What do the values of the coefficients tell you?

The coefficients of the least squares line are interpreted in a similar way to the corresponding coefficients for the simple linear model.

y  =  b0 + b1 x + b2 z

Intercept
The intercept, b0, is the predicted response when both x and z are zero.
Slope for X
The parameter b1 is the predicted increase in the response when x increases by 1, provided the value of z remains the same. For observational data, the two explanatory variables may be correlated so an increase of 1 in x may be associated with a change in z so this interpretation is of less practical use.
Slope for Z
The parameter b2 is similarly the predicted increase in y when z increases by 1, provided x does not change.

It is important to remember that the slope parameters only describe the effect of the variables if the other variable remains constant.


Energy expenditure of bees

A researcher recorded energy expenditure (joules/sec) for bees drinking water with different sucrose concentrations (%) and at different temperatures (degrees Celsius). The diagram below shows the least squares equation on the top right and displays it as a plane on the left.

At any fixed sucrose concentration, the model predicts that energy expenditure will increase by 0.394 joules/sec for each increase in temperature of 1 degree. This prediction is shown by the red equation at the bottom. Use the slider to adjust the sucrose concentration and observe that only the intercept of this line changes — its slope remains 0.394.

Click the y-x button to rotate the 3-dimensional diagram to show energy vs temperature. The diagram at the bottom is simply a cross-section of the full regression plane.


The next diagram is similar but focuses on the relationship between energy expenditure and sucrose an fixed temperatures.

At any fixed temperature, the model predicts that energy expenditure will increase by 0.206 joules/sec for each increase of 1% in sucrose concentration. This prediction is again shown at the bottom . Use the slider to adjust the temperature and observe that only the intercept of this line changes — its slope remains 0.206.

Click the y-z button to rotate the 3-dimensional diagram to show energy vs sucrose. Again the diagram at the bottom is simply a cross-section of the full regression plane.

Heart catheter length

In heart catheterisation, a catheter is passed through a major vein or artery from the leg into the heart. X-rays are used to position the tip of the catheter. The catheter length (cm), height (in) and weight (lb) were recorded from 12 children. Surgeons want to estimate the length of catheter to use, based on the height and weight of a child.

The least squares equation (and plane) for these data predicts that:

These data are observational and that the heights and weights of the children are highly correlated — taller children also tend to be heavier. Since both explanatory variables measure different aspects of the same thing, the sizes of the children, we will now investigate a model with only the variable weight.

Click the checkbox for Height to remove this variable from the model. The least squares line for predicting catheter length from weight alone has slope 0.284.


We will discuss the consequences of correlated explanatory variables in much greater depth in a later section.