Warning

It is important to note that the least squares coefficients associated with the explanatory variables do not describe their overall relationship with the response as displayed in simple scatterplots of the response against the individual explanatory variables.

The least squares coefficient associated with a variable describes the effect of changes to that variable if all other variables are held constant. This is also called the variable's conditional effect on the response.

Body fat

Percentage body fat of individuals is an important measure of their health, but is a difficult quantity to measure. Scientists accurately determined body fat from 252 men using an underwater weighing technique and recorded several other body measurements that were easier to obtain.

Response
Body fat (percent)
Explanatory variables
Weight (lbs)
Age (yrs)
Height (inches)
Neck circumference (cm)
Chest circumference (cm)
Abdomen circumference (cm)
Hip circumference (cm)
Thigh circumference (cm)
Knee circumference (cm)
Ankle circumference (cm)
Extended biceps circumference (cm)
Forearm circumference (cm)
Wrist circumference (cm)

The least squares estimates for a linear model predicting body fat from the other variables results in a prediction equation,

The negative signs of some of these coefficients might be unexpected! These coefficients are estimates of the general linear model parameters and they may not be accurate estimates — we will consider their standard errors in the next page — but this does not totally explain the signs of the coefficients.

Use the pop-up menu in the diagram below to investigate how variables that are positively correlated with body fat can have negative least squares coeffients.

For example, the least squares coefficient of weight in the full model, -0.089, is negative but it is positively correlated with body fat. There is no contradiction here:

The value of the coefficient is interpreted as follows:

Comparing men with the same other body measurements, each extra pound in weight is predicted to correspond to a decrease of 0.089 percent body fat.

The other coefficients are interpreted in a similar way. For example,

Comparing men with the same weight and other body measurements, each extra 1cm in abdomen circumference is predicted to correspond to an increase of 0.886 percent body fat.