Understanding the Angle Between Regression Lines

Regression lines are statistical tools used to model the relationship between two variables, say X and Y. The regression line of Y on X predicts the value of Y based on a given value of X, while the regression line of X on Y predicts the value of X based on Y. These two lines typically intersect at the point representing the mean of X and the mean of Y \((\bar{X}, \bar{Y})\). The angle between these lines provides insight into the correlation between the variables.

Problem Analysis

We are given the following information about two variables, X and Y:

• The standard deviation of Y is double the standard deviation of X: \(\sigma_Y = 2\sigma_X\).
• The correlation coefficient between X and Y is \(r = 0.5\).

Our goal is to find the acute angle between the line of regression of Y on X and the line of regression of X on Y.

Key Concepts for Regression Angle Calculation

To find the angle between the two regression lines, we need their slopes.

• The equation of the regression line of Y on X is typically written as \(Y - \bar{Y} = b_{YX}(X - \bar{X})\). The slope of this line is \(m_1 = b_{YX}\).
• The equation of the regression line of X on Y is typically written as \(X - \bar{X} = b_{XY}(Y - \bar{Y})\). To find the slope of this line when Y is plotted against X (as is standard in the XY plane), we rearrange the equation to express Y in terms of X: \(Y - \bar{Y} = \frac{1}{b_{XY}}(X - \bar{X})\). The slope of this line is \(m_2 = \frac{1}{b_{XY}}\).

The regression coefficients \(b_{YX}\) and \(b_{XY}\) are given by the formulas:

• \(b_{YX} = r \frac{\sigma_Y}{\sigma_X}\)
• \(b_{XY} = r \frac{\sigma_X}{\sigma_Y}\)

The acute angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by the formula:

Step-by-Step Solution

Step 1: Calculate the regression coefficients \(b_{YX}\) and \(b_{XY}\).

We are given \(r = 0.5\) and \(\sigma_Y = 2\sigma_X\).

Calculate \(b_{YX}\):

Calculate \(b_{XY}\):

Step 2: Determine the slopes of the two regression lines in the XY plane.

The slope of the regression line of Y on X is \(m_1 = b_{YX}\).

The slope of the regression line of X on Y (when plotted as Y vs X) is \(m_2 = \frac{1}{b_{XY}}\).

Step 3: Use the formula for the angle between two lines.

The slopes are \(m_1 = 1\) and \(m_2 = 4\). The formula for the acute angle \(\theta\) is:

Substitute the values of \(m_1\) and \(m_2\):

Since we are looking for the acute angle, \(\tan \theta\) is positive.

Step 4: Find the angle \(\theta\).

Taking the arctan (or inverse tangent) of both sides gives the angle:

This is the acute angle between the two lines of regression.

Revision Table: Regression Angle Formulas

Additional Information: Properties of Regression Lines and Correlation

The angle between the two regression lines is closely related to the correlation coefficient \(r\).

• If \(r = 0\) (no correlation), then \(b_{YX} = 0\) and \(b_{XY} = 0\). The slope of Y on X is \(m_1 = 0\) (a horizontal line). The slope of X on Y has \(b_{XY} = 0\), meaning \(X - \bar{X} = 0(Y - \bar{Y})\), so \(X = \bar{X}\) (a vertical line). A horizontal line and a vertical line are perpendicular, so the angle between them is \(90^\circ\). The formula gives \(\tan \theta = |\frac{m_2 - m_1}{1 + m_1 m_2}| = |\frac{\infty - 0}{1 + 0}| = \infty\), which means \(\theta = 90^\circ\).
• If \(|r| = 1\) (perfect correlation), then the two regression lines coincide. The angle between them is \(0^\circ\) or \(180^\circ\). The formula gives \(\tan \theta = |\frac{m_2 - m_1}{1 + m_1 m_2}|\). If \(|r|=1\), then \(b_{YX}b_{XY} = r \frac{\sigma_Y}{\sigma_X} \times r \frac{\sigma_X}{\sigma_Y} = r^2 = 1\). Also, \(m_1 = b_{YX}\) and \(m_2 = \frac{1}{b_{XY}}\), so \(m_1 m_2 = b_{YX} \frac{1}{b_{XY}} = \frac{b_{YX}}{b_{XY}}\). If \(b_{YX}b_{XY}=1\), then \(b_{YX} = \frac{1}{b_{XY}}\), which means \(m_1 = m_2\). If the slopes are equal, the lines are parallel or coincident, and the angle is \(0^\circ\). The formula gives \(\tan \theta = |\frac{m_1 - m_1}{1 + m_1^2}| = \frac{0}{1 + m_1^2} = 0\), so \(\theta = 0^\circ\).
• The angle between the lines decreases as the absolute value of the correlation coefficient \(|r|\) increases. A smaller angle indicates stronger correlation.
• In our problem, \(r = 0.5\). This is a moderate positive correlation, so we expect an acute angle between \(0^\circ\) and \(90^\circ\). Our result \(\arctan\left(\frac{3}{5}\right)\) confirms this, as \(3/5\) is between 0 and infinity.