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8.2.1 Interpretation of Coefficients

Regression Equation:

image

Where:
- β0: Intercept.
- β1: Coefficient for x1​ (quantitative variable).
- β2: Coefficient for x2 (categorical dummy variable).

Interpreting Coefficients with a Dummy Variable

image

Impact of β2:

image

Example with Basavaraja's Dataset:

  • Regression Equation: image

  • x1: Price of the computer.

  • x2 = 0: Dell.

  • x2 = 1: Lenovo.

  • Predictions: image

  • 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:
    image

Regression Equation:

image

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.