April 12th, 2024
Demystifying the Assumptions of Multiple Linear Regression
By Josephine Santos · 7 min read
Overview
In the realm of statistical modeling, multiple linear regression (MLR) stands as a cornerstone technique. It allows researchers to predict the value of an outcome variable based on the value of two or more predictor variables. However, like all statistical methods, MLR comes with its own set of assumptions. Ensuring these assumptions hold true is crucial for the validity of the regression model. Let's dive deep into these assumptions and understand their significance.
Assumptions:
The foundational assumption of MLR is that there exists a linear relationship between the dependent and independent variables. This means that changes in the predictor variables correspond to consistent changes in the response variable.
How to Check: Scatterplots can be a visual aid to determine the nature of the relationship. If the data points roughly form a straight line, the linearity assumption holds.
2. Multivariate Normality
This fancy term simply means that the residuals (or errors) from the regression model should follow a normal distribution.
How to Check: A histogram or a Q-Q plot of the residuals can help in visualizing their distribution. For a more formal approach, goodness-of-fit tests like the Kolmogorov-Smirnov test can be applied directly to the residuals.
3. No Multicollinearity
Multicollinearity arises when two or more independent variables in the model are highly correlated, making it difficult to isolate the individual effect of predictors.
How to Check:
- Correlation Matrix: A matrix of Pearson’s bivariate correlations among predictors can be computed. Correlation coefficients greater than 0.80 typically indicate problematic multicollinearity.
- Variance Inflation Factor (VIF): VIF quantifies how much the variance is inflated due to multicollinearity. A VIF value exceeding 10 is usually a red flag.
4. Homoscedasticity
This assumption posits that the variance of the residuals remains constant across all levels of the independent variables. In simpler terms, the spread of residuals should be roughly the same throughout the data.
How to Check: A scatterplot of residuals against predicted values is the go-to method. The absence of any distinct patterns (like a funnel shape) indicates that the data is homoscedastic.
Sample Size and Variable Types
- There should be at least two independent variables, which can be of nominal, ordinal, or interval/ratio type.
- A general rule of thumb is to have at least 20 cases for each independent variable in the analysis.
Addressing Violations
- For multicollinearity, centering the data or removing the problematic variables can help.
- If homoscedasticity is violated, considering non-linear transformations or adding quadratic terms might be the solution.
Conclusion
Multiple linear regression is a powerful tool, but its strength is derived from the validity of its assumptions. Ensuring these assumptions are met not only bolsters the reliability of the model but also enhances the insights drawn from it.
How Julius Can Assist: Navigating the intricacies of multiple linear regression can be daunting. Julius.ai simplifies this process, offering tools and solutions to check and address the assumptions of MLR. Whether it's visualizing the data, computing VIF values, or suggesting remedies for violations, Julius is here to guide you every step of the way. Dive into the world of regression with confidence, knowing Julius has got your back!
