Here’s a structured list of 100 chapter titles for Regression Analysis from beginner to advanced, focusing on mathematical concepts and applications:
- Introduction to Regression Analysis: Concepts and Overview
- The Importance of Regression in Statistics and Data Analysis
- The Difference Between Regression and Correlation
- Simple Linear Regression: The Basics
- Assumptions of Linear Regression Models
- The Concept of Dependent and Independent Variables
- The Ordinary Least Squares (OLS) Method
- The Line of Best Fit: Minimizing the Sum of Squared Errors
- Introduction to Residuals and Their Interpretation
- Interpreting the Coefficients in a Simple Linear Regression Model
- Understanding the R-squared Statistic in Regression Analysis
- Hypothesis Testing for Regression Coefficients
- Confidence Intervals for Regression Parameters
- The F-Test for Overall Significance in Regression
- Goodness-of-Fit and Its Role in Regression Analysis
- Simple Linear Regression Model: Mathematical Formulation
- Derivation of the OLS Estimators
- Assumptions of Simple Linear Regression: Linearity, Independence, Homoscedasticity, Normality
- Estimating the Parameters Using Least Squares Method
- Diagnostic Plots: Checking the Assumptions
- Model Validation: Checking Residuals for Normality
- The Role of Outliers and Influential Points in Simple Linear Regression
- Transformation of Variables for Improving Fit
- Assessing Model Accuracy and Predictive Power
- Prediction Intervals in Simple Linear Regression
- The Relationship Between Covariance and Regression
- Interpreting the Slope and Intercept in Practical Contexts
- Multicollinearity in Simple Linear Regression
- Statistical Software for Simple Linear Regression
- Applications of Simple Linear Regression in Real-World Data
- Introduction to Multiple Linear Regression
- Mathematical Formulation of Multiple Linear Regression Models
- Estimating Multiple Regression Parameters Using OLS
- Multicollinearity in Multiple Regression Models
- Assumptions of Multiple Linear Regression: Extensions from Simple Case
- Stepwise Regression: Forward, Backward, and Bidirectional Methods
- Interaction Terms and Their Inclusion in Multiple Regression
- The Influence of Categorical Variables in Multiple Regression
- Model Selection Criteria: AIC, BIC, and Adjusted R-squared
- Multivariate Regression vs. Multiple Regression
- Addressing Multicollinearity: Variance Inflation Factor (VIF)
- Assessing Model Fit: Comparing Models in Multiple Regression
- Residual Diagnostics for Multiple Regression Models
- Regularization Techniques in Multiple Regression: Ridge, Lasso, Elastic Net
- Applications of Multiple Linear Regression in Economics and Business
- Introduction to Generalized Linear Models
- Exponential Family of Distributions in GLMs
- Link Functions in Generalized Linear Models
- Logistic Regression: Binary Outcomes and Logit Link
- Poisson Regression: Modeling Count Data
- The Negative Binomial Regression Model
- Model Interpretation and Coefficients in GLMs
- Estimating Parameters in GLMs Using Maximum Likelihood Estimation (MLE)
- Assumptions and Diagnostics in GLMs
- Assessing Fit in Generalized Linear Models
- Robust Standard Errors in GLMs
- The Role of Overdispersion in Poisson Regression
- Applications of GLMs in Medical and Social Sciences
- Handling Nonlinear Relationships in GLMs
- Comparing GLMs with Traditional Linear Regression Models
- Introduction to Nonlinear Regression Models
- The Concept of Nonlinearity in Regression Analysis
- Estimation Techniques for Nonlinear Regression
- The Gauss-Newton and Levenberg-Marquardt Algorithms
- Evaluating Nonlinear Models: Goodness-of-Fit and Diagnostics
- The Role of Initial Guesses in Nonlinear Regression
- Comparison of Linear and Nonlinear Regression Models
- Parameter Interpretation in Nonlinear Regression
- Fitting Exponential, Logarithmic, and Power Models
- Using Nonlinear Regression for Growth Curves
- The Nonlinear Least Squares Method
- Constraints and Bounds in Nonlinear Regression Models
- Nonlinear Regression in Curve Fitting Problems
- Applications of Nonlinear Regression in Physical Sciences
- Handling Local Minima in Nonlinear Regression
- Polynomial Regression: An Introduction
- Fitting Polynomial Models: Why and When to Use
- Interpreting Polynomial Coefficients
- Overfitting in Polynomial Regression: How to Avoid It
- The Use of Polynomial Regression in Curve Fitting
- Diagnostics and Residual Plots in Polynomial Regression
- Higher-Degree Polynomial Regression: Advantages and Limitations
- Regularization in Polynomial Regression (Ridge and Lasso)
- Using Polynomial Regression for Predicting Nonlinear Trends
- Applications of Polynomial Regression in Engineering and Economics
- The Curse of Dimensionality in High-Degree Polynomial Regression
- Comparing Polynomial and Nonlinear Regression Models
- Smoothing and Regularizing Polynomial Regression Models
- Polynomial Regression in Data Science and Machine Learning
- Extensions of Polynomial Regression for Multiple Variables
- Mixed-Effects Models: Combining Fixed and Random Effects
- Bayesian Regression Analysis: Principles and Methods
- Markov Chain Monte Carlo (MCMC) Methods in Regression
- Nonparametric Regression: Kernel Smoothing and Splines
- Quantile Regression: Estimating Conditional Quantiles
- Robust Regression Methods: Handling Outliers
- Ridge and Lasso Regression: Techniques for Regularization
- Principal Component Regression (PCR) and Partial Least Squares (PLS)
- High-Dimensional Regression and Variable Selection Techniques
- The Future of Regression Analysis: Machine Learning and Beyond
This list takes the reader from basic concepts like simple and multiple linear regression through to more advanced topics like Bayesian regression, nonparametric methods, and machine learning techniques. Each chapter is designed to focus on the mathematical underpinnings and practical applications of regression analysis in various fields, such as economics, engineering, and data science.