8.1 - Linear Relationships

To define a useful model, we must investigate the relationship between the response and the predictor variables. As mentioned before, the focus of this Lesson is linear relationships. For a brief review of linear functions, recall that the equation of a line has the following form:

\(y=mx+b\)

where m is the slope and b is the y-intercept.

Given two points on a line, \(\left(x_1,y_1\right)\) and \(\left(x_2, y_2\right)\), the slope is calculated by:

\begin{align} m&=\dfrac{y_2-y_1}{x_2-x_1}\\&=\dfrac{\text{change in y}}{\text{change in x}}\\&=\frac{\text{rise}}{\text{run}} \end{align}

The slope of a line describes a lot about the linear relationship between two variables. If the slope is positive, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant.