4. Regression
Establishing mathematical dependency models to predict values.
While correlation simply describes how two variables move together, Regression allows us to predict the value of one variable based on another.
A. The Linear Regression Model
Uses the Principle of Least Squares to fit a straight line by minimizing the sum of squared residuals.
Y = β₀ + β₁X + ε
- Y: Dependent / Response variable (e.g., oxygen consumption).
- X: Independent / Predictor variable (e.g., running speed).
- β₀: The Y-intercept (value of Y when X = 0).
- β₁: Regression Coefficient (Slope). Change in Y per unit increase in X.
- ε: Residual/Error representing natural variation.
B. Mathematical Properties
Relationship with Correlation (r):
r is the geometric mean of the two regression coefficients (bᵧₓ and bₓᵧ):
r = ±√(bᵧₓ · bₓᵧ)
- Coefficients must share the same sign (+ or -) as r.
- If bᵧₓ > 1, then bₓᵧ must be < 1 (since r² ≤ 1).
C. Core Distinction Table
| Feature | Correlation | Regression |
|---|---|---|
| Purpose | Measures strength/direction of association. | Predicts the value of a dependent variable. |
| Nature of Variables | Symmetric; swapping X and Y changes nothing. | Asymmetric; X is cause/predictor, Y is effect/response. |
| Units | Unitless (dimensionless scale). | Expressed in actual physical units (e.g., mg of drug/kg body mass). |
Professor-Level Analytical Question
Synthesize Correlation and Regression concepts.
Scenario:
A marine biologist studies the relationship between water salinity (X) and the respiration rate of a specific crab species (Y).
- The regression coefficient of Y on X (bᵧₓ) is -0.5.
- The regression coefficient of X on Y (bₓᵧ) is -0.8.
Q1: What is the exact value of the correlation coefficient (r) between salinity and respiration rate?
Q2: Biologically speaking, what does the sign of this r value tell you about the crabs' physiological adaptation?
Hint: Remember that the sign of r is the same as the sign of the regression coefficients. Use the geometric mean formula.