Which of the statement below is true about the coefficient of determination, r2?

Question

BlackTom AI · Accepted Answer

R² is defined as the fraction of the total variance in the observed outcome explained by the model: R² = 1 − SSE/SST, where SST = Σ(y_i − ȳ)² (total sum of squares) and SSE = Σ(y_i − ŷ_i)² (sum of squared errors).

1) Relevant knowledge points and concepts
- R² measures explained variance, not correctness of model form, causation, or predictive validity. 
- R² is computed as 1 − SSE/SST, so it can equal 1 (perfect fit), be between 0 and 1 for typical OLS with an intercept when SSE ≤ SST, or be negative if SSE > SST (e.g., bad model, no intercept, or different R² definitions). 
- Goodness of fit assessment requires residual analysis, validation (cross‑validation), checks for overfitting, influential points, and model assumptions.

2) Step-by-step reasoning
- Write formula: R² = 1 − SSE/SST.
- Example 1 (good apparent fit): SST = 100, SSE = 10 → R² = 1 − 10/100 = 0.90 (high R²). But inspect residuals: if residuals show pattern or outliers drive fit, model still poor despite high R².
- Example 2 (perfect fit): SSE = 0 → R² = 1 − 0/100 = 1 (perfect fit).
- Example 3 (worse than predicting mean): SST = 100, SSE = 120 → R² = 1 − 120/100 = −0.20 (negative R² indicates model fits worse than using the mean).

3) Calculations (intermediate steps shown above)
- SST = 100, SSE = 10 → R² = 1 − 10/100 = 0.90.
- SST = 100, SSE = 0 → R² = 1 − 0 = 1.
- SST = 100, SSE = 120 → R² = 1 − 120/100 = −0.20.

4) Why the correct answer is correct
Answer b states that a reasonable R² plus careful model assessment is required. This is correct because R² alone does not check model assumptions, overfitting, influential points, or predictive performance; complementary diagnostics are necessary.

5) Why other options are incorrect
- a (High R² alone justifies goodness): False — high R² can be misleading (overfitting, nonlinearity, outliers).  
- c (R² = 1 shows a very poor fit): False — R² = 1 indicates a perfect fit (SSE = 0).  
- d (R² < 0 shows a good fit): False — negative R² means the model is worse than the mean predictor.

6) Conclusion
Therefore answer b is correct: a reasonable R² together with careful model assessment is required to judge goodness of fit. This matches the provided answer.

Which of the statement below is true about the coefficient of determination, r2?单项选择题

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