CourseKata Chapter 7

Modeling by Group Means

Mansour Abdoli, PhD

Session Goals

By the end of today you can:

• Explain what a group model does
• Fit lm(y ~ group)
• Interpret intercept and coefficients
• Explain how predictions change from the empty model

Review: The Empty Model

• DATA = Mean + Error

• Assumption:
  • Everyone is the same (One Mean)
  • Variations are by Chance (Error)

• Formula: \[\hat{y} = \bar{y}\]

• Application:
  Everyone gets the same prediction.

Example: Thumb Length

Model:

Empty-Model Residual

Residual = Observed − Predicted

Modeling with Group Means

Grouping by Gender

• Questions:
  • Does thumb length differ by gender?
  • Does one overall mean seem reasonable?

Model by Group (Gender)

• In word: \[\text{Thumb} = \text{Gender} + \text{Error}\]

• In Application:

  • Each gender gets its own mean as prediction.

Fit the Group Model

Interpreting Coefficients

• Here, Female is the reference:
  • Intercept = mean thumb for females
  • GenderMale = difference (Male − Female)

Visualizing Predictions

• Interpretation:
  • Black points: observed data
  • Red/Blue lines: predicted group means

Compare to Empty Model

Residual Comparison (Conceptual)

• Discussion:

  • Which model has smaller residual spread?
  • What does that imply?

Key Insight

• The group model:
  • Allows different means
  • Reduces prediction error
  • Uses information from explanatory variable
• But…
  • Is the reduction large enough to matter?