the variance inflation factor (VIF).
• Learn about the methods used to fix multicollinearity, including dropping correlated features.

What is Multicollinearity?

Multicollinearity is a statistical phenomenon that occurs when two or more independent variables in a regression model are highly correlated, indicating a strong linear relationship among the predictor variables. This issue complicates regression analysis by making it difficult to accurately determine the individual effects of each independent variable on the dependent variable.

The presence of multicollinearity can lead to unstable and unreliable coefficient estimates, making it challenging to interpret the results and draw meaningful conclusions from the model. Detecting and addressing multicollinearity is crucial to ensure the validity and robustness of regression models. For example, in a regression model, variables such as height and weight or household income and water consumption often show high correlation.

Example

An everyday example of multicollinearity can be illustrated with Colin, who experiences happiness from watching television while eating chips. His happiness is hard to attribute to either activity individually because the more television he watches, the more chips he eats, making the two activities highly correlated.

This makes it difficult to determine whether his happiness is more influenced by eating chips or watching television, exemplifying the multicollinearity problem. In the context of machine learning, multicollinearity, marked by a correlation coefficient close to +1.0 or -1.0 between variables, can lead to less dependable statistical conclusions. Therefore, managing multicollinearity is essential in predictive modeling to obtain reliable and interpretable results.

Issues with Multicollinearity in Regression Models

Multicollinearity can be a problem in a regression model when using algorithms such as OLS (ordinary least squares) in statsmodels. This is because the estimated regression coefficients become unstable and difficult to interpret in the presence of multicollinearity. Statsmodels is a Python library that provides a range of tools for statistical analysis, including regression analysis.

When multicollinearity is present, the estimated regression coefficients may become large and unpredictable, leading to unreliable inferences about the effects of the predictor variables on the response variable. Therefore, it is important to check for multicollinearity and consider using other regression techniques that can handle this problem, such as ridge regression or principal component regression.

Understanding the Impact of Multicollinearity

For example, let’s assume that in the following linear equation:

Y = W0+W1*X1+W2*X2

Coefficient W1 is the increase in Y for a unit increase in X1 while keeping X2 constant. But since X1 and X2 are highly correlated, changes in X1 would also cause changes in X2, and we would not be able to see their individual effect on Y.

The regression coefficient, also known as the beta coefficient, measures the strength and direction of the relationship between a predictor variable (X) and the response variable (Y). In the presence of multicollinearity, the regression coefficients become unstable and difficult to interpret because the variance of the coefficients becomes large. This results in wide confidence intervals and increased variability in the predicted values of Y for a given value of X. As a result, it becomes challenging to determine the individual contribution of each predictor variable to the response variable and make reliable inferences about their effects on Y.

“ This makes the effects of X1 on Y difficult to distinguish from the effects of X2 on Y. ”

Multicollinearity may not affect the accuracy of the machine-learning model as much. But we might lose reliability in determining the effects of individual features in your model – and that can be a problem when it comes to interpretability.

What Causes Multicollinearity?

Multicollinearity could occur due to the following problems:

• Multicollinearity could exist because of the problems in the dataset at the time of creation. These problems could be because of poorly designed experiments, highly observational data, or the inability to manipulate the data.
  For example, determining the electricity consumption of a household from the household income and the number of electrical appliances. Here, we know that the number of electrical appliances in a household will increase with household income. However, this cannot be removed from the dataset.
• Multicollinearity could also occur when new variables are created which are dependent on other variables.
  For example, creating a variable for BMI from the height and weight variables would include redundant information in the model, and the new variable will be a highly correlated variable.
• Including identical variables in the dataset.
  For example, including variables for temperature in Fahrenheit and temperature in Celsius.
• Inaccurate use of dummy variables can also cause a multicollinearity problem. This is called the Dummy variable trap.
  For example, in a dataset containing the status of marriage variable with two unique values: ‘married’, and ’single’. Creating dummy variables for both of them would include redundant information. We can make do with only one variable containing 0/1 for ‘married’/’single’ status.
• Insufficient data, in some cases, can also cause multicollinearity problems.

Detecting Multicollinearity Using a Variance Inflation Factor (VIF)

Let’s try detecting multicollinearity in a dataset to give you a flavor of what can go wrong.

I have created a dataset determining the salary of a person in a company based on the following features:

• Gender (0 – female, 1- male)
• Age
• Years of service (Years spent working in the company)
• Education level (0 – no formal education, 1 – under-graduation, 2 – post-graduation)

df=pd.read_csv(r'C:/Users/Dell/Desktop/salary.csv')
df.head()

In Python, there are several ways to detect multicollinearity in a dataset, such as using the Variance Inflation Factor (VIF) or calculating the correlation matrix of the independent variables. To address multicollinearity, techniques such as regularization or feature selection can be applied to select a subset of independent variables that are not highly correlated with each other. In this article, we will focus on the most common one – VIF (Variance Inflation Factors).

” VIF determines the strength of the correlation between the independent variables. It is predicted by taking a variable and regressing it against every other variable. “

or

VIF score of an independent variable represents how well the variable is explained by other independent variables.

R^2 value is determined to find out how well an independent variable is described by the other independent variables. A high value of R^2 means that the variable is highly correlated with the other variables. This is captured by the VIF, which is denoted below:

So, the closer the R^2 value to 1, the higher the value of VIF and the higher the multicollinearity with the particular independent variable.

# Import library for VIF
from statsmodels.stats.outliers_influence import variance_inflation_factor

def calc_vif(X):

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

    return(vif)

• VIF starts at 1 and has no upper limit
• VIF = 1, no correlation between the independent variable and the other variables
• VIF exceeding 5 or 10 indicates high multicollinearity between this independent variable and the others

X = df.iloc[:,:-1]
calc_vif(X)
view raw

We can see here that the ‘Age’ and ‘Years of service’ have a high VIF value, meaning they can be predicted by other independent variables in the dataset.

Although correlation matrix and scatter plots can also be used to find multicollinearity, their findings only show the bivariate relationship between the independent variables. VIF is preferred as it can show the correlation of a variable with a group of other variables.

Fixing Multicollinearity

Dropping one of the correlated features will help in bringing down the multicollinearity between correlated features:

X = df.drop(['Age','Salary'],axis=1)
calc_vif(X)

The image on the left contains the original VIF value for variables, and the one on the right is after dropping the ‘Age’ variable. We were able to drop the variable ‘Age’ from the dataset because its information was being captured by the ‘Years of service’ variable. This has reduced the redundancy in our dataset. Dropping variables should be an iterative process starting with the variable having the largest VIF value because other variables highly capture its trend. If you do this, you will notice that VIF values for other variables would have reduced, too, although to a varying extent.

In our example, after dropping the ‘Age’ variable, VIF values for all variables have decreased to varying degrees.

Next, combine the correlated variables into one and drop the others. This will reduce the multicollinearity.

df2 = df.copy()
df2['Age_at_joining'] = df.apply(lambda x: x['Age'] - x['Years of service'],axis=1)
X = df2.drop(['Age','Years of service','Salary'],axis=1)
calc_vif(X)

The image on the left contains the original VIF value for variables, and the one on the right is after combining the ‘Age’ and ‘Years of service’ variables. Combining ‘Age’ and ‘Years of experience’  into a single variable, ‘Age_at_joining’ allows us to capture the information in both variables.

However, multicollinearity may not be a problem every time. The need to fix multicollinearity depends primarily on the following reasons:

• When you care more about how much each individual feature rather than a group of features affects the target variable, then removing multicollinearity may be a good option
• If multicollinearity is not present in the features you are interested in, then multicollinearity may not be a problem.

Conclusion

We learned how the problem of multicollinearity could occur in regression models when two or more independent variables in a data frame have a high correlation with one another. Its presence can cause the regression coefficients to become unstable and difficult to interpret, which can lead to wide confidence intervals and increased variability in the predicted values of the dependent variable. Understanding what causes it and how to detect and fix it can help us to overcome these problems.

In this article, we explored how the Variance Inflation Factor (VIF) can be used to detect the existence of multicollinearity in our dataset and how to fix the problem by identifying and dropping the correlated variables. Remember, when assessing the statistical significance of predictor variables in a regression model, it is important to consider their individual coefficients and their standard errors, p-values, and confidence intervals. Predictor variables with high multicollinearity may have inflated standard errors and p-values, which can lead to incorrect conclusions about their statistical significance.

If you want to understand other regression models or want to understand model interpretation, I highly recommend going through the following wonderfully written articles:

• Regression Modeling
• Machine Learning Model Interpretability

As a next step, you should also check out the Fundamentals of Regression (free) course.

Key Takeaways

• Multicollinearity occurs when two or more independent variables have a high correlation with one another in a regression model, which makes it difficult to determine the individual effect of each independent variable on the dependent variable.
• Multicollinearity can occur due to poorly designed experiments, highly observational data, creating new variables that are dependent on other variables, including identical variables in the dataset, inaccurate use of dummy variables, or insufficient data.
• One method to detect multicollinearity is to calculate the variance inflation factor (VIF) for each independent variable, and a VIF value greater than 1.5 indicates multicollinearity.
• To fix multicollinearity, one can remove one of the highly correlated variables, combine them into a single variable, or use a dimensionality reduction technique such as principal component analysis to reduce the number of variables while retaining most of the information.

Frequently Asked Questions