1. Regression to the Mean (Francis Galton)
Regression to the mean explains how data points tend to move toward the average over time. Francis Galton first used this principle to study hereditary traits.
Regression Equation:
y = β₀ + β₁x + ε
Where:
y: Dependent variable (e.g., offspring’s trait)
x: Independent variable (e.g., parent’s trait)
β₀: Intercept (baseline when x = 0)
β₁: Slope (change rate of y relative to x)
ε: Error term (random variation)
2. Categorization and Subjective Weights
This example shows how subjective weighting of variables influences regression outcomes.
Categorized Groups:
- Group A: High parental education (x₁ = 90), high income (x₂ = 80)
- Group B: Moderate parental education (x₁ = 60), moderate income (x₂ = 50)
- Group C: Low parental education (x₁ = 30), low income (x₂ = 20)
Regression Model with Subjective Weights:
y = β₀ + β₁(x₁) + β₂(x₂) + ε
Weights: β₁ = 0.6 (education), β₂ = 0.4 (income)
Predicted Outcomes:
- Group A: y = β₀ + (0.6 × 90) + (0.4 × 80) = β₀ + 54 + 32 = β₀ + 86
- Group B: y = β₀ + (0.6 × 60) + (0.4 × 50) = β₀ + 36 + 20 = β₀ + 56
- Group C: y = β₀ + (0.6 × 30) + (0.4 × 20) = β₀ + 18 + 8 = β₀ + 26
This example highlights how prioritizing certain variables (like education over income) can skew predictions and reinforce biases.