Two-Factor ANOVA

Two-Factor ANOVA Overview

  • ANOVA Purpose: Compares means of three or more groups.

  • One-Way ANOVA: Examines variance with one independent variable.

  • Two-Way ANOVA: Evaluates two independent factors and their interaction on a dependent variable.

Interaction in Two-Way ANOVA

  • Interaction Definition: Occurs when levels of one factor depend on levels of another.

  • Degrees of Freedom:

    • Main Effect A: df_A = i - 1

    • Main Effect B: df_B = j - 1

    • Interaction: df_{AB} = (i - 1)(j - 1)

    • Error: df_{error} = n - ij

    • Total: N - 1

Assumptions

  • Independence of observations.

  • Normality for each treatment combination.

  • Homogeneity of variance across treatment combinations.

Hypothesis Testing Steps

  1. Determine Interaction:

    • Null: No interaction.

    • Alternative: There is an interaction.

    • Check p-value and interaction plots.

  2. Analyze Effects:

    • If significant interaction, analyze simple effects.

    • If no interaction, analyze main effects separately (for each factor).

Example Studies

Caffeine and Exam Scores

  • Study on time of day and caffeine effects on scores.

  • Null for interaction: no interaction between time and caffeine.

  • Results showed significant main effect of time.

Attendance and Game Outcome

  • Investigated relationship between game outcome and brand on attendance.

  • Found evidence of interaction; simple effects examined.

Recommendations

  • For higher exam scores: take tests in the afternoon regardless of caffeine.

  • For attendance: If previous game is a win, choose Adidas; if a loss, brand choice is irrelevant.