1. Consider the numerical examples given in Section 8.8 of Chapter 8, involving assess- ment of the relationship of the independent variables HGT, AGE, and (AGE) to the dependent variable WGT. Suppose that HGT is the independent variable of primary concern, so interest lies in evaluating the relationship of HGT to WGT, controlling for the possible confounding effects of AGE and (AGE)². a. Assuming that no interaction of any kind exists, state an appropriate regression model to use as the baseline (i.e., standard) for decisions about confounding. b. Using an appropriate regression coefficient given in part (a) as your measure of association, determine whether confounding exists due to AGE and/or (AGE)². c. Can (AGE) be dropped from your initial model in part (a) because it is not needed to control adequately for confounding? Explain your answer (using a regression coefficient as your measure of association). d. Should (AGE)² be retained in the final model for the sake of precision? Explain. e. In light of both confounding and precision, what should be your final model? Why? f. How would you modify your initial model in part (a) to allow for assessing inter- actions? g. Regarding your answer to part (f), how would you test for interaction?
1. Consider the numerical examples given in Section 8.8 of Chapter 8, involving assess- ment of the relationship of the independent variables HGT, AGE, and (AGE) to the dependent variable WGT. Suppose that HGT is the independent variable of primary concern, so interest lies in evaluating the relationship of HGT to WGT, controlling for the possible confounding effects of AGE and (AGE)². a. Assuming that no interaction of any kind exists, state an appropriate regression model to use as the baseline (i.e., standard) for decisions about confounding. b. Using an appropriate regression coefficient given in part (a) as your measure of association, determine whether confounding exists due to AGE and/or (AGE)². c. Can (AGE) be dropped from your initial model in part (a) because it is not needed to control adequately for confounding? Explain your answer (using a regression coefficient as your measure of association). d. Should (AGE)² be retained in the final model for the sake of precision? Explain. e. In light of both confounding and precision, what should be your final model? Why? f. How would you modify your initial model in part (a) to allow for assessing inter- actions? g. Regarding your answer to part (f), how would you test for interaction?
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 4E
Related questions
Question
Transcribed Image Text:1. Consider the numerical examples given in Section 8.8 of Chapter 8, involving assess-
ment of the relationship of the independent variables HGT, AGE, and (AGE)² to the
dependent variable WGT. Suppose that HGT is the independent variable of primary
concern, so interest lies in evaluating the relationship of HGT to WGT, controlling
for the possible confounding effects of AGE and (AGE)².
a. Assuming that no interaction of any kind exists, state an appropriate regression
model to use as the baseline (i.e., standard) for decisions about confounding.
b. Using an appropriate regression coefficient given in part (a) as your measure of
association, determine whether confounding exists due to AGE and/or (AGE)².
c. Can (AGE)² be dropped from your initial model in part (a) because it is not
needed to control adequately for confounding? Explain your answer (using a
regression coefficient as your measure of association).
d. Should (AGE)² be retained in the final model for the sake of precision? Explain.
e. In light of both confounding and precision, what should be your final model?
Why?
f. How would you modify your initial model in part (a) to allow for assessing inter-
actions?
g. Regarding your answer to part (f), how would you test for interaction?
![Model 1 WGT = Bo + B₁HGT + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT
Source
Model
Error
Corrected Total
DF
1
10
11
Parameter
Intercept
HGT
Source
Model
Error
Corrected Total
4 Partial Fstatistics will be discussed in Chapter 9 on hypothesis testing.
R²
Parameter
Intercept
HGT
AGE
R-Square
0.663014
Sum of Squares
588.9225232
299.3274768
888.2500000
SSY
9
11
R-Square
0.779986
Â₁
Copyright 2013 Cengage Leaming. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s
torial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions re
Coeff Var
8.718857
DF Sum of Squares
2
692.8226065
195.4273935
888.2500000
Mean Square
588.9225232
29.9327477
SSE
SSY-SSE
Coeff Var
7.426048
[Portion of output omitted]
Estimate
6.189848707
1.072230356
Estimate
6.553048251
0.722037958
2.050126352
Root MSE
5.471083
Standard Error
12.84874620
0.24173098
Model 4 WGT = Bo + B₁HGT + B₂AGE + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT and AGE
Mean Square
346.4113033
21.7141548
F statistic for overall test
Root MSE
4.659845
F Value
Standard Error
10.94482708
0.26080506
0.93722561
19.67
WGT Mean
62.75000
t Value
Pr > F
WGT Mean
62.75000
t Value
0.48
4.44 0.0013
0.6404
0.0013
8.8 Numerical Examples
Pr > It
P-value for
overall test
Pr> t
F Value Pr > F
15.95 0.0011
0.60
0.5641
2.77 0.0218
2.19 0.0565
(continued)
Test statistics and P-values for partial tests
on model parameters (see Section 9.3)
14
Model 2 WGT = Bo + B₂AGE + E
Edited SAS Output (PROC REG) for Regression of WGT on AGE
Source
Model
Error
Corrected Total
Parameter
Intercept
AGE
Source
Model
Error
Corrected Total
DF
1
R-Square
0.592618
10
11
Parameter
Intercept
HGT
AGESQ
DF
2
9
11
R-Square
0.776414
Sum of Squares
526.3928571
361.8571429
888.2500000
Coeff Var
9.586385
Estimate
30.57142857
3.64285714
Sum of Squares
689.6499511
198.6000489
888.2500000
Coeff Var
7.486084
Mean Square
526.3928571
36.1857143
Model 5 WGT = B₁ + B₁HGT + B3(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT and (AGE)²
Estimate
15.11753900
0.72597651
0.11480164
Root MSE
6.015456
Standard Error
8.61370526
0.95511512
Root MSE
4.697518
WGT Mean
62.75000
Mean Square
344.8249755
22.0666721
Standard Error
11.79690059
0.26333057
0.05373319
F Value
14.55
t Value
Pr> t
3.55 0.0053
3.81 0.0034
F Value
15.63
WGT Mean
62.75000
t Value
Pr> t
1.28 0.2321
2.76 0.0222
2.14 0.0614
Pr > F
0.0034
Pr > F
0.0012
Model 3 WGT =B₁ + B3(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on (AGE)²
150
Source
Model
Error
Corrected Total
Source
Model
Error
Parameter
Intercept
AGESQ
R-Square
0.587596
Corrected Total
DF Sum of Squares
1
10
11
521.9320473
366.3179527
888.2500000
Chapter 8 Multiple Regression Analysis: General Considerations
Copyright 2013 Cengage Leaming. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChap
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restriction
Parameter
Intercept
HGT
AGE
AGESQ
Coeff Var
9.645292
11
R-Square
0.780254
Estimate
45.99764279
0.20597161
DF Sum of Squares
3
8
693.0604634
195.1895366
888.2500000
Model 6 WGT = B₁ + B₁HGT + B₂AGE + ₂(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT, AGE, and (AGE)²
Coeff Var
7.871718
Estimate
Mean Square
521.9320473
36.6317953
Root MSE
6.052421
3.438426001
0.723690241
2.776874563
-0.041706699
Standard Error
4.76964028
0.05456692
Root MSE
4.939503
Mean Square
231.0201545
24.3986921
WGT Mean
62.75000
Standard Error
33.61081984
0.27696316
7.42727877
0.42240715
t Value Pr > It
9.64
<.0001
3.77 0.0036
Pr > F
F Value
14.25 0.0036
F Value
WGT Mean
62.75000
9.47
t Value
Pr > F
Pr> t
0.10 0.9210
2.61 0.0310
0.37
0.7182
-0.10 0.9238
(continued)
0.0052](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Fe6e739dc-f3d8-4adf-9aeb-64a7b3689b62%2Fcm3qjxe_processed.png&w=3840&q=75)
Transcribed Image Text:Model 1 WGT = Bo + B₁HGT + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT
Source
Model
Error
Corrected Total
DF
1
10
11
Parameter
Intercept
HGT
Source
Model
Error
Corrected Total
4 Partial Fstatistics will be discussed in Chapter 9 on hypothesis testing.
R²
Parameter
Intercept
HGT
AGE
R-Square
0.663014
Sum of Squares
588.9225232
299.3274768
888.2500000
SSY
9
11
R-Square
0.779986
Â₁
Copyright 2013 Cengage Leaming. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s
torial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions re
Coeff Var
8.718857
DF Sum of Squares
2
692.8226065
195.4273935
888.2500000
Mean Square
588.9225232
29.9327477
SSE
SSY-SSE
Coeff Var
7.426048
[Portion of output omitted]
Estimate
6.189848707
1.072230356
Estimate
6.553048251
0.722037958
2.050126352
Root MSE
5.471083
Standard Error
12.84874620
0.24173098
Model 4 WGT = Bo + B₁HGT + B₂AGE + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT and AGE
Mean Square
346.4113033
21.7141548
F statistic for overall test
Root MSE
4.659845
F Value
Standard Error
10.94482708
0.26080506
0.93722561
19.67
WGT Mean
62.75000
t Value
Pr > F
WGT Mean
62.75000
t Value
0.48
4.44 0.0013
0.6404
0.0013
8.8 Numerical Examples
Pr > It
P-value for
overall test
Pr> t
F Value Pr > F
15.95 0.0011
0.60
0.5641
2.77 0.0218
2.19 0.0565
(continued)
Test statistics and P-values for partial tests
on model parameters (see Section 9.3)
14
Model 2 WGT = Bo + B₂AGE + E
Edited SAS Output (PROC REG) for Regression of WGT on AGE
Source
Model
Error
Corrected Total
Parameter
Intercept
AGE
Source
Model
Error
Corrected Total
DF
1
R-Square
0.592618
10
11
Parameter
Intercept
HGT
AGESQ
DF
2
9
11
R-Square
0.776414
Sum of Squares
526.3928571
361.8571429
888.2500000
Coeff Var
9.586385
Estimate
30.57142857
3.64285714
Sum of Squares
689.6499511
198.6000489
888.2500000
Coeff Var
7.486084
Mean Square
526.3928571
36.1857143
Model 5 WGT = B₁ + B₁HGT + B3(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT and (AGE)²
Estimate
15.11753900
0.72597651
0.11480164
Root MSE
6.015456
Standard Error
8.61370526
0.95511512
Root MSE
4.697518
WGT Mean
62.75000
Mean Square
344.8249755
22.0666721
Standard Error
11.79690059
0.26333057
0.05373319
F Value
14.55
t Value
Pr> t
3.55 0.0053
3.81 0.0034
F Value
15.63
WGT Mean
62.75000
t Value
Pr> t
1.28 0.2321
2.76 0.0222
2.14 0.0614
Pr > F
0.0034
Pr > F
0.0012
Model 3 WGT =B₁ + B3(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on (AGE)²
150
Source
Model
Error
Corrected Total
Source
Model
Error
Parameter
Intercept
AGESQ
R-Square
0.587596
Corrected Total
DF Sum of Squares
1
10
11
521.9320473
366.3179527
888.2500000
Chapter 8 Multiple Regression Analysis: General Considerations
Copyright 2013 Cengage Leaming. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChap
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restriction
Parameter
Intercept
HGT
AGE
AGESQ
Coeff Var
9.645292
11
R-Square
0.780254
Estimate
45.99764279
0.20597161
DF Sum of Squares
3
8
693.0604634
195.1895366
888.2500000
Model 6 WGT = B₁ + B₁HGT + B₂AGE + ₂(AGE)² + E
Edited SAS Output (PROC REG) for Regression of WGT on HGT, AGE, and (AGE)²
Coeff Var
7.871718
Estimate
Mean Square
521.9320473
36.6317953
Root MSE
6.052421
3.438426001
0.723690241
2.776874563
-0.041706699
Standard Error
4.76964028
0.05456692
Root MSE
4.939503
Mean Square
231.0201545
24.3986921
WGT Mean
62.75000
Standard Error
33.61081984
0.27696316
7.42727877
0.42240715
t Value Pr > It
9.64
<.0001
3.77 0.0036
Pr > F
F Value
14.25 0.0036
F Value
WGT Mean
62.75000
9.47
t Value
Pr > F
Pr> t
0.10 0.9210
2.61 0.0310
0.37
0.7182
-0.10 0.9238
(continued)
0.0052
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 1 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt