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Epidemology

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Estimating Risk
Sukon Kanchanaraksa, PhD
Johns Hopkins University

Section A
Relative Risk

Risk

Incidence of Disease = Absolute Risk
(Attack Rate)

4

Attack Rates from Food-Borne Outbreak Exercise

Attack Rate (%)
Food

(1)
Ate

(2)
Not Ate

Egg salad

83

30

Macaroni

76

67

Cottage cheese

71

69

Tuna salad

78

50

Ice cream

78

64

Other

72

50
5

Attack Rates from Food-Borne Outbreak Exercise

Attack Rate (%)

Difference of Attack
Rates

Food

(1)
Ate

(2)
Not Ate

(1)–(2)

Egg salad

83

30

53

Macaroni

76

67

9

Cottage cheese

71

69

2

Tuna salad

78

50

28

Ice cream

78

64

14

Other

72

50

22
6

Attack Rates from Food-Borne Outbreak Exercise

Attack Rate (%)

Difference of Attack
Rates

Ratio of
Attack
Rates

Food

(1)
Ate

(2)
Not Ate

(1)–(2)

(1)/(2)

Egg salad

83

30

53

2.77

Macaroni

76

67

9

1.13

Cottage cheese

71

69

2

1.03

Tuna salad

78

50

28

1.56

Ice cream

78

64

14

1.21

Other

72

50

22

1.44
7

Approaches to the Measurement of Excess Risk
Ratio of risks
Risk in exposed
Risk in non− exposed
Differences in risks
(Risk in exposed) – (Risk in non-exposed)

8

Relative Risk or Risk Ratio

Relative risk (RR) =

Risk in exposed
Risk in non-exposed

9

Cohort Study
Then follow to see whether
Disease
develops

Exposed
First,
identify

Not exposed a

c

Disease does not develop b

d

Calculate and compare
Totals

Incidence of disease

a+b

a a+b c+d

c c+d a c = Incidence in exposed
= Incidence in not exposed a+b c+d
10

Cohort Study
Then follow to see whether
Disease
develops

Exposed
First,
identify

Not exposed Disease does not develop a

c

a
= Incidence in exposed a+b c
= Incidence in not exposed c+d b

d

Calculate and compare
Totals

Incidence of disease

a+b

a a+b c+d

c c+d a
Relative Risk = a + b c c+d

11

Cohort Study
Then follow to see whether
Develop
CHD

First select Smoke cigarettes Do not smoke cigarettes

84

87

Do not develop CHD
2916

4913

calculate

Totals

Incidence of disease

3000

84
3000

5000

87
5000

84
28.0
= 1.61
Relative Risk = 3000 =
87
17.4
5000

12

Interpreting Relative Risk of a Disease
If RR = 1
− Risk in exposed = Risk in non-exposed
− No association
If RR > 1
− Risk in exposed > Risk in non-exposed
− Positive association; ? causal
If RR < 1
− Risk in exposed < Risk in non-exposed
− Negative association; ? protective

13

Cross-Tabulation Table (Food-Borne Outbreak Exercise)

Attack Rates of Sore Throat

Egg Salad
Ate

Tuna
Salad

Did not eat Ate

46/53
(87%)

3/10
(30%)

Did not eat 8/12
(67%)

3/10
(30%)

14

Cross-Tabulation Table (Food-Borne Outbreak Exercise)

Relative Risk of Sore Throat

Egg Salad
Ate

Tuna
Salad

Did not eat Ate

2.9

1.0

Did not eat 2.2

1.0

The baseline group for comparison is the no exposure group—
i.e., those who did not eat tuna salad and did not eat egg salad

15

Exposure-Disease Tables Expanded from the CrossTabulation Table (Food-Borne Outbreak Exercise)
Sore Throat
Both
Tuna
Salad Ate and Did not
Egg
Salad eat either Yes

No

Total

46

7

53

3

7

10

Sore Throat

RR = (46/53)/(3/10) =2.9

Yes
Tuna Ate
Salad Did not
Only eat either No

Total

3

7

10

3

7

10

RR = (3/10)/(3/10) =1.0

Sore Throat
Yes
Ate
Egg
Salad Did not
Only eat either No

Total

8

4

12

3

7

10

RR = (8/12)/(3/10) =2.2

16

Relative Risk by Food Items
No tuna salad
Ate tuna salad

Relative Risk

2

+Tuna
1
+Egg

0

1 2
No Egg Salad

3 4
Ate Egg Salad

17

Relative Risk for MI and CHD Death in Men Aged 30–62 in Relation to Cigarette Smoking
Cholesterol Levels
5

Blood Pressure
5

Low
Relative Risk

4

High*

< 130 mmHg
4

3

3

2

2

1

1

0

130+ mmHg

0
Non-Smoker
Smoker
* High > 220 mg/100 cc

Source: Doyle et al, 1964

Non-Smoker

Smoker
18

Relationship between Serum Cholesterol Levels and Risk of
Coronary Heart Disease by Age and Sex
Serum
Cholesterol mg/dL Men
Aged 30–49

Women

Aged 50–62

Aged 30–49

Aged 50–62

Incidence Rates (per 1,000)
< 190

38.2

105.7

11.1

155.2

190–219

44.1

187.5

9.1

88.9

220–249

95.0

201.1

24.3

96.3

250+

157.5

267.8

50.4

121.5

Source: Doyle et al, 1964

19

Incidence Rates and RR of CHD in Relation to Serum
Cholesterol Levels by Age and Sex
Serum
Cholesterol mg/dL Men
Aged 30–49

Women

Aged 50–62

Aged 30–49

Aged 50–62

Incidence Rates (per 1,000)
< 190

38.2

105.7

11.1

155.2

190–219

44.1

187.5

9.1

88.9

220–249

95.0

201.1

24.3

96.3

250+

157.5

267.8

50.4

121.5

Relative Risk*
< 190

1.0

2.8

0.3

4.1

190–219

1.2

4.9

0.2

2.3

220–249

2.5

5.3

0.6

2.5

250+

4.1

7.0

1.3

3.2

* RR of 1.0 set at level for males 30–49 yrs of age with cholesterol level < 190 mg/dL.

20

Incidence Rates and RR of CHD in Relation to Serum
Cholesterol Levels by Age and Sex
Serum
Cholesterol mg/dL Men
Aged 30–49

Women

Aged 50–62

Aged 30–49

Aged 50–62

Incidence Rates (per 1,000)
< 190

38.2

105.7

11.1

155.2

190–219

44.1

187.5

9.1

88.9

220–249

95.0

201.1

24.3

96.3

250+

157.5

267.8

50.4

121.5

Relative Risk*
< 190

1.0

2.8

0.3

4.1

190–219

1.2

4.9

0.2

2.3

220–249

2.5

5.3

0.6

2.5

250+

4.1

7.0

1.3

3.2

* RR of 1.0 set at level for males 30–49 yrs of age with cholesterol level < 190 mg/dL.

21

Section B
Odds Ratio

Interpreting Odds
“Odds” is often known as the ratio of money that may be won versus the amount of money bet
In statistics, an odds of an event is the ratio of:
− The probability that the event WILL occur to the probability that the event will NOT occur
For example, in 100 births, the probability of a delivery being a boy is 51% and being a girl is 49%
The odds of a delivery being a boy is 51/49 = 1.04
In simpler term, an odds of an event can be calculated as:
− Number of events divided by number of non-events

23

Calculating Risk in a Cohort Study

Exposed
Non-exposed

Develop
Disease
a c Do Not
Develop
Disease b d

The probability that an exposed person develops disease

a
=
a+b

The probability that a non-exposed person develops disease

c
=
c+d
24

Applying Concept of Odds
Let’s borrow the concept of odds and apply it to disease and non-disease So, the odds of having the disease is the ratio of the probability that the disease will occur to the probability that the disease will not occur
Or, the odds of having the disease can be calculated as the number of people with the disease divided by the number of people without the disease
[Note: in the exposure-disease 2x2 table, the odds of having a disease in the exposed group is the same as the odds that an exposed person develops the disease]

25

Calculating Odds in a Cohort Study

Exposed
Non-exposed

Develop
Disease
a c Do Not
Develop
Disease b d

The odds that an exposed person develops disease

a
=
b

The odds that a non-exposed person develops disease

=

c d 26

Calculating Odds in a Cohort Study

Exposed
Non-exposed

Develop
Disease
a c Do Not
Develop
Disease b d

Odds ratio is the ratio of the odds of disease in the exposed to the odds of disease in the non-exposed a odds that an exposed person develops the disease
OR =
= b c odds that a non - exposed person develops the disease d 27

Disease Odds Ratio in a Cohort Study

a b = a x d = ad
OR = c b c bc d

28

Calculating Odds Ratio in a Case-Control Study
Case
History of
Exposure
No History of
Exposure

Control

a

b

c

d

a
The odds that a case was exposed = c b
The odds that a control was exposed = d 29

Calculating Odds Ratio in a Case-Control Study
Case
History of
Exposure
No History of
Exposure

Control

a

b

c

d

Odds ratio (OR) is the ratio of the odds that a case was exposed to the odds that a control was exposed a odds that a case was exposed
= c
OR = b odds that a control was exposed d 30

Exposure Odds Ratio in a Case-Control Study

a c = a x d = ad
OR = b c b bc d

31

Odds Ratio versus Relative Risk
Odds ratio can be calculated in a cohort study and in a casecontrol study
− The exposure odds ratio is equal to the disease odds ratio
Relative risk can only be calculated in a cohort study

32

When Is Odds Ratio a Good Estimate of Relative Risk?
When the “cases” studied are representative of all people with the disease in the population from which the cases were drawn, with regards to history of the exposure
When the “controls” studied are representative of all people without the disease in the population from which the cases were drawn, with regards to history of exposure
When the disease being studied is not a frequent one

33

When Is Odds Ratio a Good Estimate of Relative Risk?
If the incidence of the disease is low, then: a+b ~ b c+d ~ d
Therefore:

RR =

~

a/(a+b) c /(c + d)

ad a/b =
= OR bc c/d
34

Comparing OR to RR: Disease Is Infrequent
Develop
Disease

Do not
Develop
Disease

Exposed

200

9800

10,000

NonExposed

100

9900

10,000

200/10, 000
Relative Risk =
100/10, 000
Odds Ratio

=

200 x 9900
100 x 9800

= 2
= 2.02
35

Comparing OR to RR: Disease Is NOT Infrequent
Develop
Disease

Do not
Develop
Disease

Exposed

50

50

100

NonExposed

25

75

100

50/75
Relative Risk =
= 2
50/25
50 x 75
= 3
Odds Ratio =
50 x 25
36

Interpreting Odds Ratio of a Disease
If OR = 1
− Exposure is not related to disease
− No association; independent
If OR > 1
− Exposure is positively related to disease
− Positive association; ? causal
If OR < 1
− Exposure is negatively related to disease
− Negative association; ? protective

37

Section C
Odds Ratio in Unmatched and Matched Case-Control

Unmatched Case-Control Study: Example
CASE

CONTROL

E

N

E

E

N

N

E

N

N

E

N

N

E

N

E

E

E

N

N

N
E = Exposed
N = Not exposed

Assume a study of 10 cases and
10 unmatched controls, with these findings

39

Unmatched Case-Control Study: Example
CASE

CONTROL

E

N

E

E

N

N

E

N

N

E

N

N

E

N

E

E

E

N

N

N
E = Exposed
N = Not exposed

Thus, 6 of 10 cases were exposed, and 3 of 10 controls were exposed. In a 2x2 table, we have the following:

Case

Control

Exposed

6

3

Not
Exposed

4

7

40

Unmatched Case-Control Study: Example
CASE

CONTROL

E

N

E

E

N

N

E

N

N

E

N

N

E

N

E

E

E

N

N

N
E = Exposed
N = Not exposed

Case

Control

Exposed

6

3

Not
Exposed

4

7

6x7 ad =
= 3.5
OR =
3x4
bc

41

Quick Pause
In a hypothetical 2x2 table with the following rows and columns, is the OR calculated correctly? Control

Case

Exposed

8

3

Not
Exposed

4

7

8x7 ad =
= 4.7
OR =
3x4
bc

42

Quick Pause
Control

Case

Exposed

8

3

Not
Exposed

4

7

Incorrect!
8x7
ad
=
= 4.7
OR =
3x4
bc

Why?
43

Odds Ratio in a Case-Control Study a a d ad x
=
OR = c = b c b bc d

(# cases exposed) x (# controls not exposed)
=
(# cases not exposed) x (# controls exposed)

The numerator is the product of cases exposed and controls not exposed.

44

Case-Control Study: Example
Cases
CHD

Controls
(without disease)

Smoked cigarettes

112

176

Did not smoke cigarettes 88

224

Total

200

400

% Smoking cigarettes

112
= 56%
200

176
= 44%
400

112 x 224 ad =
= 1.62
OR =
176 x 88 bc 45

Matched Case-Control Study
In a matched case-control study, one or more controls are selected to match to a case on certain characteristics, such as age, race, and gender
When one control is matched to a case, the case and the matched control form a matched pair

46

Concordant and Discordant Pairs
We can define two types of matched pairs by the similarity or difference of the exposure of the case and control in each pair
Concordant pairs are:
1. Pairs in which both the case and the control were exposed, and
2. Pairs in which neither the case nor the control was exposed Discordant pairs are:
3. Pairs in which the case was exposed but the control was not, and
4. Pairs in which the control was exposed and the case was not 47

2x2 Table in a Matched Case-Control Study

Discordant

Controls
Not
Exposed
Exposed
Exposed

Cases

Not
Exposed
Concordant
48

2x2 Table in a Matched Case-Control Study
“aa” = number of matched pairs
2 x aa subjects in this cell

Controls
Not
Exposed
Exposed

Exposed
Cases

aa

bb

Not
Exposed

cc

dd

Total number of subjects = 2 x (aa+bb+cc+dd)

49

OR from 2x2 Table in a Matched Case-Control Study bb Odds ratio (matched) = cc Controls
Not
Exposed
Exposed

Exposed
Cases

aa

bb

Not
Exposed

cc

dd

Note: bb is not the product of b and b (not b x b); it is the number of pairs

50

Matched Case-Control Study: Example
CASE

CONTROL

E

N

E

E

N

N

E

N

N

E

N

N

E

N

E

E

E

N

N

N
E = Exposed
N = Not exposed

Assume a study of 10 cases and
10 controls in which each control was matched to a case resulting in
10 pairs.

51

Matched Case-Control Study: Example
CASE

CONTROL

E

N

E

E

N

N

N

E

N

E

N

N

N

2

4

Not
Exposed

1

3

E

E

Exposed

E

Not
Exposed

N

N

Exposed

N

E

Controls

E = Exposed
N = Not exposed

Cases

4
= 4
Matched OR =
1
52

Review: Matched Case-Control Study
Q1. How many pairs?

Controls
Exposed

Not
Exposed

Exposed

2

4

Not
Exposed

1

3

Cases

Q2. How many subjects?
Q3. What are the discordant pairs? Q4. Which is the “bb” cell?
Q5. What is the “bb” cell?

53

Review: Unmatching a Matched 2x2 Table
Matched CC

Controls
Exposed

Exposed
Cases

Not
Exposed

2

4

Not
Exposed

1

3

Disease
Unmatched
2x2

Yes

No

Exposed
Exposure
Not
Exposed
54

Section D
Attributable Risk

Attributable Risk
Attributable risk (AR) is a measure of excess risk that is attributed to the exposure
Attributable risk in the exposed group equals the difference between the incidence in the exposed group and the incidence in the non-exposed (baseline) group

56

Attack Rates from Food-Borne Outbreak Exercise

Attack Rate (%)

Difference of Attack
Rates

Egg salad

(1)
Ate
83

(2)
Not Ate
30

(1)–(2)
53

Macaroni

76

67

9

Cottage cheese

71

69

2

Tuna salad

78

50

28

Ice cream

78

64

14

Other

72

50

22

Food

57

Risk in Exposed and Non-Exposed Groups

Background
Risk

Exposed group

Non-exposed group

58

Risk in Exposed and Non-Exposed Groups
Incidence
due to exposure Attributable risk

Incidence not due to exposure Exposed group

Background
Risk

Non-exposed group

59

Risk in Exposed and Non-Exposed Groups
1. Incidence attributable to exposure (attributable risk)

(

)–(

Incidence in
= exposed group

)

Incidence in non-exposed group

60

Risk in Exposed and Non-Exposed Groups
1. Incidence attributable to exposure (attributable risk)

(

)–(

Incidence in
= exposed group

)

Incidence in non-exposed group

2. Proportion of incidence attributable to exposure
(proportional attributable risk)

(

) (

)

Incidence in
Incidence in
= exposed group – non-exposed group
Incidence in exposed group

61

Example: Cohort Study

Develop
CHD

Do not develop CHD

Totals

Smoke cigarettes 84

2916

3000

28.0 per
1,000

Do not smoke cigarettes

87

4913

5000

17.4 per
1,000

Incidence of disease

62

Attributable Risk in Smokers
1. The incidence in smokers which is attributable to their smoking =

(

Incidence in smokers )–(

)

Incidence in non-smokers = 28.0 – 17.4 = 10.6/1,000/year

63

Proportion Attributable Risk in Smokers
2. The proportion of the total incidence in the smokers which is attributable to their smoking

=

(

Incidence in smokers )–(

)

Incidence in non-smokers Incidence in smokers

=

28.0 – 17.4
28.0

10.6
= 0.379 = 37.9%
=
28.0

64

Risk in the Total Population
Population is a mix of exposed and non-exposed groups

65

Attributable Risk in the Total Population
3. Incidence attributable to exposure

(

)–(

Incidence in
= total population

)

Incidence in non-exposed group

66

Attributable Risk in the Total Population
3. Incidence attributable to exposure

(

)–(

Incidence in
= total population

)

Incidence in non-exposed group

4. Proportion of incidence attributable to exposure

(

) (

)

Incidence in
Incidence in
= total population – non-exposed group
Incidence in total population

67

Attributable Risk in the Total Population
3. Incidence attributable to smoking in the total population

(

)–(

Incidence in
= total population

)

Incidence in non-exposed group

68

Attributable Risk in the Total Population
If the incidence in the total population is unknown, it can be calculated if we know:
− Incidence among smokers
− Incidence among nonsmokers
− Proportion of the total population that smokes

69

Attributable Risk in the Total Population
We know that:
− The incidence in smokers = 28.0/1,000/year
− The incidence in nonsmokers = 17.4/1,000/year
From another source, we learn that:
− The proportion of smokers in the population is 44%
So, we know that:
− The proportion of nonsmokers in the population is 56%

70

Attributable Risk in the Total Population
Incidence in total population =

(

Percent smokers in population )(

Incidence in smokers

)+ (

Incidence in nonsmokers

Percent non-smokers in population

)(

)

(28.0/1000) (.44) + (17.4/1000) (.56)
= 22.1/1000/year

71

Attributable Risk in the Total Population
3. Incidence attributable to smoking

(

)–(

Incidence in
= total population

)

Incidence in non-smokers (22.1/1000/year) – (17.4/1000/year)
= 4.7/1000/year

72

Attributable Risk in the Total Population
4. Proportion of incidence attributable to exposure

(

) (

Incidence in
Incidence in
= total population – non-smokers Incidence in total population

)

22.1–17.4
22.1
= 21.3%

73

Lung Cancer, CHD Mortality in Male British Physicians
Age-Adjusted Death
Rates/100,000
Smokers

Non-Smokers

RR

AR

%AR

Lung cancer

140

10

14.0

130

92%

CHD

669

413

1.6

256

38%

%AR = Proportion attributable risk

Source: Doll and Peto (1976). BMJ, 2:1525.

74

Lung Cancer, CHD Mortality in Male British Physicians
Age-Adjusted Death
Rates/100,000
Smokers

Non-Smokers

RR

AR

%AR

Lung cancer

140

10

14.0

130

92%

CHD

669

413

1.6

256

38%

%AR = Proportion attributable risk

Source: Doll and Peto (1976). BMJ, 2:1525.

75

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