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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Magnetic Regression: Mathematical Example</title>
<link rel="stylesheet" href="style.css">
<style>
body {
padding-left: 40px;
font-family: Arial, sans-serif;
line-height: 1.6;
}
</style>
</head>
<body>
<header>
<h1>Magnetic Regression: Mathematical Breakdown</h1>
</header>
<main>
<section>
<h2>1. Regression to the Mean (Francis Galton)</h2>
<p>
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.
</p>
<h3>Regression Equation:</h3>
<p>
<em>y = β₀ + β₁x + ε</em>
</p>
<p>
Where:<br>
<strong>y</strong>: Dependent variable (e.g., offspring’s trait)<br>
<strong>x</strong>: Independent variable (e.g., parent’s trait)<br>
<strong>β₀</strong>: Intercept (baseline when x = 0)<br>
<strong>β₁</strong>: Slope (change rate of y relative to x)<br>
<strong>ε</strong>: Error term (random variation)
</p>
</section>
<section>
<h2>2. Categorization and Subjective Weights</h2>
<p>
This example shows how subjective weighting of variables influences regression outcomes.
</p>
<h3>Categorized Groups:</h3>
<ul>
<li><strong>Group A:</strong> High parental education (x₁ = 90), high income (x₂ = 80)</li>
<li><strong>Group B:</strong> Moderate parental education (x₁ = 60), moderate income (x₂ = 50)</li>
<li><strong>Group C:</strong> Low parental education (x₁ = 30), low income (x₂ = 20)</li>
</ul>
<h3>Regression Model with Subjective Weights:</h3>
<p>
<em>y = β₀ + β₁(x₁) + β₂(x₂) + ε</em><br>
Weights: <strong>β₁ = 0.6</strong> (education), <strong>β₂ = 0.4</strong> (income)
</p>
<h3>Predicted Outcomes:</h3>
<ul>
<li><strong>Group A:</strong> y = β₀ + (0.6 × 90) + (0.4 × 80) = β₀ + 54 + 32 = β₀ + 86</li>
<li><strong>Group B:</strong> y = β₀ + (0.6 × 60) + (0.4 × 50) = β₀ + 36 + 20 = β₀ + 56</li>
<li><strong>Group C:</strong> y = β₀ + (0.6 × 30) + (0.4 × 20) = β₀ + 18 + 8 = β₀ + 26</li>
</ul>
<p>
This example highlights how prioritizing certain variables (like education over income) can skew predictions and reinforce biases.
</p>
</section>
<nav>
<a href="magnetic-regression.html">← Back to Magnetic Regression</a>
</nav>
</main>
</body>
</html>