When evaluating multiple linear regression models for a dataset, several key metrics and diagnostics help determine the best fit. This comprehensive guide outlines crucial factors for effectively comparing two linear regression models.
Key Metrics for Model Comparison
A primary metric for comparison is the Root Mean Squared Error (RMSE), also known as the standard error of the regression. RMSE represents the square root of the average squared difference between predicted and actual values. It quantifies the model’s prediction accuracy in the original data units, providing a lower bound for forecast error standard deviation.
Mean Absolute Error (MAE) offers another perspective on error magnitude. MAE averages the absolute differences between predicted and actual values, making it less sensitive to extreme errors compared to RMSE. While not always included in standard regression outputs, MAE offers an easily interpretable measure of average error.
Mean Absolute Percentage Error (MAPE) expresses average error as a percentage, facilitating comparison across different datasets or variables. However, MAPE is only applicable to strictly positive data and can be misleading with values close to zero.
Mean Absolute Scaled Error (MASE) compares a model’s performance to a naive forecast (e.g., a random walk). A MASE value below 1 indicates superior performance compared to the naive model, especially useful for time series data.
While RMSE and MAE focus on absolute error, Mean Error (ME) and Mean Percentage Error (MPE) reveal potential bias in predictions. A non-zero ME or MPE suggests a systematic tendency to over- or under-predict.
Adjusted R-squared, reflecting the proportion of variance explained by the model, can aid comparison when using the same dependent variable and estimation period. However, it’s less reliable when comparing models with different dependent variable transformations or estimation periods.
Figure 1: Comparison of linear and quadratic models.
Beyond Numerical Metrics: Diagnostics and Qualitative Factors
Beyond these core metrics, several diagnostic tests and qualitative aspects contribute to a thorough model comparison.
Residual Analysis: Examining residual plots (residuals vs. time, predicted values, other variables) reveals potential violations of model assumptions, such as non-linearity, heteroscedasticity, or non-independence. While minor violations might suggest model refinement, major violations signal unreliability of error statistics.
Goodness-of-fit tests: Assessing the normality of residuals and checking for patterns like autocorrelation helps validate model assumptions.
Out-of-Sample Testing: Evaluating model performance on a held-out dataset (validation period) offers a realistic estimate of future prediction accuracy. However, small validation samples can be susceptible to random fluctuations.
Complexity and Interpretability: Simpler models with fewer parameters are often preferred, especially when performance differences are minimal. A parsimonious model enhances interpretability and reduces the risk of overfitting. Metrics like Mallows’ Cp, AIC, and BIC penalize model complexity.
Figure 2: Example of overfitting.
Conclusion: A Holistic Approach
Comparing linear regression models requires a comprehensive approach, considering both quantitative metrics and qualitative factors. While RMSE often serves as a primary indicator of predictive accuracy, MAE, MAPE, and MASE provide valuable complementary perspectives. Diagnostic tests, out-of-sample validation, and model complexity considerations ensure a robust and reliable model selection process. Prioritize simpler models when performance differences are marginal, and always interpret results in the context of the specific application and data characteristics.
