GUIDE TO MODEL SELECTION FOR ANALYSIS USING AIC AND BIC
Model selection is the process of evaluating and comparing different statistical models to choose the most suitable one. The goal of model selection is to find a model that best predicts outcomes and reasonably explains the data without overfitting or underfitting.
1. Model Selection Criteria
There are several criteria for evaluating and selecting models, including:
- Prediction accuracy: Measures the model’s ability to predict new values based on existing data.
- Model complexity: Refers to the number of parameters and the structure of the model. A more complex model may easily lead to overfitting, while a too-simple model may result in underfitting.
- Information criteria: Metrics like AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) help balance between prediction accuracy and model complexity.
2. Model Selection Methods
Common model selection methods include:
- Data splitting: Dividing the data into two sets, one for training the model and one for testing it. This method helps evaluate the model’s predictive power on unseen data.
- Cross-validation: This technique divides the data into multiple parts and uses each part as a test set while the remaining parts are used as a training set. This process is repeated several times to ensure the model is evaluated comprehensively.
- Information criteria: Using metrics like AIC and BIC to evaluate and compare models. The AIC focuses on predictive ability, while the BIC gives more weight to model complexity.
3. Model Selection Using Information Criteria
Two of the most common criteria for model selection are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Both criteria assess the goodness-of-fit of different statistical models to a dataset, but they have different philosophies and objectives.
3.1. Akaike Information Criterion (AIC)
AIC was developed by Hirotugu Akaike in 1974 and is designed to evaluate the goodness-of-fit of a model without assuming that the model is completely correct. Instead, AIC focuses on finding the model that is closest to reality. The formula for AIC is:
AIC = – 2ln(L) + 2p
where (L) is the maximum likelihood of the model, and (p) is the number of parameters in the model.
3.2. Bayesian Information Criterion (BIC)
BIC, introduced by Gideon E. Schwarz in 1978, is a criterion based on Bayesian theory for model selection. BIC not only considers the model’s goodness-of-fit but also penalizes the number of parameters to avoid overfitting. The formula for BIC is:
BIC = −2ln(L) + pln(n)
where (n) is the sample size.
4. Comparison between AIC and BIC
4.1. Philosophy and Objectives
- AIC: Aims to find the model with the best predictive ability. AIC leans towards predictive efficiency, meaning the model with the lowest prediction error. This makes AIC a useful tool in analyses where accurate prediction is the main goal (Aho et al., 2014).
- BIC: Focuses on consistency and finding the true model. BIC penalizes more heavily for the number of parameters, thus tending to select simpler models when the sample size is large (Aho et al., 2014).
4.2. Applications
- AIC: Suitable for situations where the goal is prediction, and the model may indeed be quite complex. AIC is often used in ecological models and fields where modeling complex processes is necessary (Aho et al., 2014).
- BIC: Suitable for confirmatory analyses, where the goal is to find the simplest and most accurate model. BIC is often used in studies where it is assumed that one of the chosen models is correct (Aho et al., 2014).
4.3. Conclusion
Both AIC and BIC are powerful tools in model selection, each with its own strengths and limitations. The choice between AIC and BIC depends on the objective of the analysis: AIC prioritizes accurate predictions, while BIC prioritizes model simplicity and correctness. Understanding these differences helps researchers and analysts choose the most appropriate tool for their study.
5. Model Selection Rule
Choose the model with the lowest AIC or BIC: Both criteria trade-off between model accuracy and simplicity. A model with a lower AIC or BIC is generally considered better.
6. How to Use AIC and BIC in Practice
- Step 1: Identify potential models that could fit your data.
- Step 2: Calculate the AIC and BIC values for each model.
- Step 3: Compare the AIC and BIC values. Choose the model with the lowest value.
7. Example Illustration
Suppose you have a dataset and are considering three different models:
| Model | Number of Parameters (p) | Log (likelihood) | Number of Observations (n) | AIC | BIC |
|---|---|---|---|---|---|
| M1 | 3 | -120 | 100 | 246 | 255 |
| M2 | 5 | -115 | 100 | 240 | 257 |
| M3 | 4 | -118 | 100 | 244 | 255 |
In this example, Model M2 has the lowest AIC value of 240, but Models M1 and M3 have lower BIC values (255). However, since both AIC and BIC are selection criteria, you may consider other factors such as parameter significance and model complexity to make the final decision.
8. Notes
- Overfitting and Underfitting: Ensure that the model is not too complex (overfitting) or too simple (underfitting).
- Residual analysis: Check the residuals of the model to ensure they follow the assumptions of regression analysis.
By applying AIC and BIC, you can objectively and scientifically choose the most appropriate model for your data.
9. References
Aho, K., Derryberry, D., & Peterson, T. (2014). Model selection for ecologists: the worldviews of AIC and BIC. Ecology, 95(3), 631–636.
Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods & Research, 33(2), 261–304.
Raftery, A. E. (1996). Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika, 83(2), 251–266.
Weakliem, D. L. (2016). Hypothesis testing and model selection in the social sciences. Guilford Publications.
