Formulas of MSE and RSS
Residual Sum of Squares (RSS) is the numerator in the formula of Mean Squared Error (MSE). That is, RSS is part of MSE formula.
\[ SSR=\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 \]
\[ MSE=\frac{SSR}{n-p-1}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n-p-1}\]
where,
\( n \) is the number of observations.
\( \hat{y_i} \) is is estimated value.
\( y_i \) is observed value.
\( p \) is the is the number of estimated parameters (excluding the intercept).
Relationship between MSE and RSS
Thus, RSS is the sum of the squares of residuals. Building on RSS, MSE takes the number of observation (i.e., \( n\) ) into consideration. That is, MSE is “mean” of squared errors (estimated by squared residuals).
Why do we use \( n-p-1 \) rather than \( n\) as the denominator in the formula of MSE? It is because \( n\) leads to a biased estimate of MSE, whereas \( n-p-1 \) leads to an unbiased estimate of MSE. (See the discussion here on Wikipedia.)
Further Reading
- Difference between Mean Squared Residuals (MSR) and Mean Square Error (MSE)
- Difference between MSD and MSE
- Calculate Sum of Squared Residuals (SSR) in R (R, Python)
- Use sklearn to Calculate SSR in Python
- Calculate Mean Squared Residuals (MSR) in R (R, Python)
- Calculate Mean Squared Error (MSE) (R, Python)
- How to Calculate Mean Squared Deviation in R