Graphical display of the equation
The linear equation to predict a response from a single explanatory variable can be displayed as a straight line on a scatterplot of Y against X.
y = b0 + b1 x
The corresponding equation for predicting Y from X and Z is
y = b0 + b1 x + b2 z
This linear model can be displayed as a plane on a 3-dimensional scatterplot of Y against X and Z.
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.