Comparison of Proportions
Part II
Instructor: 李奕慧
yihwei@mail.tcu.edu.tw
1
Lecture Overview



Study designs in epidemiology
Measures of study effect for 2
categorical variables
1. Risk difference
2. Relative risk
3. Odds ratio
McNemar’s test for Matched-pair
study
2
Epidemiologic Study Design

Analytical studies
Intervention
studies
 Clinical trials
Observational studies
 Cross-sectional studies
 Cohort studies
 Case-control studies
3
Measures of Study Effect
For cohort and cross-sectional studies only
RD  p1  p2
risk difference
RR  p1 / p2
risk ratio/relative risk
For any type of study
p1 / q1
OR 
p2 / q2
odds ratio
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Risk difference (RD, p1-p2)
戴安全帽
頭部
合計
受傷 是
否
是
12
62
74
否
88
38 126
合計 100 100 200
第一組：意外發生時機車騎士沒有戴安全帽
p1:第一組母體比例
第一組樣本
n1=100, pˆ 1 (頭部受傷比例)=0.62
第二組：意外發生時機車騎士有戴安全帽
p2: 第二組母體比例
第二組樣本：n2=100,
pˆ 2 =0.12
5
建構 p1-p2 的信賴區間(confidence interval, CI)
( pˆ 1  pˆ 2 )  Z  / 2
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
where
pˆ 1  pˆ 2  0 . 5 ,
Z 0 .025  1 . 96
0.62x0.38 0.12x0.88

 0.0584
100
100
0.5  1.96x0.0584 = (0.386, 0.614)
沒戴安全帽頭部受傷的機率，
較有戴安全帽者高出
39%~61%，戴安全帽的機
車騎士，在車禍發生時，可
以減少39%~61%的頭部受
傷的機會。
6
檢定 H0:p1-p2=0 versus Ha:p1-p20
Z 
ˆ1  p
ˆ 2 ) ( p1  p2 )
(p
ˆ1 (1 p
ˆ1 )
p
n1

ˆ 2 (1 p
ˆ2 )
p
n2
Z = 0.5/0.058 =8.56, P-value < 0.001
Reject H0, there are significant differences in the population
proportions between the two groups.
7
2 sample
proportion
test.xls
8
Example for Clinical trial and Risk Difference
BMJ 2006;333;11939
比較Intervention group與Control group病人
住院3天後的死亡率(death after day 3)




每一個病人皆觀察其死亡(=1)或存活(=0)情形
Intervention group：5/132 (5%)的人住院3天後死
亡
Control group: 8/133 (8%)的人住院3天後死亡
兩組住院3天後死亡率的差異(control-intervention)
為 2% (point estimate), 95%CI:(-3% to 8%)，表
示兩組母體死亡率差異的範圍介於-3%~8%，
Intervention不會影響病人的死亡率。
10
Cohort study / Clinical trial
Disease +
Exposed
(Intervention)
Disease -
Study population
Disease +
Non-exposed
(Control)
Disease -
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Cohort Data 2 x 2 table
ill
not ill
Exposed
a
b
Unexposed
c
d
Incidence in exposed (p1)
= a/(a+b)
Incidence in non-exposed(p2) = c/(c+d)
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Effect measures in cohort studies

Hypothesis
 Is
the incidence among exposed higher than
among unexposed

Absolute measures
 Risk

difference (RD)
Relative measures
 Relative
P1  P2
risk/Risk Ratio (RR)
P1
P2
13
Foodborne Outbreak in a Wedding, Dublin
Ate
ham
Did not
eat ham
ill
not ill
Incidence
49
49
98
50%
4
6
10
40%
53
55
108
Risk difference
0.5 - 0.4 = 0.1 (10%)
Relative risk
0.5 / 0.4 = 1.25
14
RR與OR的意義
RR=1
 RR>1
 RR<1

ln(RR)=0
ln(RR)>0
ln(RR)<0
沒有相關
危險因子
保護因子
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Relative risk
RR = 1.25
 RR = 1.13
 RR = 13
 RR= 0.8

25 % increase in risk
13 % increase in risk
13 fold increase
20% of risk reduction
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Example for Relative Risk
A Randomized Trial of Aspirin on the Risk of
Embolic Events in Patients With Infective
Endocarditis
JACC, 2003;42(5):775–80
17
aspirin * bleeding Crosstabulation
Bleeding (（果）
Yes(=1) No(=2)
Total
Count
17
42
59
Aspirin Yes(=1)
% within aspirin
28.8% 71.2% 100.0%
(因）
No(=2)
Count
8
47
55
% within aspirin
14.5% 85.5% 100.0%
Total
Count
25
89
114
% within aspirin
21.9% 78.1% 100.0%
Pearson Chi-Square
Continuity Correctionb
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear
Association
N of Valid Cases
Chi-Square Tests
Asymp. Sig. Exact Sig. (2- Exact Sig. (1(2-sided)
sided)
sided)
Value
df
a
3.385
1
.066
2.603
1
.107
3.455
1
.063
.074
.053
3.355
1
.067
114
a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 12.06.
b. Computed only for a 2x2 table
Aspirin.sav
18
Risk Estimate
95% Confidence
Interval
Odds Ratio for aspirin (yes / no)
For cohort bleeding = yes
For cohort bleeding = no
N of Valid Cases
Value
2.378
Lower
.931
Upper
6.074
1.981
.930
4.218
.833
.685
1.013
114
OR = (17/42)/(8/47) = 2.38
RR = (17/59)/(8/55) =1.98
Aspirin.sav
19
Case-control (Retrospective)
Exposed
Cases
Non-exposed
Study Population
Exposed
Controls
Non-exposed
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Odds ratio in Case-control and Cohort study
有子宮頸癌
CASE
150
無子宮頸癌
Control
50
Total
Nonsmoker
150
250
400
Total
300
300
600
Smoker
200
OR for smoking =(150/150)/(50/250) = 5
OR for cervical cancer = (150/50)/(150/250) = 5
不管是針對暴露因子，或疾病的odds ratio都相等，
odds ratio不會因為研究設計而有所改變。
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Relative Risk and Odds ratio
P( Disease | exposed) {1  P( Disease | exposed)}
OR 
P( Disease | un exposed) {1  P( Disease | un exposed)}
(cohortstudy)
P(exposed | Case) /{1  P( Disease | Case)}

P(exposed | Control) /{1  P(exposed | Control)}
(case  controlstudy)
P( Disease | exposed)
RR 
P( Disease | un exposed)
當疾病發生的機率很低時，OR  RR
22
smoking * Cervical_cancer Crosstabulation
smoking
Total
Yes
(=1)
Count
% within Cervical_cancer
Cervical_cancer
Yes (=1) No(=2)
150
50
50.0% 16.7%
No
(=2)
Count
% within Cervical_cancer
150
250
50.0% 83.3%
Count
% within Cervical_cancer
Total
200
33.3%
400
66.7%
300
300
600
100.0%
100.0%
100.0%
Risk Estimate
Odds Ratio for smoking (yes / no)
Value
5.000
For cohort Cervical_cancer = yes
2.000
For cohort Cervical_cancer = no
.400
N of Valid Cases
600
95% Confidence Interval
Lower
Upper
3.424
7.302
1.722
.311
2.323
.515
Cervical
cancer2.sav
23
Matched Pair Study Design
Journal of Stroke and Cerebrovascular Diseases, 2005: pp 174-178
24
SBP control (<=140 mmHg) status before and
after stroke admission
Before
After
Control
Not
合計
Control
20 (67%)
10
30 (47%)
Not
14 (41%)
20
34
合計
34 (53%)
30
64
H0: Pbefore=Pafter
(中風前SBP控制的比例＝中風後SBP控制的比例）
Ha: PbeforePafter
Stroke.sav
25
Paired categorical data, McNemar test
Before
After
Number of pts
Controlled
Controlled
20
Not controlled
Not controlled
20
Not Controlled
Controlled
14
Controlled
Not controlled
試問住院前後血壓控制情形有改善嗎？
10
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Chi-Square Tests
Asymp. Sig. Exact Sig. Exact Sig.
Value df (2-sided) (2-sided) (1-sided)
Pearson Chi-Square
Fisher's Exact Test
McNemar Test
N of Valid Cases
4.158a
1
.041
.049
.036
.541c
64
a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 14.06.
b. Computed only for a 2x2 table
c. Binomial distribution used.
應該使用McNemar檢定的結果，
而非一般的Chi-square檢定結果。
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Thank you!
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