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# Confidence interval for Attributable Risk

There is unfortunately no simple way to estimate the confidence limits for attributable risks, and the best way is to use exact methods. If this is not feasible, there are several approximate methods, of which one suggested by Leung & Kupper (1981) is probably the best.

Using the notation previously introduced:

 Explanatoryvariable ⇓ Response variable ⇓ Totals + - + a b a + b - c d c + d Totals a + c b + d a + b+ c + d = n

• a,b,c and d are the number of individuals in each cell,
• n is the total number of individuals.

#### Algebraically speaking -

 95% CL (θ)   = (ad-bc) exp(±u)  Nc + (ad - bc) exp(±u)
where:
 u   = √ 1.96 (a + c) (c+d) ad (N-c) + c2b  ad - bc Nc (a+c) (c+d)
• and a, b, c, d and N are as above

For our example u works out to 0.94675. Hence the upper 95% confidence limit is 0.2482 and the lower 95% confidence limit is 0.0474 Hence we are 95% certain that the attributable risk is enclosed by the limits of 4.74% to 24.82%.

The same important points to bear in mind when estimating attributable risk apply in the same way when estimating its confidence interval :

1. It is assumed 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 cannot be estimated - and nor can its confidence interval.

2. However large or 'significant' an attributable risk 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.
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