8.2.1 Interpretation of Coefficients¶
Regression Equation:¶
Where:
- β0: Intercept.
- β1: Coefficient for x1 (quantitative variable).
- β2: Coefficient for x2 (categorical dummy variable).
Interpreting Coefficients with a Dummy Variable¶
Impact of β2:¶
Example with Basavaraja's Dataset:¶
-
Regression Equation:
-
x1: Price of the computer.
- x2 = 0: Dell.
-
x2 = 1: Lenovo.
-
Predictions:
-
Interpretation: Lenovo scores 3.93 points higher on average than Dell.
Residual Analysis¶
- Residuals vs. x1 (price): Appear evenly distributed around the x-axis.
- Residuals vs. x2 (dummy variable): Clustered at the two dummy values (0 or 1).
- No major outliers: A few residuals above or below ±2.
When There are More Categories¶
- Scenario: Three brands (Lenovo, Dell, Asus).
- Use k - 1 dummy variables for \( k \) categories:
Regression Equation:¶
Interpretation of Coefficients:¶
- β0: Average satisfaction score for Asus.
- β1: Difference between Lenovo and Asus.
- β2: Difference between Dell and Asus.
Alternative Baseline:¶
- Change the reference category.
- Example: Use Lenovo as the baseline.
- β0: Average score for Lenovo.
- β1: Difference between Asus and Lenovo.
- β2: Difference between Dell and Lenovo.
Simplified Explanation¶
What is Happening?¶
- Categorical data (e.g., brands) must be converted into numbers (0s and 1s) to fit into the regression model.
Why Use Dummy Variables?¶
- Dummy variables allow categorical factors to be analyzed in regression.
How Do We Interpret Results?¶
- Each category (e.g., Dell or Lenovo) changes the regression equation slightly.
- Example: Lenovo gives an average satisfaction score \( 3.93 \) points higher than Dell.
What Happens with 3+ Categories?¶
- Use multiple dummy variables. For 3 brands:
- Assign 1s and 0s to represent two of the brands.
- The third acts as a baseline for comparison.
Key Takeaways¶
- Dummy variables allow you to analyze categorical factors like brands.
- The choice of baseline category (e.g., Asus or Lenovo) affects how results are interpreted but not the overall conclusion.
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