The Heart and Estrogen Replacement Study was a large scale study designed to examine risk factors for coronary heart disease in a cohort of post-menopausal women. The data contains a number of biomarkers such as serum cholesterol and glucose measurements, anthropometric data such as body mass index (BMI) and waist to hip ratio, and behavioral data such as exercise frequency and smoking status.

In the examples shown below, SAS output 1-5 represent example analyses from this data on N = 2608 individuals. The outcome variable in all cases is high density lipoprotein (HDL) cholesterol at year 1 of the study. In each analysis below, at least one of the following variables are used as explanatory variables:

Physact1 = physical activity level (1=low, 2=moderate, 3=frequent)

Smoking = smoking status (1=smoker, 0=nonsmoker)

TCHOL = total cholesterol (continuous variable)

BMI = body mass index (continuous variable)

Exercise = exercise frequency (1=regular, 0=irregular)

For questions 1-6, examine the computer output of the five different models provided for you in the separate packet labeled SAS Output #1-SAS Output #6. Match the output with the most appropriate analysis. You may select each letter (method) more than once, once or not at all.

(1 point each)

1.SAS Output #1 a.Independent Samples t-test

2.SAS Output #2 b. Two-way ANOVA without an interaction

3.SAS Output #3 c. Two-way ANOVA with an interaction

4.SAS Output #4 d. ANCOVA

5.SAS Output #5 e. Multiple Linear Regression

6.SAS Output #6 f. One-way ANOVA

For questions 7-22, refer to SAS Output #1.

7.What are the omnibus null and alternative hypotheses? (1 point)

a.:.H₀: ₁ = ₂ = ₃ = 0, H₁: at least one _i 0

b.H₀: y₁ = y₂ = y₃, H₁: y₁ y₂ y₃

c.H₀: ₁ = ₂ = ₃, H₁: _{1 2 3}

d.H₀: ₁ = ₂ = ₃, H₁: at least one _idiffers from another _j

e.H₀: y₁ = y₂ = y₃, H₁: at least one y_i differs from another y_j

8.What are the independent and dependent variables? (1 point)

a.Dependent = physical activity level, independent = HDL at year 1

b.Dependent = HDL at year 1, independent = physical activity level

c.Dependent is unknown, independent = physical activity level

d.Dependent is unknown, independent = HDL at year 1

e.Variable order is not a relevant concern, i.e. (x, y) is the same as (y, x). Either physical activity level or HDL at year 1 can be the independent variable. The other is then the dependent variable.

For questions 9-22 refer to the following: In the first table of SAS Output #1, several important values have been replaced with the letters A-J. You need to find these missing values.

9.Model Degrees of Freedom (labeled A) (0.5 point)

a.1

b.2

c.3

d.4

e.5

10.Error Degrees of Freedom (labeled B) (0.5 point)

a.2604

b.2605

c.2606

d.2607

e.2608

11.Total Degrees of Freedom (labeled C) (0.5 point)

a.2608

b.2607

c.2606

d.2605

e.2604

12.Sum of Square Model (labeled D) (0.5 point)

a. 4891.873

b.2445.9036

c.13.81

d.496755.0512

e.12.83

13.Mean Square Model (labeled E) (0.5 point)

a.4891.873

b.2445.9036

c.13.81

d.496755.0512

e.12.83

14.Mean Square Error (labeled F) (0.5 point)

a.4891.873

b.2445.9036

c.13.81

d.496755.0512

e.190.6929

15.What is the value of the test statistic for the omnibus null hypothesis (labeled G)? (1 point)

a. 91.63

b.5.02

c.12.83

d.25.2

e.2.48

16.What is the p-value corresponding to the test statistic in question (15)? (0.5 point)

a.<0.00005

b. <0.0001

c.0.0041

d.0.0081

e.0.0162

17.Using a 0.05 significance level, what decision and conclusion should you make regarding the omnibus null hypothesis? (1 point)

a.Because p-value < 0.05, we reject H₀ and conclude that all 3 of the physical activity groups have a different mean HDL at year 1.

b.Because p-value < 0.05, we reject H₀ and conclude that all 3 of the physical activity groups have the same mean HDL at year 1.

c.Because p-value < 0.05, we fail to reject H₀ and conclude that all 3 of the physical activity groups have a different mean HDL at year 1.

d.Because p-value < 0.05, we fail to reject H₀ and conclude that all 3 of the physical activity groups have the same mean HDL at year 1.

e.Because p-value < 0.05, we reject H₀ and conclude that at least one of the 3 physical activity groups does not have the same mean HDL level at year 1 compared to another of the 3 physical activity groups.

18.Assuming you rejected the omnibus null hypothesis in question 17 (this may or may not be true), what test should you conduct next? (1 point)

a.Multiple comparisons test

b.Bonferroni test

c.Tukey test

d.None of the above are appropriate

e.Any of the above are appropriate

19.What is the estimated HDL level at year 1 for each physical activity level (labeled H, I, and J)? (1 point)

a.y₁ = 49.75, y₂= 51.52, y₃ = 53.19

b.₁ = 49.75, ₂ = 51.52, ₃ = 53.19

c.y₁ = 49.75, y₂= 1.78, y₃ = 3.44

d.₁ = 0, ₂ = 1.78, ₃ = 3.44

e.y₁ = 53.19, y₂ = 51.52, y₃ = 49.75

f.₁ = 51.52, ₂ = 49.75, ₃ = 53.19

20.Do any of the 3 treatment groups differ significantly in their mean change in the number of DMFT? If so, which one(s)? (1 point)

a.No, ₁ = ₂ (Compare our p-value =.0225 to 0.05)

b.Yes, ₂ = ₃ (Compare our p-value = .0357 to 0.05)

c.No, ₃ = ₁ (Compare our p-value < .0001 to 0.05)

d.Both b and c

e.Yes, all 3 pairwise comparisons show a significant difference after controlling for multiple comparisons.

21.What types of variation, if any, are being analyzed in this problem? (1 point)

a.We are comparing the mean HDL level at year 1 for all 3 phyiscal activity groups, which is a question that does not involve analysis of variability.

b.We are comparing variability in the group means relative to the mean of the entire sample.

c.We are comparing variability in the individual HDL values for a given group relative to the mean HDL level of that group.

d.None of the above

e.Both b and c

22.Is there a reference cell in this problem? If so, what is it? (1 point)

a.Yes, the reference cell is physical activity level 1.

b.Yes, the reference cell is physical activity level 2.

c.Yes, the reference cell is physical activity level 3.

d.Yes, there is a reference cell but the group cannot be determined with the given output.

e.No, there is no reference cell.

23.Which of the SAS outputs may be used to assess the effects of physical activity and smoking status on HDL at year 1? (1 point)

a.SAS Output #1

b.SAS Output #2 because there is a significant interaction between smoking and physical activity

c.SAS Output #2 because there is not a significant interaction between smoking and physical activity

d.SAS Output #4

e.SAS Output #5

Practice Problem: Write out the estimated model given in SAS Output #2.

Practice Problem: Write out the estimated model given in SAS Output #3.

Practice Problem: Write out the estimated model given in SAS Output #4.

Practice Problem: Write out the estimated model given in SAS Output #5.

Practice Problem: Write out the estimated model given in SAS Output #6.

SAS OUTPUT 1

SAS OUTPUT 2

SAS OUTPUT 3

SAS OUTPUT 4

SAS OUTPUT 5

SAS OUTPUT 6