Marginal and conditional relationships
In a linear model that predicts a response from several explanatory variables, the least squares coefficient associated with any explanatory variable describes its effect on the response if all other variables are held constant. This is also called the variable's conditional effect on the response.
This may be very different from the size and even the sign of the coefficient when a linear model is fitted with only that single explanatory variable. This simple linear model describes the marginal relationship between the response and that variable.
The difference between marginal and conditional effects is most easily explained in an example.
When interpreting the value of a coefficient in a linear model, make sure that you do so correctly in terms of a marginal or conditional relationship.
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,
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−18.62 − 0.089 wt + 0.059 age − 0.037 ht − 0.441 neck − 0.017 chest + 0.886 abdomen − 0.180 hip + 0.244 thigh − 0.014 knee + 0.168 ankle + 0.154 biceps + 0.434 forearm − 1.508 wrist |
The negative signs of some of these coefficients might be unexpected! For example,
The least squares coefficient of weight, -0.089, is negative.
Does this mean that heavier men tend to have less body fat?
Use the diagram above to investigate the marginal relationships between body fat and each of the explanatory variables. As expected, body fat is positively correlated with weight. There is no contradiction with its negative coefficient in the full model:
The value of the coefficient in the full model should be 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.