Multicollinearity in Regression Analysis: Everything You Need to Know


Regression attempts to determine the character and strength of the relationship between one dependent variable and a series of other independent variables. It helps assess the strength of the relationship between different variables and make a model of the future relationships between them. “Multicollinearity” in regression refers to the predictor which correlates with the other predictors.

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What is Multicollinearity?

Whenever the correlations between two or more predictor variables are high, Multicollinearity in regression occurs. In simple words, a predictor variable, also called a multicollinear predictor, can be used to predict the other variable. This leads to the creation of redundant information, which skews the results in the regression model.

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The examples for multicollinear predictors would be the sales price and age of a car, the weight, height of a person, or annual income and years of education. 

Calculating correlation coefficients is the easiest way to detect multicollinearity for all the pairs of predictor values. If the r, that correlation coefficient is exactly +1 or -1, it is called the perfect multicollinearity. If the correlation coefficient is exactly or close to +1 or -1, then one of the variables must be discarded from the model only in case when it is possible.

It is rare with experimental data, but it is very common that multicollinearity rears its ugly head when it comes to observational studies. It can lead to unreliable and unstable estimation of regression when the condition is present. With the help of analyzing the result, a few other problems can be interfered like: 

  • The t-statistic will usually be pretty small, and the confidence intervals of the coefficient will be wide. It means that it gets difficult to reject the null hypothesis. 
  • There might be a change in magnitude and/or sign in the partial regression coefficients as they are passed from sample to sample. 
  • The standard errors can be large, and the partial regression coefficient estimation may be imprecise. 
  • It gets difficult to gauge the effect on dependent variables by independent variables due to multicollinearity. 

Read: Types of Regression Models in Machine Learning

Why is Multicollinearity a problem?

Change in a single variable can cause a change in the rest of the variables, which happens when the independent variables are correlated highly. So, the model leads to a significantly fluctuating result. Since the results of the model will be unstable and highly varying, even when even a small change occurs in the data, this will constitute the following problems: 

  • The estimation of the coefficient would be unstable and would be difficult to interpret the model. That is, you cannot predict the scale of differences in the output if even one of your factors of predicting changes by 1 unit. 
  • It would be difficult to select the significant variables’ list for the model if it gives varying results every time. 
  • Overfitting can be caused due to the unstable nature of the model. You will observe that the accuracy has dropped significantly if you apply the same model to some other sample of data compared to the accuracy you got with your training dataset.

Considering the situation, it might not be troublesome for your model if only moderate collinearity problems occur. However, it is always suggested to solve the problem if there exists a severe issue in collinearity. 

What is the cause of Multicollinearity?

There are two types:

  1. Structural Multicollinearity in regression: This usually caused by the researcher or you while creating new predictor variables.
  2. Data-based multicollinearity in regression: This is generally caused due to the experiments designed poorly, methods of collection of data which cannot be manipulated, or purely observational data. In a few cases, the variables can be highly correlated due to data collection from 100% observational studies, and there is no error from the researcher’s side. Due to this, it is always suggested to conduct the experiments whenever it is possible by setting the predictor variable’s level in advance. 

Also Read: Linear Regression Project Ideas & Topics

The other causes may also include

  1. Lack of data. In a few cases, collecting an ample amount of data can help in resolving the issue. 
  2. The variables used as dummy might be used incorrectly. For e.g., the researcher can fail in adding a dummy variable for every category or excluding one category.
  3. Considering a variable in regression, which is a combination of the other variables in the regression—for example, considering “total investment income” when it is income from saving interest + income from bonds and stocks.
  4. Including two almost or completely identical variables. For example, bond/savings income and investment income, weight in kilos, and weight in pounds.

To check whether multicollinearity has occurred

You can plot the matrix of correlation of all the independent variables. Alternatively, you can use VIF, that is, the Variance Inflation Factor for each independent variable. It measures the multicollinearity in the multiple regression set of variables. The value of VIF is proportional to the correlation between this variable and the rest. This means, the higher the VIF value, the higher the correlation. 

Related Read: Linear Regression in Machine Learning

How can we fix the problem of Multicollinearity?

  1. Selection of the Variable: The easiest way is to remove a few variables that highly correlate with each other and only leave the most significant ones in the set. 
  2. Transformation of the variable: The second method is a variable transformation, which will reduce the correlation and still manage to maintain the feature. 
  3. Analysis of Principal Component: The Principal Component Analysis is usually used to reduce the data dimension by decomposing the data into a number of independent factors. It has a lot of applications like the model calculation can be simplified by reducing the predicting factors in number. 

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Before building the regression model, you should always check the problem of multicollinearity. To look at each independent variable easily, VIF is recommended to see if they have a considerable correlation with the rest. The correlation matrix can help choose the important factors when unsure which variables you should be selecting. It also helps in understanding why a few variables have a high value of VIF.

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What is meant by the term ordinal regression in machine learning?

Ordinal regression is a type of regression analysis that belongs to the regression analysis family. Ordinal regression analyses data and explains the relationship between one dependent variable and two or more independent variables as a predictive study. Ordinal regression is used to predict the dependent variable when there are 'ordered' numerous categories and independent factors. To put it another way, it allows dependent variables with different ordered levels to interact with one or more independent variables more easily.

Does the presence of multicollinearity affect decision trees?

If two characteristics are highly associated in a specific machine learning model, the decision tree would nevertheless select just one of them while splitting up. If the data is skewed or unbalanced, a single tree leads to a greedy approach, but ensemble learning methods such as random forests and gradient boosting trees make the prediction impervious to multicollinearity. As a result, random forests and decision trees are unaffected by multicollinearity.

How is logistic regression different from linear regression?

In some aspects, linear regression differs from logistic regression. Logical regression produces discrete remarks and findings, but linear regression produces a continuous and continuing output. In linear regression, the mean squared error is calculated, but in logistic regression, the maximum likelihood estimation is calculated. Finally, the goal of linear regression is to identify the best line to match the data, but logistic regression stays ahead by fitting the data to a sigmoid curve.

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