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You work for a pharmaceutical...

You work for a pharmaceutical company that is currently testing a new antihypertensive (drug to reduce blood pressure). Your manager has recently asked you to do a quick preliminary analysis looking at the effectiveness of this drug. You have a sample of 104 patients who all had, prior to this test, a systolic blood pressure of approximately 175. These patients were given varying dosages of your drug, including some receiving a placebo. The quantity they were given is measured in mg/kg/day and is found in the column titled "dosage." After a two month treatment period, each patient had their systolic blood pressure retested; the difference between their initial blood pressure and their new blood pressure can be found in the colum "pressure." Lower (i.e. "more" negative) numbers in this column indicate a greater reduction in systolic blood pressure, and higher numbers imply that their systolic blood pressure actually increased. You also have data on patient blood type (A, B, AB, and O) in the column titled "blood type." Use this data to answer the following questions. Problem 4.1 comes from week 9 material, 4.2 comes from week 10 material, and 4.3 relates to week 11 material. Because this is due before week 11, question 4.3 is extra credit. Should you encounter any difficulties with these problems, the optional problems below are very similar to the questions in this problem set, and the answers to the optional questions can be found in the back of the textbook. You can also request that the tutor work extensively with you on the optional problems.

Problem 4.1

"The first issue your boss has asked you to address is whether or not there are differences in the effectiveness of the drug between the four different blood types. Use the 0.05 level of significance to:

a) Perform a one-way ANOVA to look for differences in changes in blood pressure between blood types.

b) If the results in (a) indicate that it is appropriate, use the Tukey-Kramer procedure to determine which blood types differ in mean blood pressure changes.

c) Briefly summarize (in plain English) your procedures and the results of (a) and (b) for your manager.

Excel Tips: When using the Data Analysis ToolPak, Excel requires that your data be formatted differently for ANOVA than for regression. The data as downloaded is formatted correctly for regression analysis, so you will have to transform your data prior to estimating the ANOVA.

"

Problem 4.2

"In addition to looking at differences between blood types, your manager also wants to know the relationship between dosage and change in systolic blood pressure. Thus, the dosage is your independent variable and the change in blood pressure is your dependent variable.

a)Construct a scatter plot of the two variables (note: the scatter plot should look non-linear. Still, do these questions and we'll address the non-linearity in 4.3)

b)Estimate a simple linear regression between these two variables.

c)Interpret the meaning of β0 and β1.

d)Predict the mean blood pressure change associated with dosages of 0, 2.5, 5, 7.5, 10, and 12.5 Are these appropriate predictions?

e)Comment briefly on the predictive power/statistical significance of your estimates. "

"EXTRA CREDIT. To answer this question, you will need to read the materials for week 11 of the class. Here, your task is directly tackle the non-linearity issue we found in problem 4.2. Your first step is to generate a quadratic term for your dosage variable (see section 15.1 in your text).

a) Estimate a multiple regression model, again using pressure as the dependent variable, however for your independent variables you will want to use both the linear and quadratic dosage variables.

b) Comment on the results from (a) in light of your results in 4.2.

c) Calculate the optimal dosage of your drug (WARNING: There are a few ways to figure this out, but dusting off your calculus skills makes this very easy! Other, less precise methods include graphine the regression equation and locating the minimum or plugging values into the regression equation until you find the dosage that minimizes your dependent variable)

"

Blood Type Pressure Dosage

4 -14 5 Note: In column A, 1=A, 2=B, 3=AB, and 4=O

1 -12 5

1 -7 0

2 -14 2

1 -13 5

3 -5 1

2 1 9

1 -1 9

1 -7 8

3 -4 0

2 -17 6

3 -13 2

3 3 9

3 -7 0

1 -16 6

3 -14 4

4 -11 7

2 3 9

1 0 0

3 -6 0

4 -15 1

2 -8 1

1 -12 2

4 -5 0

2 -5 1

3 -13 1

1 -18 5

2 -3 8

2 -14 5

1 15 10

2 0 9

1 -19 3

1 -11 2

4 -15 4

2 -8 8

2 12 10

3 -5 1

3 -8 7

2 -12 3

4 -6 0

1 3 0

2 1 8

1 -14 3

1 -14 5

1 -5 8

4 -10 1

2 -8 2

3 -12 2

2 1 0

1 -8 1

3 -4 0

1 -12 5

1 -10 2

2 -3 0

2 -12 5

3 11 10

3 -4 8

4 13 10

3 -3 0

1 -15 6

2 -6 8

3 -8 7

2 -12 2

1 -10 0

3 -8 0

4 -1 9

2 13 10

3 8 10

3 8 10

4 -7 1

2 -6 0

3 -2 0

4 -10 6

1 -19 4

2 -7 1

2 -7 7

3 -14 5

2 -9 2

3 -18 4

4 -11 2

2 -14 5

2 -19 5

3 10 10

4 -10 7

1 0 0

1 -1 0

2 16 10

3 -7 0

3 -13 2

4 15 10

1 0 0

3 -12 3

1 -15 2

3 -1 8

3 -17 2

2 -16 5

1 3 9

2 -2 8

1 -19 4

1 -10 6

3 -12 1

1 6 9

1 -8 0

1 -16 6

Problem 4.1

"The first issue your boss has asked you to address is whether or not there are differences in the effectiveness of the drug between the four different blood types. Use the 0.05 level of significance to:

a) Perform a one-way ANOVA to look for differences in changes in blood pressure between blood types.

b) If the results in (a) indicate that it is appropriate, use the Tukey-Kramer procedure to determine which blood types differ in mean blood pressure changes.

c) Briefly summarize (in plain English) your procedures and the results of (a) and (b) for your manager.

Excel Tips: When using the Data Analysis ToolPak, Excel requires that your data be formatted differently for ANOVA than for regression. The data as downloaded is formatted correctly for regression analysis, so you will have to transform your data prior to estimating the ANOVA.

"

Problem 4.2

"In addition to looking at differences between blood types, your manager also wants to know the relationship between dosage and change in systolic blood pressure. Thus, the dosage is your independent variable and the change in blood pressure is your dependent variable.

a)Construct a scatter plot of the two variables (note: the scatter plot should look non-linear. Still, do these questions and we'll address the non-linearity in 4.3)

b)Estimate a simple linear regression between these two variables.

c)Interpret the meaning of β0 and β1.

d)Predict the mean blood pressure change associated with dosages of 0, 2.5, 5, 7.5, 10, and 12.5 Are these appropriate predictions?

e)Comment briefly on the predictive power/statistical significance of your estimates. "

"EXTRA CREDIT. To answer this question, you will need to read the materials for week 11 of the class. Here, your task is directly tackle the non-linearity issue we found in problem 4.2. Your first step is to generate a quadratic term for your dosage variable (see section 15.1 in your text).

a) Estimate a multiple regression model, again using pressure as the dependent variable, however for your independent variables you will want to use both the linear and quadratic dosage variables.

b) Comment on the results from (a) in light of your results in 4.2.

c) Calculate the optimal dosage of your drug (WARNING: There are a few ways to figure this out, but dusting off your calculus skills makes this very easy! Other, less precise methods include graphine the regression equation and locating the minimum or plugging values into the regression equation until you find the dosage that minimizes your dependent variable)

"

Blood Type Pressure Dosage

4 -14 5 Note: In column A, 1=A, 2=B, 3=AB, and 4=O

1 -12 5

1 -7 0

2 -14 2

1 -13 5

3 -5 1

2 1 9

1 -1 9

1 -7 8

3 -4 0

2 -17 6

3 -13 2

3 3 9

3 -7 0

1 -16 6

3 -14 4

4 -11 7

2 3 9

1 0 0

3 -6 0

4 -15 1

2 -8 1

1 -12 2

4 -5 0

2 -5 1

3 -13 1

1 -18 5

2 -3 8

2 -14 5

1 15 10

2 0 9

1 -19 3

1 -11 2

4 -15 4

2 -8 8

2 12 10

3 -5 1

3 -8 7

2 -12 3

4 -6 0

1 3 0

2 1 8

1 -14 3

1 -14 5

1 -5 8

4 -10 1

2 -8 2

3 -12 2

2 1 0

1 -8 1

3 -4 0

1 -12 5

1 -10 2

2 -3 0

2 -12 5

3 11 10

3 -4 8

4 13 10

3 -3 0

1 -15 6

2 -6 8

3 -8 7

2 -12 2

1 -10 0

3 -8 0

4 -1 9

2 13 10

3 8 10

3 8 10

4 -7 1

2 -6 0

3 -2 0

4 -10 6

1 -19 4

2 -7 1

2 -7 7

3 -14 5

2 -9 2

3 -18 4

4 -11 2

2 -14 5

2 -19 5

3 10 10

4 -10 7

1 0 0

1 -1 0

2 16 10

3 -7 0

3 -13 2

4 15 10

1 0 0

3 -12 3

1 -15 2

3 -1 8

3 -17 2

2 -16 5

1 3 9

2 -2 8

1 -19 4

1 -10 6

3 -12 1

1 6 9

1 -8 0

1 -16 6

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