Use the Correlation Coefficient to Summarize Regression Performance?

Area of Science:

• Statistics and Data Analysis

Background:

• The correlation coefficient (r) is widely used to assess the goodness of fit in regression analyses.
• However, its application for this purpose is often debated and potentially misleading.

Purpose of the Study:

• To critically evaluate the appropriateness of using the correlation coefficient as the sole or primary indicator of regression model fit.
• To highlight the limitations and potential pitfalls of relying on the correlation coefficient.

Main Methods:

• Review of statistical principles governing regression analysis and model evaluation.
• Analysis of scenarios where the correlation coefficient can be deceptive.
• Comparison with alternative and more robust measures of model fit.

Main Results:

• The correlation coefficient measures linear association, not necessarily the quality of fit or predictive power of a regression model.
• High correlation does not guarantee a good fit, especially with non-linear relationships or omitted variables.
• Misinterpretation can lead to incorrect conclusions about model adequacy.

Conclusions:

• The correlation coefficient should not be the sole metric for assessing regression model fit.
• Researchers should consider a suite of diagnostic tools and metrics for a comprehensive evaluation.
• Alternative measures like R-squared, adjusted R-squared, and residual analysis provide a more complete picture of model performance.