7.3.3 Applications with Examples - III¶
Introduction¶
- Objective: This session expands on multiple linear regression (MLR) applications by incorporating three independent variables—first income, monthly payment, and utilities—to analyze household debt.
- Context: Manjula Naik continues her analysis using a more complex MLR model to understand how various financial factors contribute to household debt.
Enhancing the Model¶
- Previous Findings: Previous models with two variables provided adjusted R-squared values of 0.67 and 0.45, indicating significant but varying explanations of debt variability.
- New Approach: Incorporating first income, monthly payment, and utilities into one comprehensive model to capture a broader spectrum of influencing factors.
Methodology¶
- Data Preparation:
- Variables: Debt (Y), first income (X1), monthly payment (X2), and utilities (X3).
- Location in Spreadsheet: Debt in column K, first income in column H, monthly payments in column I, and utilities in column J.
Regression Analysis¶
- Tool Used: Analysis ToolPak in Excel for regression computation.
- Model Specification: A full MLR model with debt as a function of first income, monthly payment, and utilities.
- Statistical Outputs:
- R-squared: Improved to 0.708, indicating that 70.8% of the variability in household debt is now explained by the model.
- Adjusted R-squared: Slightly lower than R-squared, around 0.70, indicating minimal overfitting despite the increased number of predictors.
- Standard Error: Reduced to 18699, suggesting increased precision in the model's estimates.
Model Assessment¶
- ANOVA Results:
- SSR: Increased significantly, indicating better explanation of variance by the model.
- SSE: Decreased, which aligns with the lower standard error and higher explanatory power.
- F-Statistic: 401.6, with a very low p-value, strongly supporting the model's overall significance.
Coefficient Analysis¶
- Interpretation of Coefficients:
- B1 (First Income): Each unit increase in first income increases debt by 0.017 units.
- B2 (Monthly Payment): Each unit increase in monthly payment increases debt by 0.96 units.
- B3 (Utilities): Each unit increase in utilities increases debt by 157.95 units.
- Significance: All coefficients are significant, indicating a meaningful contribution to the model.
Residuals and Diagnostics¶
- Residuals: Standardized residuals mostly within the range of \(\pm3\), suggesting well-behaved errors.
- Residual Plots: Random patterns across all independent variables and predicted values, affirming the model's adequacy.
Conclusion¶
- Model Validation: The enhanced MLR model significantly improves understanding and prediction of household debt dynamics.
- Future Directions: Encouraged to explore additional datasets and further refine modeling techniques to handle complex business scenarios.