📘 Section 11.7 – Adequacy of the Regression Model

1. Overview

Fitting a regression model requires verifying assumptions about error terms, model structure, and variance. Section 11.7 discusses how to examine model adequacy using residual analysis and the coefficient of determination (R²).

2. Key Assumptions in Simple Linear Regression

• Errors are uncorrelated random variables with mean zero and constant variance.
• Errors are normally distributed (for hypothesis testing and confidence intervals).
• The model is correctly specified (i.e., linear if a linear model is used).

3. Residual Analysis (Section 11.7.1)

Residuals (eᵢ = yᵢ − ŷᵢ) help detect non-normality, non-constant variance, and model misspecification. Residuals should ideally appear as random scatter with no pattern when plotted against predicted values or x-values.

Key Diagnostic Plots:

• Residuals vs. Predicted Values (ŷᵢ) → Random scatter indicates good fit.
• Residuals vs. x-values → No pattern or curvature.
• Normal Probability Plot → Linear shape suggests normality.
• Standardized Residuals → ~95% should lie between −2 and +2.

Common Residual Patterns:

• (a) Random scatter – Good fit
• (b) Funnel shape – Non-constant variance
• (c) Double bow – Unequal variance
• (d) Curved – Model misspecification

Tips:

• Use transformations (e.g., √y, ln(y), 1/y) if residuals show unequal variance.
• Don’t discard outliers without investigation—they might be meaningful.

4. Example – Oxygen Purity Residuals

Model: ŷ = 74.283 + 14.947x

• Normal probability plot: Residuals align along a straight line → Normality assumed.
• Residuals vs. predicted values and x-values: No visible pattern → Model is adequate.

Sample data point: At x = 1.02, y = 89.05, predicted ŷ = 89.53 → residual = −0.48

Another example: At x = 1.55, y = 99.42, predicted ŷ = 97.45 → residual = 1.97

5. Coefficient of Determination (R²) – Section 11.7.2

• Measures how well the regression model explains variability in y.
• Ranges from 0 to 1. Higher R² = better model fit.
• Example: R² = 152.13 / 173.38 ≈ 0.877 → 87.7% of variation in y is explained.

Limitations & Misconceptions:

• R² always increases when more variables are added – use adjusted R² for fair comparison.
• High R² ≠ good model – poor fits can still yield high R².
• R² does not imply a steep slope.
• R² does not ensure accurate future predictions.