Unit 2960: Comprehensive Regression Analysis

1. Introduction to Regression Analysis

• Definition and purpose of regression analysis
• Importance in statistical modeling and decision making
• Key concepts: dependent and independent variables
• Overview of types of regression models

2. Simple Linear Regression

2.1 Concept and Model

• Definition and mathematical formulation
• Graphical representation of regression line

2.2 Assumptions

• Linearity
• Independence of errors
• Homoscedasticity
• Normality of residuals

2.3 Fitting the Regression Line

• Least squares estimation method
• Calculating slope and intercept

2.4 Interpretation of Results

• Meaning of coefficients
• Confidence intervals and hypothesis testing for slope

2.5 Evaluating Goodness of Fit

• R-squared and its interpretation
• Residual standard error

3. Multiple Linear Regression

3.1 Model Overview

• Extending simple regression to multiple predictors
• Model specification and notation

3.2 Coefficient Estimation

• Multiple least squares estimation
• Interpretation of coefficients in multiple regression

3.3 Model Interpretation

• Partial regression coefficients
• Effect of each independent variable controlling for others

3.4 Assessing Overall Model Significance

• F-test for overall regression
• Adjusted R-squared

4. Assumptions of Regression Analysis

• Detailed explanation of key assumptions:
  • Linearity
  • Independence
  • Homoscedasticity
  • Normality of residuals
  • Multicollinearity
• Importance of validating assumptions

5. Model Evaluation and Selection

5.1 Evaluation Metrics

• R-squared vs Adjusted R-squared
• Significance testing of coefficients

5.2 Residual Analysis

• Patterns in residuals
• Identifying heteroscedasticity

5.3 Model Selection Techniques

• Comparing nested models
• Using information criteria (AIC, BIC)
• Cross-validation overview

6. Residual Analysis

• Definition and calculation of residuals
• Role of residuals in model validation
• Detecting outliers and influential points (Cook's Distance, leverage)
• Diagnosing heteroscedasticity

7. Interaction Effects in Regression

7.1 Concept of Interaction

• Definition and examples

7.2 Including Interaction Terms

• Constructing interaction variables
• Model specification with interactions

7.3 Interpretation of Interaction Effects

• Understanding changes in slopes
• Graphical visualization

8. Polynomial Regression

8.1 Introduction to Non-Linearity

• Limitations of linear models

8.2 Polynomial Model Formulation

• Adding polynomial terms (quadratic, cubic, etc.)

8.3 Fitting and Interpretation

• Estimating coefficients
• Interpreting curvature and shape

8.4 Advantages and Considerations

• Flexibility vs overfitting

9. Multicollinearity and its Effects

9.1 Understanding Multicollinearity

• Definition and causes

9.2 Consequences

• Inflated variances of coefficient estimates
• Unstable estimates and interpretation difficulties

9.3 Detection Methods

• Variance Inflation Factor (VIF)
• Condition index

9.4 Remedies and Strategies

• Variable selection
• Combining variables
• Principal Component Regression

10. Practical Applications of Regression Analysis

• Case studies from:
  • Economics (e.g., demand forecasting)
  • Marketing (e.g., sales prediction)
  • Healthcare (e.g., risk factor analysis)
  • Social sciences (e.g., behavioral studies)
• Hands-on examples with datasets
• Interpretation of results in context