22.3 Regression Analysis

Regression analysis is a powerful statistical method used to understand and quantify the relationships between a dependent variable and one or more independent variables. It is a fundamental technique in predictive modeling, helping to forecast future outcomes, identify trends, and evaluate the influence of various factors on a phenomenon.

Purpose of Regression Analysis

Regression analysis serves several key purposes:

• Prediction: To estimate the value of a dependent variable based on the values of one or more independent variables.
• Relationship Understanding: To determine the strength, direction, and type of the relationship between variables.
• Impact Evaluation: To assess how changes in independent variables affect the dependent variable, especially when multiple factors are involved.

Types of Regression

Simple Linear Regression

This is the most basic form of regression, used when there is a linear relationship between a single dependent variable and a single independent variable.

Equation:

Y = a + bX

Where:

• Y: The dependent variable (the outcome you are trying to predict).
• X: The independent variable (the predictor variable).
• a: The intercept. This is the predicted value of Y when X is zero. It represents the baseline value of the dependent variable.
• b: The slope. This indicates the average change in the dependent variable (Y) for each one-unit increase in the independent variable (X). It quantifies the strength and direction of the linear relationship.

Example: Predicting a student's exam score (Y) based on the number of hours they studied (X).

Multiple Linear Regression

This type of regression is used when there are two or more independent variables that influence a single dependent variable. It allows for a more comprehensive understanding of how multiple factors contribute to an outcome.

Equation:

Y = a + b1X1 + b2X2 + ... + bnXn

Where:

• Y: The dependent variable.
• X1, X2, ..., Xn: The independent variables.
• a: The intercept, representing the predicted value of Y when all independent variables are zero.
• b1, b2, ..., bn: The coefficients (slopes) for each independent variable. Each bi represents the average change in Y for a one-unit increase in Xi, holding all other independent variables constant.

Example: Predicting a house's price (Y) based on its size (X1), number of bedrooms (X2), and distance from the city center (X3).

Non-Linear Regression

This type of regression is employed when the relationship between the dependent and independent variables is not a straight line but rather follows a curved or more complex pattern. It is used when the assumptions of linearity are violated.

Characteristics:

• The relationship between variables cannot be adequately represented by a straight line.
• Models can take various forms, such as polynomial, exponential, or logarithmic.
• The mathematical formulation of non-linear regression is more complex than linear regression.

Example: Modeling the growth of a population over time, where growth tends to slow down as it approaches a carrying capacity.

Key Terms in Regression Analysis

• Dependent Variable (Y): The variable whose value is being predicted or explained. It is the outcome of interest.
• Independent Variable (X): The variable(s) used to predict or explain the dependent variable. These are the predictor or explanatory variables.
• Intercept (a): The expected value of the dependent variable when all independent variables are equal to zero. It represents the starting point of the relationship.
• Slope (b): The coefficient associated with an independent variable. It quantifies the average change in the dependent variable for a one-unit increase in that independent variable, assuming all other variables are held constant.
• R-squared (Coefficient of Determination): A statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. A higher R-squared indicates a better fit of the model to the data.

Regression Analysis Interview Questions

• What is regression analysis and why is it used?
• What is the difference between simple and multiple linear regression?
• How do you interpret the coefficients in a regression model?
• What are the assumptions of linear regression? (e.g., linearity, independence of errors, homoscedasticity, normality of errors)
• What is the difference between linear and non-linear regression?
• How do you evaluate the performance of a regression model? (e.g., R-squared, Adjusted R-squared, Mean Squared Error, Root Mean Squared Error, P-values)
• What does R-squared tell you in regression analysis?
• How do you handle multicollinearity in multiple regression? (e.g., Variance Inflation Factor (VIF), removing correlated variables)
• Can regression be used for classification problems? Why or why not? (Regression predicts continuous values; classification predicts discrete categories. Logistic regression is a common technique for binary classification.)
• How do you perform regression analysis using Python or R? (e.g., using libraries like scikit-learn or statsmodels in Python, or lm() function in R).