Question

The angle between the regression lines is given by

• A. ${\tan }^{-1}\left[\frac{b_{yx}-\frac{1}{b_{xy}}}{1+\frac{b_{xy}}{b_{yx}}}\right]$
• B. ${\tan }^{-1}\left[\frac{b_{yx}.b_{xy}-1}{b_{yx}+b_{xy}}\right]$
• C. ${\tan }^{-1}\left[\frac{b_{yx}+\frac{1}{b_{xy}}}{1-\frac{b_{xy}}{b_{yx}}}\right]$
• D. ${\tan }^{-1}\left[\frac{b_{yx}-b_{xy}}{1+b_{yx}.b_{xy}}\right]$

Answer 1

The correct option is B ${\tan }^{-1}\left[\frac{b_{yx}.b_{xy}-1}{b_{yx}+b_{xy}}\right]$

If $\theta$ be the angle between the two regression lines then
$\theta ={\tan }^{-1}\left[\frac{m_1{m}_2-1}{m_1+m_2}\right]$ where $m_1=b_{yx}$, $m_2=b_{xy}$
Alternative Solution: The two regression lines are
$y-\overline{y}=b_{yx}(x-\overline{x})$ $\therefore m_1=b_{yx}$
$x-\overline{x}=b_{xy}(y-\overline{y})$ $\therefore m_2=b_{xy}$
$\therefore$ Required angle ${\tan }^{-1}\left[\frac{b_{yx}-\frac{1}{b_{xy}}}{1+\frac{b_{yx}}{b_{xy}}}\right]$
$={\tan }^{-1}\left[\frac{b_{yx}.b_{xy}-1}{b_{yx}+b_{xy}}\right]$