Crash Course Statistics: ANOVA and General Linear Model Framework

Introduction

• Presenter: Adriene Hill
• Topic: Testing differences between multiple groups using ANOVA (Analysis of Variance)
• Context: Previously discussed t-tests for comparing two groups; now expanding to more than two groups.

General Linear Model (GLM) Framework

• Concept: Partitions data into two piles
  • Explained by the model: Represents the way we think things work
  • Error: Amount of information the model fails to explain

ANOVA (Analysis of Variance)

• Purpose: Compare measurements of more than two groups
• Similarity to Regression: Uses categorical variables to predict a continuous one
• Example: Predicting number of yards a soccer player runs based on position
• Model Building: Similar to regression models but with categorical variables

Example: Bunny Count Model

• Model: Number of bunnies seen based on weather (rainy vs. sunny)
  • Prediction: 1 bunny on rainy days, 5 on sunny days
  • Error Calculation: Difference between prediction and actual observation
• Regression Interpretation: Slope represents the difference in means between two groups
  • Slope Calculation: Expected value on a sunny day minus rainy day value (5-1=4)

Chocolate Bar Rating Example

• Dataset Source: Kaggle.com, by Brady Brelinski
• Groups: Chocolate bars made from Criollo, Forastero, or Trinitario beans
• Rating Scale: 5 (highest) to 1 (mostly unpalatable)

ANOVA Calculation Steps

1. Sums of Squares Total (SST): Sum of squared differences between each rating and overall mean
2. Partition Variation: Divide SST into Model Sums of Squares (SSM) and Sums of Squares for Error (SSE)

• SSM: Variation explained by the model (group means)
  • SSE: Variation not explained by the model

3. F-statistic Calculation: Compare SSM and SSE adjusted by degrees of freedom

• Degrees of Freedom (Model): k-1, where k is the number of groups (3 groups -> 2 degrees)
  • Degrees of Freedom (Error): n-k, where n is sample size (790 data points - 3 groups = 787)

Interpretation of Results

• F-statistic: Value of 7.7619
• P-value: 0.000459 (small enough to reject the null hypothesis)
• Conclusion: Evidence that bean type influences chocolate ratings
• Omnibus Test: Significant F-statistic indicates some difference but not where
  • Follow-up: Conduct multiple t-tests to pinpoint significant differences
    • 3 T-tests: To compare each unique pair of categories
    • Findings: Criollo beans have lower ratings compared to Trinitario and Forastero; no significant difference between Trinitario and Forastero

Historical Example: Fisher's Potato Study

• Context: 12 different potato varieties, effect of fertilizers
• ANOVA Table: Summarizes degrees of freedom, sums of squares, mean squares, F-statistic, and p-value
• Results: Significant F-statistic indicating differences among potato varieties
  • Follow-up: Multiple tests needed to identify specific differences

Key Takeaways

1. Similarity of Statistical Models: ANOVAs and Regressions both use GLM to model the world
2. Filtering: ANOVAs help determine if there’s an overall effect before delving into specifics

• Efficiency: Saves time by avoiding unnecessary tests