RSE与MSE的区别及模型对比时的选用疑问

Great question—this is a common point of confusion when first diving into regression metrics, especially since terminology can overlap depending on the context! Let's break down when to use each metric, tied back to what you're doing with your three models.

First, a quick recap to align on definitions (since you already have the formulas right):

• RSE² = RSS/(n-2): This is the unbiased estimate of the model's error variance (σ²) for simple linear regression. The n-2 accounts for the two parameters we estimate (intercept β₀ and slope β₁)—it's the degrees of freedom adjustment.
• MSE = RSS/n: This is the raw average of squared residuals on your training data, a biased estimate of σ² but intuitive for measuring fit.

When to Use RSE² (RSS/(n-2)) #

• Statistical inference tasks: If you need to do things like calculate confidence intervals for your coefficients, run t-tests to check if predictors are statistically significant, or build prediction intervals for new observations, you need the unbiased estimate of σ². RSE² is exactly that—without the degrees of freedom adjustment, your estimate of the error variance would be too low, leading to overly narrow intervals or misleadingly significant p-values.
• Note for more complex models: This logic extends beyond simple linear regression! For a model with p parameters (including the intercept), the unbiased estimate becomes RSS/(n-p)—this is actually what many textbooks refer to as "MSE" in a statistical inference context. So if your three models aren't simple linear regressions, adjust the denominator to n-p instead of n-2.

When to Use MSE (RSS/n) #

• Model comparison and fit assessment: Since you're building three models to compare them, MSE is perfect here. It directly tells you the average squared error your model makes on the training data—lower values mean better fit to the training set. It's intuitive, easy to interpret, and works great as a relative metric to rank your models.
• Large sample sizes: When n is very big, the difference between n and n-2 (or n-p) becomes negligible. In these cases, RSE² and MSE will be almost identical, so you can use either without worrying about meaningful differences.
• Prediction-focused tasks: If your end goal is predicting new data (rather than interpreting coefficients), the raw MSE (or better yet, test-set MSE) is a more direct measure of how your model will perform, since it reflects the average error you'd expect on new observations.

Key Takeaway #

• Use RSE² (or its generalized RSS/(n-p) form) when you need unbiased error variance estimates for statistical inference.
• Use MSE when you're comparing models' fit to training data or need an intuitive measure of average prediction error.

And don't forget to watch for terminology mix-ups—some sources use "MSE" to mean the degrees-of-freedom-adjusted version, so always double-check the formula in context!

内容的提问来源于stack exchange，提问作者Piyush