📘 Section 11.9 – Regression on Transformed Variables
🔍 Why Transform Variables?
Linear regression assumes a straight-line relationship between x and y. If the data are non-linear, applying linear regression without transformation can lead to:
- Biased parameter estimates
- Low R² (explained variance)
- Non-random residuals (model inadequacy)
🧠 When to Suspect Nonlinearity?
- Scatterplot of
x vs. y is curved.
- Residuals vs. fitted values show a bow or funnel shape.
- Theoretical model suggests exponential or inverse relationships.
🔁 Common Transformations
| Original Form | Transformation | Resulting Linear Form |
|---|
| Exponential: Y = β₀e^{β₁x}ε | ln(Y) | ln Y = ln β₀ + β₁x + ln ε |
| Inverse: Y = β₀ + β₁(1/x) + ε | z = 1/x | Y = β₀ + β₁z + ε |
| Reciprocal Exponential: Y = 1 / exp(β₀ + β₁x + ε) | ln(1/Y) | ln Y* = β₀ + β₁x + ε |
| Power: Y = β₀x^β₁ε | log-log | ln Y = ln β₀ + β₁ ln x |
📌 Tip: Always check scatterplots and residual plots before and after transformation to verify model improvement.
📊 Example – Windmill Power
A research engineer wants to model the relationship between wind velocity (x) and DC output (y).
Before Transformation (Linear Fit)
After Transformation (1/x)
✅ Summary
- Always check residual plots for signs of model inadequacy.
- Try transformations if the relationship is not linear.
- Compare R² and residual plots before and after transformation.
- The best transformation is often guided by the shape of the scatterplot and theory.
Transformation = Rescue when linear regression fails!