Difference between MSD and MSE

MSD stands for Mean Squared Deviation, whereas MSE stands for Mean Squared Error. Quite often, you will find that they are synonymic. Both MSD and MSE can be used to compare estimated values and observed values in a model.

The key nuance is on the denominator of both MSD and MSE, as it will lead to biased and unbiased estimates. The following are the formulas of MSD and MSE.

MSD Formulas

p stands for the numbers of parameters you estimate in the model (excluding intercept). If you do not estimate any parameter, p will be zero.

Unbiased MSD

\[ MSD =\frac{SSR}{n-p-1}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n-p-1}\]

Biased MSD

\[ MSD =\frac{SSR}{n}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n}\]

MSE Formulas

Unbiased MSE

\[ MSE =\frac{SSR}{n-p-1}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n-p-1}\]

Biased MSE

\[ MSE =\frac{SSR}{n}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n}\]

Further Reading