EXAMPLE 10.14

Investing in one’s bone health. As we age, our bones become weaker and are more likely to break. Osteoporosis (or weak bones) is the major cause of bone fractures in older women. Various researchers have studied this problem by looking at how and when bone mass is accumulated by young women. They’ve determined that up to 90% of a person’s peak bone mass is acquired by age 18 in girls.³ This makes youth the best time to invest in stronger bones.

Figure 10.11 displays data for a measure of bone strength, called “total body bone mineral density’’ (TBBMD), and age for a sample of 256 young women.⁴ TBBMD is measured in grams per square centimeter (g/cm²), and age is recorded in years. The solid curve is the nonlinear fit, and the dashed curves are 95% prediction limits. Similar to our example of BMI and activity level, there is a large amount of scatter about the fitted curve. Although prediction intervals may be useless in this case, the researchers can draw some conclusions regarding the relationship.

The fitted nonlinear equation is

In this equation, is the predicted value of TBBMD, the response variable, and x is age, the explanatory variable. A straight line would not do a very good job of summarizing the relationship between TBBMD and age. At first, TBBMD increases with age, but then it levels off as age increases. The value of the function where it is level is called “peak bone mass’’; it is a parameter in the nonlinear model. The estimate is 1.162 and the standard error is 0.008. Software gives the 95% confidence interval as (1.146, 1.178). Other calculations could be done to determine the age by which 90% of this peak bone mass is acquired.

577