How Can I Effectively Teach My Students To Distinguish Between The Correct Application Of The Akaike Information Criterion (AIC) And The Bayesian Information Criterion (BIC) For Model Selection In Generalized Linear Mixed-effects Models, Particularly When Dealing With Nested Models And Correlated Random Effects?

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To effectively teach students the differences between AIC and BIC for model selection in generalized linear mixed-effects models, particularly with nested models and correlated random effects, consider the following structured approach:

1. Theoretical Foundations

  • AIC (Akaike Information Criterion): Explain that AIC is rooted in information theory, aiming to find the model that best approximates the true data-generating process by balancing bias and variance. It is suitable for model comparison and prediction.
  • BIC (Bayesian Information Criterion): Discuss that BIC is Bayesian, focusing on identifying the model with the highest posterior probability, favoring simpler models as sample size increases.

2. Nested Models

  • AIC: Suitable for nested models, as it can handle subsets of models effectively.
  • BIC: Also applicable but tends to be more consistent in selecting the true model with larger samples.

3. Correlated Random Effects

  • Explain that correlated random effects influence the variance structure, affecting both AIC and BIC. BIC penalizes complex models more heavily, potentially favoring simpler models, while AIC may prefer more complex models if they explain significant variance.

4. Practical Considerations

  • Use Cases: AIC for prediction, BIC for identifying the underlying process.
  • Examples: Use real-world scenarios to demonstrate when each criterion is appropriate.

5. Computational Steps

  • Software Tools: Use R's glmer function to compute log-likelihoods and manually calculate AIC/BIC or use built-in functions.
  • Exercises: Have students apply AIC and BIC to different models, discussing results.

6. Interpretation

  • Thresholds: A difference of more than 2 is significant. Lower values are better for both.
  • Visualization: Use tables or graphs to show penalties with varying sample sizes and complexities.

7. Common Pitfalls

  • Avoid comparing non-nested models.
  • Emphasize using AIC/BIC alongside other model-checking techniques.

8. Lesson Structure

  • Lecture: Cover theory, key points, and address misconceptions.
  • Exercises: Hands-on with real data, followed by discussions on implications.

9. Interactive Learning

  • Encourage critical thinking by discussing when each criterion is appropriate and their implications.

By integrating theory, practical examples, and hands-on exercises, students will gain a comprehensive understanding of AIC and BIC, enabling them to apply these tools effectively in their research.