Adding another variable to the equation predicting Y from X

We often use a straight line to describe how a response variable is related to a single explanatory variable — its equation can be used to predict Y from X.

y  =  b0 + b1 x

The simplest way to describe how a second explanatory variable Z also affects the response, is to add a further term to this equation,

y  =  b0 + b1 x + b2 z

This equation is also called a linear model for the response.

Graphical display

On a 3-dimensional display whose axes represent Y, X and Z, the equation corresponds to a plane.


Body fat

We saw earlier that abdomen circumference and height were related to body fat in the 252 men whose body fat was measured by an underwater weighing technique. (Abdomen circumference had the highest correlation with body fat.) We will now consider how a prediction of body fat might be made using both abdomen circumference and height. Consider the equation

body fat   =   30.0  +  0.53 abdomen  −  0.80 height

The diagram below represents this equation.

Click anywhere between the abdomen and height axes (in the yellow area of the diagram) to see how the plane translates any pair of values (abdomen, height) into a predicted value of body fat, via the equation.