Multiple Regression Interpretation
Interpret the coefficients, R-squared, and significance of a multiple regression model predicting house prices from square footage and number of bedrooms.
Problem Scenario
A real estate analyst fits a multiple regression model to predict house sale price (in thousands of dollars) based on square footage and number of bedrooms, using data from 50 recent sales. The regression output is: y-hat = 25.3 + 0.112(SqFt) + 8.45(Bedrooms). R-squared = 0.834, Adjusted R-squared = 0.827. For SqFt: coefficient = 0.112, SE = 0.018, t = 6.22, p < 0.001. For Bedrooms: coefficient = 8.45, SE = 3.21, t = 2.63, p = 0.011. Overall model: F(2, 47) = 118.1, p < 0.001.
Given Data
Requirements
- Interpret each regression coefficient in context
- Explain R-squared and evaluate overall model significance
- Predict the price of a 1,800 sq ft house with 3 bedrooms
Solution
Step 1:
Interpret the SqFt coefficient: Holding the number of bedrooms constant, each additional square foot of living space is associated with an increase of $112 in the predicted sale price (0.112 thousand dollars). This is statistically significant (p < 0.001).
Step 2:
Interpret the Bedrooms coefficient: Holding square footage constant, each additional bedroom is associated with an increase of $8,450 in the predicted sale price (8.45 thousand dollars). This is statistically significant (p = 0.011).
Step 3:
Interpret the intercept: When both SqFt = 0 and Bedrooms = 0, the model predicts a price of $25,300. This value is not practically meaningful (a house with 0 sq ft does not exist) but is needed mathematically to anchor the regression plane.
Step 4:
Interpret R-squared: 83.4% of the variation in house sale prices is explained by the combination of square footage and number of bedrooms. Adjusted R-squared = 82.7%, which accounts for the number of predictors in the model.
Step 5:
Evaluate overall model significance: The F-test (F(2,47) = 118.1, p < 0.001) indicates that the model as a whole is highly significant. At least one predictor is significantly related to house price.
Step 6:
Prediction: For a 1,800 sq ft house with 3 bedrooms: y-hat = 25.3 + 0.112(1800) + 8.45(3) = 25.3 + 201.6 + 25.35 = 252.25. Predicted sale price = $252,250.
Final Answer
Each additional square foot adds approximately $112 to the price (p < 0.001), and each additional bedroom adds approximately $8,450 (p = 0.011), holding other variables constant. R-squared = 0.834, meaning the model explains 83.4% of the variation in house prices. Predicted price for an 1,800 sq ft, 3-bedroom house: $252,250.
Key Takeaways
- ✓In multiple regression, each coefficient represents the change in the response variable for a one-unit increase in that predictor, holding all other predictors constant. This "holding constant" interpretation is crucial.
- ✓Adjusted R-squared penalizes for adding more predictors and is preferred over R-squared for comparing models with different numbers of predictors.
- ✓The overall F-test assesses whether the model as a whole explains a significant amount of variance, while individual t-tests assess each predictor's unique contribution.
Common Errors to Avoid
- ✗Interpreting a coefficient without the "holding other variables constant" qualifier. In multiple regression, each coefficient is a partial effect, not a total effect.
- ✗Assuming that a non-significant predictor is unimportant. It might be correlated with another predictor (multicollinearity), which can inflate the standard error and reduce the t-statistic.
- ✗Using the model to predict far outside the range of the data (e.g., predicting the price of a 10,000 sq ft mansion when the data only includes houses up to 3,000 sq ft).
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Common questions about this problem type
R-squared always increases (or stays the same) when you add more predictors, even if those predictors are not meaningful. Adjusted R-squared penalizes for the number of predictors: it increases only if a new predictor improves the model more than would be expected by chance. When comparing models with different numbers of predictors, use adjusted R-squared.
Calculate the Variance Inflation Factor (VIF) for each predictor. A VIF above 5 (some say 10) suggests problematic multicollinearity. You can also examine the correlation matrix of the predictors. If two predictors are highly correlated (|r| > 0.8), multicollinearity may be present.