. Assumptions of Linear Regression

Linear regression relies on key statistical assumptions. Violating these can lead to biased or misleading results.

1.1. Linearity

• The relationship between the independent variables (X) and the dependent variable (Y) should be linear.
• Check: Use scatter plots to visualize relationships. If the relationship appears curved, consider polynomial regression or transformations (e.g., log, square root).

1.2. Independence of Errors

• Residuals (errors) should not be correlated.
• Check: Use the Durbin-Watson test for detecting autocorrelation.
• Fix: If autocorrelation exists, consider time-series models or adding lag variables.

1.3. Homoscedasticity (Constant Variance of Errors)

• The variance of residuals should remain constant across all levels of the independent variables.
• Check: Plot residuals vs. predicted values (should look random).
• Fix: Use log transformation, Weighted Least Squares (WLS), or heteroskedasticity-robust standard errors.

1.4. Normality of Residuals

• Residuals should be normally distributed (especially for small datasets).
• Check: Use a Q-Q plot or Shapiro-Wilk test.
• Fix: Apply transformations (log, square root, etc.) if residuals are skewed.

1.5. No Perfect Multicollinearity

• Independent variables should not be highly correlated with each other.
• Check: Use Variance Inflation Factor (VIF) (values > 10 indicate multicollinearity).
• Fix: Remove or combine correlated variables, use Principal Component Analysis (PCA), or Ridge Regression.

2. Best Practices for Building Linear Regression Models

2.1. Feature Selection

• Avoid using too many irrelevant predictors (causes overfitting).
• Use stepwise selection, Lasso regression, or domain knowledge to select the best predictors.

2.2. Scaling Features

• Some models benefit from standardization (mean = 0, variance = 1).
• Normalize variables especially when using regularization (Lasso, Ridge).

2.3. Handling Outliers

• Outliers can distort the regression coefficients.
• Check: Use box plots or leverage diagnostics like Cook’s Distance.
• Fix: Transform data, remove outliers if justified, or use robust regression.

2.4. Splitting Data for Validation

• Always split data into training and test sets (e.g., 80-20 split).
• Use cross-validation (e.g., k-fold) to assess model performance.

3. Evaluating Model Performance

3.1. Metrics for Linear Regression

• R² (Coefficient of Determination): Measures how well independent variables explain the dependent variable. A high R² (close to 1) is good but does not mean causation.
• Adjusted R²: Adjusts for the number of predictors, preventing overfitting.
• Mean Squared Error (MSE) / Root Mean Squared Error (RMSE): Measures prediction error.
• Mean Absolute Error (MAE): A simpler metric that measures absolute differences.

3.2. Avoiding Overfitting

• Too many predictors → High R² but poor generalization.
• Use regularization techniques (Ridge, Lasso) to penalize unnecessary complexity.
• Compare train vs. test performance to check for overfitting.

4. Interpreting Results Properly

• P-values (< 0.05): Check significance of predictors.
• Coefficient Signs & Magnitudes: Do they align with domain knowledge?
• Confidence Intervals: Provide range estimates for coefficients.