Rare disease assumption

The rare disease assumption can be demonstrated mathematically using the definitions for relative risk and odds ratio. Relative Risk = {a/(a+b)\over c/(c+d)} and Odds Ratio = {{a/(a+c) \over c/(a+c)}\over {b/(b+d) \over d/(b+d)}} = {a/c \over b/d} = {ad \over bc} As prevalence decreases, the number of positive cases (a+c) decreases. As (a+c) approaches 0, then a and c , individually, also approaches 0. In other words, as (a+c) approaches 0, Relative Risk = {a/(a+b)\over c/(c+d)} \approx {a/(0+b)\over c/(0+d)} ={a/b\over c/d} = {ad\over bc} = Odds Ratio . == Examples ==

The following example illustrates one of the problems, which occurs when the effects are large because the disease is common in the exposed or unexposed group. Consider the following contingency table. RR = {4/(4+6)\over 5/(5+85)} = 7.2 and OR = {4/6 \over 5/85} = 11.3 While the prevalence is only 9% (9/100), the odds ratio (OR) is equal to 11.3 and the relative risk (RR) is equal to 7.2. Despite fulfilling the rare disease assumption overall, the OR and RR can hardly be considered to be approximately the same. However, the prevalence in the exposed group is 40%, which means a is not sufficiently small compared to b and therefore b \not\approx (a+b) . RR = {4/(4+96)\over 5/(5+895)} = 7.2 and OR = {4/96 \over 5/895} = 7.46 With a prevalence of 0.9% (9/1000) and no changes to the effect size (same RR as above), estimates for RR and OR converge. Sometimes the prevalence threshold for which the rare disease assumption holds may be much lower. ==References==