Compare Models: A Practical Guide Using LOO-CV

In the realm of statistical modeling, determining the “better” model is often a nuanced task. The term “better” itself is subjective, usually implying a model that strikes an optimal balance between fitting the data well, maintaining simplicity (parsimony), and avoiding overfitting. When comparing models, especially in Bayesian statistics, tools like Leave-One-Out Cross-Validation (LOO-CV) are invaluable. The output from functions like loo() in R, while seemingly straightforward, requires careful interpretation to ensure valid model comparisons. This guide will walk you through understanding and utilizing LOO-CV for comparing models, highlighting potential pitfalls and how to interpret the results effectively.

Interpreting loo() Output: Deciphering Model Fit Metrics

When you use the loo() function in R, typically from packages like rstanarm or brms, you’re presented with several key metrics designed to assess model fit. These metrics are crucial for comparing different models and deciding which one provides a superior representation of your data. The primary outputs from loo() include:

• elpd_loo (Expected Log Predictive Density for LOO): This is an estimate of the model’s predictive accuracy on new, unseen data. A higher elpd_loo generally indicates a better-predicting model. It’s essentially the average log-likelihood of each data point being predicted by a model trained on all other data points.

• p_loo (Effective Number of Parameters): This metric estimates the effective complexity of the model in the context of LOO-CV. It can be interpreted as the number of parameters that actively contribute to fitting the data, accounting for regularization and prior shrinkage in Bayesian models. Comparing p_loo across models can give insights into their relative complexity.

• looic (LOO Information Criterion): Calculated as -2 * elpd_loo, LOOIC is an information criterion analogous to AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). Lower LOOIC values indicate a better-fitting model. This is often the most readily interpretable metric for those familiar with traditional information criteria.

For initial model comparison, focusing on looic is often the easiest approach. If you run loo() on two models, say model1 and model2, and find that model2 has a lower looic than model1, it suggests that model2 provides a better fit to the data according to LOO-CV.

Utilizing loo_compare(): Assessing Statistically Significant Differences

While observing a lower looic for one model over another is informative, it’s crucial to determine if this difference is substantial enough to be considered meaningful, rather than just due to random variation. This is where the loo_compare() function becomes essential. It directly compares the elpd_loo values of different models and quantifies the difference and its uncertainty.

The output of loo_compare() presents the models ranked by their elpd_loo, with the best-fitting model (highest elpd_loo, lowest looic) listed first. Crucially, it provides the difference in elpd_loo between the best model and each of the other models, along with the standard error (SE) of this difference.

Remember that loo_compare() focuses on the difference in elpd, not looic. Since looic = -2 * elpd_loo, a positive difference in elpd corresponds to a negative difference in looic (and vice versa). The “best” model will always have a difference of 0.0 in elpd because it’s being compared to itself.

To assess whether the difference in model fit is statistically notable, we examine the difference in elpd_loo and its standard error. A common, though somewhat arbitrary, threshold for “statistical significance” is often based on a p-value of 0.05. Using this guideline, we can roughly consider a difference “significant” if it’s more than about 1.96 times its standard error (corresponding to the z-value for a two-tailed p < 0.05 test in a normal distribution approximation).

For instance, if loo_compare() shows a difference in elpd_loo of -1.50 with a standard error of 0.6 between model2 (best fit) and model1, we can calculate 1.96 * 0.6 = 1.176. Since the absolute value of the difference (|1.50|) is greater than 1.176, we might conclude that model2 provides a statistically significantly better fit than model1, based on this approximation.

Pareto K Statistics: Checking the Reliability of LOO Approximation

A critical aspect often overlooked in model comparison using LOO is the validity of the LOO approximation itself. The loo() function, for computational efficiency, typically relies on importance sampling to approximate the true LOO cross-validation. This approximation works well under certain conditions, but can become unreliable if some data points are highly influential.

The Pareto k statistic, provided as part of the loo() output, serves as a diagnostic tool to assess the reliability of the importance sampling approximation. High Pareto k values for some observations indicate that the importance sampling is unstable for those observations, and consequently, the overall LOO approximation might be untrustworthy.

While there isn’t a universally agreed-upon threshold, Pareto k values exceeding 0.7 are often considered problematic, suggesting that the LOO approximation should be treated with caution. If you observe a significant number of high Pareto k values, as in the original example, the results of loo() and loo_compare() should not be blindly accepted.

Addressing Untrustworthy LOO Results: Troubleshooting High Pareto K Values

When faced with high Pareto k statistics, signaling an unreliable LOO approximation, several strategies can be considered:

1. Exact LOO-CV: If computationally feasible, performing a true LOO cross-validation, where the model is refitted n times, each time leaving out one observation, can provide a more accurate result. However, this can be very time-consuming for complex models or large datasets.

2. Model Re-specification: High Pareto k values can sometimes indicate model misspecification. Consider revisiting your model formulation. Are there issues with model complexity, variable interactions, or distributional assumptions that might be leading to influential observations and poor LOO performance? In the original example, it was speculated that model2 might be overfitting due to added complexity, which could be a cause of high Pareto k values.

3. Robust LOO Methods: Explore more robust variants of LOO or alternative cross-validation techniques that are less sensitive to influential observations.

4. Examine Influential Observations: Investigate the data points with high Pareto k values. Are these outliers or observations with unusual characteristics? Understanding why these points are influential might provide insights into model limitations or data issues.

Packages like rstanarm and brms often provide warnings and guidance when loo() detects problematic Pareto k values, suggesting possible next steps. Consulting the documentation of these packages and resources like the Stan forums and LOO package documentation is highly recommended for in-depth troubleshooting.

Conclusion: LOO-CV as a Powerful Tool, Used with Prudence

LOO-CV, as implemented in tools like the loo package, offers a powerful framework for Bayesian model comparison. Metrics like looic and the output of loo_compare() provide valuable insights into relative model fit and predictive performance. However, it’s crucial to interpret these results in conjunction with diagnostic measures like Pareto k statistics. High Pareto k values signal potential issues with the LOO approximation and necessitate further investigation. By carefully considering both the model comparison metrics and the diagnostics, you can effectively utilize LOO-CV to select models that are not only well-fitting but also robust and reliable for prediction. Remember that model comparison is an iterative process, and tools like LOO-CV are best used as part of a broader strategy that includes model checking, domain expertise, and careful consideration of the research question.