Boyd, Delost and Holcomb (1998) conducted a study to determine if significant gender differences existed in the mean amount of calcium levels in the blood among subjects 65 years of age and older. The researchers performed a retrospective chart review of laboratory procedures performed in six different physician practices. The nonmissing data consisted of 176 subjects. The variables collected were SEX (1=male, 2=female), LAB (lab where blood was analyzed; 1=Metpath, 2=Deyor, 3=St. Elizabeth's, 4=CB Rouche, 5=YOH, 6=Horizon), AGE (years), AKPHOS (alkaline phosphatase, IU/L), and CAMMOL (raw calcium, mmol/L). Using a 0.05 significance level and the given computer output, you now need to test the claim that the patients in the six physician practices (or labs) have different mean calcium levels by answering the questions that follow.

For questions 1-5, Match the output with the most appropriate analysis. select each letter (method) more than once, once or not at all.

(1 point each)

1. SAS Output #1 a. One-way ANOVA

2. SAS Output #2 b. Two-way ANOVA with an interaction

3. SAS Output #3 c. Two-way ANOVA without an interaction

4. SAS Output #4 d. ANCOVA

5. SAS Output #5 e. Multiple linear regression

2

For questions 6-22, refer to SAS Output #1.

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

a. H0: 1 = 2 = 3 = 4 = 5 = 6 = 0, H1: at least one i 0

b. H0: y1 = y2 = y3 = y4 = y5 = y6, H1: y1 y2 y3 y4 y5 y6

c. H0: 1 = 2 = 3 = 4 = 5 = 6, H1: 1 2 3 4 5 6

d. H0: y1 = y2 = y3 = y4 = y5 = y6, H1: at least one yi differs from another yj

e. H0: 1 = 2 = 3 = 4 = 5 = 6, H1: at least one i differs from another j

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

a. Dependent = calcium level, independent = sex

b. Dependent = sex, independent = calcium level

c. Dependent = calcium level, independent = lab

d. Dependent = lab, independent = calcium level

e. Cannot be determined from the given information.

For questions 8-14 refer to the following: In the first table of SAS Output #1, several important values have been replaced with the letters A-G. find these missing values.

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

a. 1

b. 4

c. 5

d. 6

e. 7

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

a. 169

b. 170

c. 171

d. 174

e. 175

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

a. 170

b. 173

c. 174

d. 175

e. 176

3

11. Sum of Square Error (labeled D) (0.5 point)

a. 0.14

b. 1.28

c. 2.78

d. 3.73

e. 6.90

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

a. 0.067

b. 0.079

c. 0.094

d. 0.12

e. 0.47

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

a. 0.016

b. 0.16

c. 0.98

d. 1.47

e. 2.78

14. t test statistic (labeled G) (0.5 point)

a. 0.062

b. 0.085

c. 0.18

d. 1.36

e. 5.76

15. What is the value of the test statistic for the omnibus null hypothesis H0? (1 point)

a. 0.17

b. 0.20

c. 1.96

d. 5.76

e. 7.92

4

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

a. <0.0001

b. 0.05

c. 0.27

d. 0.41

e. >0.9999

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 of the labs have a significantly different mean calcium level from one another.

b. Because p-value < 0.05, we reject H0 and conclude that at least one lab has a significantly different mean calcium level than the others.

c. Because p-value < 0.05, we fail to reject H0 and conclude that the mean calcium level does not differ significantly between the 4 labs.

d. Because p-value > 0.05, we fail to reject H0 and conclude that the mean calcium level does not differ significantly between the 4 labs.

e. Because p-value > 0.05, we reject H0 and conclude that the mean calcium level does differ significantly between the 4 labs.

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. Tukey test

b. Bonferroni test

c. Multiple comparisons test

d. None of the above are appropriate

e. Any of the above are appropriate

19. What is the estimated mean calcium level for each of the six labs? (1 point)

a. 1 = -0.045, 2 = 0.062, 3 = -0.0081, 4 = 0.085, 5 = -0.058, 6 = 0

b. y1 = -0.045, y2 = 0.062, y3 = -0.0081, y4 = 0.085, y5 = -0.058, y6 = 0

c. 1 = -0.045, 2 = 0.062, 3 = -0.0081, 4 = 0.085, 5 = -0.058, 6 = 2.36

d. y1 = -0.045, y2 = 0.062, y3 = -0.0081, y4 = 0.085, y5 = -0.058, y6 = 2.36

e. 1 = 2.32, 2 = 2.42, 3 = 2.35, 4 = 2.45, 5 = 2.30, 6 = 2.36

f. y1 = 2.32, y2 = 2.42, y3 = 2.35, y4 = 2.45, y5 = 2.30, y6 = 2.36

5

20. what if any of the six labs differ significantly in their mean calcium level? If so, which one(s) differ? (1 point)

a. Yes, 1 2

b. Yes, 1 4

c. Yes, 2 5

d. Yes, 4 5

e. All of the above

f. Both a and b

g. No, the six labs have the same mean calcium level.

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

a. We are comparing the mean calcium level for 6 groups, which has nothing to do with variation.

b. Variation in calcium level between labs

c. Variation in calcium level within labs

d. Both b and c

e. None of the above

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

a. Yes, the reference cell is lab 1.

b. Yes, the reference cell is lab 2.

c. Yes, the reference cell is lab 5.

d. Yes, the reference cell is lab 6.

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

f. No, there is no reference cell.

23. Suppose the researchers are interested in comparing the mean calcium level between males and females, adjusted for the patient's age and alkaline phosphatase level. Using the most appropriate model, select which of the following statements is true. (1 point)

a. Mean calcium level differs significantly for sex, age and alkaline phosphatase.

b. Mean calcium level differs significantly for sex and alkaline phosphatase but not age.

c. Mean calcium level differs significantly for age and alkaline phosphatase but not sex.

d. Mean calcium level does not differ significantly between males and females, after adjusting for age and alkaline phosphatase.

e. Mean calcium level differs significantly between males and females, after adjusting for age and alkaline phosphatase.

6

24. The interaction plot below was generated from the model in SAS Output #3. What can you say about the following figure? (1 point)

a. As lab increases, the calcium level decreases at the same rate for both males and females.

b. There is no significant interaction between sex and lab.

c. There is a same direction interaction between sex and lab.

d. There is a reverse interaction between sex and lab.

e. Both a and b

7

OPTIONAL PRACTICE QUESTIONS: The following are optional practice questions that you can work on after you submit your quiz. They will NOT be graded; however, you should be prepared to answer these types of questions on the final exam.

Practice 1: Write out the estimated model given in SAS Output #2.

Practice 2: Write out the estimated model given in SAS Output #3.

Practice 3: Write out the estimated model given in SAS Output #4.

Practice 4: Write out the estimated model given in SAS Output #5. 8

SAS Output #1

Sum of

Source DF Squares Mean Square F Value Pr > F

Model A 0.47191070 E H <.0001

Error B D F

Corrected Total C 3.25679091

R-Square Coeff Var Root MSE CAMMOL Mean

0.144901 5.434330 0.127991 2.355227

Source DF Type III SS Mean Square F Value Pr > F

LAB 5 0.47191070 0.09438214 H <.0001

Standard

Parameter Estimate Error t Value Pr > |t|

Intercept 2.360000000 B 0.05225203 45.17 <.0001

LAB 1 -0.044597701 B 0.05402379 -0.83 0.4102

LAB 2 0.062380952 B 0.05585977 1.12 0.2657

LAB 3 -0.008125000 B 0.06127094 -0.13 0.8947

LAB 4 0.085000000 B 0.06245312 G 0.1753

LAB 5 -0.058181818 B 0.06495780 -0.90 0.3717

LAB 6 0.000000000 B . . .

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

CAMMOL LSMEAN

LAB LSMEAN Number

1 2.31540230 1

2 2.42238095 2

3 2.35187500 3

4 2.44500000 4

5 2.30181818 5

6 2.36000000 6

Least Squares Means for effect LAB

Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: CAMMOL

i/j 1 2 3 4 5 6

1 0.0002 0.9010 0.0073 0.9995 0.9625

2 0.0002 0.4211 0.9927 0.0654 0.8739

3 0.9010 0.4211 0.3534 0.9179 1.0000

4 0.0073 0.9927 0.3534 0.0662 0.7501

5 0.9995 0.0654 0.9179 0.0662 0.9471

6 0.9625 0.8739 1.0000 0.7501 0.9471

SAS Output #2

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 6 0.62472682 0.10412114 6.69 <.0001

Error 169 2.63206409 0.01557434

Corrected Total 175 3.25679091 9

R-Square Coeff Var Root MSE CAMMOL Mean

0.191823 5.298733 0.124797 2.355227

Source DF Type III SS Mean Square F Value Pr > F

SEX 1 0.15281612 0.15281612 9.81 0.0020

LAB 5 0.36544303 0.07308861 4.69 0.0005

Standard

Parameter Estimate Error t Value Pr > |t|

Intercept 2.370541395 B 0.05105927 46.43 <.0001

SEX 1 -0.063248370 B 0.02019155 -3.13 0.0020

SEX 2 0.000000000 B . . .

LAB 1 -0.012246523 B 0.05367871 -0.23 0.8198

LAB 2 0.081957829 B 0.05482336 1.49 0.1368

LAB 3 -0.002854302 B 0.05976581 -0.05 0.9620

LAB 4 0.088011827 B 0.06090239 1.45 0.1503

LAB 5 -0.045723806 B 0.06346173 -0.72 0.4722

LAB 6 0.000000000 B . . .

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

H0:LSMean1=

CAMMOL LSMean2

SEX LSMEAN Pr > |t|

1 2.32548386 0.0020

2 2.38873223 10

SAS Output #2 (continued)

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

CAMMOL LSMEAN

LAB LSMEAN Number

1 2.32667069 1

2 2.42087504 2

3 2.33606291 3

4 2.42692904 4

5 2.29319340 5

6 2.33891721 6

Least Squares Means for effect LAB

Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: CAMMOL

i/j 1 2 3 4 5 6

1 0.0015 0.9998 0.0806 0.9620 0.9999

2 0.0015 0.2017 1.0000 0.0345 0.6680

3 0.9998 0.2017 0.3528 0.9517 1.0000

4 0.0806 1.0000 0.3528 0.0901 0.6994

5 0.9620 0.0345 0.9517 0.0901 0.9792

6 0.9999 0.6680 1.0000 0.6994 0.9792 11

SAS Output #3

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 11 0.68000807 0.06181892 3.93 <.0001

Error 164 2.57678284 0.01571209

Corrected Total 175 3.25679091

R-Square Coeff Var Root MSE CAMMOL Mean

0.208797 5.322114 0.125348 2.355227

Source DF Type III SS Mean Square F Value Pr > F

SEX 1 0.10766731 0.10766731 6.85 0.0097

LAB 5 0.33628448 0.06725690 4.28 0.0011

SEX*LAB 5 0.05528125 0.01105625 0.70 0.6214

Standard

Parameter Estimate Error t Value Pr > |t|

Intercept 2.386000000 B 0.05605728 42.56 <.0001

SEX 1 -0.156000000 B 0.13731172 -1.14 0.2576

SEX 2 0.000000000 B . . .

LAB 1 -0.022071429 B 0.06085692 -0.36 0.7173

LAB 2 0.048545455 B 0.06210156 0.78 0.4355

LAB 3 -0.031833333 B 0.06672150 -0.48 0.6339

LAB 4 0.082181818 B 0.06760762 1.22 0.2259

LAB 5 -0.030285714 B 0.07339620 -0.41 0.6804

LAB 6 0.000000000 B . . .

SEX*LAB 1 1 0.084444310 B 0.14029241 0.60 0.5481

SEX*LAB 1 2 0.130454545 B 0.14266849 0.91 0.3619

SEX*LAB 1 3 0.146833333 B 0.15521557 0.95 0.3455

SEX*LAB 1 4 0.047818182 B 0.15975057 0.30 0.7651

SEX*LAB 1 5 0.007785714 B 0.15819961 0.05 0.9608

SEX*LAB 1 6 0.000000000 B . . .

SEX*LAB 2 1 0.000000000 B . . .

SEX*LAB 2 2 0.000000000 B . . .

SEX*LAB 2 3 0.000000000 B . . .

SEX*LAB 2 4 0.000000000 B . . .

SEX*LAB 2 5 0.000000000 B . . .

SEX*LAB 2 6 0.000000000 B . . . PubH 6002: Biostatistical Applications for Public Health

SAS Output #3 (continued)

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

CAMMOL LSMEAN

SEX LAB LSMEAN Number

1 1 2.29237288 1

1 2 2.40900000 2

1 3 2.34500000 3

1 4 2.36000000 4

1 5 2.20750000 5

1 6 2.23000000 6

2 1 2.36392857 7

2 2 2.43454545 8

2 3 2.35416667 9

2 4 2.46818182 10

2 5 2.35571429 11

2 6 2.38600000 12

Least Squares Means for effect SEX*LAB

Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: CAMMOL

i/j 1 2 3 4 5 6 7 8 9 10 11 12

1 0.0211 0.9996 0.9989 0.9767 1.0000 0.3553 0.0007 0.9217 0.0019 0.9824 0.9055

2 0.0211 0.9987 1.0000 0.1386 0.9633 0.9859 1.0000 0.9885 0.9830 0.9982 1.0000

3 0.9996 0.9987 1.0000 0.9235 0.9996 1.0000 0.9762 1.0000 0.8734 1.0000 1.0000

4 0.9989 1.0000 1.0000 0.9094 0.9991 1.0000 0.9982 1.0000 0.9747 1.0000 1.0000

5 0.9767 0.1386 0.9235 0.9094 1.0000 0.4566 0.0477 0.6747 0.0236 0.7659 0.6073

6 1.0000 0.9633 0.9996 0.9991 1.0000 0.9962 0.9083 0.9984 0.8051 0.9986 0.9926

7 0.3553 0.9859 1.0000 1.0000 0.4566 0.9962 0.7079 1.0000 0.4548 1.0000 1.0000

8 0.0007 1.0000 0.9762 0.9982 0.0477 0.9083 0.7079 0.8228 0.9999 0.9516 0.9998

9 0.9217 0.9885 1.0000 1.0000 0.6747 0.9984 1.0000 0.8228 0.5671 1.0000 1.0000

10 0.0019 0.9830 0.8734 0.9747 0.0236 0.8051 0.4548 0.9999 0.5671 0.7842 0.9871

11 0.9824 0.9982 1.0000 1.0000 0.7659 0.9986 1.0000 0.9516 1.0000 0.7842 1.0000

12 0.9055 1.0000 1.0000 1.0000 0.6073 0.9926 1.0000 0.9998 1.0000 0.9871 1.0000 PubH 6002: Biostatistical Applications for Public Health

SAS Output #4

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 2 0.12362 0.06181 3.42 0.0349

Error 173 3.12514 0.01806

Corrected Total 175 3.24875

Root MSE 0.13440 R-Square 0.0381

Dependent Mean 2.35551 Adj R-Sq 0.0269

Coeff Var 5.70593

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 2.34030 0.15751 14.86 <.0001

AGE 1 -0.00088138 0.00211 -0.42 0.6761

AKPHOS 1 0.00085943 0.00033677 2.55 0.0116 14

SAS Output #5

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 3 0.32577591 0.10859197 6.39 0.0004

Error 172 2.92297806 0.01699406

Corrected Total 175 3.24875398

R-Square Coeff Var Root MSE CAMMOL Mean

0.100277 5.534308 0.130361 2.355511

Source DF Type III SS Mean Square F Value Pr > F

SEX 1 0.20215745 0.20215745 11.90 0.0007

AGE 1 0.00183794 0.00183794 0.11 0.7427

AKPHOS 1 0.05937126 0.05937126 3.49 0.0633

Standard

Parameter Estimate Error t Value Pr > |t|

Intercept 2.382667955 B 0.15326557 15.55 <.0001

SEX 1 -0.069350276 B 0.02010721 -3.45 0.0007

SEX 2 0.000000000 B . . .

AGE -0.000672042 0.00204352 -0.33 0.7427

AKPHOS 0.000623752 0.00033371 1.87 0.0633

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

H0:LSMean1=

CAMMOL LSMean2

SEX LSMEAN Pr > |t|

1 2.32201833 0.0007

2 2.39136861

SAS Output #3

Source DF

Model 11 Error 164

 Sum of Squares 0.68000807 2.57678284 3.25679091 Mean Square F Value 0.06181892 3.93 0.01571209 

Pr>F <.0001

Pr>F

 0.0097 0.0011 0.6214 

|t|

Corrected Total 

Source

SEX LAB SEX*LAB 

Parameter

 Intercept SEX 1 SEX 2 LAB 1 LAB 2 LAB 3 LAB 4 LAB 5 LAB 6 SEX*LAB 1 SEX*LAB 1 SEX*LAB 1 SEX*LAB 1 SEX*LAB 1 SEX*LAB 1 SEX*LAB 2 SEX*LAB 2 SEX*LAB 2 SEX*LAB 2 SEX*LAB 2 SEX*LAB 2 

175

R-Square 0.208797 

Coeff Var Root MSE CAMMOL Mean 5.322114 0.125348 2.355227

DF Type III SS

 1 0.10766731 5 0.33628448 5 0.05528125 

Estimate

Mean Square

 0.10766731 0.06725690 0.01105625 

F Value

 6.85 4.28 0.70 Standard Error 

t

Value Pr>

1 2 3 4 5 6 1 2 3 4 5 6 

2.386000000 B 0.05605728 -0.156000000 B 0.13731172

0.000000000 B . -0.022071429 B 0.06085692

0.048545455 B 0.06210156 -0.031833333 B 0.06672150 0.082181818 B 0.06760762 -0.030285714 B 0.07339620

 0.000000000 B . 0.084444310 B 0.14029241 0.130454545 B 0.14266849 0.146833333 B 0.15521557 0.047818182 B 0.15975057 0.007785714 B 0.15819961 0.000000000 B . 0.000000000 B . 0.000000000 B . 0.000000000 B . 0.000000000 B . 0.000000000 B . 0.000000000 B . 42.56 <.0001 -1.14 0.2576 

. . -0.36 0.7173 0.78 0.4355 -0.48 0.6339 1.22 0.2259 -0.41 0.6804

. . 0.60 0.5481 0.91 0.3619 0.95 0.3455 0.30 0.7651 0.05 0.9608

. . . . . . . . . . . . . .

11

PubH 6002: Biostatistical Applications for Public Health SAS Output #3 (continued)

Least Squares Means

Adjustment for Multiple Comparisons: Tukey-Kramer

 CAMMOL LSMEAN SEX LAB LSMEAN Number 

1 1 1 2 1 3 1 4 1 5

1. 16
2. 21

2 2 2 3 2 4 2 5 2 6

2.29237288 1 2.40900000 2 2.34500000 3 2.36000000 4 2.20750000 5 2.23000000 6 2.36392857 7 2.43454545 8 2.35416667 9 2.46818182 10 2.35571429 11 2.38600000 12 

i/j 1

1

1. 20.0211
2. 30.9996
3. 40.9989
4. 50.9767
5. 61.0000
6. 70.3553
7. 80.0007
8. 90.9217
9. 100.0019
10. 110.9824
11. 120.9055

Least Squares Means for effect SEX*LAB Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: CAMMOL

2 3 4 5 6 7 8 9 10

0.0211 0.9996 0.9989 0.9767 1.0000 0.3553 0.0007 0.9217 0.0019 0.9987 1.0000 0.1386 0.9633 0.9859 1.0000 0.9885 0.9830 0.9987 1.0000 0.9235 0.9996 1.0000 0.9762 1.0000 0.8734 1.0000 1.0000 0.9094 0.9991 1.0000 0.9982 1.0000 0.9747 0.1386 0.9235 0.9094 1.0000 0.4566 0.0477 0.6747 0.0236 0.9633 0.9996 0.9991 1.0000 0.9962 0.9083 0.9984 0.8051 0.9859 1.0000 1.0000 0.4566 0.9962 0.7079 1.0000 0.4548 1.0000 0.9762 0.9982 0.0477 0.9083 0.7079 0.8228 0.9999 0.9885 1.0000 1.0000 0.6747 0.9984 1.0000 0.8228 0.5671

0.9830 0.8734 0.9747 0.0236 0.8051 0.4548 0.9999 0.5671

0.9982 1.0000 1.0000 0.7659 0.9986 1.0000 0.9516 1.0000 0.7842 1.0000 1.0000 1.0000 0.6073 0.9926 1.0000 0.9998 1.0000 0.9871

11

0.9824 0.9982 1.0000 1.0000 0.7659 0.9986 1.0000 0.9516 1.0000 0.7842 

1.0000

12

0.9055 1.0000 1.0000 1.0000 0.6073 0.9926 1.0000 0.9998 1.0000 0.9871 1.0000 

PubH 6002: Biostatistical Applications for Public Health SAS Output #4

Source DF

Model 2 Error 173 Corrected Total 175

 Root MSE Dependent Mean Coeff Var Sum of Squares Mean Square F Value 0.06181 3.42 0.01806 R-Square 0.0381 Adj R-Sq 0.0269 

Pr > F 0.0349

0.12362 3.12514 3.24875 0.13440 2.35551 5.70593 

Variable

Intercept AGE AKPHOS Parameter DF Estimate 1 2.34030 1 -0.00088138 1 0.00085943 Standard Error 

0.15751

 0.00211 0.00033677 

t Value

 14.86 -0.42 2.55 

Pr > |t|

 <.0001 0.6761 0.0116 

SAS Output #5

Source

Model Error Corrected Total 

Source

SEX AGE AKPHOS Sum of DF Squares 3 0.32577591 172 2.92297806 175 3.24875398 Mean Square F Value 0.10859197 6.39 0.01699406 

Pr>F 0.0004

Pr>F

0.0007 0.7427 0.0633 R-Square 0.100277 

Coeff Var Root MSE CAMMOL Mean 5.534308 0.130361 2.355511

DF Type III SS

Mean Square F Value 0.20215745 11.90 0.00183794 0.11 0.05937126 3.49 

1 1 1

0.20215745 0.00183794 0.05937126 Parameter Estimate 

Intercept 2.382667955 SEX 1 -0.069350276 SEX 2 0.000000000 AGE -0.000672042 AKPHOS 0.000623752

 Standard Error 

B 0.15326557 B 0.02010721 B.

 0.00204352 0.00033371 

t Value

 15.55 -3.45 . -0.33 

1.87

Tukey-Kramer 

Pr>|t|

 <.0001 0.0007 

. 0.7427 0.0633

Least Squares Means Adjustment for Multiple Comparisons:

SEX

1 2

 CAMMOL LSMEAN 2.32201833 2.39136861 H0:LSMean1= LSMean2 Pr > |t| 0.0007