Module 6 · Regression & Correlation · Topic 6
Coefficient of Determination (R²). R² is the proportion of the total variation in y that is explained by the linear relationship with x. For the usual in-sample least-squares regression with an intercept and nonconstant y, it ranges from 0 (no explanatory power) to 1 (perfect prediction).
Relationship to r. For simple linear regression, R² = r². If r = 0.8, then R² = 0.64, meaning 64% of the variation in y is explained by x.
R² = 1 − (SSE / SST), where SST = Σ(yᵢ − ȳ)² and SSE = Σ(yᵢ − ŷᵢ)²
yhat = a + bx — Line that minimizes the sum of squared residuals.r = sum((x_i-xbar)(y_i-ybar)) / sqrt(sum((x_i-xbar)^2)sum((y_i-ybar)^2)) — Measures direction and strength of a linear association.A regression of test scores on study hours gives R² = 0.72. This means 72% of the variation in test scores can be explained by the linear relationship with study hours. The remaining 28% is unexplained by this linear model.
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