Question

Some studies have suggested that a nightly glass of wine may not only take the edge off a day but also improve health. Is wine good for your health? A study of nearly 1.3 million middle‑aged British women examined wine consumption and the risk of breast cancer. The researchers were interested in how risk changed as wine consumption increased. Risk is based on breast cancer rates in drinkers relative to breast cancer rates in nondrinkers in the study, with higher values indicating greater risk. In particular, a value greater than 1 indicates a greater breast cancer rate than that of nondrinkers. Wine intake is the mean wine intake, in grams of alcohol per day (one glass of wine is approximately 10 grams of alcohol), of all women in the study who drank the same amount of wine per week. The table contains data for drinkers only and presents data on wine intake and the relative risk of breast cancer in women. Wine intake ? (grams of alcohol per day) 2.5 8.5 15.5 26.5 Relative risk ? 1.00 1.08 1.15 1.22 The correlation ? between wine intake and relative risk is 0.9851 .

(a) If wine intake is measured in ounces of alcohol per day rather than grams per day, how would the correlation change? (There is 0.035 ounce per gram.) The correlation would not change, even though correlation depends on units. None of the options are correct. The correlation would not change, as correlation does not depend on units. The correlation would drastically change, as it greatly depends on the units. (

b) How would ? change if all the relative risks were 0.25 less than the values given in the table? Does the correlation tell us that among women who drink, those who drink more wine tend to have a greater relative risk of cancer than women who do not drink at all? The correlation would not change and the strength and direction of the quadratic relationship between risk and wine intake would not change. The correlation would change and the strength and direction of the linear relationship between risk and wine intake would change dramatically. None of the options are correct. The correlation would not change and the strength and direction of the linear relationship between risk and wine intake would not change.

(c) If drinking an additional gram of alcohol each day raised the relative risk of breast cancer by exactly 0.01 , what would be the correlation between alcohol in wine intake and relative risk of breast cancer? There would be a perfect, positive linear relationship. There would be a poor quality, negative linear relationship. There would be a perfect, positive nonlinear relationship. None of the options are correct.

Answer 1

Given,

x	y
2.5	1
8.5	1.08
15.5	1.15
26.5	1.22
r=0.9851

There is a very basic property of correlation coefficient, that it is independent on the origin and scale of variables. That means if the scale (or unit) of any variable changes then it will not affect the correlation coefficient also if we add or subtract any value from the variable then it will not affect the correlation coefficient.

Answer(1):

Here we are changing the unit of x (from gram to ounces) that means we are changing the scale of x variable. As per above property it will not affect the correlation between x and y.

The correct answer is The correlation would not change, as correlation does not depend on units.

Answer(2):

Here we are changing the origin of y (subtracting 0.25 from y). As per above property it will not affect the correlation between x and y.

The correct answer is The correlation would not change and the strength and direction of the linear relationship between risk and wine intake would not change.

Answer(3):

Here we changing the origin of both the variables. For x we are adding 1 and for y we are adding 0.01. As per above property it will not affect the correlation between x and y and we already have the correlation coefficient r=0.9851 which indicates a perfect positive linear relationship between given variables.

The correct answer is There would be a perfect, positive linear relationship.