axjack's blog

axjack is said to be an abbreviation for An eXistent JApanese Cool Klutz.

自由度調整済み決定係数を決定係数で表す

やること

タイトルの通り式変形をするだけです。

式変形

 

\displaystyle R_{adj}^2 = 1 - \frac{ \frac{\sum{(y_i - \hat{y_{i}})^2}}{n-k-1}  }{  \frac{\sum{(y_i - \bar{y})^2 }}{n-1} } \\
\displaystyle = 1 - \frac{n-1}{n-k-1} \frac{ \sum{(y_i - \hat{y_{i}})^2}} {\sum{  (y_i - \bar{y} )^2 } } \\
\displaystyle = \frac{n-1}{n-k-1}\Bigl(  \frac{n-k-1}{n-1}  - \frac{ \sum{(y_i - \hat{y_{i}})^2}} {\sum{  (y_i - \bar{y} )^2 } } \Bigr) \\
\displaystyle = \frac{n-1}{n-k-1}\Bigl( \frac{n-k-1}{n-1}  + \Bigl(1 - \frac{ \sum{(y_i - \hat{y_{i}})^2}} {\sum{  (y_i - \bar{y} )^2 } } \Bigr) - 1 \Bigr) \\
\displaystyle = \frac{n-1}{n-k-1}\Bigl( \frac{n-k-1}{n-1}  + R^2 - 1 \Bigr) \\
\displaystyle = \frac{n-1}{n-k-1}\Bigl( R^2  -  \frac{k}{n-1} \Bigr) \\

axjack is said to be an abbreviation for An eXistent JApanese Cool Klutz.