lecture 16: measures of association (kanchanaraksa)
TRANSCRIPT
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Estimating Risk
Sukon Kanchanaraksa, PhDJohns Hopkins University
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Section A
Relative Risk
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Incidence of Disease Absolute Risk=(Attack Rate)
Risk
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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
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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
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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
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Approaches to the Measurement of Excess Risk
Ratio of risks
Differences in risks
(Risk in exposed) – (Risk in non-exposed)
Risk in exposedRisk in non−exposed
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Relative Risk or Risk Ratio
Relative risk (RR) =Risk in exposed
Risk in non-exposed
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Cohort Study
Then follow to see whetherCalculate
and compare
Disease develops
Disease does not develop Totals
Incidence of disease
First, identify
Exposed a b a+b
Not exposed
c d c+d
aa + b
= Incidence in exposed
cc + d
= Incidence in not exposed
aa + b
cc + d
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Cohort Study
Then follow to see whetherCalculate
and compare
Disease develops
Disease does not develop Totals
Incidence of disease
First, identify
Exposed a b a+b
Not exposed
c d c+d
aa + b
= Incidence in exposed
cc + d
= Incidence in not exposed
aa + b
cc + d
Relative Risk =
aa + b
cc + d
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Cohort Study
Then follow to see whether calculate
Develop CHD
Do not develop
CHD TotalsIncidence of disease
First select
Smoke cigarettes
84 2916 3000
Do not smoke cigarettes
87 4913 5000
843000
875000
Relative Risk =
843000
875000
= 28.017.4
= 1.61
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Interpreting Relative Risk of a Disease
If RR = 1−
Risk in exposed = Risk in non-exposed
−
No associationIf RR > 1−
Risk in exposed > Risk in non-exposed
−
Positive association; ? causalIf RR < 1−
Risk in exposed < Risk in non-exposed
−
Negative association; ? protective
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Cross-Tabulation Table (Food-Borne Outbreak Exercise)
Attack Rates of Sore Throat Egg Salad
AteDid not
eat
Tuna Salad
Ate 46/53
(87%)3/10
(30%)
Did not eat
8/12
(67%)3/10
(30%)
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Cross-Tabulation Table (Food-Borne Outbreak Exercise)
Relative Risk of Sore Throat Egg Salad
AteDid not
eat
Tuna Salad
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
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Exposure-Disease Tables Expanded from the Cross- Tabulation Table (Food-Borne Outbreak Exercise)
Sore Throat
Yes No Total
Tuna Salad
Only
Ate 3 7 10
Did not eat either
3 7 10
Sore Throat
Both Tuna Salad and Egg Salad
Yes No Total
Ate 46 7 53
Did not eat either
3 7 10
Sore Throat
Yes No Total
Egg Salad
Only
Ate 8 4 12
Did not eat either
3 7 10
RR = (46/53)/(3/10) =2.9RR = (3/10)/(3/10) =1.0
RR = (8/12)/(3/10) =2.2
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Relative Risk by Food Items
No tuna salad
Ate tuna salad
0
1
2
No Egg Salad Ate Egg Salad
Rel
ativ
e R
isk
+Tuna
+Egg
1 2 3 4
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Relative Risk for MI and CHD Death in Men Aged 30–62 in Relation to Cigarette Smoking
0
1
2
3
4
5Cholesterol Levels
Low
High*
0
1
2
3
4
5Blood Pressure
Non-Smoker Smoker SmokerNon-Smoker* High > 220 mg/100 cc
Rel
ativ
e R
isk
< 130 mmHg
130+ mmHg
Source: Doyle et al, 1964
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Relationship between Serum Cholesterol Levels and Risk of Coronary Heart Disease by Age and Sex
Men Women Serum Cholesterol
mg/dL Aged 30–49 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
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Incidence Rates and RR of CHD in Relation to Serum Cholesterol Levels by Age and Sex
Men Women Serum Cholesterol
mg/dL Aged 30–49 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.
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Incidence Rates and RR of CHD in Relation to Serum Cholesterol Levels by Age and Sex
Men Women Serum Cholesterol
mg/dL Aged 30–49 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.
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Section B
Odds Ratio
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Interpreting Odds
“Odds” is often known as the ratio of money that may be won versus the amount of money betIn 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 occurFor 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
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The probability
that an exposed person develops disease
The probability
that a non-exposed person develops disease
=
aa + b
Develop Disease
Do Not Develop Disease
Exposed a b
Non-exposed c d
=
cc + d
Calculating Risk in a Cohort Study
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Applying Concept of Odds
Let’s borrow the concept of odds and apply it to disease and non-diseaseSo, 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 occurOr, 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]
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The odds
that an exposed person develops disease
The odds
that a non-exposed person develops disease
Develop Disease
Do Not Develop Disease
Exposed a b
Non-exposed c d
=
ab
=
cd
Calculating Odds in a Cohort Study
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Odds ratio
is the ratio of the odds of disease in the
exposed to the odds of disease in the non-exposed
Odds ratio
is the ratio of the odds of disease in the
exposed to the odds of disease in the non-exposed
Calculating Odds in a Cohort Study
Develop Disease
Do Not Develop Disease
Exposed a b
Non-exposed c d
OR = odds that an exposed person develops the disease
odds that a non - exposed person develops the disease =
abcd
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Disease Odds Ratio in a Cohort Study
OR =
abcd
= ab
x dc
= adbc
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Calculating Odds Ratio in a Case-Control Study
Case ControlHistory of Exposure
a b
No History of Exposure
c d
ac
bd
The odds that a control was exposed =
The odds that a case was exposed =
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Odds ratio (OR)
is the ratio of the odds that a case was
exposed to the odds that a control was exposed
Odds ratio (OR)
is the ratio of the odds that a case was
exposed to the odds that a control was exposed
Calculating Odds Ratio in a Case-Control Study
Case ControlHistory of Exposure
a b
No History of Exposure
c d
OR = odds that a case was exposed
odds that a control was exposed =
acbd
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Exposure Odds Ratio in a Case-Control Study
OR =
acbd
= ac
x db
= adbc
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Odds Ratio versus Relative Risk
Odds ratio can be calculated in a cohort study and in a case-control study−
The exposure odds ratio is equal to the disease odds ratio
Relative risk can only be calculated in a cohort study
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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 exposureWhen 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 exposureWhen the disease being studied is not a frequent one
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When Is Odds Ratio a Good Estimate of Relative Risk?
If the incidence of the disease is low, then:
a+b ~ bc+d ~ d
Therefore:
a/(a+b)c/(c+d)
a/bc/ d
= adbc
= OR
RR
=
~
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Comparing OR to RR: Disease Is Infrequent
Develop Disease
Do not Develop Disease
Exposed 200 9800 10,000
Non-
Exposed100 9900 10,000
Relative Risk =
200/10, 000100/10, 000
= 2
Odds Ratio =
200 x 9900100 x 9800
= 2.02
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Comparing OR to RR: Disease Is NOT Infrequent
Develop Disease
Do not Develop Disease
Exposed 50 50 100
Non-
Exposed25 75 100
Relative Risk =
50/7550/25
= 2
Odds Ratio =
50 x 7550 x 25
= 3
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Interpreting Odds Ratio of a Disease
If OR = 1−
Exposure is not related to disease
−
No association; independentIf OR > 1−
Exposure is positively related to disease
−
Positive association; ? causalIf OR < 1−
Exposure is negatively related to disease
−
Negative association; ? protective
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Section C
Odds Ratio in Unmatched and Matched Case-Control
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E = Exposed
N = Not exposed
Unmatched Case-Control Study: Example
Assume a study of 10 cases and 10 unmatched controls, with these findings
Assume a study of 10 cases and 10 unmatched controls, with these findings
CASE CONTROL
E N
E E
N N
E N
N E
N N
E N
E E
E N
N N
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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
Thus, 6 of 10 cases were exposed, and 3 of 10 controls were exposed. In a 2x2 table, we have the following:
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
E = Exposed
N = Not exposed
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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
Case Control
Exposed 6 3
Not Exposed
4 7
OR =
adbc
= 6 x 73 x 4
= 3.5
E = Exposed
N = Not exposed
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Quick Pause
OR =
adbc
= 8 x 73 x 4
= 4.7
In a hypothetical 2x2 table with the following rows and columns, is the OR calculated correctly?
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
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Quick Pause
OR =
adbc
= 8 x 73 x 4
= 4.7
Incorrect!Incorrect!
Control Case
Exposed 8 3
Not Exposed
4 7
Why?Why?
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Odds Ratio in a Case-Control Study
OR =
acbd
= ac
x db
= adbc
=
(# cases exposed) x (# controls not exposed)(# cases not exposed) x (# controls exposed)
The numerator is the product of cases exposed and controls not exposed.
The numerator is the product of cases exposed and controls not exposed.
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Case-Control Study: Example
Cases
CHDControls
(without disease)
Smoked cigarettes 112 176
Did not smoke cigarettes
88 224
Total 200 400
% Smoking cigarettes
112200
= 56% 176400
= 44%
OR =
adbc
= 112 x 224176 x 88
= 1.62
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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 genderWhen one control is matched to a case, the case and the matched control form a matched pair
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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 pairConcordant 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
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2x2 Table in a Matched Case-Control Study
Controls
ExposedNot
Exposed
CasesExposed
Not Exposed
Discordant
Concordant
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49Total number of subjects = 2 x (aa+bb+cc+dd)
2x2 Table in a Matched Case-Control Study
Controls
ExposedNot
Exposed
CasesExposed aa bb
Not Exposed
cc dd
“aa”
= number of matched pairs
2 x aa
subjects in this cell
“aa”
= number of matched pairs
2 x aa
subjects in this cell
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OR from 2x2 Table in a Matched Case-Control Study
Controls
ExposedNot
Exposed
CasesExposed aa bb
Not Exposed
cc dd
Odds ratio (matched) =
bbcc
Note: bb is not the product of b and b (not b x b);
it is the number of pairs
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Matched Case-Control Study: Example
Assume a study of 10 cases and 10 controls in which each control was matched to a case resulting in 10 pairs.
Assume a study of 10 cases and 10 controls in which each control was matched to a case resulting in 10 pairs.
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
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Matched Case-Control Study: Example
Controls
ExposedNot
Exposed
Cases
Exposed 2 4
Not Exposed
1 3
Matched OR =
41
= 4
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
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Review: Matched Case-Control Study
Controls
ExposedNot
Exposed
Cases
Exposed 2 4
Not Exposed
1 3
Q1. How many pairs?
Q2. How many
subjects?
Q3. What are the discordant
pairs?
Q4. Which is the “bb”
cell?
Q5. What is the “bb”
cell?
Q1. How many pairs?
Q2. How many
subjects?
Q3. What are the discordant
pairs?
Q4. Which is the “bb”
cell?
Q5. What is the “bb”
cell?
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Review: Unmatching a Matched 2x2 Table
Matched CC Controls
ExposedNot
Exposed
Cases
Exposed 2 4
Not Exposed
1 3 Disease
Unmatched
2x2Yes No
Exposure
Exposed
Not Exposed
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Section D
Attributable Risk
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Attributable Risk
Attributable risk (AR) is a measure of excess risk that is attributed to the exposureAttributable risk in the exposed group equals the difference between the incidence in the exposed group and the incidence in the non-exposed (baseline) group
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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
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Exposed group Non-exposed group
Risk in Exposed and Non-Exposed Groups
Background
Risk
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Exposed group Non-exposed group
Risk in Exposed and Non-Exposed Groups
Background
Risk
Attributable risk
Incidence due to
exposure
Incidence not due to
exposure
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Incidence in exposed group
Incidence in non-exposed group( ) ( )–=
Risk in Exposed and Non-Exposed Groups
1.
Incidence attributable to exposure (attributable risk)
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Incidence in exposed group
Incidence in non-exposed group( ) ( )–=
Risk in Exposed and Non-Exposed Groups
1.
Incidence attributable to exposure (attributable risk)
2.
Proportion of incidence attributable to exposure (proportional attributable risk)
Incidence in exposed group
Incidence in non-exposed group( ) ( )–=
Incidence in exposed group
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Example: Cohort Study
Develop CHD
Do not develop
CHD TotalsIncidence of disease
Smoke cigarettes
84 2916 300028.0 per
1,000
Do not smoke cigarettes
87 4913 500017.4 per
1,000
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= 28.0 –
17.4 = 10.6/1,000/year
Attributable Risk in Smokers
1.
The incidence in smokers which is attributable to their smoking
Incidence in smokers
Incidence in non-smokers( ) ( )–=
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Incidence in smokers
Proportion Attributable Risk in Smokers
2.
The proportion of the total incidence in the smokers which is attributable
to their smoking
= 0.379 = 37.9%= 28.0 –
17.428.0
10.628.0=
Incidence in smokers
Incidence in non-smokers( ) ( )–=
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Risk in the Total Population
Population is a mix of exposed and non-exposed groups
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Incidence in total population
Incidence in non-exposed group) ( )–=
Attributable Risk in the Total Population
3.
Incidence attributable to exposure
(
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Incidence in total population
Incidence in non-exposed group) ( )–= (
Attributable Risk in the Total Population
3.
Incidence attributable to exposure
4.
Proportion
of incidence attributable to exposure
( ) ( )Incidence in total population
Incidence in non-exposed group–=
Incidence in total population
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Incidence in total population
Incidence in non-exposed group) ( )–=
Attributable Risk in the Total Population
3.
Incidence attributable to smoking in the total population
(
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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
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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/yearFrom 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%
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Attributable Risk in the Total Population
Incidence in total population =
Incidence
in smokers( ) Percent
smokers in
population( )+
Incidence
in non-
smokers( ) Percent
non-smokers
in population( )
(28.0/1000) (.44) + (17.4/1000) (.56)
= 22.1/1000/year
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) ( )Incidence in total population
Incidence in non-smokers–=
Attributable Risk in the Total Population
3.
Incidence attributable to smoking
((22.1/1000/year) –
(17.4/1000/year)
= 4.7/1000/year
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22.1–17.4
= 21.3%
Attributable Risk in the Total Population
4.
Proportion of incidence attributable to exposure
Incidence in total population
Incidence in non-smokers) ( )–= (
Incidence in total population
22.1
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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.
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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.