Coefficient of Determination Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Two models predict house prices: Model 1 (size only): R2=0.60R^2=0.60. Model 2 (size + neighborhood + age): R2=0.85R^2=0.85. Explain what the increase in R2R^2 means and what caution should be applied with multi-variable R2R^2.

Solution

  1. 1
    Model 1 explains 60% of price variation; Model 2 explains 85% — 25% more variation explained by adding neighborhood and age
  2. 2
    Adding predictors almost always increases R2R^2 (even random noise predictors increase it slightly)
  3. 3
    Caution: adjusted R2R^2 penalizes for adding predictors: Radj2=1−(1−R2)(n−1)n−k−1R^2_{adj} = 1 - \frac{(1-R^2)(n-1)}{n-k-1}
  4. 4
    Use adjusted R2R^2 for model comparison when the number of predictors differs

Answer

Model 2 explains 25% more variation. Use adjusted R2R^2 to avoid inflation from adding predictors.
Adding predictors always increases R2R^2 regardless of their true relationship with y. Adjusted R2R^2 penalizes for model complexity. If adjusted R2R^2 decreases when a predictor is added, that predictor does not improve the model enough to justify its inclusion.

About Coefficient of Determination

The proportion of the total variation in the response variable yy that is explained by the linear relationship with the explanatory variable xx. It equals the square of the correlation coefficient: r2r^2.

Learn more about Coefficient of Determination →

More Coefficient of Determination Examples

Example 1 medium

A regression model has [formula] (total variation) and [formula] (unexplained variation). Calculate

Example 3 easy

The correlation between study hours and test score is [formula]. Calculate [formula] and interpret i

Example 4 hard

A model has [formula]. A researcher concludes 'the model is perfect and ready for deployment.' Ident

← All Coefficient of Determination Examples Coefficient of Determination Concept