Assumptions of Linear Regression

Have you ever built a linear regression model, only to find your predictions falling flat? If your results seem off, the issue might not be with your data but with overlooked assumptions. Many analysts dive into regression without fully understanding the foundational rules that keep the model accurate and reliable. When these assumptions are ignored or violated, the results can mislead rather than inform.

This guide is here to fix that. You’ll uncover the assumptions of linear regression that hold your models together, learn how to spot common pitfalls, and discover practical solutions to handle violations. 

By the end, you’ll have the tools to make your regression analysis not just functional, but highly reliable. Let’s build models that work!

What Is Linear Regression? Key Components and Overview

Linear regression is a statistical method you can use to model the relationship between a dependent variable (the outcome you want to predict) and one or more independent variables (the factors influencing the outcome).

Here is the basic equation for a simple linear regression model:

Y = b0 + b1X + e

Here’s what each term means:

• Y: The dependent variable (outcome).
• b0: The intercept, representing the predicted value of Y when X is zero.
• b1: The slope, showing the change in Y for a one-unit increase in X.
• X: The independent variable (predictor).
• e: The error term, capturing variability not explained by X.

To effectively apply linear regression, it’s important to understand its core components:

• Dependent Variable: The primary outcome you are trying to predict or measure.
• Independent Variables: These predictors or factors influence the dependent variable. 
• Assumptions: Linear regression operates under specific assumptions of regression like linearity, independence, and homoscedasticity.

By understanding these elements, you can use linear regression not only to analyze relationships but also to make confident, data-driven predictions that hold practical value.

Also Read: What is Predictive Analysis? Why is it Important? 

Now that you have an introduction to assumptions of regression, let's break down the key components and how this model operates in its simplest form.

Understanding the Assumptions of Linear Regression

Assumptions are the foundational conditions that must hold true for linear regression analysis to deliver reliable and unbiased results. These assumptions ensure that the AI or machine learning model captures the relationships between variables accurately, without distortion from external factors. 

Violating these assumptions can compromise the integrity of your analysis, leading to misleading conclusions and flawed predictions.

Why Assumptions Matter: Linear regression relies on a specific mathematical framework, and its validity depends on adhering to these underlying assumptions. Ignoring or violating them can result in:

• Biased or incorrect estimates of coefficients.
• Reduced predictive accuracy.
• Invalid conclusions about relationships between variables.

When assumptions are not met, the following issues may arise:

• Overestimated or underestimated effects of independent variables.
• Inflated confidence intervals or p-values, leading to false significance.
• Inability to generalize findings to new data.

This awareness will enable you to conduct analyses that are both accurate and insightful.

Also Read: Linear Regression in Machine Learning: Everything You Need to Know

With a solid understanding of the basic components of linear regression, it's time to dive into the assumptions that form the bedrock of this method.

10+ Primary and Core Assumptions in Linear Regression

When building a linear regression model, there are several key assumptions that form the foundation of the analysis. These assumptions help ensure that the model produces reliable and valid results. 

Let's take a closer look at the primary assumptions of linear regression, how to check if they're met, and what to do if they're violated.

1. Linearity

For linear regression to work correctly, the relationship between the dependent (response) variable and the independent (predictor) variables must be linear. Any changes in the independent variable(s) can lead to proportional changes in the dependent variable. The violation of this assumption means the model may not adequately represent the data.

How to Check: Use scatter plots to visualize the relationship between the variables. If the plot shows a straight-line relationship, this assumption holds. You can also use residual plots to check for linearity.

What to Do if Violated: If the relationship isn’t linear, consider transforming the variables (e.g., using logarithmic or polynomial transformations) or exploring polynomial regression for more complex relationships.

Example: Predicting salary based on years of experience. If the relationship is linear, an increase in years of experience should lead to a proportional increase in salary. However, if the effect of experience on salary starts to level off after a certain point (e.g., more experience doesn’t always equate to higher salary after 20 years), a non-linear relationship is present.

Also Read: Bayesian Linear Regression: What is, Function & Real Life Applications

2. No Autocorrelation (Independence of Errors)

In a good model, the residuals (errors) from one observation should not be correlated with the residuals from another. If autocorrelation exists, it implies that there's some underlying pattern in the data that your model is missing, which can lead to misleading results.

How to Check: One of the most common tests is the Durbin-Watson test, which checks for autocorrelation in the residuals. You can also use residual plots to look for patterns.

What to Do if Violated: If autocorrelation is present, you may want to explore time series methods (if you're working with time-dependent data) or introduce lagged variables into the model.

Example: In a stock market prediction model, if the residuals (errors) from day-to-day stock price predictions are correlated, it means that the model is failing to capture some important time-dependent relationship (e.g., past stock prices influencing future ones).

3. No Multicollinearity

Here, independent variables are highly correlated with each other. When this happens, it can be difficult to determine the individual effect of each predictor on the dependent variable and can inflate the variance of the coefficient estimates.

How to Check: The Variance Inflation Factor (VIF) measures the variance of a regression coefficient, and how much it is inflated due to collinearity. You can also check the correlation matrix for strong correlations between predictors.

What to Do if Violated: If multicollinearity is present, you might need to remove one of the correlated variables, combine them into a single predictor, or apply dimensionality reduction techniques like principal component analysis (PCA).

Example: When predicting house prices, you might include both the number of bedrooms and square footage as predictors. These two variables are likely highly correlated, as houses with more bedrooms tend to have larger square footage. If both are included in the model, it may be difficult to separate their individual effects on the price.

Also Read: How to Perform Multiple Regression Analysis? 

4. Homoscedasticity (Constant Variance of Errors)

Homoscedasticity assumes that the variance of the residuals is constant across all levels of the independent variable(s). If the variance of the errors changes as the value of the independent variable(s) changes (a phenomenon called heteroscedasticity), it can lead to inefficient estimates.

How to Check: Plot the residuals against the predicted values to visually inspect for any patterns. If the spread of the residuals remains constant across the range of predicted values, the assumption holds. The Breusch-Pagan test can also be used to test for heteroscedasticity.

What to Do if Violated: If heteroscedasticity is found, you can try using robust standard errors or employ weighted least squares regression to adjust for the changing variance.

Example: In a model predicting income based on education level, if the variance of the errors increases for higher education levels (e.g., the prediction error for people with higher education levels is larger), this violates the assumption of homoscedasticity.

Also Read: Homoscedasticity in ML Homoscedasticity & Heteroscedasticity

5. Normal Distribution of Errors

For many statistical tests in linear regression (such as hypothesis testing), the residuals should follow a normal distribution. If the errors are not normally distributed, it can affect the validity of the results, especially when dealing with small sample sizes.

How to Check: A Q-Q plot is a great tool for visualizing the distribution of residuals. If the residuals are normally distributed, the points will lie along a straight line. You can also use the Shapiro-Wilk test to formally test for normality.

Example: In a model predicting exam scores, if the errors are not normally distributed and are heavily skewed (e.g., most predictions are close to the actual scores with a few large errors), this may affect the statistical significance of your model’s coefficients.

6. Primary Assumptions of Linear Regression

For a linear regression model to produce reliable results, there are several essential assumptions that need to be met. These assumptions ensure the integrity of the model and its predictions. 

Below are some primary assumptions to keep in mind when building your model.

No Outliers

Outliers can have a disproportionate impact on the results of regression analysis. They can distort the relationship between the dependent and independent variables, leading to misleading conclusions.

How to Check: Use scatter plots and box plots to identify potential outliers. Cook’s distance is a useful statistic to detect influential outliers that have a large effect on the model's coefficients.

What to Do if Violated: Investigate the outliers to determine if they are errors in data collection or genuinely unusual observations. Depending on the context, you may choose to remove the outliers, adjust them, or retain them with caution.

Example: When predicting car prices, a few luxury cars might have extremely high prices compared to the rest of the dataset. These outliers could unduly influence the regression line, distorting the true relationship between car features (like age and mileage) and price.

Also Read: Linear Regression vs Logistic Regression: A Detailed Comparison

Additivity

The additivity assumption states that the combined effect of the independent variables on the dependent variable is additive. This means the effect of each predictor is independent and doesn’t interact with others unless specified.

How to Check: Test for interaction terms in the model. If there are significant interactions between predictors, it might suggest that additivity is not holding.

What to Do if Violated: If interactions are present, add interaction terms to the model to better capture the relationship between the variables.

Example: Predicting crop yield based on water usage and fertilizer application. If the effect of water usage and fertilizer is additive, the effect of each factor on the yield is independent. If water usage and fertilizer application interact (e.g., water usage has a larger impact when more fertilizer is used), additivity is violated.

Homogeneity of Variance

Also known as homoscedasticity, this assumption states that the variance of the residuals should be consistent across all levels of the independent variables. If the variance is not constant (heteroscedasticity), it can lead to inefficient estimates and biased conclusions.

How to Check: Plot the residuals against the fitted (predicted) values. If the spread of residuals is not constant, it could indicate heteroscedasticity. Statistical tests like Breusch-Pagan or White test can also be used.

What to Do if Violated: If the variance of residuals is unequal, consider transforming the data (e.g., using logarithmic or square root transformations) or applying weighted regression techniques to adjust for the varying residuals.

Example: When predicting test scores based on study hours, if the variance of the residuals is larger for students who studied more hours (i.e., the prediction errors are more spread out for higher study hours), this violates homogeneity of variance.

Proper Functional Form

The relationship between the independent and dependent variables should be correctly specified. This could mean a linear relationship, but it might also involve polynomial or other functional forms depending on the data.

How to Check: Use residual plots to check for patterns that suggest the model is misspecified. The Ramsey RESET test is a formal statistical test that can indicate model misspecification.

What to Do if Violated: If the functional form is incorrect, re-specify the model with a different form (e.g., add polynomial terms for a quadratic relationship or try a logarithmic transformation).

Example: Predicting car fuel efficiency based on car weight. If the relationship is non-linear (e.g., heavier cars might have a disproportionately higher fuel consumption), using a linear model would violate the proper functional form assumption.

No Measurement Errors in Independent Variables

This assumption states that the independent variables should be measured accurately and without errors. Measurement errors can lead to biased coefficient estimates, affecting the validity of the model.

How to Check: Ensure that the data collection methods are robust and that the independent variables are measured with precision.

What to Do if Violated: If measurement errors are detected, consider using errors-in-variables models, which account for the measurement inaccuracies and provide more reliable estimates.

Example: In a health study, if a variable such as "age" is measured inaccurately (e.g., using incorrect birth dates), the model’s coefficient estimates may be biased, leading to misleading conclusions about the relationship between age and health outcomes.

Also Read: Know Why Generalized Linear Model is a Remarkable Synthesis Model!

7. Additional Assumptions of Regression

In addition to the core assumptions of linear regression, there are a few more assumptions that play a significant role in ensuring the validity of your model. These additional assumptions address the structure and relationship of the data points, and violations can lead to issues like overfitting or biased estimates. 

Let’s explore these in more detail.

Balance Between Observations and Predictors

One important assumption is that the number of observations (data points) should exceed the number of predictors (independent variables). If there are too many predictors relative to the number of observations, the model can become overly complex, resulting in overfitting. Overfitting means the model fits the training data very well but performs poorly on new, unseen data because it has essentially "memorized" the training data.

How to Check: Ensure that the number of observations nn is greater than the number of predictors pp. A simple rule of thumb is n>pn > p, where the sample size should always exceed the number of predictors.

What to Do If Violated: If this assumption is violated, you can reduce the number of predictors by using feature selection techniques (e.g., removing variables with low correlation to the dependent variable) or apply dimensionality reduction methods like Principal Component Analysis (PCA) to reduce the number of predictors without losing essential information.

Example: Imagine you are predicting house prices using a dataset with many variables (e.g., number of bedrooms, square footage, age of the house, etc.) but only a small number of observations. The model might fit the training data perfectly but fail to generalize well to new data.

Independence of Each Observation

The independence of observations assumes that each data point is independent and identically distributed (IID). This means there should be no hidden patterns or correlations between the observations themselves. If the observations are not independent (e.g., if data points are related or grouped), the model's assumptions are violated, leading to biased parameter estimates and incorrect conclusions.

How to Check: You can examine residuals for patterns or correlations. Ideally, residuals should appear randomly distributed with no systematic patterns. If patterns exist, it could indicate that the assumption of independence is violated.

What to Do If Violated: If this assumption is violated, you can use techniques like time-series analysis for data with temporal dependencies (e.g., stock prices or sensor data) or mixed-effect models for data that is grouped or clustered (e.g., patients from the same hospital or families). These methods account for the dependency structure in the data and help correct for violations of independence.

Example: In clinical trials, if patient data comes from the same family or is otherwise correlated (e.g., twins or siblings in the same study), treating the data as independent would violate this assumption.

Understanding these assumptions and taking steps to check and address them is critical for building a reliable and accurate linear regression model. Each assumption plays a vital role in ensuring the model is well-specified and its results are valid.

If you want to learn more, enroll in upGrad’s linear regression courses to explore how to apply it effectively and build data-driven solutions. Start learning today!

Also Read: Assumptions of Linear Regression: 5 Assumptions With Examples

So, why exactly are these assumptions so crucial? In this section, we’ll explore the consequences of ignoring them and why they are vital for accurate model predictions.

The Importance of Assumptions in Linear Regression Models

Linear regression models are built on a set of key assumptions. When these assumptions hold true, the model is more likely to produce reliable and accurate results. However, when assumptions are violated, the consequences can be significant. Violating assumptions can lead to biased estimates, unreliable predictions, and ultimately, misleading conclusions.

When the assumptions of linear regression are violated, it can lead to several issues:

• Biased Estimates: If assumptions such as linearity, independence, or homoscedasticity are violated, the coefficient estimates can become biased, meaning they no longer accurately reflect the true relationship between the variables.
• Unreliable Predictions: A model built on incorrect assumptions may fail to generalize well to new data, leading to unreliable predictions when applied to real-world scenarios.
• Misleading Results: Violations of assumptions can lead to incorrect statistical inferences, such as invalid significance tests, confidence intervals, and p-values, making the results of the model unreliable for decision-making.

Here's a table summarizing the possible consequences of violating key assumptions in linear regression:

Also Read: 6 Types of Regression Models in Machine Learning You Should Know About

Ensuring that the assumptions are met allows the regression model to perform optimally and produce valid, interpretable results. When the assumptions hold:

• Accurate Coefficient Estimates: The model provides unbiased, reliable estimates of the relationship between the predictors and the dependent variable.
• Valid Inferences: Statistical tests, such as hypothesis tests and confidence intervals, become meaningful and provide a solid foundation for decision-making.
• Reliable Predictions: The model can be trusted to make accurate predictions on new, unseen data.

Violating these assumptions compromises the model’s effectiveness and reliability. Always checking and addressing assumptions is an essential part of building a robust regression model.

Also Read: 21 Best Linear Regression Project Ideas & Topics For Beginners

Once you understand the importance of assumptions, it's essential to know what to do when they are violated. Let’s walk through strategies to address such violations.

How to Handle Violations of Linear Regression Assumptions

When linear regression assumptions are violated, there are several strategies and techniques you can use to address the issues and ensure your model remains valid. 

Below is a brief outline of the common approaches to handle assumption violations:

• Transformations: If assumptions like linearity or homoscedasticity are violated, applying transformations (e.g., log, square root, or polynomial) to the dependent or independent variables can help. This can stabilize variance or linearize the relationship.
• Robust Methods: In the presence of outliers or heteroscedasticity, robust regression methods (e.g., Huber regression or weighted least squares) can provide more reliable estimates by minimizing the influence of outliers.
• Removing Outliers: Identify and remove outliers or influential data points that distort the regression analysis. Methods like Cook’s distance or Leverage values help detect influential points.
• Adding Interaction Terms: When the assumption of additivity is violated (i.e., there are interactions between predictors), consider adding interaction terms in the model to account for these relationships.
• Respecifying the Model: If the functional form assumption is violated, try re-specifying the model. This could involve adding polynomial terms or using a different type of regression model (e.g., quadratic regression).

Also Read: Regression in Data Mining: Different Types of Regression Techniques

In some cases, linear regression may not be suitable if assumptions are heavily violated. Here are some alternative techniques to consider when linear regression assumptions cannot be met:

Each approach aims to help the model produce reliable, valid estimates while ensuring that you make sound, interpretable conclusions from your analysis.

Also Read: Top 12 Linear Regression Interview Questions & Answers [For Freshers]

Now that you have a comprehensive understanding, it’s time to take your learning to the next level.

How upGrad Can Help You Master Linear Regression and Its Assumptions

upGrad offers a range of programs designed to help you learn linear regression and its assumptions, ensuring you have the skills needed to apply regression models effectively in real-world scenarios. With over 1 million learners and 100+ free courses, you'll gain practical skills to tackle industry challenges while developing job-ready expertise. 

Here are a few relevant courses you can check out:

Also, get personalized career counseling with upGrad to shape your future in AI, or visit your nearest upGrad center and start hands-on training today!

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References:
https://medium.com/@luvvaggarwal2002/linear-regression-in-machine-learning