Why Do We Assume the Residuals in Ordinary Linear Regression Come From a Normal Distribution?

In ordinary linear regression, we often assume that the residuals—the differences between observed values and the values predicted by the model—are normally distributed. This assumption is rooted in theoretical principles, practical considerations, and the need for valid statistical inference. This article will delve into the reasons behind this assumption and explore the implications of its being violated.

Theoretical Foundations

Central Limit Theorem is a cornerstone in the justification of the normality assumption in regression. According to the Central Limit Theorem, the distribution of the sum or average of a large number of independent random variables tends to be normally distributed, regardless of the individual distributions of those variables. In regression, the residuals can be thought of as the result of many independent factors affecting the dependent variable. As the sample size increases, the impact of individual factors averages out, making the residuals more likely to follow a normal distribution.

Practical Considerations

Statistical Inference Methods such as hypothesis testing and confidence intervals for regression coefficients rely on the assumption of normally distributed residuals. If this assumption holds true, we can employ t-tests and F-tests to evaluate the significance of the predictors and the overall model. This makes these tests valid and reliable for making inferences about the model parameters. Without the normality assumption, the results from these tests would be suspect.

Model Fit is another crucial consideration. Assuming normality helps in assessing how well the linear regression model fits the data. If the residuals are normally distributed, it suggests that the model captures the underlying relationship well. Deviations from normality can indicate issues such as model misspecification or the presence of outliers. Outliers can significantly affect the regression model, leading to biased or inefficient parameter estimates.

Homogeneity of Variance

The assumption of normally distributed residuals is often paired with the assumption of homoscedasticity—the constant variance of residuals. When both assumptions are met, the model is more robust and provides reliable estimates. If the variance of residuals changes with the level of the independent variables (heteroscedasticity), it can lead to inefficient and biased estimates, making the model less reliable.

Practical Realities

In many real-world applications, especially with large datasets, the residuals tend to approximate a normal distribution due to the averaging effect of multiple influences on the dependent variable. In these cases, the normality assumption is a reasonable approximation, making the model more practical and straightforward to apply. However, it’s important to note that this assumption may not hold in smaller datasets or highly nonlinear relationships.

Diagnostic Tools

Diagnostic tools such as Q-Q plots and residual plots are used to assess the normality of residuals. These tools allow us to visually inspect the distribution of residuals and detect deviations from normality. If the residuals do not follow a normal distribution, it may prompt further investigation into the model or the data. This could involve checking for and addressing issues such as non-linear relationships, outliers, or model misspecification.

For instance, a Q-Q plot compares the distribution of residuals to a normal distribution. If the points in a Q-Q plot lie close to a straight line, the residuals are likely normally distributed. Deviations from this line indicate potential issues with the normality assumption. Similarly, a residual plot can help identify patterns or deviations in the residuals, which could suggest problems with the model.

Conclusion

Assuming normally distributed residuals in linear regression is rooted in theoretical principles, practical considerations, and the need for valid statistical inference. While this assumption is generally reasonable in many real-world scenarios, it is essential to check and verify this assumption in practice. Violations of the normality assumption can affect the validity of the regression results, leading to incorrect conclusions.

In conclusion, while the normality assumption is a standard practice in linear regression, it is critical to assess and validate this assumption to ensure the robustness and reliability of the model. By doing so, we can enhance the validity of our statistical inferences and model predictions.