Which model selection criterion is designed to assess the trade-off between goodness-of-fit and model complexity, with lower values preferable?

Which model selection criterion is designed to assess the trade-off between goodness-of-fit and model complexity, with lower values preferable?

• A. Bayesian Information Criterion (BIC)
• B. Akaike Information Criterion (AIC)
• C. Adjusted R-Squared
• D. Root Mean Squared Error

Answer: (B)

The main idea here is penalized model selection: you want a model that fits the data well but isn’t needlessly complex. The Bayesian Information Criterion does this by combining how well the model fits (via a likelihood term) with a penalty for the number of parameters that grows with the sample size. The typical form is a fit term plus k times log(n); lower values indicate a better balance between accuracy and simplicity. Because the penalty increases with both model size and sample size, BIC tends to favor simpler models as data grow, which is exactly the trade-off described in the question. In contrast, while other options also relate to model quality, they don’t embody this trade-off the same way. The Akaike Information Criterion also balances fit and complexity but uses a smaller, constant penalty per parameter that doesn’t scale with log(n), so its behavior differs, especially with larger datasets. Adjusted R-squared rewards better fit with more predictors but increases with model complexity in a way that doesn’t produce a universally lower-is-better score. Root Mean Squared Error focuses on predictive error only and doesn’t penalize added parameters, so it doesn’t capture the trade-off in the same formal way.

The main idea here is penalized model selection: you want a model that fits the data well but isn’t needlessly complex. The Bayesian Information Criterion does this by combining how well the model fits (via a likelihood term) with a penalty for the number of parameters that grows with the sample size. The typical form is a fit term plus k times log(n); lower values indicate a better balance between accuracy and simplicity. Because the penalty increases with both model size and sample size, BIC tends to favor simpler models as data grow, which is exactly the trade-off described in the question.

In contrast, while other options also relate to model quality, they don’t embody this trade-off the same way. The Akaike Information Criterion also balances fit and complexity but uses a smaller, constant penalty per parameter that doesn’t scale with log(n), so its behavior differs, especially with larger datasets. Adjusted R-squared rewards better fit with more predictors but increases with model complexity in a way that doesn’t produce a universally lower-is-better score. Root Mean Squared Error focuses on predictive error only and doesn’t penalize added parameters, so it doesn’t capture the trade-off in the same formal way.