Solution Of Hw 5 (chapter 6) Problems.docx 1h1j24
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Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers CHAPTER 6 2. A clinical trial is planned to compare an experimental medication designed to lower blood pressure to a placebo. Before starting the trial, a pilot study is conducted involving 10 participants. The objective of the study is to assess how systolic blood pressure changes over time untreated. Systolic blood pressures are measured at baseline and again 4 weeks later. Compute a 95% CI for the difference in blood pressures over 4 weeks. Answer: Baseline: 120 145
130
160
152
143
126
121
115
135
4 Weeks: 122 142
135
158
155
140
130
120
124
130
Baseline
4 Weeks
Difference = 4 Weeks – Baseline
Difference2
120
122
2
4
145
142
–3
9
130
135
5
25
160
158
–2
4
152
155
3
9
143
140
–3
9
126
130
4
16
121
120
–1
1
115
124
9
81
135
130
–5
25
9
183
Xd
s
ΣDifferenc e 9 0.9 n 10
ΣDifferenc es 2 (ΣΣDiffere ces) 2 /n 183 (9) 2 /10 19.43 4.4 n 1 9 Xd t
sd n
, df = n – 1 = 10 – 1 = 9
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
1
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers For 95% confidence, t = 2.262 Xd t
sd n
0.9 (2.262)
4.4 10
0.9 ± 3.1 (–2.2, 4.0) 3.
The main trial is conducted and involves a total of 200 patients. Patients are enrolled and randomized to receive either the experimental medication or the placebo. The data shown here are data collected at the end of the study after 6 weeks on the assigned treatment. Experimental (n = 100)
Placebo (n = 100)
Mean (SD) systolic blood pressure
120.2 (15.4)
131.4 (18.9)
% Hypertensive
14%
22%
% With side effects
6%
8%
a. Generate a 95% CI for the difference in mean systolic blood pressures between groups. Answer:
(X1 - X 2 ) ZSp
1 1 n1 n 2
Check s12/s22 = 15.42/18.92 = 0.66 Sp
Sp
(n 1 1)s 12 (n 2 1)s 22 n1 n 2 2
(100 1)15.4 2 (100 - 1)18.9 2 100 100 2
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
2
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers Sp 297.19 =17.2
(120.2 - 131.4) 1.960(17.2)
1 1 . 100 100
–11.2 ± 4.77 (–16.0, –6.4) b. Generate a 95% CI for the difference in proportions of patients with hypertension between groups. Answer:
(pˆ 1 - pˆ 2 ) Z
pˆ 1 (1 - pˆ 1 ) pˆ 2 (1 pˆ 2 ) n1 n2
Sample size is adequate with at least five successes and failures in each group.
(0.14 0.22) 1.96
0.14(1 0.14) 0.22(1 0.22) 100 100
–0.08 ± 1.96 (0.054) –0.08 ± 0.106 (–0.19, 0.03) c. Generate a point estimate for the RR of side effects in patients assigned to the experimental group as compared to placebo. Generate a 95% CI for the RR. Answer: Rˆ R = pˆ 1/ pˆ 2 = 0.06/0.08 = 0.75
Ln( Rˆ R) Z
(n 1 - x 1 )/x 1 (n 2 - x 2 )/x 2 n1 n2
Sample size is adequate with at least five successes and failures in each group.
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
3
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers
Ln(0.75) 1.96
94/6 92/8 100 100
–0.29 ± 1.96 (0.521) –0.29 ± 1.02 (–1.31, 0.73) The 95% CI for Ln(RR) is (–1.31, 0.73). To generate the CI for the RR, we take the antilog (exp) of the lower and upper limits: (exp(–1.31), exp(0.73)) (0.27, 2.08) 4.
The following data are collected as part of a study of coffee consumption among undergraduate students. The following values reflect cups per day consumed: 3 4
6
8
2
1
0
2
a. Compute the sample mean. Answer: X
X2
3
9
4
16
6
36
8
64
2
4
1
1
0
0
2
4
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
4
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers 26
134
X
ΣX 26 3.3 n 8
b. Compute the sample standard deviation. Answer:
s
ΣX 2 (ΣΣ 2 )/n 134 (26) 2 /8 134 84.5 7.07 2.7 n 1 7 7
c. Construct a 95% CI for the mean number of cups of coffee consumed among all undergraduates. Answer:
Xt
s n
, df = n – 1 = 8 – 1 = 7
For 95% confidence, t = 2.365
Xt
s n
3.3 (2.365)
2.7 8
3.3 ± 2.26 (1.0, 5.6)
5.
A clinical trial is conducted comparing a new pain medication for arthritis. Participants are randomly assigned to receive the new medication or a placebo. The outcome is pain relief within 30 minutes; the data are shown here. Pain Relief
No Pain Relief
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
5
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers New medication
44
76
Placebo
21
99
pˆ 1 = 44/120 = 0.367, pˆ 2 = 21/120 = 0.175 a. Estimate the RD in pain relief between treatments. Answer: Rˆ D = pˆ 1 – pˆ 2 = 0.367 – 0.175 = 0.192
b. Estimate the RR in pain relief between treatments. Answer: Rˆ R = pˆ 1/ pˆ 2 = 0.367/0.175 = 2.097
c. Estimate the OR in pain relief between treatments. Answer:
ˆ R = pˆ 1 /(1 pˆ 1 ) 0.367 / 0.633 0.580 = 2.74 O pˆ 2 /(1 pˆ 2 ) 0.175 / 0.825 0.212 d. Construct a 95% CI for the OR. Answer:
ˆ R) Z Ln( O
1 1 1 1 x 1 (n 1 x 1 ) x 2 (n 2 x 2 )
Sample size is adequate with at least five successes and failures in each group.
Ln(2.74) 1.96
1 1 1 1 44 76 21 99
1.01 ± 1.96 (0.306) 1.01 ± 0.60
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
6
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers (0.41, 1.61) The 95% CI for Ln(OR) is (0.41, 1.61). To generate the CI for the odds ratio, we take the antilog (exp) of the lower and upper limits: (exp(0.41), exp(1.61)) (1.51, 5.00) 10. Suppose in the study described in Problem 9, each participant is also asked if the assigned medication causes any stomach upset. Among the 50 participants, 12 report stomach upset with the experimental medication. Construct a 90% CI for the proportion of participants who might experience stomach upset with the experimental medication. Answer:
pˆ = 12/50 = 0.24 pˆ Z
pˆ (1 - pˆ ) n
Sample size is adequate with at least five successes and failures in the sample.
0.24 1.645
0.24(1 - 0.24) 50
0.24 ± 0.099 (0.14, 0.34) 16. A clinical trial is run comparing a new drug for high cholesterol to a placebo. A total of 40 participants are randomized (with equal assignment to treatments) to receive either the new drug or placebo. Their total serum cholesterol levels are measured after 8 weeks on the assigned treatment. Participants receiving the new drug reported a mean total serum
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
7
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers cholesterol level of 209.5 (SD = 21.6) and participants receiving the placebo reported a mean total serum cholesterol level of 228.1 (SD = 19.7). a. Construct a 95% CI for the difference in mean total serum cholesterol levels between participants receiving the new drug versus placebo. Answer:
(X1 - X 2 ) tSp
1 1 n1 n 2
df = n1 + n2 –2 = 20 + 20 –2 = 38 Check s12/s22 =21.62/19.72 = 1.20 Sp
Sp
(n 1 1)s 12 (n 2 1)s 22 n1 n 2 2
(20 1)21.6 2 (20 - 1)19.7 2 20 20 2
427.33 20.7
(209.5 - 228.1) 2.024(20.7)
1 1 20 20
–18.6 ± 13.25 (–31.85, –5.35) b. Is the new drug effective? Justify briefly. Answer: Yes. The 95% CI for the difference in means does not include the null value (0), and thus there is a statistically significant difference in mean total serum cholesterol levels between participants receiving the new drug versus placebo.
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
8
130
160
152
143
126
121
115
135
4 Weeks: 122 142
135
158
155
140
130
120
124
130
Baseline
4 Weeks
Difference = 4 Weeks – Baseline
Difference2
120
122
2
4
145
142
–3
9
130
135
5
25
160
158
–2
4
152
155
3
9
143
140
–3
9
126
130
4
16
121
120
–1
1
115
124
9
81
135
130
–5
25
9
183
Xd
s
ΣDifferenc e 9 0.9 n 10
ΣDifferenc es 2 (ΣΣDiffere ces) 2 /n 183 (9) 2 /10 19.43 4.4 n 1 9 Xd t
sd n
, df = n – 1 = 10 – 1 = 9
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
1
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers For 95% confidence, t = 2.262 Xd t
sd n
0.9 (2.262)
4.4 10
0.9 ± 3.1 (–2.2, 4.0) 3.
The main trial is conducted and involves a total of 200 patients. Patients are enrolled and randomized to receive either the experimental medication or the placebo. The data shown here are data collected at the end of the study after 6 weeks on the assigned treatment. Experimental (n = 100)
Placebo (n = 100)
Mean (SD) systolic blood pressure
120.2 (15.4)
131.4 (18.9)
% Hypertensive
14%
22%
% With side effects
6%
8%
a. Generate a 95% CI for the difference in mean systolic blood pressures between groups. Answer:
(X1 - X 2 ) ZSp
1 1 n1 n 2
Check s12/s22 = 15.42/18.92 = 0.66 Sp
Sp
(n 1 1)s 12 (n 2 1)s 22 n1 n 2 2
(100 1)15.4 2 (100 - 1)18.9 2 100 100 2
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
2
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers Sp 297.19 =17.2
(120.2 - 131.4) 1.960(17.2)
1 1 . 100 100
–11.2 ± 4.77 (–16.0, –6.4) b. Generate a 95% CI for the difference in proportions of patients with hypertension between groups. Answer:
(pˆ 1 - pˆ 2 ) Z
pˆ 1 (1 - pˆ 1 ) pˆ 2 (1 pˆ 2 ) n1 n2
Sample size is adequate with at least five successes and failures in each group.
(0.14 0.22) 1.96
0.14(1 0.14) 0.22(1 0.22) 100 100
–0.08 ± 1.96 (0.054) –0.08 ± 0.106 (–0.19, 0.03) c. Generate a point estimate for the RR of side effects in patients assigned to the experimental group as compared to placebo. Generate a 95% CI for the RR. Answer: Rˆ R = pˆ 1/ pˆ 2 = 0.06/0.08 = 0.75
Ln( Rˆ R) Z
(n 1 - x 1 )/x 1 (n 2 - x 2 )/x 2 n1 n2
Sample size is adequate with at least five successes and failures in each group.
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
3
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers
Ln(0.75) 1.96
94/6 92/8 100 100
–0.29 ± 1.96 (0.521) –0.29 ± 1.02 (–1.31, 0.73) The 95% CI for Ln(RR) is (–1.31, 0.73). To generate the CI for the RR, we take the antilog (exp) of the lower and upper limits: (exp(–1.31), exp(0.73)) (0.27, 2.08) 4.
The following data are collected as part of a study of coffee consumption among undergraduate students. The following values reflect cups per day consumed: 3 4
6
8
2
1
0
2
a. Compute the sample mean. Answer: X
X2
3
9
4
16
6
36
8
64
2
4
1
1
0
0
2
4
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
4
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers 26
134
X
ΣX 26 3.3 n 8
b. Compute the sample standard deviation. Answer:
s
ΣX 2 (ΣΣ 2 )/n 134 (26) 2 /8 134 84.5 7.07 2.7 n 1 7 7
c. Construct a 95% CI for the mean number of cups of coffee consumed among all undergraduates. Answer:
Xt
s n
, df = n – 1 = 8 – 1 = 7
For 95% confidence, t = 2.365
Xt
s n
3.3 (2.365)
2.7 8
3.3 ± 2.26 (1.0, 5.6)
5.
A clinical trial is conducted comparing a new pain medication for arthritis. Participants are randomly assigned to receive the new medication or a placebo. The outcome is pain relief within 30 minutes; the data are shown here. Pain Relief
No Pain Relief
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
5
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers New medication
44
76
Placebo
21
99
pˆ 1 = 44/120 = 0.367, pˆ 2 = 21/120 = 0.175 a. Estimate the RD in pain relief between treatments. Answer: Rˆ D = pˆ 1 – pˆ 2 = 0.367 – 0.175 = 0.192
b. Estimate the RR in pain relief between treatments. Answer: Rˆ R = pˆ 1/ pˆ 2 = 0.367/0.175 = 2.097
c. Estimate the OR in pain relief between treatments. Answer:
ˆ R = pˆ 1 /(1 pˆ 1 ) 0.367 / 0.633 0.580 = 2.74 O pˆ 2 /(1 pˆ 2 ) 0.175 / 0.825 0.212 d. Construct a 95% CI for the OR. Answer:
ˆ R) Z Ln( O
1 1 1 1 x 1 (n 1 x 1 ) x 2 (n 2 x 2 )
Sample size is adequate with at least five successes and failures in each group.
Ln(2.74) 1.96
1 1 1 1 44 76 21 99
1.01 ± 1.96 (0.306) 1.01 ± 0.60
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
6
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers (0.41, 1.61) The 95% CI for Ln(OR) is (0.41, 1.61). To generate the CI for the odds ratio, we take the antilog (exp) of the lower and upper limits: (exp(0.41), exp(1.61)) (1.51, 5.00) 10. Suppose in the study described in Problem 9, each participant is also asked if the assigned medication causes any stomach upset. Among the 50 participants, 12 report stomach upset with the experimental medication. Construct a 90% CI for the proportion of participants who might experience stomach upset with the experimental medication. Answer:
pˆ = 12/50 = 0.24 pˆ Z
pˆ (1 - pˆ ) n
Sample size is adequate with at least five successes and failures in the sample.
0.24 1.645
0.24(1 - 0.24) 50
0.24 ± 0.099 (0.14, 0.34) 16. A clinical trial is run comparing a new drug for high cholesterol to a placebo. A total of 40 participants are randomized (with equal assignment to treatments) to receive either the new drug or placebo. Their total serum cholesterol levels are measured after 8 weeks on the assigned treatment. Participants receiving the new drug reported a mean total serum
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
7
Essentials of Biostatistics in Public Health, Third Edition Lisa M. Sullivan End-of-Chapter Answers cholesterol level of 209.5 (SD = 21.6) and participants receiving the placebo reported a mean total serum cholesterol level of 228.1 (SD = 19.7). a. Construct a 95% CI for the difference in mean total serum cholesterol levels between participants receiving the new drug versus placebo. Answer:
(X1 - X 2 ) tSp
1 1 n1 n 2
df = n1 + n2 –2 = 20 + 20 –2 = 38 Check s12/s22 =21.62/19.72 = 1.20 Sp
Sp
(n 1 1)s 12 (n 2 1)s 22 n1 n 2 2
(20 1)21.6 2 (20 - 1)19.7 2 20 20 2
427.33 20.7
(209.5 - 228.1) 2.024(20.7)
1 1 20 20
–18.6 ± 13.25 (–31.85, –5.35) b. Is the new drug effective? Justify briefly. Answer: Yes. The 95% CI for the difference in means does not include the null value (0), and thus there is a statistically significant difference in mean total serum cholesterol levels between participants receiving the new drug versus placebo.
Copyright © 2018 by Jones & Bartlett Learning, LLC, an Ascend Learning Company
8
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