Question

Females, on average, are shorter and weigh less than males. One of your friends, who is a pre-med student, tells you that in addition, females will weigh less for a given height. To test this hypothesis, you collect height and weight of 29 female and 81 male students at your university. A regression of the weight on a constant, height, and a binary variable, which takes a value of one for females and is zero otherwise, yields the following result: Studentw = -229.21 – 6.36 × Female + 5.58 × Height , R 2 =0.50, SER = 20.99 where Studentw is weight measured in pounds and Height is measured in inches. (a) Interpret the results. Does it make sense to have a negative intercept? (b) You decide that in order to give an interpretation to the intercept you should rescale the height variable. One possibility is to subtract 5 ft. or 60 inches from your Height, because the minimum height in your data set is 62 inches. The resulting new intercept is now 105.58. Can you interpret this number now? Do you thing that the regression R 2 has changed? What about the standard error of the regression? (c) You have learned that correlation does not imply causation. Although this is true mathematically, does this always apply?

Answer 1

a)

For every additional inch in height, weight increases by roughly 5.5 pounds. Female students weigh approximately 6.5 pounds less than male students, controlling for height. The regression explains 50 percent of the weight variation among students. It does not make sense to interpret the intercept, since there are no observations close to the origin, or, put differently, there are no individuals who are zero inches tall.

b)

There are now observations close to the origin and you can therefore interpret the intercept. A student who is 5ft. tall will weight roughly 105.5 pounds, on average. The two slopes will be unaffected, as will be the regression R2. Since the explanatory power of the regression is unaffected by rescaling, and the dependent variable and the total sums of squares have remained unchanged, the sums of squared residuals, and hence the SER, must also remain the same.

c)

Although true in general, there are cases where Y cannot cause X, as is the case here. Gaining weight is not a good way for becoming taller, or put differently, weighing 250 pounds will not make students over 7 ft. tall.