Overview

Regression Analysis is the crystal ball of statistics. It allows us to predict the future (or the unknown) based on relationships between variables. It draws the “line of best fit.”

Core Idea

The core idea is relationship. How does $X$ affect $Y$? If I study one more hour ($X$), how much will my grade increase ($Y$)?

Formal Definition

Linear Regression: $Y = mx + b$.

  • $Y$: Dependent Variable (Outcome).
  • $X$: Independent Variable (Predictor).
  • $m$: Slope (Effect size).
  • $b$: Intercept (Starting point).

Intuition

  • The Scatterplot: Imagine dots on a graph. Regression draws the line that goes through the middle of the cloud, minimizing the distance to all points.
  • Correlation ($r$): How tight are the dots to the line? (1 = Perfect line, 0 = Random cloud).
  • $R^2$: How much of the variation is explained? ($R^2 = 0.8$ means “We understand 80% of what’s going on”).

Examples

  • Real Estate: Predicting house prices based on Square Footage, Bedrooms, and Zip Code.
  • Health: Predicting life expectancy based on Smoking, Diet, and Exercise.

Common Misconceptions

  • Misconception: Regression proves causation.
    • Correction: Ice cream sales correlate with shark attacks (both go up in summer). Regression finds the link, but not the cause.
  • Misconception: You can extrapolate forever.
    • Correction: If you grow 2 inches a year as a kid, regression predicts you’ll be 10 feet tall by age 40.

Applications

  • Business: Sales forecasting.
  • Algorithmic Trading: Predicting stock prices.

Criticism and Limitations

  • Overfitting: Making a model so complex it fits the noise, not the signal.
  • Linearity: The world is often curved, not linear.

Further Reading

  • The Art of Statistics by David Spiegelhalter
  • Introduction to Statistical Learning by James et al.