Response and explanatory variables
We are often interested in how a 'response' variable, Y, depends on other explanatory variables.
We earlier described how a single explanatory variable, X, affects the response with a linear equation,
y = b0 + b1 x
The equation can be used to predict Y from X.
In this section, we will extend the equation to describe how two or more explanatory variables affect the response.
By making use of information from additional explanatory variables, we should be able to predict the response more accurately.
Body fat
Percentage body fat of individuals is an important measure of their health, but is a difficult quantity to measure. Accurate measurement of a person's body fat involves weighing the individual submersed in water, so is rarely done in fitness checks.
It is however possible to estimate percentage body fat from other body measurements that are easier to obtain. In order to determine how effectively body fat could be estimated from simple body measurements, scientists accurately determined body fat from a group of 252 men. Several other easily obtained measurements were also recorded from each subject.
Measurement | Person 1 | Person 2 | Person 3 | ... |
---|---|---|---|---|
Body fat (percent) | 12.6 | 6.9 | 24.6 | ... |
Weight (lbs) | 154.25 | 173.25 | 154.00 | ... |
Age (yrs) | 23 | 22 | 22 | ... |
Height (inches) | 67.75 | 72.25 | 66.25 | ... |
Neck circumference (cm) | 36.2 | 38.5 | 34.0 | ... |
Chest circumference (cm) | 93.1 | 93.6 | 95.8 | ... |
Abdomen circumference (cm) | 85.2 | 83.0 | 87.9 | ... |
Hip circumference (cm) | 94.5 | 98.7 | 99.2 | ... |
Thigh circumference (cm) | 59.0 | 58.7 | 59.6 | ... |
Knee circumference (cm) | 37.3 | 37.3 | 38.9 | ... |
Ankle circumference (cm) | 21.9 | 23.4 | 24.0 | ... |
Extended biceps circumference (cm) | 32.0 | 30.5 | 28.8 | ... |
Forearm circumference (cm) | 27.4 | 28.9 | 25.2 | ... |
Wrist circumference (cm) | 17.1 | 18.2 | 16.6 | ... |
The diagram below shows scatterplots of body fat against the other variables.
Use the popup menu to examine the relationship between body fat and each of the variables in turn. The correlation coefficients are also shown in the diagram — they provide a useful summary of the strength of the relationship.
Which variables are most strongly related to body fat?
Click Least squares line to show the least squares line that might be used to predict body fat from any single explanatory variable.
Could we improve the prediction by using two or
more of the measurements?