Problem 5. Osteoporosis is a common age-related condition in which bone density decreases, causing bones to become more fragile. Bone density is an important measure of bone strength and a predictor of fractures. Smoking has been reported to be a risk factor for vertebral, forearm, and hip fractures, but it is unknown whether smoking increases risk of fracture by reducing bone density. A study examining the relationship between bone density and cigarette consumption was con- ducted on 40 pairs of middle-aged female twins who had different smoking histories. In addition to having their bone density measured, participants answered questionnaires to provide details about various lifestyle habits, such as tobacco use; alcohol, coffee, and tea consumption; and menopausal history.3 Tobacco consumption was expressed in terms of pack-years; one pack-year is defined as 1 pack of cigarettes per day (usually about 20 cigarettes per pack) consumed for 1 year. Investigators first identified the heavier- and lighter-smoking twins based on pack-years. The lighter-smoking twin usually had 0 pack-years (indicating she had never smoked) or occasionally either smoked very few cigarettes per day and/or smoked for only a short time. Bone-mineral density (BMD) was assessed separately at three sites: the lumbar spine (lower back), the femoral neck (hip), and the femoral shaft (hip). The dataset bone.density, saved as bone_density.Rdata, includes the following information for each set of twins. A single row in the dataframe contains information on both twins; variables ending in "1" refer to the lighter-smoking twin, and variables ending in "2" refer to the heavier- smoking twin.4 Variable age height.1, height.2 weight.1, weight.2 tea.1, tea.2 cof.1, cof.2 alc.1, alc.2 meno.1, meno.2 pack.yr.1, pack.yr.2 lum.spine.1, lum.spine.2 fem.neck.1, fem.neck.2 fem.shaft.1, fem.shaft.2 Description age, in years height, in cm weight, in kg tea consumption, in cups/week coffee consumption, in cups/week alcohol consumption, in drinks/week menopause status, coded 0-2. pack-years smoking bone density at the lumbar spine, in g/cm2 bone density at the femoral neck, in g/cm2 bone density at the femoral shaft, in g/cm2 a) From the collected data, an investigator is interested in analyzing whether there is evidence to suggest that bone density differs between lighter-smoking twins and heavier-smoking twins, for each of the three sites where bone density was assessed.

For the meno variable, 0 indicates a pre-menopausal woman and 1 indicates a post-menopausal woman. 2 refers to unknown menopausal status. i. Would you recommend that the investigator conduct tests at significance level = 0.05 or = 0.10? Justify your answer.

ii. Using the significance level you recommended, test whether bone density differs significantly between lighter and heavier-smoking twins. Conduct separate analyses for bone density at the lumbar spine, femoral neck, and femoral shaft. Be sure to state your hypotheses and report your conclusions.

iii. Suppose the investigator decides to conclude that bone density differs between lighter-smoking twins and heavier-smoker twins if there is evidence of a significant difference in at least one site. Would the overall error rate still be equal to (as used in part ii.)?

b) The difference in bone-mineral density between the twins can also be expressed as a percentage of the average bone density of the twins. For a given pair of twins,

the difference in BMD (%) = 100 BMD of lighter-smoking twin - BMD of heavier-smoking twin mean BMD of twins

i. Help make a variable ls.diff.percent that contains the difference in bone mineral density (%) at the lumbar spine for each pair of twins in the study. Help make a variable pack.yr.diff that contains the difference in tobacco use between heavier-smoking twins and lighter-smoking twins.

ii. Using graphical and numerical summaries, explore the relationship between the difference in tobacco use and the difference in BMD (%) at the lumbar spine. Describe your findings.

iii. Assess whether the difference in BMD (%) at the lumbar spine between heavier-smoking and lighter-smoking twins is significantly associated with the degree of difference in tobacco use at the = 0.05 significance level.

c) A New York Times reporter is potentially interested in writing a piece about the research you have conducted. They have requested that you help with a short statement, no more than seven sentences long. Drawing from the results of your analyses in parts a) and b), summarize your conclusions about the relationship between smoking and bone mineral density.