Approaches to the measurement of excess risk

• 1. Ratio of RISKS

• 2. Difference in RISKS:– (risk in Exposed)-(risk in Non-Exposed)

Risk in Exposed

Risk in Non-Exposed

Measures of RISK in Epidemiologic Studies

• Without an explicit Comparison– Absolute risk

• With an explicit comparison– Relative risk– Odds ratio– Attributable risk

An example comparing two ways of calculating excess risk

Population

Incidence in A B

Exposed 40% 90%

Non-exposed 10% 60%

Differences inincidence rates

30% 30%

Ratio of incidence rates 4.0 1.5

Cohort Study

Diseasedevelops

Diseasedoes notdevelop

Totals Incidencerates ofDisease

Exposed a b a+b aa+b

FirstSelect Not

Exposedc d c+d c

c+d

…then follow to see whether:

aa+c

cc+d

= Incidence in exposed

= Incidence in Non-exposed

aa+b

cc+d

Relative Risk =

Hypothetical cohort study of the 1-year incidence of Acute Myocardial Infarction in indivduals with Severe Hpertension (180 mm Hg) and Normal Systolic blood Pressure (<120 mm Hg)

Myocardial Infarction

BloodPressureStatus

Number Present Absent Probability Probability Oddsdis

Severehypertension

10,000 180 9,820 180/10,000=0.018 180/(10,000-180)=180/9820=0.01833

Normal 10,000 30 9,970 30/10,000=0.003 30/(10,000-30)=30/9970=0.00301

Relative Risk = = 6.0 Odds Ratio =0.0180.003

1809820 309970

0.018330.00301

= = 6.09

Adapted from M. Szkly & J Nieto; Epidemiology: beyond the basics.

When is the Odds Ratio a Good Estimate of the Relative Risk?

• When the “cases” studied are representative of all people with the disease in the population from which the cases were drawn, with regard 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 regard to history of exposure;

• When the disease being studied is not a frequent one.

Incidence of Local Reactions in the Vaccinated and Placebo Groups, Influenza Vaccination trial

Local Reaction

Group Number Present Absent Probability Probability Oddsdis

Vaccine 2570 650 1920 650/2570=0.2529 650/(2570-650)=650/1920=0.3385

Placebo 2410 170 2240 170/2410=0.0705 170/(2410-170)=170/2240=0.0759

Source: Data from R Seltser, PE Sartwell, and JA Bell, A Controlled test of Asian Influenza Vaccine in Population of Families, Am J. of Hygiene, 1962 (75):112-135. Adapted from M. Szkly & J Nieto; Epidemiology: beyond the basics.

Note: Based on data for individuals 40 years old or older in Seltser et al. To avoid rounding ambiguities in these and subsequent examples based on these data, the original sample sizes in Seltser et al.’s study (257 vaccinees and 241 placebo recipients) were multiplied by 10.

Cross-tabulation of exposure and disease in a cohort study

Exposure Diseased Non-Diseased

DiseaseIncidence (Risk)

Probability Odds of Disease

Present

a b q+ =

Absent

c d q- =

q+

1-q+

a

a+b

a

a+b

a

a+b

=

1-

a

b=

c

c+d=

cc+d

c

c+d1 -

q-

1-q-=

c

d

a b

c d

An expression of of the mathematical relationship between the OR on the one hand and the relative risk on the other, can be derived as follows. Assume that q+ is the incidence (probability) in exposed (e.g. vaccinated) and q- the incidence in unexposed individuals. The odds ratio is then:

q+1-q+ q-1-q-

OR = =q+1-q+ x

1-q- q- =

q+q-

1-q-

1-q+x

Notice that the term q+/q- in the equation is the relative risk. Thus the term

defines the bias responsible for the discrepancy between the relative risk and odds ratio estimates (built-in bias). If the association between the exposure and the outcome is positive, q- < q+, thus (1-q-) > (1-q+). The bias term will therefor be greater than 1.0, leading to an overestimation of the relative risk by the odds ratio. By analogy, if the factor is protective, the opposite occurs - that is, (1-q-) < (1-q+) - and the odds ratio will again overestimate the strength of the association. In general, the odds ratio tends to yield an estimate further away from 1.0 than the relative risk on both sides of the scale (above or below 1.0).

1-q-

1-q+

Do the math:1. Using the hypertension/myocardial infarct example

RR=

OR=

OR=RR x “built in bias =

2. Using the example of local reactions to the influenza vaccine

RR=

OR=

OR=RR x “built in bias =

Cohort studyA sample of exposed and non-exposed

DevelopCHD

Do NotDevelop

CHDTotal

Incidence per1,000 per

year

Smokecigarettes

84 2916 3000 28.0

Do notsmokecigarettes

87 4913 5000 17.4

Incidence among smokers = 84/3000=28.0

Incidence among non-smokers = 87/5000=17.4

Attributable risk in the exposed (ARexp)

Unexposed Exposed

Inci

denc

e (p

er 1

,000

)

ARexp

Attributable Risk

The incidence in smokers which is attributable to their smoking

Incidence in smokers - Incidence in Non-Smokers

The ARexp of CHD attributable to smoking is:

Percent ARexp

• A percent ARexp (%ARexp) is simply the ARexp expressed as a percentage of the risk in the exposed (q+). The excess risk associated with the exposure as a percentage of the total q+.

• For a binary exposure, it is

• %ARexp = q+ - q- x 100

• q+

• The %ARexp in the CHD/Smoking example is:

Population Attributable Risk

• PAR is dependent on the population prevalence of exposure.

• As the population is composed of exposed and unexposed individuals, the incidence in the population is similar to the incidence in the unexposed when the exposure is rare (A).

• Incidence in the population is closer to that in the exposed, when the exposure is common (B).

Population Attributable Risk (PAR) and its dependence on the population prevalence of the exposure

Unexposed Population Exposed

Inci

den

ce (

per

1,0

00)

ARexpPop ARexp

Unexposed Population Exposed

Inci

den

ce (

per

1,0

00)

ARexp

Pop ARexp

A.

B.

Incidence in the total population, which is due to the exposure, can be calculated by subtracting:

[Incidence in the total population] – [Incidence in the non-exposed group]

In order to calculate this, one must know:

EITHER: the incidence in the total population,

OR: the incidence among smokers, the incidence among non-smokers,

AND: the proportion of the total population with the exposure, i.e., the proportion of the population that smokes in the population under study or from which incidence can be calculated

We KNOW: incidence among smokers = 28.0/1,000/year,

And, the incidence among non-smokers = 17.4/1,000/year.

If we assume that from some other source of information, we know that the proportion of smokers in the population is 44% (and therefor the proportion of non-smokers is 56%), then the incidence in the total population can be calculated as:

[28.0/1,000] [.44] + [17.4/1,000] [.56] = 22.0/1,000

SO THAT:

[Incidence in the total population] – [Incidence in the non-exposed group]

= [22.0/1,000/year] – [17.4/1,000/year] – 4.6/1,000/year

And the proportion of the incidence in the total population, which is attributable to the exposure, can be calculated by:

[Incidence in the total population] – [Incidence in the non-exposed group]--

Incidence in the total population

= 22.0 – 17.4 = 20.9% 22.0

i.e., there should be a total reduction of 20.9% in the incidence of CHD in this population if smoking were eliminated.

Levin’s formula for the attributable risk for the Total Population

After simple arithmetic manipulation, the previous formula can be expressed as a function of the prevalence of the exposure in the population and the relative risk :

p (RR-1) x 100 p (RR-1) + 1

where p = proportion of the population with the characteristic or exposureand RR= relative risk or Odds Ratio (if applicable)

PAR: dependence on prevalence of exposure and relative risk

0

20

40

60

80

100

0 20 40 60 80 100

Exposure prevalence (%)

Popu

latio

n A

R (%

) RR=10

RR=5

RR=3

RR=2

For all values of the relative risk, the population AR increased markedly as the exposure prevalence increases.