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# Attributable Risk Proportion

The risk ratio provides a good measure of the relative importance of the risk factor in relation to the disease, but it does not give information on the overall importance of the risk factor. To do this we must also take account of the prevalence of the risk factor by working out the attributable risk proportion.

Let's take a hypothetical example of the incidence of lung cancer following exposure to asbestos dust. We will assume that a cohort of 1000 randomly selected men is followed over a ten year period, and the incidence of lung cancer is recorded.

 Exposure to asbestos dust Lung cancer Total Affected Unaffected Yes 9 (a) 41 (b) 50 No 48 (c) 902 (d) 950 Total 57 943 1000 (n)

We can readily calculate the risk ratio (λ) for asbestos as 3.563. This tells us that there is a higher than normal risk of cancer for people exposed to asbestos dust. It does not tell us how important asbestos dust is in relation to other risk factors as a possible cause of cancer in the population under study.

For this we must also take into account the probability of exposure (pE) of members of the population to asbestos dust - in this case it is quite low at only 0.05. We can work out the proportion of cases of lung cancer that are attributable to asbestos dust by working out the attributable risk proportion. Two equivalent formulae are available to do this. Note that both relative risk and probability of exposure to the risk factor are used in the first equation:

#### Algebraically speaking -

 Attributable risk proportion (θ)   = pE(λ - 1) 1 +  pE(λ - 1)
where:
• λ is the risk ratio  = a/ (a+b) c/ (c+d)
• pE is the probability of exposure to the risk factor calculated as (a+b)/N

#### Alternatively -

 Attributable risk proportion (θ)   = rO - rE rO
where:
• rO is the overall risk of disease calculated as (a+c)/N,
• rE is the risk in the unexposed group calculated as c/(c+d)

Using the first of these formulae, we have already worked out the risk ratio as 3.563, and the probability of exposure to asbestos as 0.05. This gives an attributable risk proportion of 0.1140. Using the second formula, the overall risk of disease is 0.057 and the risk in the unexposed group is 0.0505. This again gives an attributable risk proportion of 0.1140.

We can conclude from this that 11.40% of cases of lung cancer may be attributable to exposure to asbestos. Providing the relationship really is causal (see below), this is the percentage by which the rate of occurrence of lung cancer in the population would drop if exposure to asbestos dust could be eliminated.

#### Important points

There are several important points to bear in mind when estimating attributable risk:

1. The key assumption for estimating attributable risk proportion is that a random sample has been taken in order to estimate the prevalence of the risk factor in the population at large. Without such a sample, attributable risk proportion cannot be estimated - and cohorts are very seldom composed of a random sample! It is, however, possible to use an estimate of prevalence of the risk factor from another study such as a previous (or better, concurrent) population survey which used probability sampling.

2. The value we have obtained for attributable risk proportion is only an estimate of the population value. We need some indication of how precise an estimate we have managed to achieve. For this we need to estimate its confidence interval which we consider in Unit 9.
3. Unfortunately, however large or 'significant' an attributable risk proportion is, it does not mean that you have proved that the risk factor necessarily causes the disease. It is possible that both may be linked to a third confounding factor which is actually what causes the disease.
4. A final very important point - the simple formulae we have given here are not valid when risk ratios are adjusted for confounding factors. In such a situation one should refer to Rockhill (1998) for the appropriate methodology.

#### Other definitions of attributable risk

Just to complicate matters, some epidemiologists define attributable risk (a commonly used synonym for attributable risk proportion) in a different way, namely as the difference between risk in the exposed group and risk in the unexposed group. However, this is better termed the risk difference:

#### Algebraically speaking -

Risk difference    =  rE - rU

where:
• rE is the risk of disease in the exposed calculated as a/(a+b)
• rU is the risk in the unexposed group calculated as c/(c+d)

In our example the risk difference (0.18-0.0505) is 0.1295.